the acoustics module user’s guide - lost … 3 contents chapter 1: introduction acoustics module...

VERSION 4.4

User s Guide

Acoustics Module

C o n t a c t I n f o r m a t i o n

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C o n t e n t s

C h a p t e r 1 : I n t r o d u c t i o n

Acoustics Module Physics Interface Guide 18

Where Do I Access the Documentation and Model Libraries?

24

Overview of the User’s Guide 27

C h a p t e r 2 : M o d e l i n g w i t h t h e A c o u s t i c s M o d u l e

Acoustics Module Capabilities 32

What Can the Acoustics Module Do? . . . . . . . . . . . . . . . 32

What are the Application Areas? . . . . . . . . . . . . . . . . . 33

Which Problems Can You Solve? . . . . . . . . . . . . . . . . . 35

Fundamentals of Acoustics Modeling 36

Acoustics Explained . . . . . . . . . . . . . . . . . . . . . . 36

Examples of Standard Acoustics Problems . . . . . . . . . . . . . 37

Mathematical Models for Acoustic Analysis . . . . . . . . . . . . . 39

Resolving the Waves . . . . . . . . . . . . . . . . . . . . . 41

Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Artificial Boundaries . . . . . . . . . . . . . . . . . . . . . 45

A Note About Perfectly Matched Layers (PMLs) . . . . . . . . . . . 46

Evaluating the Acoustic Field in the Far-Field Region . . . . . . . . . 47

The Far Field Plots . . . . . . . . . . . . . . . . . . . . . . 48

Solving Large Acoustics Problems Using Iterative Solvers. . . . . . . . 49

About the Material Databases for the Acoustics Module . . . . . . . . 52

The Acoustics Module Study Types 53

Stationary Study . . . . . . . . . . . . . . . . . . . . . . . 53

Frequency Domain Study . . . . . . . . . . . . . . . . . . . . 54

C O N T E N T S | 3

4 | C O N T E N T S

Eigenfrequency Study . . . . . . . . . . . . . . . . . . . . . 55

Mode Analysis Study . . . . . . . . . . . . . . . . . . . . . 56

Time Dependent Study . . . . . . . . . . . . . . . . . . . . 57

Frequency Domain Modal and Time-Dependent Modal Studies . . . . . 58

Modal Reduced Order Model . . . . . . . . . . . . . . . . . . 58

Additional Analysis Capabilities . . . . . . . . . . . . . . . . . 59

Special Variables in the Acoustics Module 60

Intensity Variables . . . . . . . . . . . . . . . . . . . . . . 60

Power Dissipation Variables . . . . . . . . . . . . . . . . . . . 62

Boundary Mode Acoustics Variables . . . . . . . . . . . . . . . 64

Reference for the Acoustics Module Special Variables . . . . . . . . . 65

C h a p t e r 3 : T h e P r e s s u r e A c o u s t i c s B r a n c h

The Pressure Acoustics, Frequency Domain Interface 68

Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure

Acoustics, Frequency Domain Interface . . . . . . . . . . . . . 72

Monopole Source . . . . . . . . . . . . . . . . . . . . . . 74

Dipole Source . . . . . . . . . . . . . . . . . . . . . . . . 74

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . . 75

Sound Hard Boundary (Wall) . . . . . . . . . . . . . . . . . . 75

Normal Acceleration . . . . . . . . . . . . . . . . . . . . . 76

Sound Soft Boundary . . . . . . . . . . . . . . . . . . . . . 76

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Impedance . . . . . . . . . . . . . . . . . . . . . . . . . 78

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 78

Plane Wave Radiation . . . . . . . . . . . . . . . . . . . . . 79

Spherical Wave Radiation. . . . . . . . . . . . . . . . . . . . 80

Cylindrical Wave Radiation . . . . . . . . . . . . . . . . . . . 80

Incident Pressure Field. . . . . . . . . . . . . . . . . . . . . 81

Periodic Condition . . . . . . . . . . . . . . . . . . . . . . 82

Interior Sound Hard Boundary (Wall) . . . . . . . . . . . . . . . 84

Axial Symmetry . . . . . . . . . . . . . . . . . . . . . . . 85

Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 85

Pressure Acoustics . . . . . . . . . . . . . . . . . . . . . . 86

Poroacoustics . . . . . . . . . . . . . . . . . . . . . . . . 92

Narrow Region Acoustics . . . . . . . . . . . . . . . . . . . 94

Background Pressure Field . . . . . . . . . . . . . . . . . . . 95

Matched Boundary . . . . . . . . . . . . . . . . . . . . . . 96

Far-Field Calculation . . . . . . . . . . . . . . . . . . . . . 97

Interior Normal Acceleration . . . . . . . . . . . . . . . . . . 99

Interior Impedance/Pair Impedance . . . . . . . . . . . . . . . 100

Interior Perforated Plate . . . . . . . . . . . . . . . . . . . 101

Line Source . . . . . . . . . . . . . . . . . . . . . . . . 102

Line Source on Axis. . . . . . . . . . . . . . . . . . . . . 105

Monopole Point Source . . . . . . . . . . . . . . . . . . . 106

Point Source . . . . . . . . . . . . . . . . . . . . . . . 108

Circular Source . . . . . . . . . . . . . . . . . . . . . . 110

The Pressure Acoustics, Transient Interface 112

Domain, Boundary, Edge, and Point Nodes for the Pressure

Acoustics, Transient Interface . . . . . . . . . . . . . . . . 113

Transient Pressure Acoustics Model. . . . . . . . . . . . . . . 113

The Gaussian Pulse Source Type . . . . . . . . . . . . . . . . 115

The Boundary Mode Acoustics Interface 117

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 118

Boundary, Edge, Point, and Pair Nodes for the Boundary Mode

Acoustics Interface . . . . . . . . . . . . . . . . . . . . 119

Theory Background for the Pressure Acoustics Branch 120

The Governing Equations. . . . . . . . . . . . . . . . . . . 120

Pressure Acoustics, Frequency Domain Equations . . . . . . . . . 123

Pressure Acoustics, Transient Equations . . . . . . . . . . . . . 127

Boundary Mode Acoustics Equations . . . . . . . . . . . . . . 127

Theory for the Plane, Spherical, and Cylindrical Radiation Boundary

Conditions . . . . . . . . . . . . . . . . . . . . . . . 129

Theory for the Far-Field Calculation: The Helmholtz-Kirchhoff Integral . 131

Solving Large Acoustic Problems . . . . . . . . . . . . . . . . 135

Theory for the Pressure Acoustics Fluid Models 136

Introduction to the Pressure Acoustics Fluid Models . . . . . . . . 136

About the Linear Elastic with Attenuation Fluid Model . . . . . . . 137

C O N T E N T S | 5

6 | C O N T E N T S

About the Viscous Fluid Model . . . . . . . . . . . . . . . . 139

About the Thermally Conducting Fluid Model . . . . . . . . . . . 139

About the Thermally Conducting and Viscous Fluid Model . . . . . . 140

About the Poroacoustics Fluid Model . . . . . . . . . . . . . . 141

About the Narrow Region Acoustics Fluid Model . . . . . . . . . 144

References for the Pressure Acoustics Branch 147

C h a p t e r 4 : A c o u s t i c - S t r u c t u r e I n t e r a c t i o n

The Acoustic-Solid Interaction, Frequency Domain Interface 150

Domain, Boundary, Edge, Point, and Pair Nodes for the

Acoustic-Solid Interaction, Frequency Domain Interface. . . . . . 153

Pressure Acoustics Model . . . . . . . . . . . . . . . . . . 155

Acoustic-Structure Boundary . . . . . . . . . . . . . . . . . 163

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 164

Flow Line Source on Axis . . . . . . . . . . . . . . . . . . 164

Intensity Line Source on Axis . . . . . . . . . . . . . . . . . 165

Power Line Source on Axis . . . . . . . . . . . . . . . . . . 166

Intensity Edge Source . . . . . . . . . . . . . . . . . . . . 166

Flow Edge Source . . . . . . . . . . . . . . . . . . . . . 167

Power Edge Source . . . . . . . . . . . . . . . . . . . . . 167

Flow Circular Source . . . . . . . . . . . . . . . . . . . . 168

Intensity Circular Source . . . . . . . . . . . . . . . . . . . 168

Power Circular Source. . . . . . . . . . . . . . . . . . . . 169

Intensity Point Source . . . . . . . . . . . . . . . . . . . . 169

Flow Point Source . . . . . . . . . . . . . . . . . . . . . 170

Power Point Source . . . . . . . . . . . . . . . . . . . . . 170

The Acoustic-Solid Interaction, Transient Interface 172

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 173

Gaussian Pulse Point, Edge, Circular, and Line Sources . . . . . . . . 174

The Acoustic-Piezoelectric Interaction, Frequency Domain

Interface 175


Acoustic-Piezoelectric Interaction, Frequency Domain Interface. . . 178

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 181

The Acoustic-Piezoelectric Interaction, Transient Interface 182

The Elastic Waves and Poroelastic Waves Interfaces 185

The Elastic Waves Interface . . . . . . . . . . . . . . . . . . 185

The Poroelastic Waves Interface . . . . . . . . . . . . . . . . 187

Domain, Boundary, and Shared Nodes for the Elastic Waves and

the Poroelastic Waves Interface . . . . . . . . . . . . . . . 188

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 190

Poroelastic Material . . . . . . . . . . . . . . . . . . . . . 190

Porous, Fixed Constraint . . . . . . . . . . . . . . . . . . . 194

Porous, Free . . . . . . . . . . . . . . . . . . . . . . . 195

Porous, Pressure . . . . . . . . . . . . . . . . . . . . . . 195

Porous, Prescribed Displacement. . . . . . . . . . . . . . . . 196

Porous, Prescribed Velocity . . . . . . . . . . . . . . . . . . 198

Porous, Prescribed Acceleration . . . . . . . . . . . . . . . . 198

Porous, Roller . . . . . . . . . . . . . . . . . . . . . . . 199

Porous, Septum Boundary Load . . . . . . . . . . . . . . . . 200

Continuity . . . . . . . . . . . . . . . . . . . . . . . . 200

Theory for the Elastic Waves and Poroelastic Waves

Interfaces 203

About Elastic Waves . . . . . . . . . . . . . . . . . . . . 203

About Poroelastic Waves. . . . . . . . . . . . . . . . . . . 204

About the Boundary Conditions for Poroelastic Waves . . . . . . . 208

References for the Elastic Waves and Poroelastic Waves Interfaces . . . 210

The Acoustic-Shell Interaction, Frequency Domain Interface 212


Acoustic-Shell Interaction, Frequency Domain Interface . . . . . . 215

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 217

Initial Values (Boundary) . . . . . . . . . . . . . . . . . . . 218

Exterior Shell . . . . . . . . . . . . . . . . . . . . . . . 218

Interior Shell . . . . . . . . . . . . . . . . . . . . . . . 219

Uncoupled Shell . . . . . . . . . . . . . . . . . . . . . . 220

C O N T E N T S | 7

8 | C O N T E N T S

The Acoustic-Shell Interaction, Transient Interface 221

The Pipe Acoustics Interfaces 224

The Pipe Acoustics, Frequency Domain Interface. . . . . . . . . . 224

The Pipe Acoustics, Transient Interface . . . . . . . . . . . . . 226

Edge, Boundary, Point, and Pair Nodes for the Pipe Acoustics

Interfaces . . . . . . . . . . . . . . . . . . . . . . . 227

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 228

Fluid Properties . . . . . . . . . . . . . . . . . . . . . . 228

Pipe Properties . . . . . . . . . . . . . . . . . . . . . . 229

Closed. . . . . . . . . . . . . . . . . . . . . . . . . . 231

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . 231

Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 232

End Impedance . . . . . . . . . . . . . . . . . . . . . . 232

Theory for the Pipe Acoustics Interfaces 235

Governing Equations . . . . . . . . . . . . . . . . . . . . 235

Theory for the Pipe Acoustics Boundary Conditions . . . . . . . . 239

Solving Transient Problems . . . . . . . . . . . . . . . . . . 242

Cut-off Frequency . . . . . . . . . . . . . . . . . . . . . 243

Swirl Correction Factor References for the Pipe Acoustics Interfaces . . . . . . . . . . . 244

C h a p t e r 5 : T h e A e r o a c o u s t i c s B r a n c h

The Linearized Potential Flow, Frequency Domain Interface 246

Domain, Boundary, Edge, Point, and Pair Nodes for the Linearized

Potential Flow, Frequency Domain Interface . . . . . . . . . . 248

Linearized Potential Flow Model . . . . . . . . . . . . . . . . 249

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 250

Sound Hard Boundary (Wall) . . . . . . . . . . . . . . . . . 250

Velocity Potential . . . . . . . . . . . . . . . . . . . . . . 251

Normal Mass Flow . . . . . . . . . . . . . . . . . . . . . 252

Plane Wave Radiation . . . . . . . . . . . . . . . . . . . . 252

Incident Velocity Potential . . . . . . . . . . . . . . . . . . 253

Sound Soft Boundary . . . . . . . . . . . . . . . . . . . . 254

Normal Velocity . . . . . . . . . . . . . . . . . . . . . . 254

Impedance and Pair Impedance . . . . . . . . . . . . . . . . 255

Vortex Sheet . . . . . . . . . . . . . . . . . . . . . . . 255

Interior Sound Hard Boundary (Wall) . . . . . . . . . . . . . . 256

Continuity . . . . . . . . . . . . . . . . . . . . . . . . 257

Axial Symmetry . . . . . . . . . . . . . . . . . . . . . . 257

Mass Flow Line Source on Axis . . . . . . . . . . . . . . . . 258

Mass Flow Edge Source . . . . . . . . . . . . . . . . . . . 258

Mass Flow Point Source . . . . . . . . . . . . . . . . . . . 258

Mass Flow Circular Source . . . . . . . . . . . . . . . . . . 259

The Linearized Potential Flow, Transient Interface 260

Domain, Boundary, Edge, Point, and Pair Nodes for the Linearized

Potential Flow, Transient Interface . . . . . . . . . . . . . . 261

The Linearized Potential Flow, Boundary Mode Interface 263

Boundary, Edge, Point, and Pair Nodes for the Linearized Potential

Flow, Boundary Mode Interface . . . . . . . . . . . . . . . 265

The Compressible Potential Flow Interface 266

Domain, Boundary, and Pair Nodes for the Compressible Potential

Flow Interface . . . . . . . . . . . . . . . . . . . . . 268

Compressible Potential Flow Model . . . . . . . . . . . . . . . 268

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 269

Normal Flow . . . . . . . . . . . . . . . . . . . . . . . 269

Mass Flow . . . . . . . . . . . . . . . . . . . . . . . . 270

Slip Velocity . . . . . . . . . . . . . . . . . . . . . . . . 270

The Linearized Euler, Frequency Domain Interface 272

Domain, Boundary, and Pair Nodes for the Linearized Euler,

Frequency Domain Interface . . . . . . . . . . . . . . . . 274

Linearized Euler Model . . . . . . . . . . . . . . . . . . . 274

Rigid Wall . . . . . . . . . . . . . . . . . . . . . . . . 277

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 277

Domain Sources . . . . . . . . . . . . . . . . . . . . . . 278

Incident Acoustic Fields . . . . . . . . . . . . . . . . . . . 279

Prescribed Acoustic Fields . . . . . . . . . . . . . . . . . . 279

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 280

C O N T E N T S | 9

10 | C O N T E N T S

Impedance . . . . . . . . . . . . . . . . . . . . . . . . 280

Moving Wall. . . . . . . . . . . . . . . . . . . . . . . . 281

Interior Wall . . . . . . . . . . . . . . . . . . . . . . . 281

The Linearized Euler, Transient Interface 282

Domain, Boundary, and Pair Nodes for the Linearized Euler,

Transient Interface . . . . . . . . . . . . . . . . . . . . 283

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 283

Moving Wall. . . . . . . . . . . . . . . . . . . . . . . . 284

The Linearized Navier-Stokes, Frequency Domain Interface 285

Domain, Boundary, and Pair Nodes for the Linearized Navier-Stokes,

Frequency Domain and Transient Interfaces . . . . . . . . . . 287

Linearized Navier-Stokes Model . . . . . . . . . . . . . . . . 288

No Slip . . . . . . . . . . . . . . . . . . . . . . . . . 290

Isothermal . . . . . . . . . . . . . . . . . . . . . . . . 291

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 291

First Order Material Parameters . . . . . . . . . . . . . . . . 292

Domain Sources . . . . . . . . . . . . . . . . . . . . . . 293

Pressure (Adiabatic). . . . . . . . . . . . . . . . . . . . . 293

Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

Prescribed Velocity . . . . . . . . . . . . . . . . . . . . . 294

Prescribed Pressure . . . . . . . . . . . . . . . . . . . . . 294

Normal Stress . . . . . . . . . . . . . . . . . . . . . . . 295

Stress . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Adiabatic . . . . . . . . . . . . . . . . . . . . . . . . . 295

Prescribed Temperature . . . . . . . . . . . . . . . . . . . 296

Heat Flux. . . . . . . . . . . . . . . . . . . . . . . . . 296

The Linearized Navier-Stokes, Transient Interface 297

Theory Background for the Aeroacoustics Branch 299

General Governing Equations . . . . . . . . . . . . . . . . . 300

Linearized Potential Flow . . . . . . . . . . . . . . . . . . . 302

Compressible Potential Flow . . . . . . . . . . . . . . . . . 306

Linearized Euler . . . . . . . . . . . . . . . . . . . . . . 308

Linearized Navier-Stokes . . . . . . . . . . . . . . . . . . . 311

References for the Aeroacoustics Branch Interfaces 313

C h a p t e r 6 : T h e T h e r m o a c o u s t i c s B r a n c h

The Thermoacoustics, Frequency Domain Interface 316


Thermoacoustics, Frequency Domain Interface . . . . . . . . . 320

Thermoacoustics Model . . . . . . . . . . . . . . . . . . . 322

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 326

Heat Source. . . . . . . . . . . . . . . . . . . . . . . . 326

Sound Hard Wall . . . . . . . . . . . . . . . . . . . . . . 327

Isothermal . . . . . . . . . . . . . . . . . . . . . . . . 327

Acoustic-Thermoacoustic Boundary . . . . . . . . . . . . . . 327

Pressure (Adiabatic). . . . . . . . . . . . . . . . . . . . . 328

Wall. . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 329

Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 329

Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

Stress . . . . . . . . . . . . . . . . . . . . . . . . . . 330

No Stress . . . . . . . . . . . . . . . . . . . . . . . . 331

Normal Stress . . . . . . . . . . . . . . . . . . . . . . . 331

Normal Impedance . . . . . . . . . . . . . . . . . . . . . 331

Adiabatic . . . . . . . . . . . . . . . . . . . . . . . . . 332

Temperature Variation . . . . . . . . . . . . . . . . . . . . 332

The Thermoacoustic-Solid Interaction, Frequency Domain

Interface 333


Thermoacoustic-Solid Interaction, Frequency Domain Interface . . . 335

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 338

Continuity . . . . . . . . . . . . . . . . . . . . . . . . 338

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 339

The Thermoacoustic-Shell Interaction, Frequency Domain

Interface 340


C O N T E N T S | 11


Thermoacoustic-Shell Interaction, Frequency Domain Interface . . . 343

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 346

Initial Values (Boundary) . . . . . . . . . . . . . . . . . . . 347

Exterior Shell . . . . . . . . . . . . . . . . . . . . . . . 347

Interior Shell . . . . . . . . . . . . . . . . . . . . . . . 348

Uncoupled Shell . . . . . . . . . . . . . . . . . . . . . . 349

Theory Background for the Thermoacoustics Branch 350

The Viscous and Thermal Boundary Layers . . . . . . . . . . . . 350

General Linearized Compressible Flow Equations . . . . . . . . . 352

Formulation for Eigenfrequency Studies . . . . . . . . . . . . . 357

Formulation for Mode Analysis . . . . . . . . . . . . . . . . 358

Solver Suggestions for Large Thermoacoustic Models . . . . . . . . 359

References for the Thermoacoustics, Frequency Domain Interface . . . 360

C h a p t e r 7 : T h e S t r u c t u r a l M e c h a n i c s B r a n c h

The Solid Mechanics Interface 364

Domain, Boundary, Edge, Point, and Pair Nodes for Solid Mechanics . . 366

Linear Elastic Material . . . . . . . . . . . . . . . . . . . . 368

Change Thickness . . . . . . . . . . . . . . . . . . . . . 371

Damping . . . . . . . . . . . . . . . . . . . . . . . . . 372

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 375

Body Load . . . . . . . . . . . . . . . . . . . . . . . . 375

Free. . . . . . . . . . . . . . . . . . . . . . . . . . . 376

Boundary Load . . . . . . . . . . . . . . . . . . . . . . 376

Fixed Constraint . . . . . . . . . . . . . . . . . . . . . . 377

Prescribed Displacement . . . . . . . . . . . . . . . . . . . 378

Roller . . . . . . . . . . . . . . . . . . . . . . . . . . 379

Periodic Condition . . . . . . . . . . . . . . . . . . . . . 380

Edge Load . . . . . . . . . . . . . . . . . . . . . . . . 381

Point Load . . . . . . . . . . . . . . . . . . . . . . . . 381

Initial Stress and Strain. . . . . . . . . . . . . . . . . . . . 382

Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 383

Prescribed Velocity . . . . . . . . . . . . . . . . . . . . . 383

Prescribed Acceleration . . . . . . . . . . . . . . . . . . . 384

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 385

Antisymmetry . . . . . . . . . . . . . . . . . . . . . . . 386

Spring Foundation . . . . . . . . . . . . . . . . . . . . . 386

Thin Elastic Layer. . . . . . . . . . . . . . . . . . . . . . 388

Pre-Deformation . . . . . . . . . . . . . . . . . . . . . . 390

Added Mass . . . . . . . . . . . . . . . . . . . . . . . . 390

Low-Reflecting Boundary . . . . . . . . . . . . . . . . . . . 391

Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 392

Rotating Frame . . . . . . . . . . . . . . . . . . . . . . 393

Theory for the Solid Mechanics Interface 395

Material and Spatial Coordinates . . . . . . . . . . . . . . . . 395

Coordinate Systems. . . . . . . . . . . . . . . . . . . . . 396

Lagrangian Formulation . . . . . . . . . . . . . . . . . . . 397

About Linear Elastic Materials . . . . . . . . . . . . . . . . . 398

Strain-Displacement Relationship . . . . . . . . . . . . . . . . 405

Stress-Strain Relationship. . . . . . . . . . . . . . . . . . . 408

Plane Strain and Plane Stress Cases . . . . . . . . . . . . . . . 408

Axial Symmetry . . . . . . . . . . . . . . . . . . . . . . 409

Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 411

Pressure Loads . . . . . . . . . . . . . . . . . . . . . . 413

Equation Implementation . . . . . . . . . . . . . . . . . . . 413

Setting up Equations for Different Studies . . . . . . . . . . . . 414

Damping Models . . . . . . . . . . . . . . . . . . . . . . 416

Initial Stresses and Strains . . . . . . . . . . . . . . . . . . 419

About Spring Foundations and Thin Elastic Layers . . . . . . . . . 420

About Added Mass . . . . . . . . . . . . . . . . . . . . . 422

Geometric Nonlinearity Theory for the Solid Mechanics Interface . . . 423

About the Low-Reflecting Boundary Condition . . . . . . . . . . 427

Cyclic Symmetry and Floquet Periodic Conditions . . . . . . . . . 428

Theory for Rotating Frame and Gravity . . . . . . . . . . . . . 429

Calculating Reaction Forces 432

Using Predefined Variables to Evaluate Reaction Forces . . . . . . . 432

Using Weak Constraints to Evaluate Reaction Forces . . . . . . . . 433

Using Surface Traction to Evaluate Reaction Forces . . . . . . . . . 434

Evaluating Surface Traction Forces on Internal Boundaries . . . . . . 435



Geometric Nonlinearity, Frames, and the ALE Method 436

Reference for Geometric Nonlinearity. . . . . . . . . . . . . . 438

Springs and Dampers 439

Damping and Loss 441

Overview of Damping and Loss . . . . . . . . . . . . . . . . 441

Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . 445

Rayleigh Damping. . . . . . . . . . . . . . . . . . . . . . 445

Equivalent Viscous Damping. . . . . . . . . . . . . . . . . . 446

Loss Factor Damping . . . . . . . . . . . . . . . . . . . . 447

Explicit Damping . . . . . . . . . . . . . . . . . . . . . . 447

C h a p t e r 8 : T h e P i e z o e l e c t r i c D e v i c e s I n t e r f a c e

The Piezoelectric Devices Interface 450

Domain, Boundary, Edge, Point, and Pair Nodes for the Piezoelectric

Devices Interface . . . . . . . . . . . . . . . . . . . . 452

Piezoelectric Material . . . . . . . . . . . . . . . . . . . . 454

Electrical Material Model . . . . . . . . . . . . . . . . . . . 455

Electrical Conductivity (Time-Harmonic) . . . . . . . . . . . . . 457

Damping and Loss . . . . . . . . . . . . . . . . . . . . . 458

Remanent Electric Displacement . . . . . . . . . . . . . . . . 459

Dielectric Loss. . . . . . . . . . . . . . . . . . . . . . . 459

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 460

Periodic Condition . . . . . . . . . . . . . . . . . . . . . 460

Theory for the Piezoelectric Devices Interface 462

The Piezoelectric Effect . . . . . . . . . . . . . . . . . . . 462

Piezoelectric Constitutive Relations . . . . . . . . . . . . . . . 463

Piezoelectric Material . . . . . . . . . . . . . . . . . . . . 465

Piezoelectric Dissipation . . . . . . . . . . . . . . . . . . . 465

Initial Stress, Strain, and Electric Displacement. . . . . . . . . . . 465

Geometric Nonlinearity for the Piezoelectric Devices Interface . . . . 466

Damping and Losses Theory . . . . . . . . . . . . . . . . . 468

References for the Piezoelectric Devices Interface . . . . . . . . . 471

Piezoelectric Damping 473

About Piezoelectric Materials . . . . . . . . . . . . . . . . . 473

Piezoelectric Material Orientation . . . . . . . . . . . . . . . 474

Piezoelectric Losses. . . . . . . . . . . . . . . . . . . . . 480

References for Piezoelectric Damping . . . . . . . . . . . . . . 483

C h a p t e r 9 : G l o s s a r y

Glossary of Terms 486


1

I n t r o d u c t i o n

The Acoustics Module is an optional package that extends the COMSOL Multiphysics® environment with customized interfaces and functionality optimized for the analysis of acoustics and vibration problems.

This module solves problems in the general areas of acoustics, acoustic-structure interaction, aeroacoustics, thermoacoustics, pressure and elastic waves in porous materials, and vibrations. The physics interfaces included are fully multiphysics enabled, making it possible to couple them to any other physics interface in COMSOL Multiphysics. Explicit demonstrations of these capabilities are supplied with the product in a library (the Acoustics Module model library) of ready-to-run models that make it quicker and easier to get introduced to discipline-specific problems. One example being a model of a loudspeaker involving both electromechanical and acoustic-structural couplings.

This chapter is an introduction to the capabilities of this module. A summary of the physics interfaces and where you can find documentation and model examples is also included. The last section is a brief overview with links to each chapter in this guide.

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A c ou s t i c s Modu l e Ph y s i c s I n t e r f a c e Gu i d e

The Acoustics Module extends the functionality of the physics interfaces of the COMSOL Multiphysics base package. The details of the physics interfaces and study types for the Acoustics Module are listed in the table below.

In the COMSOL Multiphysics Reference Manual:

• Studies and Solvers

• The Physics Interfaces

• For a list of all the core physics interfaces included with a COMSOL Multiphysics license, see Physics Guide.

INTERFACE ICON TAG SPACE DIMENSION

AVAILABLE PRESET STUDY TYPE

Acoustics

Pressure Acoustics

Pressure Acoustics, Frequency Domain1

acpr all dimensions eigenfrequency; frequency domain; frequency-domain modal; mode analysis (2D and 1D axisymmetric models only)

Pressure Acoustics, Transient

actd all dimensions eigenfrequency; frequency domain; frequency-domain modal; time dependent; time-dependent modal; modal reduced order model; mode analysis (2D and 1D axisymmetric models only)

Boundary Mode Acoustics

acbm 3D, 2D axisymmetric

mode analysis

1 : I N T R O D U C T I O N

Acoustic-Structure Interaction

Acoustic-Solid Interaction, Frequency Domain

acsl 3D, 2D, 2D axisymmetric

eigenfrequency; frequency domain; frequency-domain modal

Acoustic-Solid Interaction, Transient

astd 3D, 2D, 2D axisymmetric

eigenfrequency; frequency domain; frequency-domain modal; time dependent; time-dependent modal; modal reduced order model

Acoustic-Shell Interaction, Frequency Domain2

acsh 3D eigenfrequency; frequency domain; frequency-domain modal

Acoustic-Shell Interaction, Transient2

acshtd 3D eigenfrequency; frequency domain; frequency-domain modal; time dependent; time-dependent modal; modal reduced order model

Acoustic-Piezoelectric Interaction, Frequency Domain

acpz 3D, 2D, 2D axisymmetric


Acoustic-Piezoelectric Interaction, Transient

acpztd 3D, 2D, 2D axisymmetric

eigenfrequency; frequency domain; frequency-domain modal; time dependent; time-dependent modal; modal reduced order model

Elastic Waves elw 3D, 2D, 2D axisymmetric


Poroelastic Waves elw 3D, 2D, 2D axisymmetric


Pipe Acoustics, Frequency Domain3

pafd 3D, 2D eigenfrequency; frequency domain

Pipe Acoustics, Transient3

patd 3D, 2D time dependent



A C O U S T I C S M O D U L E P H Y S I C S I N T E R F A C E G U I D E | 19

20 | C H A P T E R

Aeroacoustics

Linearized Potential Flow, Frequency Domain

ae all dimensions frequency domain; mode analysis (2D and 1D axisymmetric models only)

Linearized Potential Flow, Transient

aetd all dimensions frequency domain; time dependent mode analysis (2D and 1D axisymmetric models only)

Linearized Potential Flow, Boundary Mode

aebm 3D, 2D axisymmetric

mode analysis

Compressible Potential Flow

cpf all dimensions stationary; time dependent

Linearized Euler, Frequency Domain

lef 3D, 2D axisymmetric, 2D, and 1D

frequency domain; eigenfrequency

Linearized Euler, Transient

let 3D, 2D axisymmetric, 2D, and 1D

time dependent

Linearized Navier-Stokes, Frequency Domain4

lnsf 3D, 2D axisymmetric, 2D, and 1D

frequency domain; eigenfrequency

Linearized Navier-Stokes, Transient4

lnst 3D, 2D axisymmetric, 2D, and 1D

time dependent

Thermoacoustics

Thermoacoustics, Frequency Domain

ta all dimensions eigenfrequency; frequency domain; frequency domain modal; mode analysis (2D and 1D axisymmetric models only)

Thermoacoustic-Solid Interaction, Frequency Domain

tas 3D, 2D, 2D axisymmetric

eigenfrequency; frequency domain; frequency domain modal

Thermoacoustic-Shell Interaction2

tash 3D eigenfrequency; frequency domain; frequency domain modal




S H O W M O R E P H Y S I C S O P T I O N S

There are several general options available for the physics interfaces and for individual nodes. This section is a short overview of these options, and includes links to additional information.

To display additional options for the physics interfaces and other parts of the model tree, click the Show button ( ) on the Model Builder and then select the applicable option.

After clicking the Show button ( ), additional sections are displayed on the settings window when a node is clicked and additional nodes are made available.

Structural Mechanics

Solid Mechanics1 solid 3D, 2D, 2D axisymmetric

stationary; eigenfrequency; prestressed analysis, eigenfrequency; time dependent; time-dependent modal; frequency domain; frequency-domain modal; prestressed analysis, frequency domain; modal reduced order model

Piezoelectric Devices pzd 3D, 2D, 2D axisymmetric

stationary; eigenfrequency; time dependent; time-dependent modal; frequency domain; frequency domain modal; modal reduced order model

1This physics interface is included with the core COMSOL package but has added functionality for this module.2 Requires both the Structural Mechanics Module and the Acoustics Module.3 Requires both the Pipe Flow Module and the Acoustics Module.4 These physics interfaces are available as a technology preview feature.



The links to the features described in the COMSOL Multiphysics Reference Manual (or any external guide) do not work in the PDF, only from the online help in COMSOL Multiphysics.


22 | C H A P T E R

Physics nodes are available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

The additional sections that can be displayed include Equation, Advanced Settings, Discretization, Consistent Stabilization, and Inconsistent Stabilization.

You can also click the Expand Sections button ( ) in the Model Builder to always show some sections or click the Show button ( ) and select Reset to Default to reset to display only the Equation and Override and Contribution sections.

For most nodes, both the Equation and Override and Contribution sections are always available. Click the Show button ( ) and then select Equation View to display the Equation View node under all nodes in the Model Builder.

Availability of each node, and whether it is described for a particular node, is based on the individual selected. For example, the Discretization, Advanced Settings, Consistent

Stabilization, and Inconsistent Stabilization sections are often described individually throughout the documentation as there are unique settings.

In general, to add a node, go to the Physics toolbar, no matter what operating system you are using.

SECTION CROSS REFERENCE

Show More Options and Expand Sections

Advanced Physics Sections

The Model Builder

Discretization Show Discretization

Discretization (Node)

Discretization—Splitting of complex variables

Compile Equations

Consistent and Inconsistent Stabilization Show Stabilization

Numerical Stabilization

Constraint Settings Weak Constraints and Constraint Settings

Override and Contribution Physics Exclusive and Contributing Node Types


O T H E R C O M M O N S E T T I N G S

At the main level, some of the common settings found (in addition to the Show options) are the Interface Identifier, Domain Selection, Boundary Selection, Edge Selection, Point Selection, and Dependent Variables.

At the node level, some of the common settings found (in addition to the Show options) are Domain Selection, Boundary Selection, Edge Selection, Point Selection, Material Type, Coordinate System Selection, and Model Inputs. Other sections are common based on application area and are not included here.

SECTION CROSS REFERENCE

Coordinate System Selection Coordinate Systems

Domain, Boundary, Edge, and Point Selection (geometric entity selection)

About Geometric Entities

About Selecting Geometric Entities

The Geometry Entity Selection Sections

Equation Physics Nodes—Equation Section

Interface Identifier Predefined and Built-In Variables

Variable Naming Convention and Namespace

Viewing Node Names, Identifiers, Types, and Tags

Material Type Materials

Model Inputs About Materials and Material Properties

Selecting Physics

Model Inputs and Multiphysics Couplings

Pair Selection Identity and Contact Pairs

Continuity on Interior Boundaries


24 | C H A P T E R

Whe r e Do I A c c e s s t h e Do cumen t a t i o n and Mode l L i b r a r i e s ?

A number of Internet resources provide more information about COMSOL, including licensing and technical information. The electronic documentation, topic-based (or context-based) help, and the Model Libraries are all accessed through the COMSOL Desktop.

T H E D O C U M E N T A T I O N A N D O N L I N E H E L P

The COMSOL Multiphysics Reference Manual describes all core physics interfaces and functionality included with the COMSOL Multiphysics license. This book also has instructions about how to use COMSOL and how to access the electronic Documentation and Help content.

Opening Topic-Based HelpThe Help window is useful as it is connected to many of the features on the GUI. To learn more about a node in the Model Builder, or a window on the Desktop, click to highlight a node or window, then press F1 to open the Help window, which then displays information about that feature (or click a node in the Model Builder followed by the Help button ( ). This is called topic-based (or context) help.

If you are reading the documentation as a PDF file on your computer, the blue links do not work to open a model or content referenced in a different guide. However, if you are using the Help system in COMSOL Multiphysics, these links work to other modules (as long as you have a license), model examples, and documentation sets.

To open the Help window:

• In the Model Builder, click a node or window and then press F1.

• On any toolbar (for example, Home or Geometry), hover the mouse over a button (for example, Browse Materials or Build All) and then press F1.

• From the File menu, click Help ( ).

• In the upper-right part of the COMSOL Desktop, click the ( ) button.


Opening the Documentation Window

T H E M O D E L L I B R A R I E S W I N D O W

Each model includes documentation that has the theoretical background and step-by-step instructions to create the model. The models are available in COMSOL as MPH-files that you can open for further investigation. You can use the step-by-step instructions and the actual models as a template for your own modeling and applications. In most models, SI units are used to describe the relevant properties, parameters, and dimensions in most examples, but other unit systems are available.

Once the Model Libraries window is opened, you can search by model name or browse under a module folder name. Click to highlight any model of interest and a summary of the model and its properties is displayed, including options to open the model or a PDF document.

To open the Help window:

• In the Model Builder, click a node or window and then press F1.

• On the main toolbar, click the Help ( ) button.

• From the main menu, select Help>Help.

To open the Documentation window:

• Press Ctrl+F1.

• From the File menu select Help>Documentation ( ).

To open the Documentation window:

• Press Ctrl+F1.

• On the main toolbar, click the Documentation ( ) button.

• From the main menu, select Help>Documentation.

The Model Libraries Window in the COMSOL Multiphysics Reference Manual.

W H E R E D O I A C C E S S T H E D O C U M E N T A T I O N A N D M O D E L L I B R A R I E S ? | 25

26 | C H A P T E R

Opening the Model Libraries WindowTo open the Model Libraries window ( ):

C O N T A C T I N G C O M S O L B Y E M A I L

For general product information, contact COMSOL at [email protected].

To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to [email protected]. An automatic notification and case number is sent to you by email.

C O M S O L WE B S I T E S

• From the Home ribbon, click ( ) Model Libraries.

• From the File menu select Model Libraries.

To include the latest versions of model examples, from the File>Help menu, select ( ) Update COMSOL Model Library.

• On the main toolbar, click the Model Libraries button.

• From the main menu, select Windows>Model Libraries.

To include the latest versions of model examples, from the Help menu select ( ) Update COMSOL Model Library.

COMSOL website www.comsol.com

Contact COMSOL www.comsol.com/contact

Support Center www.comsol.com/support

Product Download www.comsol.com/support/download

Product Updates www.comsol.com/support/updates

COMSOL Community www.comsol.com/community

Events www.comsol.com/events

COMSOL Video Gallery www.comsol.com/video

Support Knowledge Base www.comsol.com/support/knowledgebase


http://www.comsol.com

http://www.comsol.com/contact/

http://www.comsol.com/support/

http://www.comsol.com/support/download/

http://www.comsol.com/support/updates

http://www.comsol.com/community

http://www.comsol.com/events/

http://www.comsol.com/video/

http://www.comsol.com/support/knowledgebase/

Ove r v i ew o f t h e U s e r ’ s Gu i d e

The Acoustics Module User’s Guide gets you started with modeling using COMSOL Multiphysics. The information in this guide is specific to this module. Instructions on how to use COMSOL in general are included with the COMSOL Multiphysics Reference Manual.

TA B L E O F C O N T E N T S , G L O S S A R Y, A N D I N D E X

To help you navigate through this guide, see the Contents, Glossary, and Index.

M O D E L I N G W I T H T H E A C O U S T I C S M O D U L E

The Modeling with the Acoustics Module chapter introduces you to Acoustics Module Capabilities and the Fundamentals of Acoustics Modeling including illustrative models and information that serves as a reference source for more advanced modeling. The Acoustics Module Study Types briefly describes the available study types and the Fundamentals of Acoustics Modeling has some background theory and equations to help get you started.

T H E P R E S S U R E A C O U S T I C S B R A N C H

The Pressure Acoustics Branch chapter describes the following interfaces.

The Pressure Acoustics, Frequency Domain Interface is the core interface which models the sound waves in the frequency domain and The Pressure Acoustics, Transient Interface is the core interface which models the sound waves in the time domain. The Boundary Mode Acoustics Interface solves for modes that propagate through a cross section of your geometry. The physics interfaces solve for the acoustic variations in pressure. The Helmholtz equation is solved in the frequency domain and the scalar wave equation is solved in the time domain. Domain conditions exist for modeling losses in a homogenized way in porous materials as well as in narrow regions. Sources such as background fields are also available. Boundary conditions include sources, nonreflecting radiation conditions, impedance conditions, periodic conditions, far-field calculation conditions, as well as interior boundaries such as walls or perforated plates.

As detailed in the section Where Do I Access the Documentation and Model Libraries? this information can also be searched from the COMSOL Multiphysics software Help menu.

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28 | C H A P T E R

T H E A C O U S T I C - S T R U C T U R E I N T E R A C T I O N B R A N C H

Acoustic-Structure Interaction chapter describes these following interfaces.

The Acoustic-Solid Interaction, Frequency Domain Interface is a combination of pressure acoustics and solid mechanics with predefined couplings and The Acoustic-Solid Interaction, Transient Interface is a combination of transient pressure acoustics and solid mechanics with predefined couplings.

The Acoustic-Piezoelectric Interaction, Frequency Domain Interface is a combination of pressure acoustics and piezoelectric effects with a predefined coupling for the boundary between the acoustic domain and the piezoelectric device. The Acoustic-Piezoelectric Interaction, Transient Interface combines Pressure Acoustics, Transient, Solid Mechanics, Electrostatics, and the Piezoelectric Devices interface features.

The Elastic Waves Interface combines pressure acoustics and solid mechanics to connect the fluid pressure with the structural deformation in solids. It also features The Poroelastic Waves Interface, which can be seen as linear elastic waves coupled to pressure waves in porous elastic materials damped by a pore fluid.

The Acoustic-Shell Interaction, Frequency Domain Interface requires a Structural Mechanics Module license and is found under the Acoustic-Structure Interaction

branch. It uses the features from the Pressure Acoustics, Frequency Domain and the Shell interfaces to connect the acoustics pressure waves in a fluid domain with the structural deformation in a shell. The interface is available for 3D geometry only.

The Acoustic-Shell Interaction, Transient Interface, which also requires a Structural Mechanics Module license, uses the features from the Pressure Acoustics, Transient and the Shell interfaces to connect the transient pressure acoustics in a fluid domain with the structural deformation of shell boundary. The interface is available for 3D geometry only.

The Pipe Acoustics Interfaces, which require both the Pipe Flow Module and the Acoustics Module, have the equations and boundary conditions for modeling the propagation of sound waves in flexible pipe systems. The equations are formulated in a general way to include the possibility of a stationary background flow. There are two interfaces, one for transient analysis and one for frequency domain studies.

T H E A E R O A C O U S T I C S B R A N C H

The following interfaces are under The Aeroacoustics Branch.


The Linearized Potential Flow, Frequency Domain Interface models acoustic waves in potential flow in the frequency domain and The Linearized Potential Flow, Transient Interface models acoustic waves in potential flow in the time domain. The Linearized Potential Flow, Boundary Mode Interface solves for modes that propagate through a cross section of your geometry.

The Compressible Potential Flow Interface models irrotational flow used as input for the background flow in the linearized potential flow interfaces.

The Linearized Euler, Frequency Domain Interface and The Linearized Euler, Transient Interface models the acoustic variations in density, velocity, and pressure in the presence of a stationary background mean-flow that is well approximated by an ideal gas flow. This physics interfaces is used for aeroacoustic simulations that can be described by the linearized Euler equations.

The Linearized Navier-Stokes, Frequency Domain Interface and The Linearized Navier-Stokes, Transient Interface models the acoustic variations in pressure, velocity, and temperature in the presence of any stationary isothermal or non-isothermal background mean-flow. This physics interface is used for aeroacoustic simulations that can be described by the linearized Navier-Stokes equations.

T H E T H E R M O A C O U S T I C S B R A N C H

The Thermoacoustics Branch chapter describes The Thermoacoustics, Frequency Domain Interface, which is necessary when modeling acoustics accurately in geometries with small dimensions. Near walls viscosity and thermal conduction become important because they create a viscous and a thermal boundary layer where losses are significant.

The Thermoacoustic-Solid Interaction, Frequency Domain Interface is also described here. This physics interface combines features from pressure acoustics, thermoacoustics and solid mechanics with predefined couplings between all three physics.

The Thermoacoustic-Shell Interaction, Frequency Domain Interface requires a Structural Mechanics Module license. The interface uses the features from the Thermoacoustics, Frequency Domain and the Shell interfaces to connect wave propagation in pressure acoustic domains and thermoacoustic domains with the structural deformation of shell boundaries.

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30 | C H A P T E R

T H E S T R U C T U R A L M E C H A N I C S B R A N C H

The Structural Mechanics Branch chapter provides information about the Solid Mechanics interface for modeling, for example, the structural part of acoustic-structure interaction. This is an extension of the Solid Mechanics interface in COMSOL Multiphysics, and you find it under the Structural Mechanics branch. The theory for this interface is also included.

P I E Z O E L E C T R I C D E V I C E S

The Piezoelectric Devices Interface chapter provides information about modeling piezoelectric effects using the Piezoelectric Devices interface. You find it under the Structural Mechanics branch.


2

M o d e l i n g w i t h t h e A c o u s t i c s M o d u l e

This chapter introduces you to the Acoustic Module modeling stages, including a short description of the module capabilities and application areas. Information that serves as a reference source for more advanced modeling is presented, along with the basic governing equations and the different study types.

In this chapter:

• Acoustics Module Capabilities

• Fundamentals of Acoustics Modeling

• The Acoustics Module Study Types

• Special Variables in the Acoustics Module

31

32 | C H A P T E R

A c ou s t i c s Modu l e C apab i l i t i e s

In this section:

• What Can the Acoustics Module Do?

• What are the Application Areas?

• Which Problems Can You Solve?

What Can the Acoustics Module Do?

The Acoustics Module is a collection of physics interfaces for COMSOL Multiphysics adapted to a broad category of acoustics simulations in fluids and solids. This module is useful even if you are not familiar with computational techniques. It can serve equally well as an excellent tool for educational purposes.

The module supports time-harmonic (frequency domain), modal, and transient studies for fluid pressure as well as static, transient, eigenfrequency, and frequency-response analyses for structures. The available physics interfaces include the following functionality:

• Pressure acoustics: model the propagation of sound waves (pressure waves) in the frequency domain and in the time domain.

• Acoustic-structure interactions: combine pressure waves in the fluid with elastic waves in the solid. The interface provides predefined multiphysics couplings at the fluid-solid interface.

• Boundary mode acoustics: find propagating and evanescent modes in ducts and waveguides.

• Thermoacoustics: model the detailed propagation of sound in geometries with small length scales. This is acoustics including thermal and viscous losses explicitly. Also known as visco-thermal acoustics, thermo-viscous acoustics, or linearized compressible Navier-Stokes.

• Aeroacoustics: model the influence a background mean flow has on the propagation of sound waves in the flow, so-called, flow borne noise/sound. Interfaces exist to solve the linearized potential flow, the linearized Euler, and the linearized Navier-Stokes (the latter is available as a technology preview) in both time and frequency domain.

2 : M O D E L I N G W I T H T H E A C O U S T I C S M O D U L E

• Compressible potential flow: determine the flow of a compressible, irrotational, and inviscid fluid.

• Solid mechanics: solve structural mechanics problems.

• Piezoelectricity: model the behavior of piezoelectric materials in a multiphysics environment solving for the electric field and the coupling to the solid structure.

• Poroelastic and elastic waves: model the propagation of elastic waves in solids and in porous materials. In the latter case the coupled behavior of the porous matrix and the saturating fluid is modeled in detail.

• Pipe acoustics: use this interface to model the propagation of sound waves in pipe systems including the elastic properties of the pipe. The equations are formulated in 1D for fast computation and can include a stationary background flow. This functionality requires the addition of the Pipe Flow Module.

All the physics interfaces include a large number of boundary conditions. For the pressure acoustics applications you can choose to analyze the scattered wave in addition to the total wave. Perfectly matched layers (PMLs) provide accurate simulations of open pipes and other models with unbounded domains. The modeling domain can include dipole sources as well as monopole sources and it is easy to specify a monopole point source. The module also includes modeling support for several types of damping and losses that occur in porous materials (poroacoustics) or that are due to viscous and thermal losses (narrow region acoustics). For results evaluation of pressure acoustics models, you can compute the far field (phase and magnitude) and plot it in predefined far-field plots.

What are the Application Areas?

The Acoustics Module can be used in all areas of engineering and physics to model the propagation of sound waves in fluids. The module also includes several multiphysics interfaces because it is common for many application areas involving sound to also have interaction between fluid and solid structures, have electric fields in piezo materials, have heat generation, or require modeling of electro-acoustic transducers. Typical application areas for the Acoustics Module include:

• Automotive applications such as mufflers, particulate filters, and car interiors.

• Sound scattering and sound emission problems.

• Civil engineering applications such as characterization of sound insulation and sound scatterers. Vibration control and sound transmission problems. Pipe acoustics for HVAC type of systems.

A C O U S T I C S M O D U L E C A P A B I L I T I E S | 33

34 | C H A P T E R

• Modeling of loudspeakers, microphones, and other transducers.

• Mobile applications such as feedback analysis, optimized transducer placement, and directivity assessment.

• Aeroacoustics for jet engine noise, muffler systems with non-isothermal flow, and flow meters.

• Ultrasound piezo transducers for sonar applications.

• Musical instruments.

• Bioacoustic applications with ultrasound and more.

• Underwater acoustics, for example, ultrasound.

• Pressure waves in geophysics.

• Advanced multiphysics applications such as photoacoustics, optoacoustics, thermoacoustic cooling, acoustofluidics, acoustic streaming and radiation, and combustion instabilities.

Figure 2-1: An application example is the modeling of mufflers. Here a pressure isosurface plot from the Absorptive Muffler model from the COMSOL Multiphysics model library.

Using the full multiphysics couplings within the COMSOL Multiphysics environment, you can couple the acoustic waves to, for example, an electromagnetic analysis or a


structural analysis for acoustic-structure interaction. The module smoothly integrates with all of the COMSOL Multiphysics functionality.

Which Problems Can You Solve?

The Acoustics Module physics interfaces handle acoustics in fluids (both quiescent and moving background flows) and solids. The physics interfaces for acoustics in fluids support transient, eigenfrequency, frequency domain, mode analysis, and boundary mode analysis in pressure acoustics and linearized potential flow. Thermoacoustic problems, that involve thermal and viscous losses, have support for eigenfrequency and frequency domain analysis. The study of elastic and poroelastic waves in solids also has support for eigenfrequency and frequency domain analysis. The physics interfaces for solids support static, transient, eigenfrequency, and frequency response analysis. Further, by using the predefined couplings between fluid and solid physics interfaces, you can solve problems involving acoustic-structure interaction including the coupling to piezoelectric materials.

All categories are available as 2D, 2D axisymmetric, and 3D models, with the following differences.

• The Acoustic-Shell Interaction interfaces are only supported in 3D and also require the addition of the Structural Mechanics Module.

• The Pipe Acoustics interfaces, which require the Pipe Flow Module, exist in edges in 2D and 3D.

• In 2D the module offers in-plane physics interfaces for problems with a planar symmetry as well as axisymmetric physics interfaces for problems with a cylindrical symmetry.

• Use the fluid acoustics physics interfaces with 1D and 1D axisymmetric geometries.

When using the axisymmetric models, the horizontal axis represents the r direction and the vertical axis the z direction. The geometry in the right half plane; that is, the geometry must be created and is valid only for positive r.

A C O U S T I C S M O D U L E C A P A B I L I T I E S | 35

36 | C H A P T E R

Fundamen t a l s o f A c ou s t i c s Mode l i n g

There are certain difficulties that often arise when modeling acoustics, such as the rather severe requirements on the mesh resolution, the modeling of artificial boundaries, and the modeling of real-world damping materials. This section also includes a brief introduction to acoustics, gives some examples of standard acoustics problems, and provides a short introduction to the mathematical formulation of the governing equations.

In this section:

• Acoustics Explained

• Examples of Standard Acoustics Problems

• Mathematical Models for Acoustic Analysis

• Resolving the Waves

• Damping

• Artificial Boundaries

• A Note About Perfectly Matched Layers (PMLs)

• Evaluating the Acoustic Field in the Far-Field Region

• The Far Field Plots

• Solving Large Acoustics Problems Using Iterative Solvers

• About the Material Databases for the Acoustics Module

Acoustics Explained

Acoustics is the physics of sound. Sound is the sensation, as detected by the ear, of very small rapid changes in the air pressure above and below a static value. This static value is the atmospheric pressure (about 100,000 pascals), which varies slowly. Associated with a sound pressure wave is a flow of energy—the intensity. Physically, sound in air is a longitudinal wave where the wave motion is in the direction of the movement of

The Physics Interfaces and Building a COMSOL Model in the COMSOL Multiphysics Reference Manual


energy. The wave crests are the pressure maxima, while the troughs represent the pressure minima.

Sound results when the air is disturbed by some source. An example is a vibrating object, such as a speaker cone in a sound system. It is possible to see the movement of a bass speaker cone when it generates sound at a very low frequency. As the cone moves forward it compresses the air in front of it, causing an increase in air pressure. Then it moves back past its resting position and causes a reduction in air pressure. This process continues, radiating a wave of alternating high and low pressure propagating at the speed of sound.

The propagation of sound in solids happens through small-amplitude elastic oscillations of its shape. These elastic waves are transmitted to surrounding fluids as ordinary sound waves. The elastic sound waves in the solid are the counter part to the pressure waves or compressible waves propagating in the fluid.

Examples of Standard Acoustics Problems

Depending on the basic dependent variable used to model the acoustic field, the acoustical interfaces can be divided into the following main categories.

• Pressure acoustics—The dependent variable is the acoustic pressure p.

• Acoustic-solid interaction—The dependent variables are the pressure p and the displacement field u in the solid.

• Poroelastic waves—The dependent variables are the pressure p inside the saturating fluid and the total displacement u of the porous matrix.

• Aeroacoustics—The dependent variables are the acoustic perturbations to the background mean flow fields. In the linearized potential flow interface it is the potential for the acoustic particle-velocity field v = . In the linearized Euler interface the dependent variables are the acoustic variations in pressure p, density , and velocity field u. In the linearized Navier-Stokes they are the pressure p, velocity field u, and temperature T. In the typical situation, the background fluid is in motion with, for example, a total velocity utot u0 u, split into a stationary background-flow velocity u0 and the particle velocity u associated with the acoustic waves.

• Thermoacoustics—The dependent variables are the acoustic pressure p, the particle-velocity field v, and the acoustic temperature variation T. This is a detailed acoustic model solving the full set of linearized equations for a compressible flow:

F U N D A M E N T A L S O F A C O U S T I C S M O D E L I N G | 37

38 | C H A P T E R

Navier-Stokes (momentum conservation), continuity (mass conservation), and energy conservation equations.

These standard problems or scenarios occur frequently when analyzing acoustics:

T H E R A D I A T I O N P R O B L E M

A vibrating structure (a speaker, for example) radiates sound into the surrounding space. A radiation boundary condition or a PML (perfectly matched layer) is necessary to model the unbounded open domain.

T H E S C A T T E R I N G P R O B L E M

An incident wave impinges on a body and creates a scattered wave. A radiation boundary condition or a PML is necessary. This could be a sonar application in underwater acoustics or an analysis of the scattered sound field around a human head.

T H E S O U N D F I E L D I N A N I N T E R I O R S P A C E

The acoustic waves stay in a finite volume so no radiation condition is necessary. For example, this case represents the sound inside a room or a car interior. A more advanced example is the sound inside a transducer like a microphone; in this case, the acoustic field should be solved with the Thermoacoustics interface.

C O U P L E D F L U I D - E L A S T I C S T R U C T U R E I N T E R A C T I O N ( S T R U C T U R A L

A C O U S T I C S )

If the radiating or scattering structure consists of an elastic material, the interaction must be considered between the body and the surrounding fluid. In the multiphysics coupling, the acoustic analysis provides a load (the sound pressure) to the structural analysis, and the structural analysis provides accelerations to the acoustic analysis.

T H E TR A N S M I S S I O N P R O B L E M

An incident sound wave propagates into a body, which can have different acoustic properties. Pressure and acceleration are continuous on the boundary. A typical transmission problem is that of modeling the behavior of mufflers.

A E R O A C O U S T I C S P R O B L E M S

The sound (noise) field is influenced by a background flow. This could be the propagating sound from a jet engine or the acoustic damping properties of a muffler with flow.


PO R O E L A S T I C WA V E S P R O B L E M

If the acoustic waves are propagating inside the saturating fluid of porous material the detailed coupling between the fluid pressure and the solid displacement need to be taken into account. In cases where only the fluid pressure is of interest, the porous material can be modeled using an equivalent fluid model.

TR A N S D U C E R P R O B L E M S

Transducers are devices for transformation of one form of energy to another (electrical, mechanical, or acoustical). This type of problem is common in acoustics and is a true multiphysics problem involving electric, structural, and acoustic interfaces. Typical problems of this type involve modeling loudspeakers, microphones, and piezo transducers.

Mathematical Models for Acoustic Analysis

Standard acoustic problems involve solving for the small acoustic pressure variations p on top of the stationary background pressure p0. Mathematically this represents a linearization (small parameter expansion) around the stationary quiescent values.

The governing equations for a compressible lossless (no thermal conduction and no viscosity) fluid flow problem are the momentum equation (Euler's equation) and the continuity equation. These are given by:

where is the total density, p is the total pressure, and u is the velocity field. In classical pressure acoustics all thermodynamic processes are assumed reversible and adiabatic, known as an isentropic process. The small parameter expansion is performed on a stationary fluid of density 0 (SI unit: kg/m3) and at pressure p0 (SI unit: Pa) such that:

where the primed variables represent the small acoustic variations. Inserting these into the governing equations and only retaining terms linear in the primed variables yields:

ut------ u u+

1---p–=

t------ u + 0=

p p0 p'+=

0 '+=

u 0 u'+=

withp' p0«

' 0«


40 | C H A P T E R

One of the dependent variables, the density, is removed by expressing it in terms of the pressure using a Taylor expansion (linearization):

where cs is recognized as the (isentropic) speed of sound (SI unit: m/s) at constant entropy s. The subscripts s and 0 are dropped in the following. Finally, rearranging the equations (divergence of momentum equation inserted into the continuity equation) and dropping the primes yields the wave equation for sound waves in a lossless medium:

(2-1)

The speed of sound is related to the compressibility of the fluid where the waves are propagating. The combination c

2 is called the bulk modulus, commonly denoted K(SI unit: N /m2). The equation is further extended with two optional source terms: the dipole source qd (SI unit: N/m3) and the monopole source Qm (SI unit: 1/s2).

A special case is a time-harmonic wave, for which the pressure varies with time as

where 2f (SI unit: rad/s) is the angular frequency and f (SI unit: Hz) is denoting the frequency. Assuming the same harmonic time-dependence for the source terms, the wave equation for acoustic waves reduces to an inhomogeneous Helmholtz equation:

(2-2)

With the two source terms removed, this equation can also be treated as an eigenvalue PDE to solve for eigenmodes and eigenfrequencies.

u't------- 1

0------p'–=

't------- 0 u' + 0=

'0p---------

sp'

1

cs2

-----p'= =

1

c2--------

t2

2

p 1---– p qd–

+ Qm=

p x t p x eit=

1---– p qd–

2p

c2---------– Qm=


Typical boundary conditions for the wave equation and the Helmholtz equation are:

• Sound-hard boundaries (walls)

• Sound-soft boundaries

• Impedance boundary conditions

• Radiation boundary conditions

These are described in more detail in the next sections.

In lossy media, an additional term of first order in the time derivative needs to be introduced to model attenuation of the sound waves:

where da is the damping coefficient. Note also that even when the sound waves propagate in a lossless medium, attenuation frequently occurs by interaction with the surroundings at the boundaries of the system.

Resolving the Waves

Solutions to acoustic problems are wavelike. The waves are characterized by a wavelength in space, whose value depends on the frequency and speed of sound c in the medium according to cf. This wavelength has to be resolved by the mesh.

For the solution on the discrete grid to have any meaning at all there has to be at least two degrees of freedom (DOFs) per wavelength in the direction of propagation, but such coarse a solution is useless in practice. In reality, the lower limit for a fully reliable solution lies at about ten to twelve degrees of freedom per wavelength.

Because the direction of propagation is generally not known beforehand, it is good practice to aim for an isotropic mesh with about twelve DOFs per wavelength on

1

c2--------

t2

2

pda tp

– 1---– p qd–

+ Qm=

A detailed derivation of the governing equations for the propagation of compressional (acoustic) waves in a viscous and thermally conducting fluid is given in Theory Background for the Thermoacoustics Branch and in moving media in Theory Background for the Aeroacoustics Branch.


42 | C H A P T E R

average, independently of the direction. Therefore the number of DOFs in a sufficiently resolved mesh is about

Before starting a new model, try to estimate the required number of DOFs using these guidelines. The maximum number of DOFs that can be solved for differs between computer systems, but a 32-bit system can usually deal with somewhere from a few hundred thousand up to a million DOFs. Even on a 64-bit system models with more than a few million DOFs are cumbersome to handle.

U S I N G L A G R A N G E E L E M E N T S

When creating an unstructured mesh for use with the default 2nd-order Lagrange elements, set the maximum element size hmax to about or smaller. Because all elements in the constructed mesh are smaller than hmax, the limit is set larger than the actual required element size. After meshing the model, check the total number of DOFs against the model volume and the above guidelines. If the mesh turns out, on average, to be too coarse or too fine, try to change hmax accordingly.

• 1728 times the model volume measured in wavelengths cubed in 3D

• 144 times the model area measured in wavelengths squared in 2D

• 12 times the model length measured in wavelengths in 1D

Unstructured meshes are generally better than structured meshes for wave problems where the direction of wave propagation is not known everywhere in advance. The reason is that in a structured mesh, the average resolution typically differs significantly between directions parallel to the grid lines and directions rotated 45 degrees about one of the axes.

Meshing in the COMSOL Multiphysics Reference Manual


Damping

Fluids with a dynamic viscosity in the same range as air or water—by far the most common media in acoustics simulations—exhibit practically no internal damping over the number of wavelengths that can be resolved on current computers. Instead, damping takes place through interaction with solids, either because of friction between the fluid and a porous material filling the domain, or because acoustic energy is transferred to a surrounding solid where it is absorbed. In systems with small length scales, significant losses can occur in the viscous and thermal acoustic boundary layer at walls.

PO R O U S A B S O R B I N G M A T E R I A L S

For frequency-domain modeling, the most convenient and compact description of a damping material (material refers to the homogenization of a fluid and a porous solid) is given by its complex wave number k and complex impedance Z, both functions of frequency. Knowing these properties, define a complex speed of sound as cck and a complex density as ckZ. Defining c and cc results in a so-called equivalent-fluid model or fluid model.

It is possible to directly measure the complex wave number and impedance in an impedance tube in order to produce curves of the real and imaginary parts (the resistance and reactance, respectively) as functions of frequency. These data can be used directly as input to COMSOL Multiphysics interpolation functions to define k and Z.

Sometimes acoustic properties cannot be obtained directly for a material you want to try in a model. In that case you must resort to knowledge about basic material properties independent of frequency. Several empirical or semi-empirical models exist in COMSOL and can estimate the complex wave number and impedance as function of material parameters. These models are defined in the Poroacoustics domain feature of the Pressure Acoustics interface, they are the Johnson-Champoux-Allard model and the Delany-Bazley-Miki models, the latter uses frequency and flow resistivity as input.

COMSOL includes a series of fluid models that are described in Pressure Acoustics and Theory for the Pressure Acoustics Fluid Models. In addition, The Poroelastic Waves Interface can be used for detailed modeling of the propagation of coupled pressure and elastic waves in porous materials.


44 | C H A P T E R

B O U N D A R Y L A Y E R A B S O R P T I O N ( T H E R M O A C O U S T I C S )

In systems of small dimensions (or at low frequencies) the size of the acoustic boundary layer (the viscous and thermal acoustic penetration depth) that exists at all walls can become comparable to the physical dimensions of the modeled system. In air the boundary layer thickness is 0.22 mm at 100 Hz. This is typically the case inside miniature transducers, condenser microphones, in MEMS systems, in tubing for hearing aids, or in narrow gaps of vibrating structures.

For such systems it is often necessary to use a more detailed model for the propagation of the acoustics waves. This model is implemented in the Thermoacoustics interface in COMSOL. In simple cases for sound propagating in long ducts of constant cross sections, the losses occurring at the boundaries can be smeared out on the fluid using one of the fluid models of the Narrow Region Acoustics domain feature.

D A M P I N G A T B O U N D A R I E S

Acoustics in closed ducts and cavities appears to be easier to deal with than exterior problems because no artificial boundary condition is necessary. On the other hand, real-world cavity walls are usually either treated in some way (lined) or elastic in themselves.

The problem is that a liner typically reflects part of the wave and does so not at the interface with the domain but somewhere inside the liner or at its back wall against whatever structure is outside. This means that a liner boundary condition must contain more information about the outside world than an absorbing boundary. It also means that a real-world liner cannot be adequately described by a local boundary condition because waves at oblique incidence cause waves to propagate in the tangential direction inside the liner layer.

In fact, there seems to be no final answer as to how the process inside a porous liner is most accurately modeled. Various assumptions can be made about the interaction between the fluid pressure waves and the liner material and about boundary conditions between liner and free fluid and at the back of the liner. The most accurate ways to deal with the situation include modeling the actual liner layer. It is only possible to use a general impedance boundary condition for thin liners and when the angle of incidence

More details on the detailed acoustic model for viscous and thermal losses are described in The Thermoacoustics Branch. See the boundary layer absorption fluid models in Narrow Region Acoustics.


is known for a liner that cannot be assumed locally reacting, an assumption that rarely holds with any justification.

Artificial Boundaries

In most cases, the acoustic wave pattern that is to be simulated is not contained in a closed cavity. That is, there are boundaries in the model that do not represent a physical wall or limit of any kind. Instead, the boundary condition has to represent the interaction between the wave pattern inside the model and everything outside. Conditions of this kind are generically referred to as artificial boundary conditions (ABCs).

Such conditions should ideally contain complete information about the outside world, but this is not practical. After all, the artificial boundary was introduced to avoid spending degrees of freedom (DOFs) on modeling whatever is outside. The solution lies in trying to approximate the behavior of waves outside the domain using only information from the boundary itself. This is difficult in general for obvious reasons.

One particular case that occurs frequently in acoustics concerns boundaries that can be assumed to let wave energy propagate out from the domain without reflections. This leads to the introduction of a particular group of artificial boundary conditions known as non-reflecting boundary conditions (NRBCs), of which two kinds are available in this module: matched boundary conditions and radiation boundary conditions. The radiation boundary conditions apply primarily to wave guide ports connected to a cavity, while the matched boundary conditions approximate the boundary at infinity in an exterior problem. A drawback to these boundary conditions is that they are not perfectly nonreflecting when subjected to a general incoming wave. This is described in more detail in Theory for the Plane, Spherical, and Cylindrical Radiation Boundary Conditions and related sections.

Another way to model an open nonreflecting boundary is to add a so-called perfectly matched layer (PML) domain. This domain dampens all outgoing waves with no or

Porous Absorber: model library path Acoustics_Module/Industrial_Models/porous_absorber


46 | C H A P T E R

minimal reflections. See A Note About Perfectly Matched Layers (PMLs) for more information.

A Note About Perfectly Matched Layers (PMLs)

The perfectly matched layer (PML) is a domain or layer that is added to an acoustic model to mimic an open and non-reflecting infinite domain. It sets up a perfectly absorbing domain as an alternative to non-reflecting boundary conditions. The PML works with all types of waves, not only plane waves. It is also efficient at very oblique angles of incidence. The PML imposes a complex-valued coordinate transformation to the selected domain that effectively makes it absorbing at a maintained wave impedance, and thus eliminating reflections at the interface.

A Perfectly Matched Layers node is added to the model from the Model>Definitions node. The PMLs can be used for the Pressure Acoustics, Acoustic-Structure Interaction, Aeroacoustics, and Thermoacoustics interfaces.

The Absorptive Muffler and Muffler with Perforates models both use a nonreflecting boundary condition of the radiation type.

Infinite Element Domains and Perfectly Matched Layers in the COMSOL Multiphysics Reference Manual

The PMLs damp a certain wavelength existing in the system. The wavelength is deducted from the frequency and a reference wave speed cref. The wave speed is defined in the Typical Wave Speed section. Set cref equal to the speed of sound of the material in the PML.

If the PML spans both a solid and a fluid domain try to set an average value of the reference wave speed cref based on the speed in the fluid and the speed for compressional waves in the solid.


Evaluating the Acoustic Field in the Far-Field Region

The Acoustics Module has functionality to evaluate the acoustic pressure field in the far-field region outside of the computational domain. This is the Far-Field Calculation feature available for pressure acoustics problems. This section gives some general advice for analyzing the far field.

T H E N E A R - F I E L D A N D F A R - F I E L D R E G I O N S

The solution domain for a scattering or radiation problem can be divided into two zones, reflecting the behavior of the solution at various distances from objects and sources. In the far-field region, scattered or emitted waves are locally planar, velocity and pressure are in phase with each other, and the ratio between pressure and velocity approaches the free-space impedance of a plane wave.

Moving closer to the sources into the near-field region, pressure and velocity gradually slide out of phase. This means that the acoustic field contains energy that does not travel outward or radiate. These evanescent wave components are effectively trapped close to the source. Looking at the sound pressure level, local maxima and minima are apparent in the near-field region.

Naturally, the boundary between the near-field and far-field regions is not sharp. A general guideline is that the far-field region is that beyond the last local energy maximum, that is, the region where the pressure amplitude drops monotonously at a rate inversely proportional to the distance from any source or object, R.

A similar definition of the far-field region is the region where the radiation pattern—the locations of local minima and maxima in space—is independent of the distance to the wave source. This is equivalent to the criterion for Fraunhofer diffraction in optics, which occurs for Fresnel numbers, Fa2/R, much smaller than 1. For engineering purposes, this definition of the far-field region can be applied:

(2-3)

In Equation 2-3, a is the radius of a sphere enclosing all objects and sources, is the wavelength, and k is the wave number. Another way to write the expression leads to the useful observation that the size of the near-field region expressed in source-radius units is proportional to the dimensionless number k a, with a prefactor slightly larger than one.

R 8a2

---------- 8

2------ka2=


48 | C H A P T E R

Knowing the extent of the near-field region is useful when applying radiation boundary conditions because these are accurate only in the far-field region. PMLs, on the other hand, can be used to truncate a domain already inside the near-field region.

T H E H E L M H O L T Z - K I R C H H O F F I N T E G R A L R E P R E S E N T A T I O N

In many cases, solving the acoustic Helmholtz equation everywhere in the domain where results are requested is neither practical nor necessary. For homogeneous media, the solution anywhere outside a closed surface containing all sources and scatterers can be written as a boundary integral in terms of quantities evaluated on the surface. To evaluate this Helmholtz-Kirchhoff integral, it is necessary to know both Dirichlet and Neumann values on the surface. Applied to acoustics, this means that if the pressure and its normal derivative is known on a closed surface, the acoustic field can be calculated at any point outside, including amplitude and phase. This functionality is included in the Far-Field Calculation feature. The feature has two options for the evaluation, one full integral and one that only looks in the extreme far field. See the section Theory for the Far-Field Calculation: The Helmholtz-Kirchhoff Integral for further details.

F U L L I N T E G R A L

To evaluate the full Helmholtz-Kirchhoff integral use the Full integral option in the settings for the far-field variables. The full Helmholtz-Kirchhoff integral gives the pressure at any point at a finite distance from the source surface, but the numerical integration tends to lose accuracy at large distances. See Far-Field Calculation.

T H E F A R - F I E L D L I M I T

In many applications, the quantity of interest is the far-field radiation pattern, which can be defined as the limit of r | p | when r goes to infinity in a given direction. To evaluate the pressure in the far-field limit use the Integral approximation at r option in the settings for the far-field variables See Far-Field Calculation.

The Far Field Plots

Evaluating the acoustic pressure in the far field is essential for the development of several acoustic devices. This is especially true as it is essential to reduce the computational domain while still being able to determine the pressure in the near-field to far-field. Application areas range from underwater acoustic transducers and loudspeakers, to determining the spatial sensitivity of microphone systems (for example, using reciprocity).


The Far Field plots are specially designed for easy evaluation of the far-field variables, that is, the acoustics far-field pressure and the far-field sound pressure level.

The variables are plotted for a selected number of angles on a unit circle (in 2D) or a unit sphere (in 3D). The angle interval and the number of angles can be manually specified. Also the circle origin and radius of the circle (2D) or sphere (3D) can be specified. For 3D Far Field plots you also specify an expression for the surface color.

The main advantage with the Far Field plot, as compared to making a Line Graph, is that the unit circle/sphere that you use for defining the plot directions, is not part of your geometry for the solution. Thus, the number of plotting directions is decoupled from the discretization of the solution domain.

Solving Large Acoustics Problems Using Iterative Solvers

This section has some guidance for solving large acoustics problems. For smaller problems using a direct solver like MUMPS is often the best choice. For larger problems, especially in 3D, the only option is often to use an iterative method such as multigrid.

P U R E P R E S S U R E A C O U S T I C P R O B L E M S

The underlying equation for many of the problems within acoustics is the Helmholtz equation. For high frequencies (or wave numbers) the matrix resulting from a finite-element discretization becomes highly indefinite. In such situations, it can be

Default Far Field plots are automatically added to any model that uses far-field calculations.


• Far Field and Results Analysis and Plots in the COMSOL Multiphysics Reference Manual

• For a 3D example, see Bessel Panel: model library path Acoustics_Module/Tutorial_Models/bessel_panel.

• For a 2D axisymmetric example, see Cylindrical Subwoofer: model library path Acoustics_Module/Tutorial_Models/cylindrical_subwoofer.


50 | C H A P T E R

problematic to use geometric multigrid (GMG) with simple smoothers such as Jacobi or SOR (the default smoother). Fortunately, there exist robust and memory-efficient approaches that circumvent many of the difficulties associated with solving the Helmholtz equation using geometric multigrid.

When using a geometric multigrid as a linear system solver together with simple smoothers, the Nyquist criterion must be fulfilled on the coarsest mesh. If the Nyquist criterion is not satisfied, the geometric multigrid solver might not converge. One way to get around this problem is to use GMRES or FGMRES as a linear system solver with geometric multigrid as a preconditioner. The default preconditioner is the incomplete LU, right-click the Iterative solver node and select Multigrid. Even if the Nyquist criterion is not fulfilled for the coarse meshes of the multigrid preconditioner, such a scheme is more likely to converge. For problems with high frequencies this approach might, however, lead to a large number of iterations. Then it might be advantageous to use either:

• Geometric multigrid as a linear system solver (set the Solver selection to Use

preconditioner) with GMRES as a smoother. Under the Multigrid node right-click the Presmoother and Postsmoother nodes and select the Krylov Preconditioner with the Solver selection to GMRES.

• FGMRES as a linear system solver (set the Solver selection to FGMRES) with geometric multigrid as a preconditioner (where GMRES is used as a smoother, as above).

Using GMRES/FGMRES as an outer iteration and smoother removes the requirements on the coarsest mesh. When GMRES is used as a smoother for the multigrid preconditioner, FGMRES must be used for the outer iterations because such a preconditioner is not constant.

Use GMRES as a smoother only if necessary because GMRES smoothing is very time- and memory-consuming on fine meshes, especially for many smoothing steps.

When solving large acoustics problems, the following options, in increasing order of robustness and memory requirements, can be of use:

• If the Nyquist criterion is fulfilled on the coarsest mesh, try to use geometric multigrid as a linear system solver (set Multigrid as preconditioner and set the linear

See Y. Saad, “A Flexible Inner-outer Preconditioned GMRES Algorithm,” SIAM J. Sci. Statist. Comput., vol. 14, pp. 461–469, 1993.


system solver to Use preconditioner) with default smoothers. The default smoothers are fast and have small memory requirements.

• An option more robust than the first point is to use GMRES as a linear system solver with geometric multigrid as a preconditioner (where default SOR smoothers are used). GMRES requires memory for storing search vectors. This option can sometimes be used successfully even when the Nyquist criterion is not fulfilled on coarser meshes. Because GMRES is not used as a smoother, this option might find a solution faster than the next two options even if a large number of outer iterations are needed for convergence.

• If the above suggestion does not work, try to use geometric multigrid as a linear system solver with GMRES as a smoother.

• If the solver still has problems converging, try to use FGMRES as a linear system solver with geometric multigrid as a preconditioner (where GMRES is used as a smoother).

• Try to use as many multigrid levels as needed to produce a coarse mesh for which a direct method can solve the problem without using a substantial amount of memory.

• If the coarse mesh is still too fine for a direct solver, try using an iterative solver with 5–10 iterations as coarse solver.

A C O U S T I C - S T R U C T U R E I N T E R A C T I O N P R O B L E M S

In models that involve acoustic-structure interaction, the strategy for solving large problems involves solving the system in a segregated way. That is, the system is not solved fully coupled in one step but iterations are used solving one physics at the time.

To set up such a solver right-click on the Stationary Solver step and select Segregated. In the first Segregated Step solve for the structural dependent variables (displacement). Set up a second segregated step where you select the pressure dependent variable. Under each of the steps select the solver of choice for solving the single physics problem. For example, in the case where only a small structural domain is included, use an iterative multigrid approach for the acoustics (see above) and a direct solver for the structure.

This strategy is only readily applicable when the coupling between the solid and the acoustic domain is done via Neumann conditions; this is the case for all models where the acoustic domain uses Pressure Acoustics. In, for example, models with Thermoacoustic-Structure interaction the coupling is based on a Dirichlet condition


52 | C H A P T E R

(a pointwise constraint) and required reformulating the continuity condition using weak constraints.

About the Material Databases for the Acoustics Module

The Acoustics Module includes two material databases: Liquids and Gases, with temperature-dependent fluid dynamic and thermal properties, and a Piezoelectric Materials database with over 20 common piezoelectric materials.


• Studies and Solvers

• Multigrid

For detailed information about Materials, the Liquids and Gases Material Database, and the Piezoelectric Materials Database see the COMSOL Multiphysics Reference Manual.


Th e A c ou s t i c s Modu l e S t u d y T yp e s

The Acoustics Module is primarily designed for frequency-domain simulations, including related eigenvalue and mode analysis problems. Transient analysis is possible but less efficient from the computational point of view. The Thermoacoustics interfaces only support the frequency-domain type analysis. The Compressible Potential Flow interface is tailored to model a stationary background flow to be used in a subsequent time-harmonic aeroacoustics simulation. In the Solid Mechanics interface, the static analysis type is also included and can be use to model the stationary state of pre-stressed systems subject to time harmonic vibrations.

The analysis types require different solvers and equations. The following study types, briefly discussed in this section, can help you find good candidates for the application:

• Stationary Study

• Frequency Domain Study

• Eigenfrequency Study

• Mode Analysis Study

• Time Dependent Study

• Frequency Domain Modal and Time-Dependent Modal Studies

• Modal Reduced Order Model

• Additional Analysis Capabilities

Stationary Study

A stationary analysis solves for stationary displacements or a steady-state condition. All loads and constraints are constant. For a stationary analysis, use a Stationary study type ( ). For all pure acoustic and vibration problems this type of analysis yields the

Studies and Solvers and Harmonic Perturbation, Prestressed Analysis, and Small-Signal Analysis in the COMSOL Multiphysics Reference Manual

T H E A C O U S T I C S M O D U L E S T U D Y TY P E S | 53

54 | C H A P T E R

zero solution as, by definition, these represent and describe propagating varying fields—either time-dependent or time harmonic in the frequency domain.

Frequency Domain Study

Wave propagation is modeled by equations from linearized fluid dynamics (pressure waves) and structural dynamics (elastic waves). The full equations are time-dependent, but noting that a harmonic excitation of the field u has a time dependence of the form

gives rise to an equally harmonic response with the same frequency; the time can be eliminated completely from the equations. Instead the angular frequency f, enters as a parameter where f is the frequency.

This procedure is often referred to as working in the frequency domain or Fourier domain as opposed to the time domain. From the mathematical point of view, the time-harmonic equation is a Fourier transform of the original time-dependent equations and its solution as function of is the Fourier transform of a full transient solution. It is therefore possible to synthesize a time-dependent solution from a frequency-domain simulation by applying an inverse Fourier transform.

COMSOL Multiphysics and the Acoustics Module are based on the finite element method; a frequency domain simulation suits this method very well. Therefore, choose the Frequency Domain study type ( ) over a Time Dependent study whenever possible. Certain important software features, notably PMLs and damping due to porous media or boundary layer absorption, are only present when using the frequency domain physics interfaces.

The result of a frequency domain analysis is a complex time-dependent field u, which can be interpreted as an amplitude uamp = abs(u) and a phase angle uphase = arg(u). The actual displacement at any point in time is the real part of the solution:

Visualize the amplitudes and phases as well as the solution at a specific angle (time). When using the Solution data sets, the solution at angle (phase) parameter makes this

Stationary in the COMSOL Multiphysics Reference Manual

u ueit=

u uamp 2f t uphase+ cos=


task easy. When plotting the solution, COMSOL Multiphysics multiplies it by ei, where is the angle in radians that corresponds to the angle (specified in degrees) in the Solution at angle field. The plot shows the real part of the evaluated expression:

The angle is available as the variable phase (in radians) and is allowed in plot expressions. Both the frequency freq and angular frequency omega are available variables.

Eigenfrequency Study

If all sources are removed from a frequency-domain equation, its solution becomes zero for all but a discrete set of angular frequencies , where the solution has a well-defined shape but undefined magnitude. These solutions are known as eigenmodes and the corresponding frequencies as eigenfrequencies.

The eigenmodes and eigenfrequencies have many interesting mathematical properties, but also direct physical significance because they identify the resonance frequency (or frequencies) of the structure. When approaching a resonance frequency in a harmonically-driven problem, a weaker and weaker source is needed to maintain a given response level. At the actual eigenfrequency, the time-harmonic problem breaks down and lacks solution for a nonzero excitation.

Select the Eigenfrequency study type ( ) when you are interested in the resonance frequencies of the acoustic domain or the structure, whether you want to exploit them, as in a musical instrument, or avoid them, as in a reactive muffler or inside a hifi speaker system. To an engineer, the distribution of eigenfrequencies and the shape of eigenmodes can also give a good first impression about the behavior of a system.

u uamp uphase+ cos=

In a frequency domain study almost everything is treated as harmonic—prescribed pressures and displacements, velocities, and accelerations—not only the forces and dependent fields. Notable exceptions are certain quantities, such as the sound pressure level, which by definition are time averages.

Frequency Domain and Solution (data sets) in the COMSOL Multiphysics Reference Manual


56 | C H A P T E R

An eigenfrequency analysis solves for the eigenfrequencies and the shape of the eigenmodes. When performing an eigenfrequency analysis, specify whether to look at the mathematically more fundamental eigenvalue (available as the variable lambda) or the eigenfrequency f which is more commonly used in an acoustics context:

Mode Analysis Study

The Mode Analysis study ( ) is available with The Pressure Acoustics, Frequency Domain Interface, The Linearized Potential Flow, Frequency Domain Interface, and The Thermoacoustics, Frequency Domain Interface in plane 2D and axially symmetric 1D acoustics interfaces.

The Boundary Mode Acoustics Interface and The Linearized Potential Flow, Boundary Mode Interface are special interfaces for more advanced Mode Analysis studies on boundaries in 3D and 2D axisymmetry. Acoustic waves can propagate over large distances in ducts and pipes, with a generic name referred to as waveguides. After some distance of propagation in a waveguide of uniform cross section, such guided waves can be described as a sum of just a few discrete propagating modes, each with its own shape and phase speed. The equation governing these modes can be obtained as a spatial Fourier transform of the time-harmonic equation in the waveguide axial z direction, or more easily by inserting the assumption that the mode is harmonic in space,

and eliminating all out-of-plane z dependence.

f –2i---------=

Eigenfrequency in the COMSOL Multiphysics Reference Manual

The axial wave number kz is a parameter in the 2D acoustics physics interfaces.

u ue ikz z–=


Similar to the full time-harmonic equation, the transformed equation can be solved at a given frequency with a nonzero excitation for most axial wave numbers kz. But at certain discrete values the equation breaks down. These values are the propagation constants or wave numbers of the propagating or evanescent waveguide modes. The eigenvalue solver can solve for these propagation constants together with the corresponding mode shapes.

The most common use for the Mode Analysis is to define sources for a subsequent time-harmonic simulation. If there is a component with one or more waveguide connections, its behavior can be described by simulating its response to the discrete set of propagating modes on the waveguide port cross sections. In thermoacoustics a Mode Analysis study also provides information about the absorption coefficient for the propagating modes, which is the imaginary part of the wave number.

Time Dependent Study

The complete equations behind the theory of acoustic wave propagation are time-dependent, as discussed in the Frequency Domain Study section. Solving time-domain equations is more complicated from a numerical point of view and should therefore be avoided when possible. Short-term transient processes like step and

The propagating wave number is a function of the frequency. The relation between the two is commonly referred to as a dispersion curve.

Mode Analysis in the COMSOL Multiphysics Reference Manual

Jet Pipe: model library path Acoustics_Module/Tutorial_Models/jet_pipe


58 | C H A P T E R

impulse responses can benefit from modeling in the time domain, if not for efficiency so for convenience.

Frequency Domain Modal and Time-Dependent Modal Studies

The Frequency Domain Modal study type ( ) is used to do modal analysis in the frequency domain and the Time-Dependent Modal study ( ) is used to do time-dependent modal analysis.

Modal Reduced Order Model

The Modal Reduced Order Model study type ( ) is used to obtain the data necessary to construct reduced-order models from a COMSOL Multiphysics simulation. This study step is added after an existing Eigenvalue study step by right-clicking the Study 1 node and selecting Study Steps>Modal Reduced Order Model. After solving the model, right-click the Derived Values node (under Results) and select System Matrix. In the output section select the Matrix to display and the Format. Using the Matrix settings it is possible to access the stiffness, damping, and mass matrices of the system, for example.

Some central modeling techniques, such as the use of PMLs, are not available for the Time Dependent study type. Be careful when defining your sources to avoid, as far as possible, exciting waves at frequencies that the mesh cannot resolve.

Time Dependent in the COMSOL Multiphysics Reference Manual

Frequency Domain Modal and Time-Dependent Modal in the COMSOL Multiphysics Reference Manual


• Modal Reduced Order Model and Introduction to Solvers and Studies

• System Matrix and Results Analysis and Plots


Additional Analysis Capabilities

In a multiphysics interface you might want to use different analysis types for the different dependent variables. This can be done by adding an Empty Study ( ), and then adding different study steps to this study. Also perform parametric analyses by using the Parametric Sweep study node ( ). Typical parameters to vary include geometric properties, the frequency, and the out-of-plane or axial wave number.

Parametric Sweep in the COMSOL Multiphysics Reference Manual


60 | C H A P T E R

S p e c i a l V a r i a b l e s i n t h e A c ou s t i c s Modu l e

Several specialized variables specific to acoustics are predefined in the Acoustics Module and can be used when analyzing the results of an acoustic simulation. The variables are available from the expressions selection menus when plotting. In this section:

• Intensity Variables

• Power Dissipation Variables

• Boundary Mode Acoustics Variables

• Reference for the Acoustics Module Special Variables

Intensity Variables

The propagation of an acoustic wave is associated with a flow of energy in the direction of the wave motion, the intensity vector I. The sound intensity in a specific direction (through a specific boundary) is defined as the time average of energy flow per unit area in the direction of the normal to that area.

Knowledge of the intensity is important when characterizing the strength of a sound source—that is, the power emitted by the source. The power is given by the integral of n·I on a surface surrounding the source, where n is the surface normal. The intensity is also important when characterizing transmission phenomena, for example, when determining transmission loss or insertion loss curves.

The acoustic intensity vector I (SI unit: W/m2) is defined as the time average, or root mean square (RMS), of the instantaneous energy flow per unit area pu, such that


• Results Analysis and Plots

• Operators, Functions, and Constants

I1T---- pu td

0

T

=


where p is the pressure and u the particle velocity. In the frequency domain (harmonic time dependence) the integral reduces to

The instantaneous value of the intensity is defined as

Both the intensity (RMS) and the instantaneous intensity are available as results and analysis variables and can be selected from the expressions menus when plotting. The variables are defined for The Pressure Acoustics, Frequency Domain Interface, The Linearized Potential Flow, Frequency Domain Interface, and The Thermoacoustics, Frequency Domain Interface. The variables are defined in Table 2-1, Table 2-2, and Table 2-3. In the variable names, phys_id represents the physics interface identifier (for example, acpr for a Pressure Acoustics interface or ta for Thermoacoustics).

TABLE 2-1: INTENSITY VARIABLES IN 3D

VARIABLE DESCRIPTION

phys_id.I_rms Magnitude of the intensity vector

phys_id.Ix x-component of the intensity vector

phys_id.Iy y-component of the intensity vector

phys_id.Iz z-component of the intensity vector

phys_id.I_inst Magnitude of the instantaneous intensity vector

phys_id.Iix x-component of the instantaneous intensity vector

phys_id.Iiy y-component of the instantaneous intensity vector

phys_id.Iiz z-component of the instantaneous intensity vector

TABLE 2-2: INTENSITY VARIABLES IN 2D AXISYMMETRIC



phys_id.Ir r-component of the intensity vector

phys_id.Iz z-component of the intensity vector


phys_id.Iir r-component of the instantaneous intensity vector

phys_id.Iiz z-component of the instantaneous intensity vector

I12---Re pu 1

4--- pu pu+ = =

Iinst pu Re p Re u = =

S P E C I A L V A R I A B L E S I N T H E A C O U S T I C S M O D U L E | 61

62 | C H A P T E R

Power Dissipation Variables

Common to all the pressure acoustics viscous and thermally conducting fluid models (and The Thermoacoustics, Frequency Domain Interface) is that all the interfaces model some energy dissipation process, which stem from viscous and thermal dissipation processes. The amount of dissipated energy can be of interest as a results analysis variable or as a source term for a multiphysics problem. An example could be to determine the amount of heating in the human tissue when using ultrasound.

The energy conservation-dissipation corollary describes the transport and dissipation of energy in a system (see Ref. 1 pp. 516). In linear acoustics, this equation is derived by taking the dot product (scalar product) of the momentum and the velocity v, adding it to the continuity equation, and then adding the entropy. After some manipulation and integration, the use of Gauss’ theorem yields Equation 2-4

TABLE 2-3: INTENSITY VARIABLES IN 2D



phys_id.Ix x-component of the intensity vector

phys_id.Iy y-component of the intensity vector


phys_id.Iix x-component of the instantaneous intensity vector

phys_id.Iiy y-component of the instantaneous intensity vector


• Results Analysis and Plots

• Expressions and Predefined Quantities


(2-4)

where:

• w is the disturbance energy of the control volume

• v = |v| is the velocity

• T is the temperature variations

• p is the acoustic pressure

• p0 is the static pressure

• T0 the static temperature

• 0 the static density

• c0 the isentropic speed of sound

• Cp the heat capacity at constant pressure

• k the coefficient of thermal conduction

• I is the flux of energy out of a control volume

• is the dissipated energy per unit volume and time (SI unit: Pa/s = J/(m3s) = W/m3)

• s is the entropy

• is the viscous dissipation function, and

• v and t are the viscous and thermal contributions to the dissipation function.

In the Acoustics Module special variables exist for the dissipation term .

For the case of a plane wave propagating in the bulk of a fluid (the general thermal and viscous fluid models described in About the Thermally Conducting and Viscous Fluid Model) the dissipation is

t w Vd I n Ad

+ Vd

–= or w

t------- I+ –=

w 12---0v2 1

2--- p2

0c02

----------- 12---0T0Cp

-------------s2+ +=

I pv ejviijk

T0------T T–

ij–=

:vk

T0------ T 2+ v t+= =

:v


64 | C H A P T E R

and in the frequency domain after averaging over one period

(2-5)

where * in Equation 2-5 is the complex conjugate operator.

For the case of the Thermoacoustics, Frequency Domain interface, the dissipation term is directly given by the RMS value of the tensor expression

(2-6)

where: in Equation 2-5 is the double dot operator. In the above expressions, the time averaged expressions for a product in the frequency domain is defined as:

The power dissipation variables are defined in Table 2-4. In the variable names, phys_id represents the physics interface identifier (acpr, for example, for a Pressure Acoustics interface).

Boundary Mode Acoustics Variables

A series of special variables exist for postprocessing after solving a boundary mode acoustics problem. They include in-plane and out-of-plane components of the velocity v and acceleration a.

The in-plane (ip) and out-of-plane (op) components to the acceleration and velocity are defined as

TABLE 2-4: POWER DISSIPATION VARIABLES


phys_id.diss_therm Thermal power dissipation density

phys_id.diss_visc Viscous power dissipation density

phys_id.diss_tot Total thermal and viscous power dissipation density

1

0c02

2------------------- 4

3---

b------+

k 1– Cp

--------------------+pt------ 2

=

1

0c02

2------------------- 4

3---

b------+

k 1– Cp

--------------------+2

2------pp=

v :v 14--- :v : v + = =

AB Re Aeit Re Beit 14--- AB AB+ = =


where n is the normal to the surface being modeled. The velocity and acceleration are defined in terms of the gradient of the pressure p as follows

and

where kn is the out-of-plane wave number solved for, m is a possible radial wave mode number, and is the tangential derivative along the boundary.

The boundary mode acoustics variables are defined in Table 2-5. In the variable names, phys_id represents the physics interface identifier (acbm, for example, for a Boundary Mode Acoustics interface).

TABLE 2-5: BOUNDARY MODE ACOUSTICS VARIABLES IN 3D


phys_id.vipx In-plane velocity, x-component

phys_id.vipy In-plane velocity, y-component

phys_id.vipz In-plane velocity, z-component

phys_id.vip_rms In-plane velocity RMS value

phys_id.aipx In-plane acceleration, x-component

phys_id.aipy In-plane acceleration, y-component

phys_id.aipz In-plane acceleration, z-component

phys_id.aip_rms In-plane acceleration RMS value

phys_id.vopx Out-of-plane velocity, x-component

phys_id.vopy Out-of-plane velocity, y-component

phys_id.vopz Out-of-plane velocity, z-component

phys_id.vop_rms Out-of-plane velocity RMS value

aip a a n n–=

vip v v n n–=

aop a n n=

vop v n n=

p tp iknpn–= in 3D

p tp iknp nr 0 nz – p 0 imr-----– 0

+= in 2D axisymmetry

vi-------p= and a iv=

t


66 | C H A P T E R

Reference for the Acoustics Module Special Variables

1. A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustic Society of America, Melville, New York, 1991.

phys_id.aopx Out-of-plane acceleration, x-component

phys_id.aopy Out-of-plane acceleration, y-component

phys_id.aopz Out-of-plane acceleration, z-component

phys_id.aop_rms Out-of-plane acceleration RMS value

TABLE 2-5: BOUNDARY MODE ACOUSTICS VARIABLES IN 3D



3

T h e P r e s s u r e A c o u s t i c s B r a n c h

This chapter describes the Acoustics Module background theory and physics interfaces found under the Pressure Acoustics branch ( ) when adding a physics interface.

• The Pressure Acoustics, Frequency Domain Interface

• The Pressure Acoustics, Transient Interface

• The Boundary Mode Acoustics Interface

• Theory Background for the Pressure Acoustics Branch

• Theory for the Pressure Acoustics Fluid Models

• References for the Pressure Acoustics Branch

65

66 | C H A P T E R

Th e P r e s s u r e A c ou s t i c s , F r e qu en c y Doma i n I n t e r f a c e

The Pressure Acoustics, Frequency Domain (acpr) interface ( ), found under the Pressure Acoustics branch ( ) when adding a physics interface, is used to compute the pressure variation for propagation of acoustic waves in fluids at quiescent background conditions. It is suited for all frequency-domain simulations with harmonic variations of the pressure field.

The physics interface can be used for linear acoustics described by a scalar pressure variable. It includes domain conditions to model losses in a homogenized way, so-called fluid models for porous materials, as well as losses in narrow regions. Domain features also include background incident acoustic fields, as well as domain monopole and dipole sources. The plane wave attenuation behavior of the acoustic waves may be entered as a user-defined quantity, or defined to be bulk viscous and thermal losses.

The physics interface solves the Helmholtz equation in the frequency domain for given frequencies, or as an eigenfrequency or modal analysis study.

An acoustics model can be part of a larger multiphysics model that describes, for example, the interactions between structures and acoustic waves. This physics interface is suitable for modeling acoustics phenomena that do not involve fluid flow.

The sound pressure p, which is solved for in pressure acoustics, represents the acoustic variations (or excess pressure) to the ambient pressure. In the absence of flow, the ambient pressure is simply the static absolute pressure. In the presence of a background acoustic pressure wave pb the total acoustic pressure pt is the sum of the pressure solved for p and the background pressure wave. The governing equations are formulated using the total pressure in a a so-called scattered field formulation. The equations hence contain the information about the background pressure, which for example could be a user-defined incident wave or a plane wave.

When the geometrical dimensions of the acoustic problems are reduced from 3D to 2D (planar symmetry or axisymmetric) or to 1D axisymmetric, it is possible to specify an out-of-plane wave number kz and a circumferential wave number m, when applicable. The wave number used in the equations keq contains both the ordinary wave number k as well as the out-of-plane wave number and circumferential wave number, when applicable.

3 : T H E P R E S S U R E A C O U S T I C S B R A N C H

The Pressure Acoustics interface solves the full acoustic problem including a priori knowledge about the acoustic problem, in the form of background pressure fields and symmetries.

The following table lists the names and SI units for the most important physical quantities in the Pressure Acoustics, Frequency Domain interface:

In the following descriptions of the functionality in this physics interface, the subscript c in c and cc (the density and speed of sound, respectively) denotes that these can be complex-valued quantities in models with damping.

When this physics interface is added, these default nodes are also added to the Model

Builder— Pressure Acoustics Model, Sound Hard Boundary (Wall), and Initial Values.

Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and point conditions. You can also right-click Pressure Acoustics to select physics from the context menu.

I N T E R F A C E I D E N T I F I E R

The interface identifier is used primarily as a scope prefix for variables defined by the physics interface. Refer to such interface variables in expressions using the pattern

TABLE 3-1: PRESSURE ACOUSTICS, FREQUENCY DOMAIN INTERFACE PHYSICAL QUANTITIES

QUANTITY SYMBOL SI UNIT ABBREVIATION

Pressure p pascal Pa

Density kilogram/meter3 kg/m3

Frequency f hertz Hz

Wave number k 1/meter 1/m

Dipole source qd newton/meter3 N/m3

Monopole source Qm 1/second2 1/s2

Speed of sound c meter/second m/s

Acoustic impedance Z pascal-second/meter Pa·s/m

Normal acceleration an meter/second2 m/s2

Source location r0 meter m

Wave direction nk (dimensionless) 1

Physics Nodes—Equation Section in the COMSOL Multiphysics Reference Manual

T H E P R E S S U R E A C O U S T I C S , F R E Q U E N C Y D O M A I N I N T E R F A C E | 67

68 | C H A P T E R

<identifier>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter.

The default identifier (for the first interface in the model) is acpr.

D O M A I N S E L E C T I O N

The default setting is to include All domains in the model to define a sound pressure field and the associated acoustics equation. To choose specific domains, select Manual from the Selection list.

E Q U A T I O N

Expand the Equation section to see the equations solved for with the Equation form specified. The default selection is Equation form is set to Study controlled. The available studies are selected under Show equations assuming.

P R E S S U R E A C O U S T I C S E Q U A T I O N S E T T I N G S

When the Equation form is set to Study controlled, the scaling and non-reflecting boundary settings are optimized for the numerical performance of the different solvers.

To display the Pressure Acoustics Equation Settings section for 2D and 1D models, click to expand the Equation section, then select Frequency domain as the Equation form and enter the settings as described in Scaling Factor and Non-reflecting Boundary Condition Approximation.

For 1D axisymmetric models, the Circumferential wave number m (dimensionless) default is 0 and the Out-of-plane wave number kz (SI unit: rad/m) default is 0 rad/m. Enter different values or expressions as required.

For 2D axisymmetric models, the Circumferential wave number m (dimensionless) default is 0. Enter a different value or expression as required.


Scaling Factor and Non-reflecting Boundary Condition ApproximationFor all component dimensions, and if required, click to expand the Equation section, then select Frequency domain as the Equation form and enter the settings as described below.

The default Scaling factor is 1/2 and Non-reflecting boundary condition

approximation is Second order. These values correspond to the equations for a Frequency Domain study when the equations are study controlled.

To get the equations corresponding to an Eigenfrequency study, change the Scaling

factor to 1 and the Non-reflecting boundary conditions approximation to First order.

S O U N D P R E S S U R E L E V E L S E T T I N G S

The zero level on the dB scale varies with the type of fluid. That value is a reference pressure that corresponds to 0 dB. This variable occurs in calculations of the sound pressure level Lp based on the root mean square (rms) pressure prms, such that

where pref is the reference pressure and the star (*) represents the complex conjugate. This is an expression valid for the case of harmonically time-varying acoustic pressure p.

Based on the fluid type, select a Reference pressure for the sound pressure level. Select:

• Use reference pressure for air to use a reference pressure of 20 Pa (20·106 Pa).

• Use reference pressure for water to use a reference pressure of 1 Pa (1·106 Pa).

• User-defined reference pressure to enter a reference pressure pref, SPL (SI unit: Pa). The default value is the same as for air, 20 Pa.

TY P I C A L WAV E S P E E D

Enter a value or expression for the Typical wave speed for perfectly matched layers cref (SI unit m/s). The default is 343 m/s.

For 2D models, the Out-of-plane wave number kz (SI unit: rad/m) default is 0 rad/m. Enter a different value or expression as required.

Lp 20prmspref---------- log= with prms

12--- p p=


70 | C H A P T E R

D E P E N D E N T V A R I A B L E S

This physics interface defines one dependent variable (field), the Pressure p. If required, edit the name, which changes both the field name and the dependent variable name. If the new field name coincides with the name of another pressure field in the model, the interfaces share degrees of freedom and dependent variable name. The new field name must not coincide with the name of a field of another type, or with a component name belonging to some other field.

D I S C R E T I Z A T I O N

To display this section, click the Show button ( ) and select Discretization. Select Quadratic (the default), Linear, Cubic, Quartic, or Quintic for the Pressure. Specify the Value type when using splitting of complex variables—Real or Complex (the default).

Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure Acoustics, Frequency Domain Interface

The Pressure Acoustics, Frequency Domain Interface has these domain, boundary, edge, point, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

• Show More Physics Options

• Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure Acoustics, Frequency Domain Interface


Eigenmodes of a Room: model library path COMSOL_Multiphysics/Acoustics/eigenmodes_of_room

This model also requires the Particle Tracing Module—Acoustic Levitator: model library path Acoustics_Module/Tutorial_Models/acoustic_levitator



• Background Pressure Field

• Circular Source

• Continuity

• Cylindrical Wave Radiation

• Dipole Source

• Destination Selection in the COMSOL Multiphysics Reference Manual

• Far-Field Calculation

• Impedance

• Incident Pressure Field

• Interior Normal Acceleration

• Interior Impedance/Pair Impedance

• Interior Perforated Plate

• Interior Sound Hard Boundary (Wall)

• Initial Values

• Line Source

• Line Source on Axis

• Matched Boundary

• Monopole Point Source

• Monopole Source

• Narrow Region Acoustics

• Normal Acceleration

• Periodic Condition

• Plane Wave Radiation

• Point Source

• Poroacoustics

• Pressure Acoustics

• Pressure

• Sound Hard Boundary (Wall)

• Sound Soft Boundary

• Spherical Wave Radiation

• Symmetry

Continuity in the total pressure is the default condition on interior boundaries.

The Pressure Acoustics, Transient Interface also shares these nodes, with some additional features described in Domain, Boundary, Edge, and Point Nodes for the Pressure Acoustics, Transient Interface.

The Boundary Mode Acoustics Interface also shares these nodes, with one additional feature described in Boundary, Edge, Point, and Pair Nodes for the Boundary Mode Acoustics Interface. For the Boundary Mode Acoustics interface, apply the feature to boundaries instead of domains for 3D models.


72 | C H A P T E R

Monopole Source

Use the Monopole Source node to add a the domain source term Qm to the governing equation. A monopole source added to a domain has a uniform strength in all directions. In advanced models this source term can, for example, be used to represent a domain heat source causing pressure variations. Add this node from the More> submenu.


From the Selection list, choose the domains to define.

M O N O P O L E S O U R C E

Enter a Monopole source Qm (SI unit: 1/s2). The default is 0 1/s2.

Dipole Source

Use the Dipole Source node to add the domain source term qd to the governing equation. This source is typically stronger in two opposite directions. In advanced models this term can, for example, be used to represent a uniform constant background flow convecting the sound field. Add this node from the More> submenu.



For axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r = 0) into account and automatically adds an Axial Symmetry node to the model that is valid on the axial symmetry boundaries only.

In a transient model the Monopole Source can be used to add nonlinearities to the governing equation. See the model Nonlinear Acoustics: Modeling of the 1D Westervelt Equation for such an example: model library path Acoustics_Module/Tutorial_Models/nonlinear_acoustics_westervelt_1d.


D I P O L E S O U R C E

Enter coordinates for the Dipole source qd (SI unit: N/m3). These are the individual components of the dipole source vector. The defaults are 0 N/m3.

Initial Values

The Initial Values node adds initial values for the sound pressure and the pressure time derivative that can serve as an initial guess for a nonlinear solver. If more than one initial value is needed, from the Physics toolbar click to add more Initial Values nodes.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or select All domains as required.

I N I T I A L V A L U E S

Enter a value or expression for the initial values for the Pressure p (SI unit: Pa) and the Pressure, first time derivative, p/t (SI unit: Pa/s). The defaults are 0 Pa and 0 Pa/s, respectively.

Sound Hard Boundary (Wall)

The Sound Hard Boundary (Wall) adds a boundary condition for a sound hard boundary or wall, which is a boundary at which the normal component of the acceleration is zero:

For zero dipole source and constant fluid density, this means that the normal derivative of the pressure is zero at the boundary:

Sound-hard boundaries are available for all study types. Note that this condition is identical to the Symmetry condition.

n–10------– p qd–

0=

np 0=


74 | C H A P T E R

B O U N D A R Y S E L E C T I O N

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required.

Normal Acceleration

The Normal Acceleration adds an inward normal acceleration an:

Alternatively, specify the acceleration a0 of the boundary. The part in the normal direction is used to define the boundary condition:

This feature represents an external source term. It can also be used to manually couple acoustics with a structural analysis for modeling acoustic-structure interaction.


From the Selection list, choose the boundaries to define.

N O R M A L A C C E L E R A T I O N

Select a Type—Inward Acceleration (the default) or Acceleration.

• If Inward Acceleration is chosen, enter the value of the Inward acceleration an (SI unit: m/s2). The default is 0 m/s2. Use a positive value for inward acceleration or a negative value for outward acceleration.

• If Acceleration is chosen, enter values for the components of the Acceleration a0 (SI unit: m/s2). The defaults are 0 m/s2.

Sound Soft Boundary

The Sound Soft Boundary adds a boundary condition for a sound soft boundary, where the acoustic pressure vanishes: p = 0. It is an appropriate approximation for a liquid-gas interface and in some cases for external waveguide ports.

n–10------– p qd–

an=

n10------– p qd–

n a 0=



From the Selection list, choose the boundaries to define. If the node is selected from the Pairs submenu, this list cannot be edited and it shows the boundaries in the selected pairs.

P A I R S E L E C T I O N

If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect.

C O N S T R A I N T S E T T I N G S

To display this section, click the Show button ( ) and select Advanced Physics Options. To Apply reaction terms on all dependent variables, select All physics (symmetric). Otherwise, select Current physics (internally symmetric) or Individual dependent

variables to restrict the reaction terms as required. Select the Use weak constraints check box to replace the standard constraints with a weak implementation.

Pressure

The Pressure node creates a boundary condition that acts as a pressure source at the boundary, which means a constant acoustic pressure p0 is specified and maintained at the boundary: p = p0. In the frequency domain, p0 is the amplitude of a harmonic pressure source.

The node is also available from the Pairs submenu as an option at interfaces between parts in an assembly.



P R E S S U R E

Enter the value of the Pressure p0 (SI unit: Pa). The default is 0 Pa.




These are the same settings as for Sound Soft Boundary.


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Impedance

The Impedance node adds an impedance boundary condition, which is a generalization of the sound-hard and sound-soft boundary conditions:

In the Pressure Acoustics, Transient interface using a Time Dependent study, the impedance boundary condition is the following:

Here Zi is the acoustic input impedance of the external domain and it has the unit of a specific acoustic impedance. From a physical point of view, the acoustic input impedance is the ratio between the local pressure and local normal particle velocity.

The Impedance boundary condition is a good approximation for a locally reacting surface—a surface for which the normal velocity at any point depends only on the pressure at that exact point.



I M P E D A N C E

Enter the value of the Impedance Zi (SI unit: Pa·s/m). The default value is set to the specific impedance of air 1.2 kg/m3·343 m/s.

Symmetry

The Symmetry node adds a boundary condition where there is symmetry in the pressure. Use this condition to reduce the size of a model by cutting it in half where there are symmetries. In pressure acoustics this boundary condition is mathematically identical to the Sound Hard Boundary (Wall) condition.

n–1c-----– pt qd–

iptZi

-----------–=

n–1c-----– p qd–

1Zi-----

tpt=

In the two opposite limits Zi and Zi0, this boundary condition is identical to the Sound Hard Boundary (Wall) condition and the Sound Soft Boundary condition, respectively.






Plane Wave Radiation

The Plane Wave Radiation node adds a radiation boundary condition for a plane wave. If required, from the Physics toolbar, add an Incident Pressure Field to model an incoming wave. This radiation condition allows an outgoing plane wave to leave the modeling domain with minimal reflections, when the angle of incidence is near to normal.

The plane wave type is suitable for both far-field boundaries and ports. Because many waveguide structures are only interesting in the plane-wave region, it is particularly relevant for ports. When using the radiation condition on an open far-field boundary it is recommended to construct the boundary such that the incidence angle is near to normal, this of course requires a priory knowledge of the problem and the solution.

See the theory section Theory for the Plane, Spherical, and Cylindrical Radiation Boundary Conditions for details about the equations and the formulation of this non-reflecting boundary condition.

An estimate of the reflection coefficient Rs, for the spurious waves reflecting off the plane wave radiation boundary, is, for incident plane waves at angle , given by the expression:

where N is the order of the boundary condition (here 1 or 2). So at normal incidence (= 0) there are no spurious reflections, while, for example, at an incidence angle of 30o for N = 2 (plane wave radiation in the frequency domain) the amplitude of the spurious reflected wave is 0.5 % of the incident.

Rs 1–cos 1+cos

----------------------N

=


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Spherical Wave Radiation

The Spherical Wave Radiation node adds a radiation boundary condition for a spherical wave, for which you define the source location. If required, from the Physics toolbar add an Incident Pressure Field to model an incoming wave. This radiation condition allows an outgoing spherical wave to leave the modeling domain with minimal reflections. The geometry of the modeling domain should be adapted to have the outgoing spherical waves coincide with the boundary, this is in order to minimize spurious reflections.




S P H E R I C A L WAV E R A D I A T I O N

Enter coordinates for the Source location r0 (SI unit: m). The defaults are 0 m.

Cylindrical Wave Radiation

The Cylindrical Wave Radiation node adds a radiation boundary condition for a cylindrical wave, for which you define the source location and the source axis direction If required, from the Physics toolbar add an Incident Pressure Field to model an incoming wave. This radiation condition allow an outgoing cylindrical wave to leave

• Acoustics of a Muffler: model library path COMSOL_Multiphysics/Acoustics/automotive_muffler

• Absorptive Muffler: model library path Acoustics_Module/Industrial_Models/absorptive_muffler

Bessel Panel: model library path Acoustics_Module/Tutorial_Models/bessel_panel


the modeling domain with minimal reflections. The geometry of the modeling domain should be adapted to have the outgoing cylindrical waves coincide with the boundary, this is in order to minimize spurious reflections.




C Y L I N D R I C A L WAV E R A D I A T I O N

Enter coordinates for the Source location r0 (SI unit: m) (the defaults are 0 m) and the Source axis direction raxis (dimensionless) (the defaults are 0).

Incident Pressure Field

The Incident Pressure Field node is a subnode to all non-reflecting boundary conditions (plane, cylindrical, spherical wave radiation, and matched boundary). From the Physics toolbar, add Matched Boundary, Plane Wave Radiation, Spherical Wave Radiation, or Cylindrical Wave Radiation nodes. If the incident pressure field pi is a predefined plane wave, it is of the type:

Acoustic Cloaking: model library path Acoustics_Module/Tutorial_Models/acoustic_cloaking

pi p0ei k r –

p0eikeq

r e k

ek

------------- –

= =


80 | C H A P T E R

where p0 is the wave amplitude, k is the wave vector (with amplitude keq=|k| and wave direction vector ek), and r is the location on the boundary. The incident pressure field can also be a user-defined value or expression.


From the Selection list, choose the boundaries to include an incident pressure field pi in the boundary condition. By default, this feature node inherits the selection from its parent node, and only a selection that is a subset of the parent node’s selection can be used.

I N C I D E N T P R E S S U R E F I E L D

From the Incident pressure field type list, select Plane wave to define an incident pressure field of plane wave type. Then enter a Pressure amplitude p0 (SI unit: Pa) (the default is 0 Pa) and Wave direction ek (SI unit: m).

Select User defined to enter the expression for the Incident pressure field pi (SI unit: Pa) as a function of space. The default is 0 Pa.

Periodic Condition

The Periodic Condition node adds a periodic boundary condition that can be used to reduce the model size by using symmetries and periodicities in the geometry and physics being modeled.

In transient analysis the incident pressure field is only of the user-defined type. In this case the incident pressure field needs to be defined as a traveling wave of the form

where is the angular frequency and k is the wave vector. The function f is any function, for example, a sine function. This is a requirement for the radiation boundary condition to work properly.

f t k x–

The Porous Absorber model uses Floquet periodic boundary conditions to model an infinite porous absorber used for sound proofing. The model library path is Acoustics_Module/Industrial_Models/porous_absorber



From the Selection list, choose the boundaries to define. The software automatically identifies the boundaries as either source boundaries or destination boundaries.

PE R I O D I C I T Y S E T T I N G S

Select a Type of periodicity—Continuity (the default), Floquet periodicity, Cyclic

symmetry, User defined, or Antiperiodicity. For the Aeroacoustics interfaces, only Continuity and Antiperiodicity are available as the Type of periodicity.

• If Floquet periodicity is selected, enter a k-vector for Floquet periodicity kF (SI unit: rad/m) for the x, y, and z coordinates (3D models), or the r and z coordinates (2D axisymmetric models), or x and y coordinates (2D models). This condition is used for modeling infinite periodic structures with non-normal incident pressure fields or excitations. Use this condition to model, for example, a large perforated plate with an oblique incident wave with wave vector k (and set kF = k) by only analyzing one hole or one subset of holes that is periodic.

• If Cyclic symmetry is selected, select a Sector angle—Automatic (the default), or User

defined. If User defined is selected, enter a value for S (SI unit: rad). For any selection, also enter an Azimuthal mode number m (dimensionless). This condition is used to model any geometry that has a cyclic periodic structure like, for example, a microphone or a loudspeaker driver. Setting the azimuthal mode number m determines which mode is analyzed. The response of the full system to an external excitation is in general a linear combination of many different modes.

For the case of the Acoustic-Solid Interaction, Frequency Domain, the Acoustic-Piezoelectric Interaction, Frequency Domain, and the Thermoacoustic-Solid Interaction, Frequency Domain interfaces:

• If User defined is selected, select the Periodic in u, Periodic in v (for 3D and 2D models), and Periodic in w (for 3D and 2D axisymmetric models) check boxes as

This feature works well for cases like opposing parallel boundaries. In other cases use a Destination Selection subnode to control the destination. By default it contains the selection that COMSOL Multiphysics identifies.

When an acoustics interface uses a Solid Mechanics or Piezoelectric Devices interface, User defined is often available as a Type of periodicity.


82 | C H A P T E R

required. Then for each selection, choose the Type of periodicity—Continuity (the default) or Antiperiodicity.

The available check boxes for the User defined option are based on the interface

• Acoustic-Solid Interaction, Frequency Domain: Periodic in p, and Periodic in u (component wise).

• Acoustic-Piezoelectric Interaction, Frequency Domain and Acoustic-Piezoelectric Interaction, Transient: Periodic in p, Periodic in V, and Periodic in u (component wise).

• Thermoacoustic-Solid Interaction, Frequency Domain: Periodic in p, Periodic in

u_fluid (component wise), Periodic in T, and Periodic in u_solid (component wise).



Interior Sound Hard Boundary (Wall)

The Interior Sound Hard Boundary (Wall) node adds a boundary condition for a sound hard boundary or wall on interior boundaries. A sound-hard boundary is a boundary at which the normal component of the acceleration is zero:

In the time domain both the Cyclic symmetry and the Floquet periodicity boundary conditions reduce to the continuity condition.

To optimize the performance of the Floquet periodicity and the Cyclic

symmetry conditions it is recommended that the source and destination meshes are identical. This can be achieved by first meshing the source boundary or edge and then copy the mesh to the destination boundary or edge.


• Periodic Condition and Destination Selection

• Periodic Boundary Conditions


where the subscripts 1 and 2 represent the two sides of the boundary. For zero dipole charge and constant fluid density, this means that the normal derivative of the pressure is zero at the boundary. On an interior sound hard boundary the pressure is not continuous but is treated as a so-called slit feature.



Axial Symmetry


Continuity

Continuity is available as an option at interfaces between parts in a pair.

This condition gives continuity in total pressure and in the normal acceleration over the pair (subscripts 1 and 2 in the equation refer to the two sides in the pair):

n–1c----- pt qd– –

1 0= n–

1c-----–

pt qd–

2 0=

The Axial Symmetry node is a default node added for all 2D and 1D axisymmetric models. The boundary condition is active on all boundaries on the symmetry axis.

The boundaries section shows on which boundaries the node is active. All boundaries on the symmetry axis are automatically selected.


• Continuity on Interior Boundaries

• Identity and Contact Pairs


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When this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect.



Pressure Acoustics

The Pressure Acoustics node adds the equations for time-harmonic and eigenfrequency acoustics modeling in the frequency domain. In the settings window, define the properties for the acoustics model and model inputs including the background pressure and temperature.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from

n–1c----- pt qd–

1–

1c-----– pt qd–

2

– 0=

This list cannot be edited. It shows the boundaries in the selected pairs.

For more details about each of the available fluid models, see Theory for the Pressure Acoustics Fluid Models.

For more information about using variables during the results analysis, see Special Variables in the Acoustics Module.

These fluid models are defined using individual physics feature nodes: Poroacoustics and Narrow Region Acoustics


the Selection list to choose specific domains to compute the acoustic pressure field and the equation that defines it, or select All domains as required.

M O D E L I N P U T S

For all fluid models enter a Temperature T and an Absolute pressure p:

• Select User defined to enter a value or an expression for the absolute pressure (SI unit: Pa) and the temperature (SI unit: K) in the field. This input is always available.

• In addition, select a temperature field defined by, for example, a heat transfer interface or a non-isothermal fluid flow interface (if any).

• If applicable, select a pressure defined by a fluid-flow interface present in the model. For example, select Pressure (spf/fp1) to use the pressure defined by the Fluid

Properties node fp1 in a Single-Phase Flow interface spf. Selecting a pressure variable also activates a check box for defining the reference pressure, where 1[atm] is the default value. This makes it possible to use a system-based (gauge) pressure, while automatically including the reference pressure in the absolute pressure.

The input to these fields influences the value of the material parameters in the model. Typically, the density and the speed of sound c in the model are dependent on the absolute pressure and/or the temperature. Picking up any of those from another interface typically results in xand c = cx to be specially varying.

P R E S S U R E A C O U S T I C S M O D E L

To define the properties of the bulk fluid, select a Fluid model from the list. The fluid models represent different loss mechanisms applied in a homogenized way to the bulk of the fluid. This type of fluid model is sometimes referred to as an equivalent fluid model. The model may be a theoretical model or a model based on measurement data for the attenuation in the fluid.

Non-isothermal flow requires the addition of the Heat Transfer Module or CFD Module.

Detailed aeroacoustic models that take into account the full background flow (including the movement of the fluid) are available under The Aeroacoustics Branch.


86 | C H A P T E R

Use the Linear elastic selection (the default), to specify a linearly elastic fluid using either the density and speed of sound or the impedance and wave number. When the material parameters are real values this corresponds to a lossless compressible fluid. Go to Defining a Linear Elastic Fluid Model.

The fluid models may be roughly divided into these two categories:

• General fluids:

- Linear elastic. Go to Defining a Linear Elastic Fluid Model.

- Linear elastic with attenuation. Go to Defining a Linear Elastic with Attenuation Fluid Model.

- Ideal gas. Go to Defining an Ideal Gas Fluid Model.

• Fluids with bulk viscous and/or thermal losses:

- Viscous. Go to Defining a Viscous Fluid Model.

- Thermally conducting. Go to Defining a Thermally Conducting Fluid Model.

- Thermally conducting and viscous. Go to Defining a Thermally Conducting and Viscous Fluid Model.

Defining a Linear Elastic Fluid ModelThis is the default fluid model that lets you define the fluid properties. From the Specify list, select Density and speed of sound (the default) or Impedance and wave number. To add user defined losses, in a general way, specify the properties as complex-valued data.

• If Density and speed of sound is selected, the default Speed of sound c (SI unit: m/s) and Density (SI unit: kg/m3) values are taken From material. Select User defined to enter other values or expressions.

• If Impedance and wave number is selected, enter a Wave number k (SI unit: rad/m). By default the Characteristic acoustic impedance Z (SI unit: Pa·s/m) is the value taken From material for the fluid. Select User defined to enter other values or expressions.

The theory for the fluid models is in the section Theory for the Pressure Acoustics Fluid Models.


Defining a Linear Elastic with Attenuation Fluid ModelThis model adds a user defined attenuation to the fluid, this data is typically based on experimental data for the attenuation coefficient . Adding attenuation will make the wavenumber k complex valued. A plane wave p(x) moving in the x-direction is then

The wave thus has a spatial exponential decay governed by the attenuation coefficient.

The default Speed of sound c (SI unit: m/s) and Density (SI unit: kg/m3) values are taken From material. Select User defined to enter other values or expressions for one or both options.

Select an Attenuation type—Attenuation coefficient Np per unit length to define an attenuation coefficient in Np/m (nepers per meter), Attenuation coefficient dB per

unit length to define an attenuation coefficientin dB/m (decibel per meter), or Attenuation coefficient dB per wavelength to define an attenuation coefficient in dB/ (decibel per wavelength). For any selection, enter a value or expression in the Attenuation coefficient field.

Defining a Viscous Fluid ModelThis fluid model adds the attenuation due to bulk viscous losses. This type of model is relevant in highly viscous fluids or when acoustic waves are traveling over large distances. For each of the following, the default values are taken From material. Select User defined to enter other values or expressions for any or all options.

• Speed of sound c (SI unit: m/s)

• Density (SI unit: kg/m3)

k c---- i–=

p x e ikx– e i c x– e x–= =

About the Linear Elastic with Attenuation Fluid Model


88 | C H A P T E R

• Dynamic viscosity (SI unit: Pa·s)

• Bulk viscosity B (SI unit: Pa·s)

Defining a Thermally Conducting Fluid ModelThis fluid model adds the attenuation due to thermal conduction effects in the bulk of the fluid. This type of model is relevant in fluids that have high thermal conduction or when acoustic waves are traveling over large distances. For each of the following, the default values are taken From material. Select User defined to enter other values or expressions for any or all options.



• Heat capacity at constant pressure Cp (SI unit: J/(kg·K))

• Ratio of specific heats (SI unit: 1)

• Thermal conductivity k (SI unit: W/(m·K))

Defining a Thermally Conducting and Viscous Fluid ModelThis fluid model adds the bulk attenuation that is due to the combined effect of viscous losses and thermal conduction. For each of the following, the default values are taken From material. Select User defined to enter other values or expressions for any or all options.






About the Viscous Fluid Model

About the Thermally Conducting Fluid Model




Defining an Ideal Gas Fluid ModelSpecify the fluid properties by selecting a gas constant type and selecting between entering the heat capacity at constant pressure or the ratio of specific heats.

• Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K) or Mean molar

mass Mn (SI unit: kg/mol). For both options, the default values are taken From

material. Select User defined to enter other values or expressions for one or both options. If Mean molar mass is selected, the molar gas constant (universal gas constant) R 8.314 J/(mol·K), is used as the built-in physical constant.

• From the Specify Cp or list, select Heat capacity at constant pressure Cp (SI unit: J /(kg·K)) or Ratio of specific heats . For both options, the default values are taken From material. Select User defined to enter other values or expressions for one or both options. For common diatomic gases such as air, 1.4 is the standard value.

It is possible to assess the magnitude of the losses due to thermal conduction and viscosity, that is, the power dissipation density (SI unit: W/m3). This is done during the analysis process by plotting the variables for:

• the viscous power dissipation density (diss_visc),

• the thermal power dissipation density (diss_therm), or

• the combined total power dissipation density (diss_tot).

About the Thermally Conducting and Viscous Fluid Model


90 | C H A P T E R

Poroacoustics

The Poroacoustics node defines a fluid domain with porous materials modeled in a homogenized way using a, so-called, equivalent fluid model. The poroacoustics models are either the Delany-Bazley-Miki or the Johnson-Champoux-Allard models.




Enter a Temperature T and an Absolute pressure pA. Select User defined (the default) to enter a value or an expression for the absolute pressure (SI unit: Pa) and the temperature (SI unit: K) in the field.

PO R O A C O U S T I C S M O D E L

Choose a Poroacoustics model—Delany-Bazley-Miki (the default) or Johnson-Champoux-Allard.

F L U I D P R O P E R T I E S

Enter the properties of the saturating fluid that is inside of the porous material. By default the Fluid material uses Domain material.

The default Speed of sound c (SI unit: m/s, default is 0) and Density f (SI unit: kg/m3, default is 0) are taken From material. Choose User defined to enter a different value or expression than the defaults.

About the Poroacoustics Fluid Model

These poroacoustics models were previously available from the Pressure

Acoustics feature node and have been renamed:

• Delany-Bazley-Miki was previously called Macroscopic empirical porous

models.

• Johnson-Champoux-Allard was previously called Biot equivalents.


If Johnson-Champoux-Allard is selected as the Poroacoustics model, the following also require defining.

• Heat capacity at constant pressure Cp (SI unit: J/(kg·K)). The default is 0 J/(kg·K).

• Ratio of specific heats (dimensionless). The default is 0.

• Thermal conductivity k (SI unit: W/(m·K)). The default is 0 W/(m·K).

• Dynamic viscosity (SI unit: Pa·s). The default is 0 Pa·s.

PO R O U S M A T R I X P R O P E R T I E S

In this section, enter the properties that describe the porous material. By default the Porous elastic material uses the Domain material (the material defined for the domain). Select another material as required. Then based on the poroacoustics model selected, enter the following settings.

Delany-Bazley-MikiIf Delany-Bazley-Miki is selected as the Poroacoustics model, by default, the Flow

resistivity Rf (SI unit: Pa·s/m2) uses values From material. Or choose User defined and enter a value or expression in the field. The default is 0 Pa·s/m2.

Select an option from the Constants list—Delany-Bazely (the default), Miki, or User

defined. If User defined is selected, enter values in the C1 to C8 fields.

Johnson-Champoux-AllardIf Johnson-Champoux-Allard is selected as the Poroacoustics model, select a Porous matrix

approximation—Rigid (the default) or Limp. See the About the Poroacoustics Fluid Model for details.

• If Limp is selected, the default value for Drained density of porous material d (SI unit: kg/m3) is taken From material. Select User defined to enter another value or expression.

• For both Limp and Rigid porous matrix approximations, the Porosity p (dimensionless) value is taken From material. Select User defined to enter other values

Flow resistivity is easy to measure and is independent of frequency.


92 | C H A P T E R

or expressions for one or both options. Then enter values or expressions for each of the following:

- Flow resistivity Rf (SI unit: Pa·s/m2). The default is 0 Pa·s/m2.

- From the Specify list, select Viscous characteristic length parameter to enter a value for s (dimensionless) (the default is 1), or select Viscous characteristic length to directly enter an expression for Lv (SI unit: m). The default expression for the Viscous characteristic length is sqrt(acpr.mu*acpr.tau*8/(acpr.Rf*acpr.epsilon_p)), which corresponds to s = 1.

- Thermal characteristic length Lth (SI unit: m). The default expression is 2*acpr.Lv.

- Tortuosity factor (dimensionless). The default is 1.

Narrow Region Acoustics

The Narrow Region Acoustics node is a fluid model for viscous and thermal boundary-layer induced losses in channels and ducts. The losses due to the viscous and thermal dissipation in the acoustic boundary-layer are here homogenized and smeared on the fluid. This equivalent-fluid model may be used in long tubes of constant cross section instead of a full detailed thermoacoustic model.



The Johnson-Champoux-Allard (JCA) model with a motionless skeleton is sometimes referred to as one of the Biot equivalent fluid models with a rigid porous matrix.

About the Narrow Region Acoustics Fluid Model

This feature was previously available from the Pressure Acoustics feature node as a fluid model where it was named Boundary-layer absorption.


N A R R O W R E G I O N A C O U S T I C S

Select a Duct type—Wide duct approximation (the default) or Very narrow circular duct

(isothermal).

• If Wide duct approximation is selected, enter a Hydraulic diameter Hd (SI unit: m). The default is 0 m.

• If Very narrow circular duct (isothermal) is selected, enter a Radius a (SI unit: m). The default is 0 m.

Then for each of the following, the default values are taken From material. Select User

defined to enter other values or expressions for any or all options.





The following are available for Wide duct approximation only:



Background Pressure Field

Add a Background Pressure Field node to model an incident pressure wave or to study the scattered pressure field ps, which is defined as the difference between the total acoustic pressure pt and the background pressure field pb:

This feature sets up the equations in a so-called scattered field formulation where the dependent variable is the scattered field p = ps. In a model where the background pressure field is not defined on all acoustic domains (or it is different) continuity is automatically applied in the total field pt on internal boundaries between domains.

The background pressure field can be a function of space and, for The Pressure Acoustics, Transient Interface using a Time Dependent study, a function of time.

pt pb ps+=

For a Frequency Domain study type, the frequency of the background pressure field is the same as for the dependent variable p.


94 | C H A P T E R



B A C K G R O U N D P R E S S U R E F I E L D

Select a Background pressure field type—Plane wave (the default) or User defined. The Plane wave option results in a background pressure field of the type:

where r is the spatial coordinate, ek specifies the wave direction, p0 is the wave pressure amplitude, is the angular frequency, c the speed of sound, and is equal to the wave number k of the background plane wave.

• If Plane wave is selected, enter values for the Pressure amplitude p0 (SI unit: Pa) and Wave direction, ek (dimensionless). Select to define the Speed of sound c (SI unit: m/s) either From material or User defined (default is 0 m/s).

• Select User defined to enter the expression for the Background pressure field pb (SI unit: Pa). The default is 0 Pa.

Matched Boundary

The Matched Boundary node adds a matched boundary condition. Like the radiation boundary conditions, it belongs to the class of non-reflecting boundary conditions (NRBCs). If required, right-click the main node to add an Incident Pressure Field subnode.

Properly set up, the matched boundary condition allows one mode with wave number k1 (set k2k1), or two modes with wave numbers k1 and k2, to leave the modeling domain with minimal reflections. The equation is given by

pp p0ei

c----

r ek ek

----------------–

=

c

• Acoustic Cloaking: model library path Acoustics_Module/Tutorial_Models/acoustic_cloaking

• Acoustic Scattering off an Ellipsoid: model library path Acoustics_Module/Tutorial_Models/acoustic_scattering


Here T, for a given point on the boundary, refers to the Laplace operator in the tangential plane at that point, while pi is the amplitude of an optional incoming plane wave with wave vector k. In addition to pi, specify the propagation direction, nk, whereas the wave number is defined by keq /cc in 3D and in 2D.

The matched boundary condition is particularly useful for modeling acoustic waves in ducts and waveguides at frequencies below the cutoff frequency for the second excited transverse mode. In such situations set k1 /cc and k2 1/cc, where 12f1, and f1 is the cutoff frequency for the first excited mode. The cutoff frequency or wave number may be found using a Boundary Mode Acoustics model. When k1 k2 /cc, the matched boundary condition reduces to the time-harmonic plane-wave radiation boundary condition.



M A T C H E D B O U N D A R Y

From the Match list, select One mode (the default) or Two modes. Then enter a Wave

number (SI unit rad/m) based on the selection: k1 if One mode is selected and k1 and k2 if Two modes is selected.

Far-Field Calculation

Use the Far-Field Calculation node to apply the source boundaries for the near-to-far-field transformation and to specify a name for the acoustic far-field variable. This feature allows the calculation of the pressure field outside the computational domain. The far-field boundary need to enclose all sources and scatterers.

n–1c-----– p qd–

i

cc---- 2

k1k2+ p iTp+

c k1 k2+ -------------------------------------------------------------+

i cc---- 2

k1k2+ pi iT pi+

c k1 k2+ ---------------------------------------------------------------- n

1c----- pi+=

keq2

cc---- 2

kz2

–=


96 | C H A P T E R


From the Selection list, choose the boundaries to specify the source aperture for the far field.

F A R - F I E L D C A L C U L A T I O N

Enter a Far-field variable name for the far-field acoustic pressure field (the default is pfar)

Select a Type of integral—Integral approximation for r (the default) to compute the value in The Far-Field Limit or Full integral to compute The Helmholtz-Kirchhoff Integral Representation.

If required, use symmetry planes in your model when calculating the far-field variable. The symmetry planes have to coincide with one of the Cartesian coordinate planes. For each of these planes, select the type of symmetry check boxes—Symmetry in the x=0

plane, Symmetry in the y=0 plane, or Symmetry in the z=0 plane. This selection should match the boundary condition used for the symmetry boundary. With these settings, the parts of the geometry that are not in the model for symmetry reasons can be included in the far-field analysis.

A D V A N C E D S E T T I N G S

To display this section, click the Show button ( ) and select Advanced Physics Options.

The option Use polynomial-preserving recovery for the normal gradient is selected per default. This means that the far-field feature automatically uses the polynomial-preserving recovery operator ppr() to get an enhanced evaluation of the normal derivative of the pressure, on internal boundaries. This increases the precision of the far-field calculation. If you click to clear this check box this removes all instances of the operator from the equations.

The ppr() operator is not added when the far-field calculation is performed on an external boundary or a boundary adjacent to a perfectly matched layer (PML) domain. In the latter case, the down() or up() operator is added in order to retrieve values of variables from the physical domain only.

In these cases, use a single boundary layer mesh on the inside of the outer boundary or on the inside of the PML layer to enhance the precision of the far-field calculation.


Interior Normal Acceleration

The Interior Normal Acceleration node adds a normal acceleration on an interior boundary and ensures that the pressure is non-continuous here. The pressure has a so-called slit condition on this boundary. This boundary condition can be used to model sources as, for example, the movement of a speaker cone modeled as a boundary. The condition adds the normal part of an acceleration a0:

Alternatively, specify the inward acceleration an. The normal of the boundary is interpreted as pointing outward.



• The Far Field Plots


• ppr and pprint and up and down (operators) in the COMSOL Multiphysics Reference Manual

• Acoustic Scattering off an Ellipsoid: model library path Acoustics_Module/Tutorial_Models/acoustic_scattering

• Bessel Panel: model library path Acoustics_Module/Tutorial_Models/bessel_panel

• Cylindrical Subwoofer: model library path Acoustics_Module/Tutorial_Models/cylindrical_subwoofer

n–1c-----– pt qd–

1

n a 0= n–1c-----– pt qd–

2

n a 0=

n–1c-----– pt qd–

1

an= n–1c-----– pt qd–

2

an=


98 | C H A P T E R

I N T E R I O R N O R M A L A C C E L E R A T I O N

Select a Type—Acceleration (the default) or Inward acceleration.

• If Acceleration is chosen, enter values for the components of the Acceleration a0 (SI unit: m/s2). The defaults are 0 m/s2.

• If Inward Acceleration is chosen, enter the value of the Inward acceleration an (SI unit: m/s2). The default is 0 m/s2. Use a positive value for inward acceleration or a negative value for outward acceleration. The normal of the boundary points outward.

Interior Impedance/Pair Impedance

The Interior Impedance and Pair Impedance nodes add an impedance boundary condition on interior boundaries or boundaries between the parts of pairs. This condition is a generalization of the sound-hard and sound-soft boundary conditions. The condition corresponds to a transfer impedance condition, relating the pressure drop across the boundary to the velocity at the boundary. In the frequency domain, it imposes the following equations:

For a Time Dependent study (time domain), the boundary condition uses the following equations:

• Cylindrical Subwoofer: model library path Acoustics_Module/Tutorial_Models/cylindrical_subwoofer

• Lumped Loudspeaker Driver: model library path Acoustics_Module/Industrial_Models/lumped_loudspeaker_driver

pt1 pt2–

n–1c-----– pt qd–

1

pt1 pt2– i– Z

---------=

n–1c-----– pt qd–

2

pt1 pt2– i– Z

---------=

n–1---– pt qd–

1

1Z--- t---- pt1 pt2– =

n–1---– pt qd–

2

1Z--- t---- pt1 pt2– =


Z is the impedance, which from a physical point is the ratio between pressure and normal particle velocity.


From the Selection list, choose the boundaries to define. For the Pair Impedance node, this list is not editable and shows the boundaries in the selected pairs.


If Pair Impedance is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect.

I N T E R I O R I M P E D A N C E / P A I R I M P E D A N C E

Enter the value of the Impedance Zi (SI unit: Pa·s/m). The default is 0 Pa·s/m.

Interior Perforated Plate

The Interior Perforated Plate node provide the possibility of specifying the characteristic properties for a perforated plate. COMSOL Multiphysics then calculates the transfer impedance using the following model expression (Ref. 3):


From the Selection list, choose the boundaries to define. For the Pair Perforated Plate node, this list is not editable and shows the boundaries in the selected pairs.


If Pair Perforated Plate is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect.

In the two opposite limits Z and Z0, this boundary condition is identical to the Sound Hard boundary condition and the Sound Soft boundary condition, respectively. Additional information is found in Identity and Contact Pairs in the COMSOL Multiphysics Reference Manual.

Zc cc----------- 1

---

8keq

ccc-------------- 1

tpdh-----+

f+

ikeq

------- tp h+ +=


100 | C H A P T E

I N T E R I O R P E R F O R A T E D P L A T E

The equation above includes the properties listed for the perforated plate. Specify the properties of the perforated plate and enter the following:

• Dynamic viscosity (SI unit: Pa·s). The default is 1.8·105 Pa·s.

• Area porosity , that is, the holes’ fraction of the boundary surface area—a dimensionless number between 0 and 1. The default is 0.1; that is, 10% of the plate’s area consists of holes.

• Plate thickness tp (SI unit: m). The default is 0 m.

• Hole diameter dh (SI unit: m). The default is 1 mm (103 m).

• End correction h to the reactance (SI unit: m). The default is 0.25dh (a quarter of the hole diameter).

• Flow resistance f a contribution to the resistive part of the impedance that can be used, for example, to include the effects of a mean flow or nonlinear effect at large sound pressure levels. The default is 0.

Line Source

Use the Line Source node to add a source on a line/edge in 3D models. This type of source corresponds a radially vibrating cylinder in the limit where its radius tends to zero. The line source adds a source term to the right hand side of the governing Helmholtz equation such that:

The transfer impedance model implemented here is only one of many engineering relations that exist for perforated plates (perforates in general). Use the Interior Impedance/Pair Impedance condition to enter a user-defined model.

1c-----– pt qd–

keq

2pt

c-------------–

1c-----4S 3

r r0– dl=

R 3 : T H E P R E S S U R E A C O U S T I C S B R A N C H

where is the delta function in 3D that adds the source on the edge where rr0 and dl is the line element along the edge (SI unit: m). The monopole amplitude S (SI unit: N/m2) depends on the source type selected, as discussed below.

E D G E S E L E C T I O N

From the Selection list, choose the edges to define.

L I N E S O U R C E

Select a Type—Flow (the default), Intensity, Power, or User defined. If User defined is selected, enter a Monopole amplitude, S = Suser (SI unit: N/m2). The default is 0 N/m2. Otherwise, follow these instructions.

FlowSelect Flow to add an edge source located at rr0 defined in terms of the volume flow rate per unit length out from source QS and the phase of the source. The flow edge source defines the following monopole amplitude:

3 r r0–


• Solution (data sets) in the COMSOL Multiphysics Reference Manual

For the Pressure Acoustics, Transient interface, only the Flow (no phase specification), User defined, and the Gaussian pulse source types are available. The Gaussian pulse source type has no effect in the frequency domain.

S ei ic4

------------QS=


102 | C H A P T E

A flow edge source with the strength QS represents an area flow out from the source (the source is a very thin cylinder with a surface that pulsates).

• Enter a Volume flow rate per unit length out from source, QS (SI unit: m2/s) for the source-strength amplitude in the field. The default is 0 m2/s.

• Enter a Phase (SI unit: rad). The default is 0 rad.

IntensitySelect Intensity to add an edge source located at rr0 defined in terms of the source intensity radiated Irms and the phase of the source. Set a desired free space reference intensity (RMS) Irms at a specified distance dsrc from the source. In a homogeneous medium, the specified intensity is obtained when the edge is a straight line (this is the reference). With other objects and boundaries present, or if the edge is curved, the actual radiated intensity is different. This source type defines the following monopole amplitude:

where Ledge is the length of the source line (automatically determined), dsrc is the distance from the source where free space reference intensity (RMS) Irms is specified. Enter values or expressions for:

• Free space reference intensity (RMS), Irms (SI unit: W/m2). The default is 0 W/m2.

• Distance from source center dsrc (SI unit: m). The default is 0 m.

• Phase (SI unit: rad). The default is 0 rad.

PowerSelect Power to add an edge source located at rr0 specified in terms of the source’s reference RMS strength by stating the total power Prms a straight line source would radiate into a homogeneous medium. This source type defines the following monopole amplitude:

When defining a Solution data set and plotting the results, specify a nonzero phase to produce a nonzero result when visualizing the resulting pressure field using the default value (0) in the Solution at angle

(phase).

S ei dsrcLedge-------------- 2Irefccc=


where Ledge is the length of the source line (automatically determined) and Prms denotes the free space reference RMS power (in the reference homogeneous case) per unit length measured in W/m. Enter values or expressions for:

• Free space reference power (RMS), Prms (SI unit: W). The default is 0 W.


Line Source on Axis

Use the Line Source on Axis node to add a source on the axis of symmetry in 2D axisymmetric models. This type of source corresponds a radially vibrating cylinder in the limit where its radius tends to zero. The line source adds a source term to the right-hand side of the governing Helmholtz equation such that:

where is the delta function in 3D that adds the source on the axis of symmetry where zz0 and r = 0, and dz is the line element along the z-axis (SI unit: m). The monopole amplitude S (SI unit: N/m2) depends on the source type selected and is the same as discussed in the 3D case for a Line Source.



S ei

Ledge--------------

cccPrms2

----------------------=

1c-----– pt qd–

keq

2pt

c-------------–

1c-----4S 3 z z0– dz=

3 z z0–





104 | C H A P T E

L I N E S O U R C E O N A X I S

Select a Type—Flow (the default), Intensity, Power, or User defined. See the options and expression for Line Source. The sources are the same but in 2D axisymmetric models they are only applicable on the symmetry axis at r = 0.

Monopole Point Source

Use the Monopole Point Source node to as a monopole point source in 3D models on any point and in 2D axisymmetric models on points on the axis of symmetry. This is a source that is uniform and equally strong in all directions. A monopole represents a radially pulsating sphere in the limit where the radius tends to zero. The monopole point source adds a point source term to the right hand side of the governing Helmholtz equation such that:

where is the delta function in three dimensions and adds the source at the point where rr0. The monopole amplitude S (SI unit: N/m2) depends on the source type selected, as discussed below.

PO I N T S E L E C T I O N

From the Selection list, choose the points to define.

M O N O P O L E PO I N T S O U R C E

Select a Type—Flow (the default), Intensity, Power, or User defined. If User defined is chosen, enter a Monopole amplitude, S = Suser (SI unit: N/m). The default is 0 N/m.

FlowSelect Flow to add an monopole point source located at rr0 defined in terms of the volume flow rate out from source QS and the phase of the source. The source defines the following monopole amplitude:

1c-----– pt qd–

keq

2pt

c-------------–

1c-----4S 3

r r0– =

3 r r0–



Enter values or expressions for:

• Volume flow rate out from source, QS (SI unit: m3/s). The default is 0 m3/s.


IntensitySelect Intensity to define the source in terms of the free space reference RMS intensity Irms it radiates. In a homogeneous medium the specified intensity is obtained (the reference), but with other objects and boundaries present the actual intensity is different. The source defines the following monopole amplitude:

where dsrc is the distance from the source where the intensity Irms is specified and is the phase of the source. Enter values or expressions for:




PowerWhen Power is selected, specify the source’s reference RMS strength by stating the power it radiates. In a homogeneous medium the specified power is obtained (the reference), but with other objects and boundaries present the actual power is different. The source defines the following monopole amplitude:

S ei ic4

------------QS=

S eidsrc 2cccIrms=

S ei cccPrms2

----------------------=


106 | C H A P T E

where Pref denotes the radiated RMS power per unit length measured in W/m. Enter values or expressions for:

• Free space reference power (RMS), Prms (SI unit: W). The default is 0 W.


Point Source

Use the Point Source node to add a point source to a 2D model. This source corresponds to an infinite line source in the out-of-plane direction. The source is uniform and equally strong in all the in-plane directions. The point source adds a point source term to the right hand side of the governing Helmholtz equation such that:

where is the delta function in 2D and adds the source at the point where (x,y) = rr0. The monopole amplitude S (SI unit: N/m2) depends on the source type selected, as discussed below.



M O N O P O L E PO I N T S O U R C E

Select a Type—Flow (the default), Intensity, Power, or User defined. If User defined is chosen, enter a Monopole amplitude, Suser (SI unit: N/m). The default is 0 N/m.

• Bessel Panel: model library path Acoustics_Module/Tutorial_Models/bessel_panel

• Hollow Cylinder: model library path Acoustics_Module/Tutorial_Models/hollow_cylinder

1c-----– pt qd–

keq

2pt

c-------------–

1c-----4S 2

r r0– =

2 r r0–



FlowSelect Flow to add an monopole point source located at rr0 defined in terms of the volume flow rate per unit length out from source QS and the phase of the source. The source defines the following monopole amplitude:


• Volume flow rate out from source, QS (SI unit: m3/s). The default is 0 m3/s.


IntensitySelect Intensity to define the source in terms of the free space reference RMS intensity Irms it radiates. In a homogeneous medium the specified intensity is obtained (the reference), but with other objects and boundaries present the actual intensity is different. The source defines the following monopole amplitude:

where dsrc is the distance from the source where the intensity Irms is specified and is the phase of the source. Enter values or expressions for:




PowerWhen Power is selected, specify the source’s reference RMS strength by stating the power per unit length it radiates. In a homogeneous medium the specified power is obtained (the reference), but with other objects and boundaries present the actual source power is different. The source defines the following monopole amplitude:

S ei ic4

------------QS=

S ei 2cIrmsdsrc2

---------------------------------=

S ei 2cPrms

2 2-------------------------=


108 | C H A P T E

where Prms denotes the free space RMS reference power per unit length measured in W/m and is the phase of the source. Enter values or expressions for:

• Free space reference power (RMS) per unit length, Prms (SI unit: W/m). The default is 0 W.


Circular Source

Use the Circular Source node to add a source in a 2D axisymmetric models on points off the axis of symmetry. Such points correspond to circular sources or ring sources. This type of source is, for example, used to mimic source terms from rotors. The circular source adds a point source term to the right hand side of the governing Helmholtz equation such that:

where is the delta function that adds the source at the point where rr0 and rd is the line element around the circular source (SI unit: m). The monopole amplitude S (SI unit: N/m2) depends on the source type selected, as discussed below



Select a Type—Flow (the default) or User defined. If User defined is selected, enter a Monopole amplitude, S = Suser (SI unit: N/m2). The default is 0 N/m2.

FlowWhen Flow is selected the source is defined in terms of the volume flow rate QS per unit length out form the source. The flow circular source defined the following monopole amplitude:

1c-----– pt qd–

keq

2pt

c-------------–

1c-----4S 3

r r0– rd=

3 r r0–

For the Pressure Acoustics, Transient interface the Flow (no phase specification), User defined, and the Gaussian pulse source types are available. The Gaussian pulse source type has no effect in the frequency domain.

S ei ic4

------------QS=



• Volume flow rate per unit length out from source, QS (SI unit: m2/s). The default is 0 m2/s.



110 | C H A P T E

Th e P r e s s u r e A c ou s t i c s , T r a n s i e n t I n t e r f a c e

The Pressure Acoustics, Transient (actd) interface ( ), found under the Acoustics>Pressure Acoustics branch ( ) when adding a physics interface, is used to compute the pressure variation when modeling the propagation of acoustic waves in fluids at quiescent background conditions. It is suited for time-dependent simulations with arbitrary time-dependent fields and sources.

The physics interface can be used to model linear and nonlinear acoustics that can be well described by the scalar pressure variable. Domain conditions also include background incident acoustic fields. User-defined sources can be added to, for example, include certain nonlinear effects such as a square pressure dependency of the density variations.

The physics interface solves the scalar wave equation in the time domain. Studies for performing time-dependent modal and modal reduced-order models also exist. The physics interface also solves in the frequency domain with the available boundary conditions.

When this interface is added, these default nodes are also added to the Model Builder— Transient Pressure Acoustics Model, Sound Hard Boundary (Wall), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and source. You can also right-click Pressure Acoustics, Transient to select physics from the context menu.


The interface identifier is used primarily as a scope prefix for variables defined by the physics interface. Refer to such interface variables in expressions using the pattern <identifier>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter.


The default identifier (for the first interface in the model) is actd.

Domain, Boundary, Edge, and Point Nodes for the Pressure Acoustics, Transient Interface

The Pressure Acoustics, Transient Interface shares most of its nodes with the Pressure Acoustics, Frequency Domain interface, except the following:

• Transient Pressure Acoustics Model

• The Gaussian Pulse Source Type

Transient Pressure Acoustics Model

The Transient Pressure Acoustics Model node adds the equations for primarily time-dependent (transient) acoustics modeling. This is the scalar wave equation

The remainder of the settings window is shared with The Pressure Acoustics, Frequency Domain Interface.


• Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure Acoustics, Frequency Domain Interface

• Domain, Boundary, Edge, and Point Nodes for the Pressure Acoustics, Transient Interface


Transient Gaussian Explosion: model library path Acoustics_Module/Tutorial_Models/gaussian_explosion

Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure Acoustics, Frequency Domain Interface

T H E P R E S S U R E A C O U S T I C S , TR A N S I E N T I N T E R F A C E | 111

112 | C H A P T E

where pt is the total acoustic pressure, is the fluid density, c is the speed of sound, qd is the Dipole Source, and Qm is the Monopole Source.

In the settings window, define the properties for the acoustics model and model inputs including temperature.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains to compute the acoustic pressure field and the equation that defines it, or select All domains as required.

TR A N S I E N T P R E S S U R E A C O U S T I C S M O D E L

Select a Fluid model—Linear elastic (the default), Viscous, Thermally conducting, Thermally conducting or viscous, or Ideal Gas. Then see the descriptions for The Pressure Acoustics, Frequency Domain Interface:

• Defining a Linear Elastic Fluid Model

• Defining a Viscous Fluid Model

• Defining a Thermally Conducting Fluid Model

• Defining a Thermally Conducting and Viscous Fluid Model

• Defining an Ideal Gas Fluid Model

1

c2---------

2pt

t2----------- 1

--- pt qd– –+ Qm=

See Pressure Acoustics for details of the fluid model equations.


M O D E L I N P U T S ( I D E A L G A S O N L Y )

If Ideal gas is selected as the Fluid model, enter a Temperature T (which can be a constant temperature or a temperature field from a heat transfer interface) and an Absolute

pressure pA:

• Select User defined to enter a value or an expression for the absolute pressure (SI unit: Pa) in the field that appears. This input is always available.

• In addition, select a pressure defined by a fluid-flow interface present in the model (if any). For example, then select Pressure (spf/fp1) to use the pressure defined by the Fluid Properties node fp1 in a Single-Phase Flow interface spf. Selecting a pressure variable also activates a check box for defining the reference pressure, where 1 [atm] has been automatically included. This allows the use of a system-based (gauge) pressure, while automatically including the reference pressure in the absolute pressure.

The Gaussian Pulse Source Type

In transient models the Gaussian pulse exists as a source type in a Line Source, Line Source on Axis, Monopole Point Source, Point Source, and Circular Source. This type adds a source with a Gaussian time profile defined in terms of its amplitude A, its frequency bandwidth f0, and the pulse peak time tp. Using this source type results in solving a wave equation of the type:

where S is the source strength and the superscript n in the delta function depends on the dimension, n = 2 in 2D and n = 3 in 3D models.

In 3D models (for Line Source), in 2D axisymmetric models (for Line Source on Axis and Circular Source), and 2D models (for Point Source), all of which are effectively line sources, enter the following values or expressions:

• The value of the pulse Amplitude A (SI unit: m2/s).

• Frequency bandwidth f0 (SI unit: Hz).

• Pulse peak time tp (SI unit: s) for the duration of the pulse.

1

c2---------

2pt

t2----------- 1

--- pt qd– –+

4

------S n r r0– =

S 4------

t----- Ae

2f02 t tp– 2– A–

2---f0

2 t tp– e 2f02 t tp– 2–

= =

T H E P R E S S U R E A C O U S T I C S , TR A N S I E N T I N T E R F A C E | 113

114 | C H A P T E

In 3D and 2D axisymmetric models for the Monopole Point Source, enter the following values or expressions:

• The value of the pulse Amplitude A (SI unit: m3/s).

• Frequency bandwidth f0 (SI unit: Hz).

• Pulse peak time tp (SI unit: s) for the duration of the pulse.

Transient Gaussian Explosion: model library path Acoustics_Module/Tutorial_Models/gaussian_explosion


Th e Bounda r y Mode A c ou s t i c s I n t e r f a c e

The Boundary Mode Acoustics (acbm) interface ( ), found under the Acoustics>Pressure Acoustics branch ( ) when adding a physics interface, is used to compute and identify propagating and nonpropagating modes in waveguides and ducts by performing a boundary mode analysis on a given boundary. The study is useful, for example, when specifying sources at inlets or analyzing transverse acoustic modes in ducts.

The physics interface solves the Helmholtz eigenvalue equation on boundaries, searching for the out-of-plane wave numbers at a given frequency.

When this interface is added, these default nodes are also added to the Model Builder— Pressure Acoustics Model, Sound Hard Boundary (Wall), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Boundary Mode Acoustics to select physics from the context menu.



The default identifier (for the first interface in the model) is acbm.


The default setting is to include All boundaries in the model to define a sound pressure field and the associated acoustic boundary mode equation. To choose specific boundaries, select Manual from the Selection list.

T H E B O U N D A R Y M O D E A C O U S T I C S I N T E R F A C E | 115

116 | C H A P T E

E Q U A T I O N


See the settings for Sound Pressure Level Settings for the Pressure Acoustics, Frequency Domain interface.


This interface defines one dependent variable (field), the Pressure p. The name can be changed but the names of fields and dependent variables must be unique within a model.


To display this section, click the Show button ( ) and select Discretization. Select Quadratic (the default), Linear, Cubic, or Quartic for the Pressure. Specify the Value type

when using splitting of complex variables—Real or Complex (the default).

Initial Values

The Initial Values node adds initial values for the sound pressure. Add more Initial

Values nodes from the Physics toolbar.

For 2D axisymmetric models, the Circumferential wave number m is by default 0. It is an integer entering the axisymmetric expression for the pressure:

Change the value as required. Also see Physics Nodes—Equation Section in the COMSOL Multiphysics Reference Manual.

p r z p r ei kzz m+ –

=


• Initial Values

• Boundary, Edge, Point, and Pair Nodes for the Boundary Mode Acoustics Interface

• Boundary Mode Acoustics Equations





Enter a value or expression for the Pressure p (SI unit: Pa) initial value. The default is 0 Pa.

Boundary, Edge, Point, and Pair Nodes for the Boundary Mode Acoustics Interface

Special post-processing variables exist for the Boundary Mode Acoustics interface. They are described in Boundary Mode Acoustics Variables.

Except for Initial Values, The Boundary Mode Acoustics Interface shares all of its feature nodes with the Pressure Acoustics, Frequency Domain interface. See Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure Acoustics, Frequency Domain Interface.

Also, for the Boundary Mode Acoustics interface, apply the features to boundaries instead of domains for 3D models.

T H E B O U N D A R Y M O D E A C O U S T I C S I N T E R F A C E | 117

118 | C H A P T E

Th eo r y Ba c k g r ound f o r t h e P r e s s u r e A c ou s t i c s B r an c h

This section describes the governing equations and the mathematical formulation of the governing equations as used in the Pressure Acoustics branch of the Acoustics Module. Details are also given regarding some of the boundary conditions, among which the radiation boundary conditions as well as the far-field calculation feature. The section starts with a general introduction to the governing equations used in pressure acoustics.

In this sections:

• The Governing Equations

• Pressure Acoustics, Frequency Domain Equations

• Pressure Acoustics, Transient Equations

• Boundary Mode Acoustics Equations

• Theory for the Plane, Spherical, and Cylindrical Radiation Boundary Conditions

• Theory for the Far-Field Calculation: The Helmholtz-Kirchhoff Integral

• Solving Large Acoustic Problems

The Governing Equations

Pressure acoustic problems involve solving for the small acoustic pressure variations p on top of the stationary background pressure p0. Mathematically, this represents a linearization (small parameter expansion) of the dependent variables around the stationary quiescent values.

The governing equations for a compressible lossless (no thermal conduction and no viscosity) fluid flow problem are the momentum conservation equation (Euler's equation) and the mass conservation equation (continuity equation). These are given by:

ut------ u u+

1---p–=

t------ u + 0=


where is the total density, p is the total pressure, and u is the velocity field. In classical pressure acoustics, all thermodynamic processes are assumed to be reversible and adiabatic, that is, isentropic processes. The small parameter expansion is performed on a stationary fluid (u0 = 0) of density 0 (SI unit: kg/m3) and at pressure p0 (SI unit: Pa) such that:

where the primed variables represent the small acoustic variations. Inserting these into the governing equations and only retaining terms linear in the primed variables yields:

(3-1)

One of the dependent variables, the density, is removed by expressing it in terms of the pressure using a Taylor expansion (linearization) in the small parameters:

(3-2)

where cs is recognized as the (isentropic) speed of sound (SI unit: m/s) at constant entropy s. This expression gives a useful condition that needs to be fulfilled for the linear acoustic equations to hold:

The subscript s is dropped in the following a long with the subscript 0 on the background density 0. Finally, rearranging Equation 3-1 and Equation 3-2 (divergence of momentum equation inserted into the continuity equation) and dropping the primes yields the wave equation for sound waves in a lossless medium:

(3-3)

Here (SI unit: kg/m3) refers to the density, and c (SI unit: m/s) denotes the speed of sound. The equation is further extended with two optional source terms: The dipole source qd (SI unit: N/m3) and the monopole source Qm (SI unit: 1/s2).

p p0 p'+=

0 '+=

u 0 u'+=

withp' p0«

' 0«

u't------- 1

0------p'–=

't------- 0 u' + 0=

'0p---------

sp'

1

cs2

-----p'= =

p' 0cs2

«

1

c2--------

t2

2

p 1---– p qd–

+ Qm=

T H E O R Y B A C K G R O U N D F O R T H E P R E S S U R E A C O U S T I C S B R A N C H | 119

120 | C H A P T E

The combination c2 is called the adiabatic bulk modulus, commonly denoted K (SI unit: Pa). The bulk modulus is equal to the one over the adiabatic compressibility coefficient 1/K (SI unit: 1/Pa).

In Equation 3-3 both the speed of sound c = c(x) and the density (x)may be dependent on the spatial coordinates x while they are independent of time, or only slowly varying.

An important special case is a time-harmonic wave, for which the pressure varies with time as

where = 2f (rad/s) is the angular frequency and f (SI unit: Hz) is the frequency. Assuming the same harmonic time dependence for the source terms, the wave equation for acoustic waves reduces to an inhomogeneous Helmholtz equation:

(3-4)

In this equation the subscript c on the density and the speed of sound refers to that they may be complex valued. Lossy media, like porous materials or highly viscous fluids, can be modeled by using the complex valued speed of sound and density. A selection of such fluid models is available in The Pressure Acoustics, Frequency Domain Interface. The attenuation in these fluid models is frequency dependent in different ways, depending on the physical origin of the damping. A description of the different fluid models is given in Theory for the Pressure Acoustics Fluid Models.

In the time domain, only certain frequency dependencies can be modeled, which limits the number of fluid models that can be used in The Pressure Acoustics, Transient Interface. One way to model damping in the time domain is to introduce an additional term of first order in the time derivative to account for attenuation of the sound waves:

(3-5)

Some classical references on acoustics for further reading are found in Ref. 4, Ref. 5, Ref. 6, Ref. 7, and Ref. 8.

p x t p x eit

=

1c-----– p qd–

2p

c cc2

-------------– Qm=

1

c2

--------t2

2

pda tp

– 1---– p qd–

+ Qm=


The damping term in Equation 3-5 is absent from the standard PDE formulations in the Pressure Acoustics, Transient interface, but it corresponds to a monopole source proportional to the time derivative of the pressure. This approach is, however, not used in the viscous and thermally conducting fluid models for transient acoustics (see About the Viscous Fluid Model, About the Thermally Conducting Fluid Model, and About the Thermally Conducting and Viscous Fluid Model). The damping is here introduced via a dipole like source.

Alternatively. treat the Helmholtz Equation 3-4 as an eigenvalue PDE to solve for eigenmodes and eigenfrequencies, see the Eigenfrequency and Mode Analysis in 2D and 1D axisymmetric sections below.

In order to solve the governing equations, boundary conditions are necessary. Typical boundary conditions used in acoustics are:


• Sound Soft Boundary (zero acoustic pressure)

• Specified acoustic Pressure

• Specified Normal Acceleration





Pressure Acoustics, Frequency Domain Equations

The Pressure Acoustics, Frequency Domain Interface exists for several types of studies. Here the equations are presented for the frequency domain, eigenfrequency, and modal studies. All the interfaces solve for the acoustic pressure p. It is available in all space dimensions—for 3D, 2D, and 1D Cartesian geometries as well as for 2D and 1D axisymmetric geometries.

Even when sound waves propagate in a lossless medium, attenuation can occur by interaction with the surroundings at the system boundaries. In particular, this applies to the impedance boundary conditions.


122 | C H A P T E

F R E Q U E N C Y D O M A I N

The frequency domain, or time-harmonic, formulation uses the inhomogeneous Helmholtz equation:

(3-6)

This is Equation 3-4 repeated with the introduction of the wave number keq used in the equations. It contains both the ordinary wave number k as well as out-of-plane and circumferential contributions, when applicable. Note also that the pressure is here the total pressure pt which is the sum of a possible Background Pressure Field pb and the scattered field ps. This enables for a so-called scattered field formulation of the equations. If no background field is present pt = ps = p.

In this equation, p p (x,) = p(x)eit (the dependence on is henceforth not explicitly indicated). Compute the frequency response by doing a parametric sweep over a frequency range using harmonic loads and sources.

When there is damping, c and cc are complex-valued quantities. The available damping models and how to apply them is described in the sections Pressure Acoustics and Theory for the Pressure Acoustics Fluid Models.

Equation 3-6 is the equation that the software solves for 3D geometries. In lower-dimensional and axisymmetric cases, restrictions on the coordinate dependence mean that the equations differ from case to case. Here is a brief summary of the situation.

2DIn 2D, the pressure is of the form

which inserted in Equation 3-6 gives

(3-7)

The out-of-plane wave number kz can be set on the Pressure Acoustics page. By default its value is 0. In the mode analysis type ikz is used as the eigenvalue .

1c-----– pt qd–

keq

2pt

c-------------– Qm=

p r p x y ei– kz z=

1c----- pt qd– –

keq

2

c-------- pt– Qm=

keq2 2

cc2

------- kz2

–=


2D AxisymmetryFor 2D axisymmetric geometries the independent variables are the radial coordinate r and the axial coordinate z. The only dependence allowed on the azimuthal coordinate is through a phase factor,

(3-8)

where m denotes the circumferential wave number. Because the azimuthal coordinate is periodic m must be an integer. Just like kz in the 2D case, m can be set on the Pressure Acoustics settings window.

As a result of Equation 3-8, the equation to solve for the acoustic pressure in 2D axisymmetric geometries becomes

1D AxisymmetryIn 1D axisymmetric geometries,

leading to the radial equation

where both the circumferential wave number m, and the axial wave number kz, appear as parameters.

1DThe equation for the 1D case is obtained by letting the pressure depend on a single Cartesian coordinate x:

p r z p r z e im–=

r r

c-----–

rp qr– r

z 1

c-----–

zp qz– keq

2

c--------rp–+ rQm=

keq2

cc------ 2 m

r----- 2

–=

p r z p r ei kz z m+ –

=

r r

c-----–

rp qr– keq

2

c--------rp– rQm=

keq2

cc------ 2 m

r----- 2

– kz2

–=


124 | C H A P T E

E I G E N F R E Q U E N C Y

In the eigenfrequency formulation, the source terms are absent; the eigenmodes and eigenfrequencies are solved for:

(3-9)

The eigenvalue introduced in this equation is related to the eigenfrequency f, and the angular frequency , through i2fi. Because they are independent of the pressure, the solver ignores any dipole and monopole sources unless a coupled eigenvalue problem is being solved.

Equation 3-9 applies to the 3D case. The equations solved in eigenfrequency studies in lower dimensions and for axisymmetric geometries are obtained from their time-harmonic counterparts, given in the previous subsection, by the substitution 22.

Switch between specifying the eigenvalues, the eigenfrequencies, or the angular frequencies by selecting from the Eigenvalue transformation list in the solver sequence’s Eigenvalue feature node’s settings window.

M O D E A N A L Y S I S I N 2 D A N D 1 D A X I S Y M M E T R I C

See Mode Analysis Study in the Boundary Mode Acoustics Equations section. The mode analysis study type is only available for the Pressure Acoustics, Frequency Domain interface in 2D and 1D axisymmetric models. Where the solver solves for the eigenvalues =ikz for a given frequency. Here kz is the out-of-plane wave number of a given mode and the resulting pressure field p represents the mode on the cross section of an infinite wave guide or duct.

xdd 1

c-----

xddp qd– –

keq2

c-------- p– Qm=

keq2

cc------ 2

=

1c-----– p

2p

c cc2

-----------+ 0=

Vibrations of a Disk Backed by an Air-Filled Cylinder: model library path Acoustics_Module/Verification_Models/coupled_vibrations_acsh


Pressure Acoustics, Transient Equations

Use the Time Dependent study type to model transient acoustic phenomena in a stationary fluid and to solve the wave equation

for the acoustic pressure, p px, t. Here c is the speed of sound and denotes the equilibrium density, while qd and Qm are dipole and monopole sources, respectively. The density and speed of sound can both be non constant in space. In contrast, they are assumed to vary with time on scales much larger than the period for the acoustic waves and are therefore considered time independent in the previous equation. This interface is available for 3D, 2D, and 1D Cartesian geometries as well as for 2D and 1D axisymmetric geometries.

Boundary Mode Acoustics Equations

When an acoustic wave of a given angular frequency is fed into a waveguide or a duct, only a finite set of shapes, or modes, for the transverse pressure field can propagate over long distances inside the structure. The higher the frequency, the higher the number of sustainable modes.

Take, as an example, a uniform straight duct whose axis is in the z-direction. The acoustic field in such a duct can be written as a sum of the form

The constant kzj is the axial wave number of the jth propagating transverse mode, pj(x, y). These transverse modes and their associated axial wave numbers are solutions to an eigenvalue problem defined on the duct’s cross section. The mode analysis capabilities in The Boundary Mode Acoustics Interface makes it possible to solve such eigenvalue problems. The interface is available for 3D Cartesian and 2D axisymmetric geometries and solves for the transverse eigenmodes for the acoustic pressure p and the associated propagation constants kz. The Mode Analysis Study is briefly discussed.

M O D E A N A L Y S I S S T U D Y

The eigenvalue solver computes a specified number of solutions pj, j to the equation

1c2---------

2p

t2

--------- 1--- p qd– –

+ Qm=

p r pj x y eikzjz–

j 0=

N

=


126 | C H A P T E

(3-10)

defined on a 2D boundary of the modeling domain (in 3D) or on the 2D domain itself, with =ikn as the eigenvalue. In this equation, p is the in-plane pressure, c is the density, cc is the speed of sound, is the angular frequency, and kn is the propagation constant in the direction normal to the surface, in this context also referred to as the out-of-plane wave number.

Notice that the above equation is identical to the time-harmonic equation for pressure acoustics, except that kn is interpreted as an eigenvalue and not as a parameter.

For axisymmetric geometries, the relevant eigenvalue equation to solve for the radial pressure modes and the eigenvalues is

Here m, the circumferential wave number, is an integer-valued parameter. The equation is defined on the interval r1rr2. The eigenvalue is defined in terms of the axial wave number kz through the equation ikz

1c----- p qd– –

2

ccc2

-----------kn

2

c-----–

p– Qm=

The out-of-plane wave number is denoted kn, and is in the normal direction to the two-dimensional surface on which Equation 3-10. As for a mode analysis study in the frequency domain the propagation direction does not necessarily have to be normal to the z-axis for 3D geometries.

Special post-processing variables exist for the Boundary Mode Acoustics interface. They are described in Boundary Mode Acoustics Variables.

rdd r

c-----

rddp

cc------ 2

2 mr----- 2

–+rpc-----+ 0=

Absorptive Muffler: model library path Acoustics_Module/Industrial_Models/absorptive_muffler


Theory for the Plane, Spherical, and Cylindrical Radiation Boundary Conditions

Specify a Plane Wave Radiation, Spherical Wave Radiation, or Cylindrical Wave Radiation boundary condition to allow an outgoing wave to leave the modeling domain with minimal reflections. The condition can be adapted to the geometry of the modeling domain. The plane wave type is suitable for both far-field boundaries and ports. Because many waveguide structures are only interesting in the plane-wave region, it is particularly relevant for ports.

Radiation boundary conditions are available for all types of studies. For the Frequency domain study, Givoli and Neta’s reformulation of the Higdon conditions (Ref. 1) for plane waves has been implemented to the second order. For cylindrical and spherical waves, COMSOL Multiphysics uses the corresponding 2nd-order expressions from Bayliss, Gunzburger, and Turkel (Ref. 2). The Transient, Mode analysis, and Eigenfrequency studies implement the same expansions to the first order.

The first-order radiation boundary conditions in the frequency domain read

where k is the wave number and ( r ) is a function whose form depends on the wave type:

• Plane wave: ( r )0

• Cylindrical wave: ( r )1(2 r)

• Spherical wave: ( r )1r

In the cylindrical and spherical wave cases, r is the shortest distance from the point r(x, y, z) on the boundary to the source. The right-hand side of the equation represents an optional incoming pressure field pi (see Incident Pressure Field).

The second-order radiation boundary conditions in the frequency domain are defined below. In these equations, T at a given point on the boundary denotes the Laplace operator in the tangent plane at that particular point.

P L A N E W AV E

n–1c-----– pt qd–

ik r + pc-----+ ik r +

pi

c----- n

pi

c--------- +=

n–1c-----– pt qd–

i kc-----p

i2kc------------T p+ +

i2kc------------T pi i k

c-----pi n

1c----- pi+ +=


128 | C H A P T E

In the notation of Givoli and Neta (Ref. 1), the above expressions correspond to the parameter choices C0 C1 C2 /k. For normally incident waves, this gives a vanishing reflection coefficient.

C Y L I N D R I C A L WA V E

The cylindrical wave boundary condition is based on a series expansion of the outgoing wave in cylindrical coordinates (Ref. 2), and it assumes that the field is independent of the axial coordinate. Specify the axis of this coordinate system by giving an orientation (nx, ny, nz) and a point (x0, y0, z0) on the axis. In axisymmetric geometries, the symmetry axis is the natural and only choice.

S P H E R I C A L W A V E

Use a spherical wave to allow a radiated or scattered wave—emanating from an object centered at the point (x0, y0, z0) that is specified—to leave the modeling domain without reflections. The boundary condition is based on an expansion in spherical coordinates from Bayliss, Gunzburger, and Turkel (Ref. 2), implemented to the second order.

TR A N S I E N T A N A L Y S I S

The transient radiation boundary condition is the first-order expression

n–1c-----– pt qd–

=

ikeq12r------ 1

8r 1 ikeqr+ ---------------------------------–+

pi p– c

------------------ n1c----- pi

rT pi rT p– 2 1 ikeqr+ c---------------------------------------+ +

n–1c-----– pt qd–

ikeq1r---+

pc-----

rTp

2c ikeqr 1+ ------------------------------------–+

rTpi

20c ikeqr 1+ ---------------------------------------– ikeq

1r---+

pi

c----- n

1c----- pi+ +=

n–1---–

pt qd– 1

--- 1

c---

tp r p+

+1--- 1

c---

tpi r pi n pi+ +

=


where ( r ) is the same wave-type dependent function as for the eigenfrequency case and pi the optional Incident Pressure Field.

Theory for the Far-Field Calculation: The Helmholtz-Kirchhoff Integral

The Acoustics Module has functionality to evaluate the acoustic pressure field in the far-field region. This section gives the relevant definitions and mathematical background as well as some general advice for analyzing the far field. Details about how to use the far-field functionality is described in Far-Field Calculation.

T H E N E A R - F I E L D A N D F A R - F I E L D R E G I O N S

The solution domain for a scattering or radiation problem can be divided into two zones, reflecting the behavior of the solution at various distances from objects and sources. In the far-field region, scattered or emitted waves are locally planar, velocity and pressure are in phase with each other, and the ratio between pressure and velocity approaches the free-space impedance of a plane wave.

Moving closer to the sources into the near-field region, pressure and velocity gradually slide out of phase. This means that the acoustic field contains energy that does not travel outward or radiate. These evanescent wave components are effectively trapped close to the source. Looking at the sound pressure level, local maxima and minima are apparent in the near-field region.

Naturally, the boundary between the near-field and far-field regions is not sharp. A general guideline is that the far-field region is that beyond the last local energy

An estimate of the refection coefficient Rs for spurious waves off the plane wave radiation boundary, for incident plane waves at angle is given by the expression:

where N is the order of the boundary condition (here 1 or 2). So at normal incidence (= 0) there are no spurious refections, while, for example, at an incidence angle of 30o for N = 2 (plane wave radiation in the frequency domain) the amplitude of the spurious reflected wave is 0.5 % of the incident.

Rs 1–cos 1+cos

----------------------N

=


130 | C H A P T E

maximum, that is, the region where the pressure amplitude drops monotonously at a rate inversely proportional to the distance from any source or object, R.

A similar definition of the far-field region is the region where the radiation pattern—the locations of local minima and maxima in space—is independent of the distance to the wave source. This is equivalent to the criterion for Fraunhofer diffraction in optics, which occurs for Fresnel numbers, Fa2/R, much smaller than 1. For engineering purposes, this definition of the far-field region can be applied:

(3-11)

In Equation 3-11, a is the radius of a sphere enclosing all objects and sources, is the wavelength, and k is the wave number. Another way to write the expression leads to the useful observation that the size of the near-field region, expressed in source-radius units, is proportional to the dimensionless number k a, with a prefactor slightly larger than one.

Knowing the extent of the near-field region is useful when applying radiation boundary conditions because these are accurate only in the far-field region. Perfectly matched layers (PMLs), on the other hand, can be used to truncate a domain already inside the near-field region.

T H E H E L M H O L T Z - K I R C H H O F F I N T E G R A L R E P R E S E N T A T I O N

In many cases, solving the acoustic Helmholtz equation everywhere in the domain where results are requested is neither practical nor necessary. For homogeneous media, the solution anywhere outside a closed surface containing all sources can be written as a boundary integral in terms of quantities evaluated on the surface. To evaluate this Helmholtz-Kirchhoff integral, it is necessary to know both Dirichlet and Neumann values on the surface. Applied to acoustics, this means that if the pressure and its normal derivative (which is related to the normal velocity) is known on a closed surface, the acoustic field can be calculated at any point outside.

In general, the solution p to Helmholtz’ equation

in the homogeneous domain exterior to a closed surface, S, can be explicitly expressed in terms of the values of p and its normal derivative on S:

R 8a2

---------- 8

2------ka2=

p– k2 p– 0=

p R G R r p r G R r p r – n SdS=


Here the coordinate vector r parameterizes S. The unit vector n is the outward normal to the exterior infinite domain; thus, n points into the domain that S encloses. The function G (R, r) is a Green’s function satisfying

This essentially means that the Green’s function, seen as a function of r, is an outgoing traveling wave excited by a simple source at R. In 3D, the Green’s function is therefore:

In 2D, the Green’s function contains a Hankel function instead of the exponential:

Inserting the 3D Green’s function in the general representation formula gives:

(3-12)

while in 2D, the Hankel function leads to a slightly different expression:

(3-13)

For axially symmetric geometries, the full 3D integral must be evaluated. The Acoustics Module uses an adaptive numerical quadrature in the azimuthal direction on a fictitious revolved geometry in addition to the standard mesh-based quadrature in the rz-plane.

To evaluate the full Helmholtz-Kirchhoff integral in Equation 3-12 and Equation 3-13, use the Full integral option in the settings for the far-field variables. See Far-Field Calculation.

T H E F A R - F I E L D L I M I T

The full Helmholtz-Kirchhoff integral gives the pressure at any point at a finite distance from the source surface, but the numerical integration tends to lose accuracy at large distances. At the same time, in many applications, the quantity of interest is the

G R r – k2G R r – 3 R r– =

G R r e ik r R––

4 r R–-----------------------=

G R r i4---H0

2 k r R– =

p R 14------ e ik r R––

r R–---------------------- p r p r 1 ik r R–+

r R– 2------------------------------------ r R– +

n SdS=

p R i4---– H0

2 k r R– p r kp r H1

2 k r R– r R–

------------------------------------ r R– +

n SdS=


132 | C H A P T E

far-field radiation pattern, which can be defined as the limit of r | p | when r goes to infinity in a given direction.

Taking the limit of Equation 3-12 when | R | goes to infinity and ignoring the rapidly oscillating phase factor, the far field, pfar is defined as

Because Hankel functions asymptotically approach exponential, the limiting 2D integral is remarkably similar to that in the 3D case:

For axially symmetric geometries, the azimuthal integral of the limiting 3D case can be handled analytically, which leads to a rather complicated expression but avoids the numerical quadrature required in the general case. For the circumferential wave number m0, the expression is:

(3-14)

In this integral, r and z are the radial and axial components of r, while R and Z are the radial and axial components of R.

To evaluate the pressure in the far-field limit according to the equations in this section, use the Integral approximation at r option in the settings window for the far-field variables See Far-Field Calculation.

T H E E L K E R N E L E L E M E N T

These integrals can be implemented as integration coupling variables in COMSOL Multiphysics. However, such an approach is very inefficient because then the simple

pfar R 14------– e

ikr RR

-----------p r ikp r R

R-------–

n SdS=

The relevant quantity is | pfar| rather than pfar because the phase of the latter is undefined. For the same reason, only the direction of R is important, not its magnitude.

pfar R 1 i–

4 k-------------- e

ikr RR

-----------p r ikp r R

R-------–

n SdS=

pfar R 12---– re

ikzZR-------

J 0krR

R----------- p r n –

S

ikp r R

----------------- inr RJ1krR

R----------- nzZJ0

krRR

----------- +

dS


structure of the integration kernels cannot be exploited. In the Acoustics Module, convolution integrals of this type are therefore evaluated in optimized codes that hides all details from the user.

Solving Large Acoustic Problems

For solving larger problems, especially in 3D, the only option is often to use an iterative method such as multigrid. For smaller problems using a direct solver like MUMPS is often the best choice. Details about solving large acoustic models are found in Solving Large Acoustics Problems Using Iterative Solvers in the Modeling with the Acoustics Module chapter. That chapter also includes comments on Resolving the Waves.


134 | C H A P T E

Th eo r y f o r t h e P r e s s u r e A c ou s t i c s F l u i d Mode l s

In this section:

• Introduction to the Pressure Acoustics Fluid Models

• About the Linear Elastic with Attenuation Fluid Model

• About the Viscous Fluid Model

• About the Thermally Conducting Fluid Model

• About the Thermally Conducting and Viscous Fluid Model

• About the Poroacoustics Fluid Model

• About the Narrow Region Acoustics Fluid Model

Introduction to the Pressure Acoustics Fluid Models

It is possible to define the properties of a fluid in several ways in pressure acoustics. In a Pressure Acoustics domain feature attenuation properties for the bulk fluid may be specified. Acoustic losses in porous materials, that are modeled in a homogenized way, are defined by the Poroacoustics domain feature (frequency domain only). The viscous and thermal losses that occur in the acoustic boundary layer can be modeled in a homogenized way using the Narrow Region Acoustics domain feature (frequency domain only). The different ways of defining the properties of a fluid are called fluid models. They are also often referred to as equivalent fluid models.

The fluid model models losses in a homogenized way. Losses and damping occur when acoustic waves propagate in a porous material (material refers to the homogenization of a fluid and a porous solid), because of bulk viscous and thermal properties, or because of thermal and viscous losses in the acoustic boundary layer at walls in narrow ducts. The purpose of the fluid model is to mimic a special loss behavior by defining a complex valued density c and speed of sound cc. These are often frequency dependent.

In a Pressure Acoustics domain feature, the default Linear elastic fluid model (see Defining a Linear Elastic Fluid Model) enables you to specify a linearly elastic fluid using either the density and speed of sound c or the impedance Z and wave number k. When any of these material parameters are complex valued, damping is introduced.


It is possible to directly measure the complex wave number and impedance in an impedance tube in order to produce curves of the real and imaginary parts (the resistance and reactance, respectively) as functions of frequency. These data can be used directly as input to COMSOL Multiphysics interpolation functions to define k and Z.

The linear elastic fluid model is thus the most general fluid model as all user-defined expressions may be entered here; analytical expressions or measurement data.

The following more specific fluid models are described in this section (settings options detailed for the Pressure Acoustics node). They can be divided into these categories:

• Propagation in general fluid with bulk losses:

- Linear elastic, define density and speed of sound or impedance and wavenumber.

- Linear elastic with attenuation, define an attenuation parameter for the fluid.

- Ideal gas is also available as an option, but is not described here. This fluid model is used to specify the fluid properties by selecting a gas constant type and selecting between entering the heat capacity at constant pressure or the ratio of specific heats. See Defining an Ideal Gas Fluid Model for details.

- Viscous and/or thermally conducting fluids (bulk losses). The losses are here due to viscous losses, thermal conduction, or the combined thermal conduction and viscous losses in the bulk of the fluid.

• Propagation in porous materials (see Poroacoustics and About the Poroacoustics Fluid Model):

- The Delany-Bazely-Miki model, which is also referred to as an empirical porous model as it is based on fitting curves to experimental data. This is a one parameter empirical model.

- The Johnson-Champoux-Allard (JCA) model is sometimes referred to as a Biot equivalent model as it mimics Biot’s theory in the limiting cases of a limp porous matrix or a rigid porous matrix. This is a 5 parameter semi-empirical model.

• Propagation in narrow regions, narrow tubes, or waveguides (see Narrow Region Acoustics and About the Narrow Region Acoustics Fluid Model). The losses are here due to absorption/dissipation in the acoustic boundary layer (thermal and viscous losses). The losses are smeared on the domain.


Use the linear elastic attenuation fluid model to specify a linearly elastic fluid with attenuation using the density and speed of sound and to account for damping of

T H E O R Y F O R T H E P R E S S U R E A C O U S T I C S F L U I D M O D E L S | 135

136 | C H A P T E

acoustic waves using an attenuation coefficient . There are different attenuation types to choose from as the fluid model—Attenuation coefficient Np/m, Attenuation coefficient dB/m, or Attenuation coefficient dB/.

Select Attenuation coefficient Np/m to define an attenuation coefficient in Np/m (nepers per meter):

Select Attenuation coefficient dB/m to define an attenuation coefficient in dB/m (decibel per meter):

Select Attenuation coefficient dB/ to define an attenuation coefficient in dB/ (decibel per wavelength):

Defining a Linear Elastic with Attenuation Fluid Model

k c---- i–=

cck----=

cc2

cc2

---------=

k c---- i 10

20------ln–=

cck----=

cc2

cc2

---------=

k c---- 1 i 10

2 20------------------ln–

=

cck----=

c =



The viscous model is an equivalent-fluid model that mimics the propagation of sound in a fluid including viscous losses occurring in the bulk of the fluid. The elastic fluid model with viscous losses is defined by:

where is the dynamic viscosity and B is the bulk viscosity (see Ref. 4 or Ref. 10 chapter 9). This choice is only appropriate for situations where the damping takes place in free space and is not related to interaction between the fluid and a solid skeleton or a wall. These losses, in most fluids, occur over long distances or at very high frequencies.


The thermally conducting model is an equivalent-fluid model that mimics the propagation of sound in a fluid including losses due to thermal conduction in the bulk. The elastic fluid model with thermal losses is defined by:

where is the ratio of specific heats, Cp is the specific heat at constant pressure, and k is the thermal conductivity (see Ref. 10 chapter 9). This choice is only appropriate for

c 1 ib

c2----------+

1–=

cc c 1 ib

c2----------+

12---

=

b 43--- B+ =

Defining a Viscous Fluid Model

c 1 ib

c2----------+

1–=

cc c 1 ib

c2----------+

12---

=

b 1– kCp

-------------------- =


138 | C H A P T E

situations where the damping takes place in free space and is not related to interaction between the fluid and a solid skeleton or a wall.


The thermally conducting and viscous model is an equivalent-fluid model that mimics the propagation of sound in a fluid including losses due to thermal conduction and viscosity in the bulk of the fluid. The elastic fluid model with thermal and viscous losses is defined by:

where is the dynamic viscosity and B is the bulk viscosity, is the ratio of specific heats, Cp is the specific heat at constant pressure, and k is the thermal conductivity (see Ref. 10 chapter 9). This choice is only appropriate for situations where the damping takes place in free space and is not related to interaction between the fluid and a solid skeleton or a wall.

Defining a Thermally Conducting Fluid Model

c 1 ib

c2----------+

1–=

cc c 1 ib

c2----------+

12---

=

b 43--- B+ 1– k

Cp--------------------+

=

Defining a Thermally Conducting and Viscous Fluid Model


About the Poroacoustics Fluid Model

The Poroacoustics node has two poroacoustics models to choose from—Delany-Bazely-Miki and Johnson-Champoux-Allard.

D E L A N Y - B A Z E L Y - M I K I

The Delany-Bazley-Miki model is an equivalent fluid model that mimics the bulk losses in certain porous/fibrous materials. The model represents a porous medium with the following complex propagation constants:

Two predefined sets of the coefficients Ci exist, one representing the Delany-Bazley model, and one set representing the Miki model (see Ref. 11, section 2.5). This is an empirical models based on fitting the two complex functions above to measured data.

These poroacoustics models were previously available from the Pressure

Acoustics feature node and have been renamed:

• Delany-Bazley-Miki was previously called Macroscopic empirical porous

models.

• Johnson-Champoux-Allard was previously called Biot equivalents.

kcc---- 1 C1

0fRf--------

C2–iC3

0fRf--------

C– 4

–+=

Zc 0c 1 C50fRf--------

C6–iC7

0fRf--------

C– 8

–+=

There are restrictions on the applicability of the Delany-Bazley model:

• The porosity of the material should be close to 1.

• The value of should lie between 0.01 and 1.0.

• The flow resistivity Rf should lie between 1000 and 50,000 Pa·s/m2.

Absorptive Muffler: model library path Acoustics_Module/Industrial_Models/absorptive_muffler

0f Rf


140 | C H A P T E

J O H N S O N - C H A M P O U X - A L L A R D

The Johnson-Champoux-Allard models is an equivalent fluid model that mimic two limiting behaviors of the full Poroelastic Material model defined by Biot’s theory. The first is the rigid porous matrix model and the second is the limp porous matrix model. An equivalent fluid models is computationally less demanding than the full poroelastic model. However, they are only physically correct for certain choices of material parameters. Both models are based on describing the, frequency dependent, effective density () and the effective bulk modulus K() of the saturating fluid inside the porous matrix.

Limp and Rigid Porous Matrix ModelsThe limp porous matrix model models materials whose solid phases (the porous matrix) are so weak that they cannot support free, structure-borne wave propagation (neither longitudinal nor transverse). That is, their drained bulk stiffness (in vacuo bulk stiffness) is very small compared to air such that the solid phase motion becomes acoustically significant. If it is light enough, the solid phase still moves because it is “dragged along” by the fluid motion. So, a limp porous material model is also an equivalent fluid model because it only features a single longitudinal wave type. Typically, the limp model can be used to model very light weight fibrous materials (say less than 10 kg/m3) if they are not specifically stiffened by the injection of binder material. In the limp case, one assumes that the stress tensor vanishes and that Biot-Willis coefficient B = 1 in Biot’s theory. This in turn yields a wave equation.

The rigid porous matrix model is at the opposite end of the limp model, in that the matrix is assumed to be so stiff that it does not move (sometimes referred to as a motionless skeleton model). In this case one assumes u = 0 in Biot's theory which yields a wave equation with complex density and bulk modulus.

The limp (subscript “limp”) and rigid (subscript “rig”) porous matrix models are defined by the following equivalent densities, ()and equivalent bulk moduli, K():


Here is the tortuosity factor, f is the fluid density, p is the porosity, Rf is the flow resistivity, is the dynamic viscosity, P0 is the quiescent pressure, is the ratio of specific heats, Lv is the viscous characteristic length, Lth is the thermal characteristic length, Pr is the Prandtl number, d is the drained porous matrix density, Sp and Vp are the surface area and volume of the pores, s is a pore geometry dependent factor between 0.3 and 3.0 (for example 1 for circular pores, 0.78 for slits), av is the average effective density, and limp is the effective limp density. See Ref. 11 and Chapter 5 in Ref. 12 for further details.

This results in the following propagation parameters

rigfp------- 1

Rfpif-------------- 1

4i2f

Rf2Lv

2p2

-------------------------++=

KeqP0p

--------- 1– 1 8

iLth2

Prf

-------------------------- 1iLth

2Prf

16--------------------------++

1–

–

1–

=

limprigav f

2–

av rig 2– f+-----------------------------------= av d pf+=

Lv1s--- 8

pRf-----------=

Lth2VpSp

---------- 2Lv=

kc, rig rig Keq= or kc, limp limp Keq=

c rig= or c limp=

cckc-----=

keq2

cc---- 2

kz2

–=

The expression given for the geometry dependent pore factor s is only valid for values of s close to 1. If this is not the case, enter the viscous characteristic length Lv directly into the model (the default selection).


142 | C H A P T E

About the Narrow Region Acoustics Fluid Model

The Narrow Region Acoustics fluid models are used to mimic the thermal and viscous losses that exist in narrow tubes where the tube cross-section length-scale is comparable to the thermal and viscous boundary layer thickness (boundary-layer absorption). Including these losses is essential in order to get correct results.

These models are commonly used in situations where solving a full detailed thermoacoustic model becomes computationally costly. This is, for example, the case in long narrow ducts/tubes of constant cross section where it is possible to add or smear the losses associated with the boundary layer onto the bulk of the fluid—that is an equivalent fluid model.

Two fluid models exist: one for wide ducts, where the duct width is larger than the acoustic boundary layer thickness, and one for very narrow circular ducts (in the isothermal limit), where the duct width is much smaller than the acoustic boundary layer thickness.

W I D E D U C T S

For a relatively wide duct, the losses introduced in the acoustic boundary layer may be studied by adding them as an effective wall shear force. This approach is used in

Porous Absorber: model library path Acoustics_Module/Industrial_Models/porous_absorber

This feature was previously available from the Pressure Acoustics feature node as a fluid model where it was named Boundary-layer absorption.

In more complex geometries where thermal and viscous losses are important, see The Thermoacoustics, Frequency Domain Interface, which is more fundamental and detailed.

Narrow Region Acoustics


Blackstock (Ref. 10) and results in equivalent fluid complex wave number kc defined by

(3-15)

where Hd is the hydraulic diameter of the duct, S is the duct cross-section area, C is the duct circumference, is the dynamic viscosity, is the ratio of specific heats, Cp is the specific heat at constant pressure, k is the thermal conductivity, and Pr is the Prandtl number. For a cylindrical duct example, Hd = 2a, where a is the radius. The approximation in Equation 3-15 is only valid for systems where the effective radius Hd/2 is larger than the boundary layer, but not so small that mainstream thermal and viscous losses are important. Thus requiring

where dvisc is the characteristic thickness of the viscous boundary layer (the viscous penetration depth), c0 is the speed of sound, and is the angular frequency.

VE R Y N A R R O W C I R C U L A R D U C T S

In the other limit where the duct diameter is sufficiently small or the frequency sufficiently low, the boundary layer thickness becomes much larger than the duct cross section a. This is the case when

where dtherm is the characteristic thickness of the thermal boundary layer (thermal penetration depth). In this case, see Pierce (Ref. 5), the system may be seen as isothermal and the acoustic temperature variation is zero everywhere in the duct T = 0. In this case the fluid complex wave number kc is defined by

kcc0----- 1

1 B i------–

----------------------------- c0----- 1 B

2----

i------+

=

B 4Hd-------

0--------- 1 1–

Pr-----------+

= PrCp

k----------= Hd 4S

C----=

dviscHd2

-------c0

2

2------ 1

dvisc---------- dvisc

-------=

a dvisc«0a2

----------------- 1«

a dtherm«0a2Cp

k------------------------ 1«


144 | C H A P T E

(3-16)

where cT is the isothermal speed of sound and a is the tube radius. Note that setting a = Hd/2 is a further approximation. The theory is derived for ducts of circular cross section and, thus, the model is only applicable for systems with small variations away from a circular cross section.

For both fluid models, the relation between the complex wave number and the complex density and speed of sound is given by the usual

kccT------ 4

0a2-----------------

12---

i–4

0cT2 a2

-------------------

12---

=

aHd2

-------= cTc0

------=

cckc-----= c

c0cc-----

2= keq

2 cc---- 2

kz2

–=


Re f e r e n c e s f o r t h e P r e s s u r e A c ou s t i c s B r an c h

1. D. Givoli and B. Neta, “High-order Non-reflecting Boundary Scheme for Time-dependent Waves,” J. Comput. Phys., vol. 186, pp. 24–46, 2004.

2. A. Bayliss, M. Gunzburger, and E. Turkel, “Boundary Conditions for the Numerical Solution of Elliptic Equations in Exterior Regions,” SIAM J. Appl. Math., vol. 42, no. 2, pp. 430–451, 1982.

3. A.B. Bauer, “Impedance Theory and Measurements on Porous Acoustic Liners,” J. Aircr., vol. 14, pp. 720–728, 1977.

4. S. Temkin, Elements of Acoustics, Acoustical Society of America, 2001.

5. A.D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America (second print), 1991.

6. D.T. Blackstock, Fundamentals of Physical Acoustics, John Wiley and Sons, 2000.

7. P.M. Morse and K.U. Ignard, Theoretical Acoustics, Princeton University Press, 1986.

8. L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, Volume 6, Butterworth-Heinemann, 2003.

9. S. Temkin, Elements of Acoustics, Acoustical Society of America, 2001.

10. D.T. Blackstock, Fundamentals of Physical Acoustics, John Wiley & Sons, 2000.

11. J.F. Allard and N. Atalla, Propagation of Sound in Porous Media, John Wiley & Sons, 2009.

12. R. Panneton, “Comment on the Limp Frame Equivalent Fluid Model for Porous Media,” J. Acoust. Soc. Am. vol. 122, no. 6, pp. EL217–EL222, 2007.

R E F E R E N C E S F O R T H E P R E S S U R E A C O U S T I C S B R A N C H | 145

146 | C H A P T E

4

A c o u s t i c - S t r u c t u r e I n t e r a c t i o n

This chapter describes the multiphysics interfaces that are used for modeling acoustic-structure interaction. These interfaces are selected from the Acoustic-Structure Interaction branch ( ) when adding a physics interface.

• The Acoustic-Solid Interaction, Frequency Domain Interface

• The Acoustic-Solid Interaction, Transient Interface

• The Acoustic-Piezoelectric Interaction, Frequency Domain Interface

• The Acoustic-Piezoelectric Interaction, Transient Interface

• The Elastic Waves and Poroelastic Waves Interfaces

• Theory for the Elastic Waves and Poroelastic Waves Interfaces

• The Acoustic-Shell Interaction, Frequency Domain Interface

• The Acoustic-Shell Interaction, Transient Interface

• The Pipe Acoustics Interfaces

• Theory for the Pipe Acoustics Interfaces

143

144 | C H A P T E

Th e A c ou s t i c - S o l i d I n t e r a c t i o n , F r e qu en c y Doma i n I n t e r f a c e

The Acoustic-Solid Interaction, Frequency Domain (acsl) interface ( ), found under the Acoustics>Acoustic-Structure Interaction branch ( ) when adding a physics interface, combines the Pressure Acoustics, Frequency Domain and Solid Mechanics interfaces to connect the acoustics pressure variations in the fluid domain with the structural deformation in the solid domain. It can, for example, be used to determine the transmission of sound through an elastic structure or solve for the coupled vibroacoustics phenomena present in a loudspeaker.

Acoustic-structure interaction refers to a multiphysics phenomenon where the acoustic pressure causes a fluid load on the solid domain, and the structural acceleration acts on the fluid domain as a normal acceleration across the fluid-solid boundary.

Special interface conditions are readily defined at the fluid-solid boundary and set up the fluid loads on the solid domain and the effect of the structural accelerations on the fluid. The interface exists for frequency domain and eigenfrequency studies.

The interface is available for 3D, 2D, and 2D axisymmetric geometries and has the capability to model pressure acoustics and solid mechanics in the frequency domain, including a special acoustic-solid boundary condition for the fluid-solid interaction.

When this interface is added, these default nodes are also added to the Model Builder— Pressure Acoustics Model, Sound Hard Boundary (Wall), Free, Acoustic-Structure Boundary, Linear Elastic Material, and Initial Values. For 2D axisymmetric models an Axial

Symmetry node is also added. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Acoustic-Solid Interaction, Frequency Domain to select physics from the context menu.

On the Constituent Physics InterfacesThe Pressure Acoustics, Frequency Domain interface is used to compute the pressure variations when modeling the propagation of acoustic waves in fluids at quiescent background conditions. It solves the Helmholtz equation and is suited for all linear frequency-domain acoustic simulations with harmonic variations of the pressure field. The physics interface includes domain conditions to model losses in a homogenized

R 4 : A C O U S T I C - S T R U C T U R E I N T E R A C T I O N

way, so-called fluid models for porous materials as well as losses in narrow regions. The domain features also include background incident acoustic fields.

The Solid Mechanics interface is intended for general structural analysis of 3D, 2D or axisymmetric bodies. In 2D, plane stress or plane strain assumptions can be used. The physics interface is based on solving Navier's equations, and results such as displacements, stresses, and strains are computed.

The following sections provide information about all nodes specific to this multiphysics interface, but all other acoustics- and solid mechanics-specific features are found under the Pressure Acoustics and Solid Mechanics interfaces, respectively.



The default identifier (for the first interface in the model) is acsl.


The default setting is to include All domains in the model to define the dependent variables and the equations. To choose specific domains, select Manual from the Selection list.


The settings are the same for Sound Pressure Level Settings for the Pressure Acoustics, Frequency Domain interface.


Enter a value or expression for the Typical wave speed for perfectly matched layers cref (SI unit m/s). The default value is the speed of sound in air, 343 m/s.

T H E A C O U S T I C - S O L I D I N T E R A C T I O N , F R E Q U E N C Y D O M A I N I N T E R F A C E | 145

146 | C H A P T E

T H I C K N E S S

R E F E R E N C E PO I N T F O R M O M E N T C O M P U T A T I O N

Enter the coordinates for the Reference point for moment computation xref (SI unit: m). The resulting moments (applied or as reactions) are then computed relative to this reference point. During the results and analysis stage, the coordinates can be changed in the Parameters section in the result nodes.


To display this section, click the Show button ( ) and select Discretization. Select Linear, Quadratic (the default), Cubic, or Quartic as the element order for the Pressure

and Displacement field. Specify the Value type when using splitting of complex variables—Real or Complex (the default).


This interface defines these dependent variables (fields), the Pressure p and the Displacement field u and its components. The name can be changed but the names of fields and dependent variables must be unique within a model.

For 2D models, define the thickness d by entering a value or expression (SI unit: m). The default value of 1 m is suitable for plane strain models, where it represents a a unit-depth slice, for example. In rare cases, when the thickness needs to be changed in parts of the geometry; then use the Change Thickness node.


• Domain, Boundary, Edge, Point, and Pair Nodes for the Acoustic-Solid Interaction, Frequency Domain Interface

• Theory for the Solid Mechanics Interface


• Acoustic-Structure Interaction: model library path Acoustics_Module/

Tutorial_Models/acoustic_structure

• Loudspeaker Driver: model library path Acoustics_Module/

Industrial_Models/loudspeaker_driver


Domain, Boundary, Edge, Point, and Pair Nodes for the Acoustic-Solid Interaction, Frequency Domain Interface

The Acoustic-Solid Interaction, Frequency Domain Interface has these domain, boundary, edge, point, and pair nodes available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

Suggestions for setting up the solvers for solving large acoustic-structure interaction problems are given in Solving Large Acoustics Problems Using Iterative Solvers in the Modeling with the Acoustics Module chapter.


Some functionality differs slightly between the pure Pressure Acoustics interfaces and the functionality in the multiphysics interfaces. This mainly concerns the acoustic point and edge sources as well as some of the fluid models.

The Acoustic-Solid Interaction, Transient Interface also shares the same nodes as listed below. The exception being the Initial Values node and some of the fluid models (under the Transient Pressure Acoustics Model domain node) which do not exist in the transient interface.


148 | C H A P T E

These nodes (listed in alphabetical order) are described in this section:

When using the Far-Field Calculation feature in one of the Acoustic-Structure Interaction interfaces an additional data set is added with default name Solution 1a and identifier {ffc1}. This data set should be used when evaluating the far field in any of the dedicated far-field plots.

If the far-field calculation feature is added subsequently a data set may have to be created manually to evaluate the far-field variables. Simply duplicate the existing data set and change the Frame to Spatial (x, y, z). Use this new data set with far-field plots and evaluations.

• Acoustic-Structure Boundary

• Initial Values

• Pressure Acoustics Model

• Flow Line Source on Axis

• Intensity Line Source on Axis

• Power Line Source on Axis

• Intensity Edge Source

• Flow Edge Source

• Power Edge Source

• Flow Circular Source (in 2D axisymmetry)

• Intensity Circular Source (in 2D axisymmetry)

• Power Circular Source (in 2D axisymmetry)

• Intensity Point Source

• Flow Point Source

• Power Point Source


These nodes (listed in alphabetical order) are described for the Pressure Acoustics, Frequency Domain:

These nodes are described for the Solid Mechanics interface (listed in alphabetical order):

Pressure Acoustics Model

The Pressure Acoustics Model node adds the equations for time-harmonic and eigenfrequency acoustics modeling in the frequency domain. In the settings window,


• Continuity


• Dipole Source


• Impedance






• Monopole Source




• Pressure




• Added Mass

• Antisymmetry

• Body Load

• Boundary Load

• Edge Load

• Fixed Constraint

• Free

• Gravity

• Linear Elastic Material

• Point Load

• Pre-Deformation

• Prescribed Acceleration

• Prescribed Displacement

• Prescribed Velocity

• Roller

• Rotating Frame

• Spring Foundation

• Symmetry

• Thin Elastic Layer


150 | C H A P T E

define the properties for the acoustics model and model inputs including the background pressure and temperature.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains to compute the acoustic pressure field and the equation that defines it, or select All domains as required.


For all fluid models enter a Temperature T and an Absolute pressure p:

• Select User defined to enter a value or an expression for the absolute pressure (SI unit: Pa) and the temperature (SI unit: K) in the field. This input is always available.

• In addition, select a temperature field defined by, for example, a heat transfer interface or a non-isothermal fluid flow interface (if any).

• If applicable, select a pressure defined by a fluid-flow interface present in the model. For example, select Pressure (spf/fp1) to use the pressure defined by the Fluid

Properties node fp1 in a Single-Phase Flow interface spf. Selecting a pressure variable also activates a check box for defining the reference pressure, where 1[atm] is the default value. This makes it possible to use a system-based (gauge) pressure, while automatically including the reference pressure in the absolute pressure.

The input to these fields influences the value of the material parameters in the model. Typically, the density and the speed of sound c in the model are dependent on the

For more details about each of the available fluid models, see Theory for the Pressure Acoustics Fluid Models.

For more information about using variables during the results analysis, see Special Variables in the Acoustics Module.

Non-isothermal flow requires the addition of the Heat Transfer Module or CFD Module.


absolute pressure and/or the temperature. Picking up any of those from another interface typically results in xand c = cx to be specially varying.

P R E S S U R E A C O U S T I C S M O D E L

To define the properties of the bulk fluid, select a Fluid model from the list. The fluid models represent different loss mechanisms applied in a homogenized way to the bulk of the fluid. This type of fluid model is sometimes referred to as an equivalent fluid model. The model can be a theoretical or a phenomenological model that accounts for the losses due to viscosity and thermal conduction, for example, when acoustic waves propagate in porous materials.

Use the Linear elastic selection, which is already the default, to specify a linearly elastic fluid using either the density and speed of sound or the impedance and wave number. When the material parameters are real values this corresponds to a lossless compressible fluid. Go to Defining a Linear Elastic Fluid Model.

The fluid models can be roughly divided into these categories:

• General fluids:

- Linear elastic. Go to Defining a Linear Elastic Fluid Model.

- Linear elastic with attenuation. Go to Defining a Linear Elastic with Attenuation Fluid Model.

- Ideal gas. Go to Defining an Ideal Gas Fluid Model.

• Porous material fluid models:

- Macroscopic empirical porous models (Delany-Bazely or Miki models). Go to Defining Macroscopic Empirical Porous Fluid Models.

- Biot equivalents (limp porous matrix or rigid porous matrix, the latter is also known as the Johnson-Champoux-Allard model). Go to Defining a Biot Equivalent Fluid Model.


152 | C H A P T E

• Fluids with bulk viscous and thermal losses:

- Viscous. Go to Defining a Viscous Fluid Model.

- Thermally conducting. Go to Defining a Thermally Conducting Fluid Model.

- Thermally conducting and viscous. Go to Defining a Thermally Conducting and Viscous Fluid Model.

• Fluids models for viscous and thermal boundary-layer induced losses in channels and ducts:

- Boundary-layer absorption (narrow or wide duct). Go to Defining a Boundary-Layer Absorption Fluid Model.

Defining a Linear Elastic Fluid ModelTo specify the properties as complex-valued data, from the Specify list, select Density

and speed of sound (the default) or Impedance and wave number.

• If Density and speed of sound is selected, the default Speed of sound c (SI unit: m/s) and Density (SI unit: kg/m3) values are taken From material. Select User defined to enter other values or expressions.

• If Impedance and wave number is selected, enter a Wave number k (SI unit: rad/m). By default the Characteristic acoustic impedance Z (SI unit: Pa·s/m) is the value taken From material for the fluid. Select User defined to enter other values or expressions.

Defining a Linear Elastic with Attenuation Fluid ModelThe default Speed of sound c (SI unit: m/s) and Density (SI unit: kg/m3) values are taken From material. Select User defined to enter other values or expressions for one or both options.

Select an Attenuation type—Attenuation coefficient Np per unit length to define an attenuation coefficient in Np/m (nepers per meter), Attenuation coefficient dB per

unit length to define an attenuation coefficientin dB/m (decibel per meter), or Attenuation coefficient dB per wavelength to define an attenuation coefficient in dB/ (decibel per wavelength). For any selection, enter a value or expression in the Attenuation coefficient field.



Defining a Viscous Fluid ModelFor each of the following, the default values are taken From material. Select User defined to enter other values or expressions for any or all options.





Defining a Thermally Conducting Fluid ModelFor each of the following, the default values are taken From material. Select User defined to enter other values or expressions for any or all options.






Defining a Thermally Conducting and Viscous Fluid ModelFor each of the following, the default values are taken From material. Select User defined to enter other values or expressions for any or all options.









154 | C H A P T E



Defining an Ideal Gas Fluid ModelSpecify the fluid properties by selecting a gas constant type and selecting between entering the heat capacity at constant pressure or the ratio of specific heats.

• Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K) or Mean molar

mass Mn (SI unit: kg/mol). For both options, the default values are taken From

material. Select User defined to enter other values or expressions for one or both options. If Mean molar mass is selected, the molar gas constant (universal gas constant) R 8.314 J/(mol·K), is used as the built-in physical constant.

• From the Specify Cp or list, select Heat capacity at constant pressure Cp (SI unit: J /(kg·K)) or Ratio of specific heats . For both options, the default values are taken From material. Select User defined to enter other values or expressions for one or both options. For common diatomic gases such as air, 1.4 is the standard value.

Defining Macroscopic Empirical Porous Fluid ModelsThe default Speed of sound c (SI unit: m/s) and Density (SI unit: kg/m3) values are taken From material. Select User defined to enter other values or expressions for one or both options.

Enter a Flow resistivity Rf (SI unit: Pa·s/m2). Flow resistivity is easy to measure and is independent of frequency.

It is possible to assess the magnitude of the losses due to thermal conduction and viscosity, that is, the power dissipation density (SI unit: W/m3). This is done during the analysis process by plotting the variables for:

• the viscous power dissipation density (diss_visc),

• the thermal power dissipation density (diss_therm), or

• the combined total power dissipation density (diss_tot).



Select an option from the Constants list—Delany-Bazely (the default), Miki, or User

defined. If User defined is selected, enter values in the C1 to C8 fields.

Defining a Biot Equivalent Fluid ModelBy default, the Fluid material uses the Domain material.

For each of the following, the default values are taken From material. Select User defined to enter other values or expressions for any or all options.


• Density f (SI unit: kg/m3)





The section Porous Model is also displays when Biot Equivalents is selected as the fluid model. The default Porous elastic material uses the Domain material (the material defined for the domain). Select another material as required.

Select a Porous matrix approximation—Limp (the default) or Rigid.

• If Limp is selected, the default value for Drained density of porous material d (SI unit: kg/m3) is taken From material. Select User defined to enter another value or expression.

• For both Limp and Rigid porous matrix approximations, the Porosity p (dimensionless) value is taken From material. Select User defined to enter other values or expressions for one or both options. Then enter values or expressions for each of the following:

- Flow resistivity Rf (SI unit: Pa·s/m2). The default is 0 Pa·s/m2.

- From the Specify list, select Viscous characteristic length parameter to enter a value for s (dimensionless) (the default is 1), or select Viscous characteristic length to directly enter an expression for Lv (SI unit: m). The default expression for the

For The Pressure Acoustics, Frequency Domain Interface and The Boundary Mode Acoustics Interface, the Poroacoustics node replaces this option (the Delany-Bazley-Miki fluid model). See About the Poroacoustics Fluid Model for background theory.


156 | C H A P T E

Viscous characteristic length is sqrt(acpr.mu*acpr.tau*8/(acpr.Rf*acpr.epsilon_p)), which corresponds to s = 1.

- Thermal characteristic length Lth (SI unit: m). The default expression is 2*acpr.Lv.

- Tortuosity factor (dimensionless). The default is 1.

Defining a Boundary-Layer Absorption Fluid ModelBy default, the Fluid material uses the Domain material.

Select a Duct type—Wide duct (the default) or Narrow duct. For either selection, enter a Hydraulic diameter Hd (SI unit: m). The default is 0 m. Then for each of the following, the default values are taken From material. Select User defined to enter other values or expressions for any or all options.





The following are available for Wide duct only:



The Biot equivalents model with a rigid porous matrix is also often referred to as the Johnson-Champoux-Allard (JCA) model with a motionless skeleton.

For The Pressure Acoustics, Frequency Domain Interface and The Boundary Mode Acoustics Interface, the Poroacoustics node replaces this option. See About the Poroacoustics Fluid Model for background theory.

The boundary-layer absorption model adds the viscous and thermal losses effect of the acoustic boundary layer to the bulk of the fluid. This equivalent-fluid model can be used in long tubes of constant cross section instead of a full detailed thermoacoustic model.


Acoustic-Structure Boundary

The Acoustic-Structure Boundary node is available for The Acoustic-Solid Interaction, Frequency Domain Interface and The Acoustic-Solid Interaction, Transient Interface. This boundary condition includes the fluid load and structural acceleration for use on the fluid-solid boundaries, where it is the default boundary condition.

For the acoustic-solid interaction, this boundary condition is the default on the boundaries between the fluid and the solid. It is not applicable on other boundaries. This boundary condition includes the following interaction from fluid to solid and vice versa:

• A pressure load (force per unit area) Fpn p on the boundaries where the fluid interacts with the solid. In this expression, n is the outward-pointing unit normal vector seen from inside the solid domain.

• A structural acceleration acting on the boundaries between the solid and the fluid. This makes the normal acceleration for the acoustic pressure on the boundary equal to the acceleration based on the second derivatives of the structural displacements u with respect to time: ann · utt.



For The Pressure Acoustics, Frequency Domain Interface and The Boundary Mode Acoustics Interface, the Narrow Region Acoustics node replaces this option. See About the Narrow Region Acoustics Fluid Model for background theory.

For the node that contains the acoustic-structure boundary condition as a default condition, the boundary selection automatically becomes the boundaries between the fluid domain and the solid domain, and manual selection is not available.

If additional Acoustic-Structure Boundary nodes are added, which is normally not necessary, select the boundaries where this condition then becomes active.


158 | C H A P T E

Initial Values

The Initial Values node adds initial values for the sound pressure and the displacement field. Add more Initial Values nodes from the Physics toolbar.




Enter a value or expression for the initial values for Pressure p (SI unit: Pa) and the Displacement field u (SI unit: m). The default values are 0.

Flow Line Source on Axis

For the Flow Line Source on Axis node, both the source amplitude and its complex phase can be specified. This is done by defining the source amplitude as a complex number. This can be useful if there are two or more sources mutually out of phase. The following term is added to the right-hand side of the equation for the acoustic pressure:

where QS is the source amplitude defined as the flow rate out from the source per unit length.


From the Selection list, choose the boundaries on the symmetry axis to define the flow line source.

Use the Flow Line Source on Axis node to add a line source along the symmetry axis in 2D axisymmetry.

i QS2

r r0–


F L O W L I N E S O U R C E O N A X I S

Enter a Flow rate out from source per unit length QS (SI unit: m2/s) for the value for the source-strength amplitude.

Intensity Line Source on Axis

Using the Intensity Line Source on Axis node, set a desired reference root mean square (RMS) intensity Iref at a specified distance dsrc from the source. In a homogeneous medium, the specified RMS intensity is obtained (this is the reference), but when other objects and boundaries are present, the actual intensity is different. The term that this feature node adds to the right-hand side of the equation is given by

where Ledge is the length of the source line (automatically determined). The reference RMS intensity Iref (SI unit: W/m2). The delta function has the dimension of a length element dl (Si unit m).


From the Selection list, choose the boundaries on the symmetry axis to define the intensity line source.

When defining a Solution data set and plotting the results, specify an imaginary source-strength to produce a nonzero result when visualizing the resulting pressure field using the default value (0) in the Solution at

angle (phase).



Use the Intensity Line Source on Axis node to add a line source located along the symmetry axis in 2D axisymmetry.

4dsrcLedge------------- 2

Irefccc

------------ 2 r r0–

2 r r0–


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I N T E N S I T Y L I N E S O U R C E O N A X I S

Enter the Reference intensity (RMS) Iref (SI unit: W/m2) of the source and the Distance

from source center dsrc (SI unit: m).

Power Line Source on Axis

In this case, specify the source’s strength by stating the total power it would radiate into a homogeneous medium. The Power Line Source on Axis node adds the following term to the right-hand side of the equation:

where Ledge is the length of the source line (automatically determined) and Pref now denotes the radiated RMS power (in the reference homogeneous case) per unit length measured in W/m.


From the Selection list, choose the boundaries on the symmetry axis to define the power line source.

POW E R L I N E S O U R C E O N A X I S

Enter the Reference power per unit length (RMS) Pref (SI unit: W/m) radiating from the line source.

Intensity Edge Source

Use an Intensity Edge Source node to add an edge source located at rr0 for 3D models. Set a desired reference RMS intensity Iref at a specified distance dsrc from the source. In a homogeneous medium the specified intensity is obtained when the edge is a straight line (this is the reference). With other objects and boundaries present, or if the edge is curved, the actual intensity is different. This node adds the following term to the right-hand side of the equation for the acoustic pressure:

Use the Power Line Source on Axis node to add a line source along the symmetry axis in 2D axisymmetry.

2Ledge------------- 2

LedgePrefccc

--------------------------- 2 r r0–


where Ledge is the length of the source line (automatically determined). The reference RMS intensity Iref (SI unit: W/m2). The delta function has the dimension of a length element dl (Si unit m).



I N T E N S I T Y E D G E S O U R C E

Enter the Free space reference intensity (RMS) Iref (SI unit: W/m2) of the source and the Distance from source center dsrc (SI unit: m).

Flow Edge Source

Use an Flow Edge Source node to add an edge source located at rr0 for 3D models. Set a desired flow rate (per unit length) QS out from the source from the source. This node adds the following term to the right-hand side of the equation for the acoustic pressure:

The delta function has the dimension of a length element dl (Si unit m).




Enter the Flow rate out from source per unit length QS (SI unit: m2/s) of the source.

Power Edge Source

Use a Power Edge Source node to add an edge source located at rr0 for 3D models. Specify the source’s reference RMS strength by stating the total power a straight line source would radiate into a homogeneous medium. The node adds the following term to the right-hand side of the equation:

4dsrcLedge------------- 2

Irefccc

------------ 2 r r0–

2 r r0–

iQS 2

r r0–

2 r r0–


162 | C H A P T E

where Ledge is the length of the source line (automatically determined) and Pref denotes the radiated RMS power (in the reference homogeneous case) per unit length measured in W/m.



PO W E R E D G E S O U R C E

Specify the power radiating from the edge source. Enter the value for the Free space

reference power (RMS) per unit length Pref (SI unit: W/m).

Flow Circular Source

In 2D axisymmetric models adding a point in the domain (away from the axis of symmetry) corresponds to adding a circular source. Define the flow rate out from the source.




Enter the Flow rate out from source per unit length QS (SI unit: m2/s) of the source.

Intensity Circular Source

In 2D axisymmetric models adding a point in the domain (away from the axis of symmetry) corresponds to adding a circular source.




Enter the Free space reference intensity (RMS) Iref (SI unit: W/m2) of the source and the Distance from source center dsrc (SI unit: m).

2Ledge------------- 2

LedgePrefccc

--------------------------- 2 r r0–


Power Circular Source

In 2D axisymmetric models adding a point in the domain (away from the axis of symmetry) corresponds to adding a circular source.



PO W E R C I R C U L A R S O U R C E

Specify the power radiating from the circular source. Enter the value for the Free space

reference power (RMS) per unit length Pref (SI unit: W/m).

Intensity Point Source

Set a desired reference RMS intensity Iref at a specified distance dsrc from the source. In a homogeneous medium (the reference) the specified intensity is obtained, but with other objects and boundaries present the actual intensity is different. The term that this node adds to the right-hand side of the equation for the acoustic pressure differs depending on the space dimension.

For 3D and 2D axisymmetric points, the following term defines the source:

For a point in 2D the corresponding terms is



For 3D and 2D models, use an Intensity Point Source node to add a point source located at rr0.

For 2D axisymmetric models, it is added on the symmetry axis at r(z,r) (z0,

4dsrc 2Irefcc

c------------- 3

r r0–

4 2f Irefdsrcc

------------------- 2 r r0–


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I N T E N S I T Y P O I N T S O U R C E

Enter the Reference intensity (RMS) Iref (SI unit: W/m2) and the Distance from source

center dsrc (SI unit: m).

Flow Point Source

Specify the source’s strength by stating the volume flow rate pout from the source QS (SI unit m3/s). The feature adds the following term to the right hand side of the governing equation:



POW E R PO I N T S O U R C E

Enter the Free space reference intensity (RMS) Pref (SI unit: W).

Power Point Source

Specify the source’s strength by stating the total RMS power it would radiate into a homogeneous medium (the reference). The expression for the source differs slightly between 2D points and 3D points.

For a point in 3D, the feature adds the following term:

Here Pref is the reference radiated RMS power (SI unit: W).

For a point in 2D, the feature uses the following term:

i 3 r r0–

For 3D and 2D models, use a Power Point Source node to add a point source located at rr0.

For 2D axisymmetric models, it is added on the symmetry axis at r(z,r) (z0,

22Prefcc

c--------------------- 3

r r0–


where Pref now denotes the reference radiated RMS power per unit length measured in W/m.



PO W E R PO I N T S O U R C E

Enter the Free space reference intensity (RMS) Pref (SI unit: W).

22Prefc

----------------- 2 r r0–

Specify the power radiating from the point source. In 2D, enter the value for the Reference power (RMS) per unit length Pref (SI unit: W/m).

For 3D and 2D axisymmetric models, specify the power radiating from the point source. Enter the value for the Reference power (RMS), Pref (SI unit: W).

Hollow Cylinder: model library path Acoustics_Module/Tutorial_Models/

hollow_cylinder


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Th e A c ou s t i c - S o l i d I n t e r a c t i o n , T r a n s i e n t I n t e r f a c e

The Acoustic-Solid Interaction, Transient (astd) interface ( ), found under the Acoustics>Acoustic-Structure Interaction branch ( ) when adding a physics interface, combines the Pressure Acoustics, Transient and Solid Mechanics interfaces to connect the acoustics pressure variations in the fluid domain with the structural deformation in the solid domain. It can, for example, be used to determine the transmission of sound through an elastic structure or solve for the coupled vibroacoustics phenomena present in a loudspeaker.

Acoustic-structure interaction refers to a multiphysics phenomenon where the acoustic pressure causes a fluid load on the solid domain, and the structural acceleration acts on the fluid domain as a normal acceleration across the fluid-solid boundary.

Special interface conditions are readily defined at the fluid-solid boundary and set up the fluid loads on the solid domain and the effect of the structural accelerations on the fluid. The physics interface allows for transient studies but also solves in the frequency domain with the available boundary conditions.

When this interface is added, these default nodes are also added to the Model Builder— Transient Pressure Acoustics Model, Sound Hard Boundary (Wall), Free, Acoustic-Structure

Boundary, Linear Elastic Material, and Initial Values. For 2D axisymmetric models an Axial Symmetry node is also added. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Acoustic-Solid Interaction, Transient to select physics from the context menu.

On the Constituent Physics InterfacesThe Pressure Acoustics, Transient interface is used to compute the pressure variations when modeling the propagation of acoustic waves in fluids at quiescent background conditions. It solves the scalar wave equation and is suited for time-dependent simulations with arbitrary time-dependent fields and sources. Domain conditions also include background incident acoustic fields. User-defined sources can be added to, for example, include certain nonlinear effects such as a square pressure dependency of the density variations.

The Solid Mechanics interface is intended for general structural analysis of 3D, 2D, or axisymmetric bodies. In 2D, plane stress or plane strain assumptions can be used. The


interface is based on solving Navier's equations, and results such as displacements, stresses, and strains are computed.



The default identifier (for the first interface in the model) is astd.

Initial Values

The Initial Values node adds initial values for the pressure, the displacement field, the structural velocity field and the pressure, first time derivative. Add more Initial Values nodes from the Physics toolbar.

For modeling of acoustic-structure interaction in the frequency domain, The Acoustic-Solid Interaction, Frequency Domain Interface contains additional functionality that is not applicable for modeling in the time domain.

See The Acoustic-Solid Interaction, Frequency Domain Interface for the rest of the settings for this interface. Links to all the feature nodes (except Initial Values in this section) are found in Domain, Boundary, Edge, Point, and Pair Nodes for the Acoustic-Solid Interaction, Frequency Domain Interface. The Gaussian Pulse sources are described in Gaussian Pulse Point, Edge, Circular, and Line Sources.




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168 | C H A P T E




Enter a value or expression for the initial values.

• Pressure p (SI unit: Pa). The default is 0 Pa.

• Pressure, first time derivative p/t (SI unit: Pa/s). The default is 0 Pa/s.

• Displacement field u (SI unit: m). The default is 0 m.

• Structural velocity field u/t (SI unit: m/s). The default is 0 m/s.

Gaussian Pulse Point, Edge, Circular, and Line Sources

In transient models a class of point and line sources exist that define a Gaussian pulse in time. These are the Gaussian Pulse Point Source, the Gaussian Edge Source, the Gaussian Circular Source, and the Gaussian Line Source. These sources define a source of the type:

where A is the source amplitude (SI unit: m2/s), f0 is the frequency bandwidth (SI unit: Hz), and tp is the pulse peak time (Si unit: s). The sources are either added on a point or an edge (all space dimensions), a point in 2D axisymmetry represents a circular source, or an on axis line source in 2D axisymmetry.

G A U S S I A N P U L S E PO I N T , E D G E , C I R C U L A R , A N D L I N E S O U R C E

Enter the Amplitude A, Frequency bandwidth f0, and Pulse peak time tp.

Qm A22f02 t tp– e 2f0

2 t tp– 2––=


Th e A c ou s t i c - P i e z o e l e c t r i c I n t e r a c t i o n , F r e qu en c y Doma i n I n t e r f a c e

The Acoustic-Piezoelectric Interaction, Frequency Domain (acpz) interface ( ), found when adding a physics interface under the Acoustics>Acoustic-Structure Interaction

branch ( ), combines the Pressure Acoustics, Frequency Domain and Piezoelectric Devices interfaces to connect and solve for the acoustic pressure variations in fluids with the structural deformation in both solids and piezoelectric solid domains. The physics interface also includes features from Electrostatics to solve for the electric field in the piezoelectric material. It may for example be used for modeling piezoelectric transducers for sonar or medical applications and, for example, enhancing the impedance matching layers as well as the far-field radiation patterns of the transducer.

The Helmholtz equation is solved in the fluid domain and the structural equations in the solid together with the constitutive relationships required to model piezoelectrics. Both the direct and inverse piezoelectric effects can be modeled, and the piezoelectric coupling can be formulated using the strain-charge or stress-charge forms.

When this interface is added, these default nodes are also added to the Model Builder— Pressure Acoustics Model, Piezoelectric Material, Sound Hard Boundary (Wall), Free, Acoustic-Structure Boundary, Zero Charge, and Initial Values. For 2D axisymmetric models an Axial Symmetry node is also added.

Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Acoustic-Piezoelectric

Interaction, Frequency Domain to select physics from the context menu.

On the Constituent Physics InterfacesThe Pressure Acoustics, Frequency Domain interface computes the pressure variations when modeling the propagation of acoustic waves in fluids at quiescent background conditions. The physics interface solves the Helmholtz equation and is suited for all linear frequency-domain acoustic simulations with harmonic variations of the pressure field. It includes domain conditions to model losses in a homogenized way, so-called fluid models for porous materials as well as losses in narrow regions. The domain features also include background incident acoustic fields.

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The Solid Mechanics interface is intended for general structural analysis of 3D, 2D, or axisymmetric bodies. In 2D, plane stress or plane strain assumptions can be used. The physics interface is based on solving Navier's equations, and results such as displacements, stresses, and strains are computed.

The following sections provide information about all nodes specific to this multiphysics interface, but all other acoustics-, solid mechanics-, and electrostatics-specific nodes are found under the Acoustical, Structural, and Electrical menus, respectively.



The default identifier (for the first interface in the model) is acpz.


The default setting is to include All domains in the model. To choose specific domains, select Manual from the Selection list.



The equations solved in the solid and fluid domains can be found in Theory for the Piezoelectric Devices Interface and Theory Background for the Pressure Acoustics Branch, respectively.


T H I C K N E S S




To display this section, click the Show button ( ) and select Discretization. Select Linear, Quadratic (the default), Cubic, or Quartic as the element order for the Pressure, Displacement field, and Electric potential. Specify the Value type when using splitting of

complex variables—Real or Complex (the default).


This interface defines these dependent variables (fields), the Pressure p, the Displacement field u and its components, and the Electric potential V. The name can be changed but the names of fields and dependent variables must be unique within a model.

For 2D models, define the thickness d by entering a value or expression (SI unit: m). The default value of 1 m is suitable for plane strain models, where it represents a a unit-depth slice, for example.

In rare cases, if the thickness needs to be changed in parts of the geometry; then use the Change Thickness node.


• Initial Values

• Domain, Boundary, Edge, Point, and Pair Nodes for the Acoustic-Piezoelectric Interaction, Frequency Domain Interface

• Theory for the Piezoelectric Devices Interface




172 | C H A P T E

Domain, Boundary, Edge, Point, and Pair Nodes for the Acoustic-Piezoelectric Interaction, Frequency Domain Interface

The Acoustic-Piezoelectric Interaction, Frequency Domain Interface has these domain, boundary, edge, point, and pair nodes available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

Piezoacoustic Transducer: model library path Acoustics_Module/

Tutorial_Models/piezoacoustic_transducer


Almost every node, except Initial Values, is shared with, and described for, other interfaces.

The Acoustic-Piezoelectric Interaction, Transient Interface also shares the same nodes except for some fluid models that are not available in the Transient Pressure Acoustics Model.

The point end edge sources are described in The Acoustic-Solid Interaction, Frequency Domain Interface section.

The continuity condition between the fluid domain and the solid domain is described in the Acoustic-Structure Boundary section under the Acoustic-Solid Interaction interface.


A C O U S T I C A L M E N U

These nodes are described for the Pressure Acoustics, Frequency Domain interface (listed in alphabetical order):


• Continuity


• Dipole Source


• Impedance






• Monopole Source



• Pressure







If the far-field calculation feature is added subsequently a data set may have to be created manually to evaluate the far-field variables. Simply duplicate the existing data set and change the Frame to Spatial (x,y,z). Use this new data set with far-field plots and evaluations.


174 | C H A P T E

S T R U C T U R A L M E N U


P I E Z O E L E C T R I C D E V I C E S M E N U

These nodes are described for the Piezoelectric Devices interface:

• Electrical Material Model

• Piezoelectric Material

E L E C T R I C A L M E N U

These nodes are described for the Electrostatics interface in the COMSOL Multiphysics Reference Manual (listed in alphabetical order):

• Added Mass

• Antisymmetry

• Body Load

• Boundary Load

• Edge Load


• Gravity

• Free


• Point Load




• Roller

• Rotating Frame


• Symmetry


• Electric Potential

• Electric Displacement Field

• Ground

• Line Charge

• Point Charge

• Space Charge Density

• Thin Low Permittivity Gap

• Zero Charge

The links to the COMSOL Multiphysics Reference Manual do not work in the PDF, only from the on line help in COMSOL Multiphysics.


Initial Values

The Initial Values node adds an initial value for the pressure, displacement field, and the electric potential. Add more Initial Values nodes from the Physics toolbar.




Enter the initial values as values or expressions for the Pressure p (SI unit: Pa), Displacement field u (SI unit: m) and the Electric potential V (SI unit: V).


176 | C H A P T E

Th e A c ou s t i c - P i e z o e l e c t r i c I n t e r a c t i o n , T r a n s i e n t I n t e r f a c e

The Acoustic-Piezoelectric Interaction, Transient (acpztd) interface ( ), found when adding a physics interface under the Acoustics>Acoustic-Structure Interaction

branch ( ), combines the Pressure Acoustics, Transient and Piezoelectric Devices interfaces to connect and solve for the acoustic pressure variations in fluids with the structural deformation in both solids and piezoelectric solid domains. The physics interface also includes features from Electrostatics to solve for the electric field in the piezoelectric material. Examples include modeling piezoelectric transducers for sonar or medical applications and, for example, enhancing the impedance matching layers.

The pressure wave equation is solved in the fluid domain and the structural dynamic equations in the solid together with the constitutive relationships required to model piezoelectrics. Both the direct and inverse piezoelectric effects can be modeled, and the piezoelectric coupling can be formulated using the strain-charge or stress-charge forms.

When this interface is added, these default nodes are also added to the Model Builder— Transient Pressure Acoustics Model, Sound Hard Boundary (Wall), Free, Zero Charge, Acoustic-Structure Boundary, Initial Values, and Piezoelectric Material. For 2D axisymmetric models an Axial Symmetry node is also added. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Acoustic-Piezoelectric Interaction, Transient to select physics from the context menu.

On the Constituent Physics InterfacesThe Pressure Acoustics, Transient interface computes the pressure variations when modeling the propagation of acoustic waves in fluids at quiescent background conditions. The interface solves the scalar wave equation and is suited for time-dependent analyses with arbitrary time-dependent fields and sources. Domain conditions also include background incident acoustic fields. User-defined sources can be added to, for example, include certain nonlinear effects such as a square pressure dependency of the density variations.

The Solid Mechanics interface is intended for general structural analysis of 3D, 2D, or axisymmetric bodies. In 2D, plane stress or plane strain assumptions can be used. The


interface is based on solving Navier's equations, and results such as displacements, stresses, and strains are computed.



The default identifier (for the first interface in the model) is acpztd.





T H I C K N E S S



TO T A L E N E R G Y A N D L O S S

This section contains the variables Wtot, the total energy (SI unit: J), and Qh,tot, the total loss (SI unit: W). The default variables are pzd.W_int and pzd.Qh_int,

For 2D models, define the thickness d by entering a value or expression (SI unit: m). The default value of 1 m is suitable for plane strain models, where it represents a a unit-depth slice, for example.

In rare cases, where the thickness needs to be changed in parts of the geometry; then use the Change Thickness node.

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178 | C H A P T E

respectively. A model might contain other components of loss and stored energy, which do not appear in the integral over the interface. Add those contributions in the fields for Wtot and Qh,tot to get the quality factor WtotQh,tot correct.


This interface defines these dependent variables (fields), the Pressure p, the Displacement field u and its components, and the Electric potential V. The name can be changed but the names of fields and dependent variables must be unique within a model.


To display this section, click the Show button ( ) and select Discretization. Select Linear, Quadratic (the default), Cubic, or Quartic as the element order for the Pressure, Displacement field, and Electric potential. Specify the Value type when using splitting of

complex variables—Real or Complex (the default).


• Domain, Boundary, Edge, Point, and Pair Nodes for the Acoustic-Piezoelectric Interaction, Frequency Domain Interface

• Gaussian Pulse Point, Edge, Circular, and Line Sources.





Th e E l a s t i c Wav e s and Po r o e l a s t i c Wav e s I n t e r f a c e s

In this section:

• The Elastic Waves Interface

• The Poroelastic Waves Interface

• Domain, Boundary, and Shared Nodes for the Elastic Waves and the Poroelastic Waves Interface

The Elastic Waves Interface

The Elastic Waves (elw) interface ( ), found under the Acoustics>Acoustic-Structure

Interaction branch ( ) when adding a physics interface, is used to compute the displacement field in solids with propagating elastic waves. The physics interface allows adding fluid or porous domains adjacent to the default solid domain. Dedicated boundary conditions define the loads between fluid, solid, and porous domains, including special sets of boundary conditions that are automatically applied for the fluid-solid, fluid-porous, and porous-solid interactions.

The interface is valid for modeling the propagation of linear elastic and linear acoustic waves in the frequency domain. Harmonic variations of the displacement field and sources are assumed.

The Helmholtz equation is solved in the fluid domain and the structural dynamic equations in the solid. In porous domains, Biot’s equations are solved, accounting for the coupled propagation of elastic waves in an elastic porous matrix and pressure waves in the saturating fluid. This includes the damping effect of the pore fluid.

Also see The Poroelastic Waves Interface.

When the Elastic Waves interface is added, these default nodes are also added to the Model Builder— Linear Elastic Material, Sound Hard Boundary (Wall), Free, Continuity, and Initial Values. For 2D axisymmetric models an Axial Symmetry node is also added.

Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Elastic Waves to select physics from the context menu.

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180 | C H A P T E

The following sections provide information about all nodes specific to these multiphysics interfaces, see details about other features under the Pressure Acoustics,

Frequency Domain or Solid Mechanics submenus, respectively.



The default identifier for the first interface in the model is elw.








To display this section, click the Show button ( ) and select Discretization. Select Linear, Quadratic (the default), Cubic, or Quartic for the Pressure and Displacement field. Specify the Value type when using splitting of complex variables—Real or Complex (the default).



This interface defines these dependent variables (fields), the Pressure p and the Displacement field u and its components. The name can be changed but the names of fields and dependent variables must be unique within a model.

The Poroelastic Waves Interface

The Poroelastic Waves (elw) interface ( ), found under the Acoustics>Acoustic-Structure Interaction branch ( ) when adding a physics interface, is used to compute the displacement field and acoustic pressure fluctuation in porous materials with propagating poroelastic waves. The physics interface allows adding fluid or solid domains adjacent to the default porous domain. Dedicated boundary conditions define the loads between fluid, solid, and porous domains, including special sets of boundary conditions that are automatically applied for the fluid-solid, fluid-porous, and porous-solid interactions. Examples of applications range from the propagation of elastic waves in rocks and soils to modeling the acoustic attenuation properties of particulate filters and sound absorbers.

The physics interface is valid for modeling the propagation of linear elastic and linear acoustic waves in the frequency domain. Harmonic variations of the displacement field and the sources are assumed.

In the porous domains, Biot’s equations are solved accounting for the coupled propagation of elastic waves in the elastic porous matrix and pressure waves in the saturating pore fluid. This includes the damping effect of the pore fluid. The Helmholtz equation is solved in the fluid domain and the structural dynamic equations in the solid.

This interface has most of the same settings as The Elastic Waves Interface.

When the Poroelastic Waves interface is added, these default nodes are also added to the Model Builder— Poroelastic Material, Sound Hard Boundary (Wall), Free, Continuity, and Initial Values. For 2D axisymmetric models an Axial Symmetry node is also added.



• The Poroelastic Waves Interface



182 | C H A P T E

Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Porelastic Waves to select physics from the context menu.

The following sections provide information about all nodes specific to these multiphysics interfaces, see details about other nodes under the Pressure Acoustics,

Frequency Domain or Solid Mechanics submenus, respectively.

Domain, Boundary, and Shared Nodes for the Elastic Waves and the Poroelastic Waves Interface

Because The Elastic Waves and Poroelastic Waves Interfaces are multiphysics interfaces, almost every node is shared with, and described for, other interfaces. Below are links to the domain, boundary, edge, pair, and point nodes as indicated.

See Domain, Boundary, and Shared Nodes for the Elastic Waves and the Poroelastic Waves Interface for all settings and to follow the links for domain, boundary, pair, edge, and point conditions.



• The Elastic Waves Interface


Acoustics of a Particulate-Filter-Like System: model library path Acoustics_Module/Tutorial_Models/acoustics_particulate_filter

Some functionality differs slightly between the pure Pressure Acoustics interfaces and the functionality in the multiphysics interfaces. This mainly concerns the acoustic point and edge sources, which are described in The Acoustic-Solid Interaction, Frequency Domain Interface.


These porous material nodes are described in this section (listed in alphabetical order):

These nodes are described for the Pressure Acoustics, Frequency Domain (listed in alphabetical order):

• Continuity

• Initial Values

• Porous, Fixed Constraint

• Porous, Free

• Poroelastic Material

• Porous, Pressure

• Porous, Prescribed Acceleration

• Porous, Prescribed Displacement

• Porous, Prescribed Velocity

• Porous, Roller

• Porous, Septum Boundary Load



• Dipole Source


• Impedance






• Monopole Source




• Pressure



• Sound Hard Boundary (Wall) (the default)





184 | C H A P T E


Initial Values

The Initial Values node adds initial values for the pressure and displacement field. Add more Initial Values nodes from the Physics toolbar.




Enter a value or expression for the Pressure p (SI unit: Pa) and Displacement field u (SI unit: m) initial values. The default is 0 Pa for the pressure and 0 m for the displacement field.

Poroelastic Material

Use the Poroelastic Material node to define the poroelastic material and fluid properties, that is the properties of the porous matrix and the saturating fluid. Also right-click the node to add an Initial Stress and Strain subnode.

• Antisymmetry

• Body Load

• Boundary Load

• Edge Load


• Free (the default boundary condition for Linear elastic materials)


• Point Load




• Roller

• Symmetry

Destination Selection in the COMSOL Multiphysics Reference Manual





This section contains field variables that appear as model inputs, if the current settings include such model inputs. By default, this section is empty. If a linear temperature relation for the conductivity is added, then define the source for the temperature T. From the Temperature list, select an existing temperature variable (from another physics interface) if available, or select User defined to define a value or expression for the temperature (SI unit: K) in the field that appears underneath the list.

C O O R D I N A T E S Y S T E M S E L E C T I O N

The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes.

PO R O E L A S T I C M A T E R I A L

The default Porous elastic material uses the Domain material (the material defined for the domain). Select another material as required.

Select a Porous model—Drained matrix, isotropic, Drained matrix, orthotropic, or Drained

matrix, anisotropic. Then enter or select the settings as described.

Porous Model for Drained Matrix, IsotropicIf Drained matrix, isotropic is selected from the Porous model list, select a pair of elastic properties to describe an isotropic drained porous material. From the Specify list, select:

• Bulk modulus and shear modulus (the default) to specify the drained bulk modulus K (SI unit: Pa) and the drained shear modulus G (SI unit: Pa). The bulk drained modulus is a measure of the solid porous matrix’s resistance to volume changes. The shear modulus is a measure of the solid porous matrix’s resistance to shear deformations.

• Young’s modulus and Poisson’s ratio to specify drained Young’s modulus (elastic modulus) E (SI unit: Pa) and Poisson’s ratio (dimensionless). For an isotropic material Young’s modulus is the spring stiffness in Hooke’s law, which in 1D form is Ewhere is the stress and is the strain. Poisson’s ratio defines the normal

To define the Absolute Pressure, see the settings for the Heat Transfer in Fluids node as described in the COMSOL Multiphysics Reference Manual.


186 | C H A P T E

strain in the perpendicular direction, generated from a normal strain in the other direction and follows the equation =

• Lamé parameters to specify the drained Lamé parameters (SI unit: Pa) and (SI unit: Pa).

• Pressure-wave and shear-wave speeds to specify the drained pressure-wave speed cp (SI unit: m/s) and the shear-wave speed cs (SI unit: m/s).

For each pair of properties, select from the applicable list to use the value From material or enter a User defined value or expression. Each of these pairs define the drained elastic properties and it is possible to convert from one set of properties to another.

Porous Model for Drained Matrix, OrthotropicWhen Drained matrix, orthotropic is selected from the Porous model list, the material properties of the solid porous matrix vary in orthogonal directions only.

The default properties take values From material. Select User defined to enter values or expressions for the drained Young’s modulus E (SI unit: Pa), the drained Poisson’s ratio (dimensionless), and the drained Shear modulus G (SI unit: Pa).

Porous Model for Drained Matrix, AnisotropicWhen Drained matrix, anisotropic is selected from the Porous model list, the material properties of the solid porous matrix vary in all directions, and the stiffness comes from the symmetric Elasticity matrix, D (SI unit: Pa). The default uses values From material. Select User defined to enter values in the 6-by-6 symmetric matrix that displays.

Parameters for All Porous ModelsEnter the following remaining parameters necessary to defined the properties of a porous material for all the Porous models selected above. The defaults use values From

material. Select User defined to enter other values or expressions as required.

• Drained density of porous material to specify the drained density of the porous material in vacuum d (SI unit: kg/m3). The drained density d is equal to (1 p) s where s is the density of the solid material from which the matrix is made and p is the porosity.

• Permeability to specify the permeability of the porous material p (SI unit: m2). The permeability is a measure of the ability of the porous material to let fluid pass through it. It hence gives some measure of the pore size and thus correlates to the viscous damping experienced by pressure waves propagating in the saturating fluid.


• Porosity to specify the porosity of the material p (dimensionless). It defines the amount of void volume inside the porous matrix and takes values between 0 (no porous material only fluid) and 1 (fully solid material no fluid).

• Biot-Willis coefficient to specify the Biot-Willis coefficient B (dimensionless). This coefficient relates the bulk modulus (compressibility) of the drained porous matrix to a block of solid material. It is defined as

where Kd is the drained bulk modulus (named K here) and Ks is the bulk module of a block of solid material. The drained bulk modulus is related to the stiffness of the porous matrix, while the solid bulk modulus is related to the compressibility of the material or grains from which the porous matrix is made. The Biot-Willis coefficient is bound by . A rigid porous matrix (Voigt upper bound) has

and a soft or limp porous matrix (Reuss lower bound) has .

• Tortuosity factor or the structural form factor (dimensionless). This is a purely geometrical factor that depends on the microscopic geometry and distribution of the pores inside the porous material. It is independent of the fluid and solid properties and is normally >1. The default is 2. The more complex the propagation path through the material, the higher is the absorption. The tortuosity partly represents this complexity.


Define the properties of the saturating fluid in terms of its density, viscosity and compressibility but also the viscosity model. The defaults use values for the material parameters are From material. Select User defined to enter other values or expressions as required.

• Density defines the density of the saturating fluid f (SI unit: kg/m3).

• Dynamic viscosity to define the dynamic viscosity of the saturating fluid f (SI unit: Pa·s). The parameter is important for the amount of viscous damping experienced by the acoustic waves.

• Compressibility of the saturating fluid f (SI unit: 1/Pa). The compressibility of the fluid enters the expression for Biot’s module M, give by

B 1KdKs-------–=

p B 1 B p= B 1=


188 | C H A P T E

It should be noted that Biot-Willis coefficient only depends on the properties of the porous matrix while Biot’s module depends on both fluid and porous matrix properties.

Select a Viscosity Model, either Biot’s low frequency range or Biot’s high frequency range.

• Biot’s low frequency range models damping at low frequencies where the acoustic boundary layer (the viscous penetration depth) is assumed to span the full width of the pores.

• If Biot’s high frequency range is selected, then also enter a Reference frequency fc (SI unit: Hz). This model implements a correction factor to the viscosity that accounts for the relative scale difference between a typical pore diameter and the acoustic boundary layer thickness. The modified viscosity is of the form

where fc is the reference frequency and a is a characteristic size of the pores. The expression for fc is one typically used in literature but it is often measured or empirically determined. The expression for fc corresponds to finding the frequency at which the viscous boundary layer thickness is of the scale a.

Porous, Fixed Constraint

The Porous, Fixed Constraint node adds a condition that makes the porous matrix fixed (fully constrained); that is, the displacements are zero in all directions. This boundary condition also sets an impervious (sound-hard) boundary for the fluid pressure.

MKs

1 p–KdKs------- p+ Ksf–

----------------------------------------------------= KsKd

1 B–----------------=

f fFffc----

= fcf

2a2f

------------------=

See High Frequency Correction for more details.




Porous, Free

The Porous, Free node is the default boundary condition. It means that there are no constraints and no loads acting on the porous matrix, and a sound-soft boundary for the fluid pressure.



Porous, Pressure

The Porous, Pressure node creates a boundary condition that acts as a pressure source at the boundary, which means a constant acoustic pressure p = p0 is specified. In the frequency domain, p0 is the amplitude of a harmonic pressure source.



P R E S S U R E

Enter the value of the Pressure p0 (SI unit: Pa) at the boundary.



Porous, Fixed Constraint (Sound-Hard Boundary) Equations

Porous, Free (Sound-Soft Boundary) Theory


190 | C H A P T E


Porous, Prescribed Displacement

The Porous, Prescribed Displacement node adds a condition where the displacements are prescribed in one or more directions to the porous matrix boundary.

If a displacement is prescribed in one direction, this leaves the porous matrix free to deform in the other directions. Also define more general displacements as a linear combination of the displacements in each direction.

• If a prescribed displacement is not activated in any direction, this is the same as a Free constraint.

• If a zero displacement is applied in all directions, this is the same as a Fixed

Constraint.





P R E S C R I B E D D I S P L A C E M E N T

Define the prescribed displacements using a Standard notation or a General notation.

Porous, Pressure Equations


Standard NotationTo define the displacements individually, click the Standard notation button (the default).

General NotationTo specify the displacements using a General notation that includes any linear combination of displacement components, click the General notation button.

Enter values in the H matrix and R vector fields. For the H matrix, also select an Isotropic, Diagonal, Symmetric, or Anisotropic matrix and enter values as required.




To define a prescribed displacement for each space direction (x, y, and z for 3D), select one or more of the Prescribed in x direction, Prescribed in y

direction, and Prescribed in z direction check boxes. Then enter a value or expression for the prescribed displacements u0, v0, or w0 (SI unit: m).

For 2D axisymmetric models and to define a prescribed displacement for each space direction (r and z), select one or both of the Prescribed in r

direction and Prescribed in z direction check boxes. Then enter a value or expression for the prescribed displacements u0, or w0 (SI unit: m).

Porous, Prescribed Displacement Equations


192 | C H A P T E

Porous, Prescribed Velocity

The Porous, Prescribed Velocity node adds a boundary condition where the velocity of the porous matrix is prescribed in one or more directions. With this boundary condition it is possible to prescribe a velocity in one direction, leaving the solid free in the other directions.




The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. Coordinate systems with directions which change with time should not be used.

PO R O U S , P R E S C R I B E D VE L O C I T Y

Porous, Prescribed Acceleration

The Porous, Prescribed Acceleration node adds a boundary condition, where the acceleration of the porous matrix is prescribed in one or more directions. With this boundary condition, it is possible to prescribe a acceleration in one direction, leaving the solid free in the other directions.

To define a porous, prescribed velocity for each space direction (x, y, and z for 3D), select one or all of the Prescribed in x direction, Prescribed in y

direction, and Prescribed in z direction check boxes. Then enter a value or expression for the components vx, vy, and vz (SI unit: m/s).

For 2D axisymmetric models and to define a prescribed velocity for each space direction (r and z), select one or both of the Prescribed in r direction and Prescribed in z direction check boxes. Then enter a value or expression for vr and vz (SI unit: m/s).

Porous, Prescribed Velocity Equations





The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. Coordinate systems with directions which change with time should not be used.

PO R O U S , P R E S C R I B E D A C C E L E R A T I O N

Porous, Roller

The Porous, Roller node adds a roller (sliding wall) constraint as the boundary condition; that is, the porous matrix displacement is zero in the direction perpendicular (normal) to the boundary, but the porous matrix is free to move in the tangential direction. This boundary condition also sets an impervious (sound-hard) boundary for the fluid pressure.



To define a porous, prescribed acceleration for each space direction (x, y, and z for 3D), select one or all of the Prescribed in x direction, Prescribed

in y direction, and Prescribed in z direction check boxes. Enter a value or expression for the prescribed acceleration ax, ay, and az (SI unit: m/s2).

For 2D axisymmetric models and to define a porous, prescribed acceleration for each space direction (r and z), select one or both of the Prescribed in r direction and Prescribed in z direction check boxes. Then enter a value or expression for the prescribed acceleration ar and az (SI unit: m/s2).

Porous, Prescribed Acceleration Equations


194 | C H A P T E




Porous, Septum Boundary Load

Add a Poroelastic Septum Boundary Load to boundaries for a pressure acting on the porous matrix through a septum layer. Right-click the node to add a Phase subnode.



PO R O U S , S E P T U M B O U N D A R Y L O A D

Enter a Surface density sep (SI unit: kg/m3). The default is 0 kg/m3. Enter coordinates for the Load FA (SI unit: N/m2).

Continuity

The Continuity boundary condition includes the fluid load and structural acceleration for use on the fluid-solid, fluid-porous, and solid-porous boundaries, where it is the default boundary condition. It is not applicable on other boundaries.

Porous, Roller Equations

Porous, Septum Boundary Load Equations


For fluid-solid boundaries, this boundary condition includes the following interaction between fluid and solid domains:

• A pressure load (force per unit area) Fpn p on the boundaries where the fluid interacts with the solid. In this expression, n is the outward-pointing unit normal vector seen from inside the solid domain.

• A structural acceleration acting on the boundaries between the solid and the fluid. This makes the normal acceleration for the acoustic pressure on the boundary equal to the acceleration based on the second derivatives of the structural displacements u with respect to time: ann · -2u.

For fluid-porous boundaries, this boundary condition includes the following interaction between fluid and porous domains:

• Continuity of the fluid pressure on the boundaries where the fluid interacts with the porous domain. The pore pressure in the porous domain is set equal to the total pressure in the fluid domain: ppore = pt.

• A pressure load for the elastic waves in the porous material

where pt is the total acoustic pressure in the fluid domain and the left hand side represents the total stress for the saturated porous domain.

• The pressure acoustic domain experiences a normal acceleration that depends both on the acceleration of the porous matrix skeleton but also on the pore pressure. Because of the pressure boundary condition, which is a bidirectional constraint, this condition reduces to the fluid experiencing a normal acceleration: an = (i)2u.

For solid-porous boundaries, this boundary condition includes the following interaction between solid and porous domains:

• Continuity of the displacement field on the boundaries where the solid interacts with the porous domain.

• A normal acceleration acting on the boundaries between the porous material and the fluid. This makes the normal acceleration for the acoustic pressure on the boundary equal to the acceleration based on the second derivatives of the structural displacements u with respect to time: ann · -2u.

n d BpI– npt–=


196 | C H A P T E


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the boundary selection automatically becomes the default selection between the fluid, solid or porous domains. Manual selection is not available.


Th eo r y f o r t h e E l a s t i c Wav e s and Po r o e l a s t i c Wav e s I n t e r f a c e s

The Elastic Waves and Poroelastic Waves Interfaces theory is described in this section:

• About Elastic Waves

• About Poroelastic Waves

• About the Boundary Conditions for Poroelastic Waves

• References for the Elastic Waves and Poroelastic Waves Interfaces

About Elastic Waves

The most general linear relation (see About Linear Elastic Materials) between the stress and strain tensors in solid materials can be written as

here, is the Cauchy’s stress tensor, is the strain tensor, and cijkl is a fourth-order elasticity tensor. For small deformations, the strain tensor is defined as

where u represents the displacement vector.

The elastic wave equation is then obtained from Newton’s second law

here, is the medium density, and s0 and F represent source terms.

An important case is the time-harmonic wave, for which the displacement varies with time as

with f (SI unit: Hz) denoting the frequency and 2f (SI unit: rad/s) the angular frequency. Assuming the same time-harmonic dependency for the source terms s0 and

ij cijklkl=

u 12--- u uT+ =

2

t2--------u u s0– – F=

u x t u x eit=

T H E O R Y F O R T H E E L A S T I C W A V E S A N D P O R O E L A S T I C W A V E S I N T E R F A C E S | 197

198 | C H A P T E

F, the wave equation for linear elastic waves reduces to an inhomogeneous Helmholtz equation:

(4-1)

Alternatively, treat this equation as an eigenvalue PDE to solve for eigenmodes and eigenfrequencies as described in Setting up Equations for Different Studies. Also add damping as described in Damping Models.

About Poroelastic Waves

In his seminal work, Biot extended the classical theory of linear elasticity to porous media saturated with fluids (Ref. 1, Ref. 2, Ref. 3).

In Biot’s theory, the bulk moduli and compressibilities are independent of the wave frequency, and can be treated as constant parameters. The porous matrix is described by linear elasticity and damping is introduced by considering the viscosity of the fluid in the pores, which can be frequency dependent.

Consider Biot’s expressions for poroelastic waves (Ref. 3, Ref. 4, Ref. 6)

(4-2)

here, u is the displacement of the porous material, is the total stress tensor (fluid and porous material), w is the fluid displacement with respect to the porous matrix, f and f are the fluid’s density and viscosity, is the tortuosity, p is the porosity, pf is the fluid pore pressure, is the permeability and av the average density. The average density is the total density (porous material plus pore fluid) av = dr + pf.

Assuming a time-harmonic dependency for the variables, u(x,t) = u(x)eit, w(x,t) = w(x)eit, the time derivatives can be removed, so the system in Equation 4-2 becomes

(4-3)

– 2u u s0– – F=

av2

t2--------u f

2

t2--------w –+ 0=

f2

t2--------u

f----- t-----w

p-----f

2

t2--------w pf+ + + 0=

av– 2u f2w – + 0=

– f2u 2c– w pf+ 0=


here, the complex density c() (Ref. 5) accounts for the tortuosity, porosity and fluid density, and the viscous drag on the porous matrix

(4-4)

H I G H F R E Q U E N C Y C O R R E C T I O N

When Biot’s high frequency range is selected from the Viscosity model list, Equation 4-4 is implemented with a frequency-dependent viscosity c(f) (Ref. 2, Ref. 3, Ref. 5)

here, fc is a reference frequency (SI unit: Hz) which determines the low-frequency range f << fc and the high-frequency range f >> fc.

The reference frequency fc can be interpreted as the limit when viscous forces equal inertial forces in the fluid motion. In the low-frequency limit, viscous effects dominate, while in the high-frequency limit, inertial effects dominate fluid motion in the pores. In Biot’s low frequency range, 0 and Fc = 1.

In order to account for a frequency dependence on the viscous drag, Biot defined the operator Fc() as

here, T() is related to the Kelvin functions Ber() and Bei()

and J0 and J1 are Bessel functions of the first kind.

U - P F O R M U L A T I O N

The formulation in terms of the displacements u and w is not optimal from the numerical viewpoint, since it requires to solve for two displacement fields (Ref. 7, Ref. 8, Ref. 9). The Poroelastic Waves interface solves for the fluid pore pressure variable pf instead of the fluid displacement field w.

The second row in Equation 4-3 is simplified to

c p-----f

fi----------+=

c f f Fcffc----

=

Fc 14--- T

1 2iT +-------------------------------------- =

T Ber' iBei' +Ber iBei +-----------------------------------------------

i– J– 1 i–

J0 i– --------------------------------------= =


200 | C H A P T E

so the first row in Equation 4-3 becomes

(4-5)

The total stress tensor is then divided into the contributions from the elastic porous (drained) matrix and from the pore fluid

here, the identity tensor I means that the pore pressure pf only contributes to the diagonal of the total stress tensor . The parameter B is the so-called Biot-Willis coefficient. The drained, elastic stress tensor is written as drc: when is the strain tensor of the porous matrix, and the elasticity tensor c contains the drained porous matrix’s elastic properties (see About Linear Elastic Materials).

Finally, arrange Equation 4-5 in terms of the variables u and p:

(4-6)

The next Biot’s equation comes from taking the divergence of the second row in Equation 4-3, previously divided by c()

(4-7)

Using the expressions for the volumetric strain vol·u and fluid displacement (Ref. 3, Ref. 4),

Biot’s modulus M is calculated from the porosity p, fluid compressibility f, Biot-Willis coefficient B and the drained bulk modulus of the porous matrix Kdr

so Equation 4-7 simplifies to

w1

2c ---------------------- pf f

2– u =

av– 2uf

c --------------– pf f– 2u – 0=

u pf dr u B– pfI=

avf

2

c --------------–

– 2u – dr u B– pfI f

c --------------pf=

2f

c --------------u 2 w+ +

1c --------------–

pf 0=

– wpfM----- Bvol+=

1M----- pf

B p–

Kdr------------------ 1 B– +=


(4-8)

and Biot’s wave equations (Equation 4-6 and Equation 4-8) can be written in terms of the variable u and pf as

(4-9)

Further arranging the first row in Equation 4-9 to fit the formulation in the Elastic Waves interface (Equation 4-1) gives

(4-10)

The body load F depends on the angular frequency and the gradient of fluid pressure and the fluid pressure acts as a spherical contribution to the diagonal of Cauchy stress tensor

Arranging the second row in Equation 4-9 to fit the implementation of the Pressure Acoustics, Frequency Domain interface gives (see Theory Background for the Pressure Acoustics Branch)

(4-11)

2f

c --------------u 2 1

M-----pf Bvol+ – 1

c --------------–

pf+ 0=

– 2 avf

2

c --------------–

u – dr u B– pfI f

c --------------pf=

2

M-------– pf 1

c --------------– pf 2fu– + 2Bvol=

The saturated (also called Gassmann) modulus can be obtained from the drained bulk modulus Kdr, Biot modulus M, and Biot-Willis coefficient B as KsatKdr B

2M (Ref. 5).

2– avf

2

c --------------–

u dr u s0– – F=

Ff

c --------------pf=

s0 BpfI=

2

M-------– pf 1

c -------------- pf qd– –+ Qm=


202 | C H A P T E

The monopole source Qm (SI unit: 1/s2) and the dipole source qd (SI unit: N/m3) depend on the angular frequency , the displacement of the porous matrix u, the fluid density and Biot-Willis coefficient B

About the Boundary Conditions for Poroelastic Waves

Although boundary conditions can be set up for the porous matrix and fluid independently of each other, there exist a few common boundary conditions which deserve special attention. The following sections refer to the boundary conditions for the system written in Equation 4-10 and Equation 4-11. See derivation in Ref. 7, Ref. 8, and Ref. 9.

PO R O U S , F R E E ( S O U N D - S O F T B O U N D A R Y ) T H E O R Y

The free boundary condition is the default for the Poroelastic Waves interface. It means that the displacement of the porous matrix in Equation 4-10 is unconstrained, so it can move freely without experiencing any loads.

The sound soft boundary condition for acoustics creates a boundary condition for Equation 4-11 where the acoustic pressure vanishes, so it sets pf0.

PO R O U S , F I X E D C O N S T R A I N T ( S O U N D - H A R D B O U N D A R Y ) E Q U A T I O N S

For simulating a poroelastic medium bounded by a rigid impervious wall, impose a Fixed Constraint node for the porous matrix displacement in Equation 4-10, u and a sound-hard boundary condition for the pore pressure in Equation 4-11:

PO R O U S , P R E S S U R E E Q U A T I O N S

For a given fluid pressure p0 on the boundary, set the pressure in Equation 4-11 to . Since the fluid pressure is set to p0, the normal stress on the porous matrix

in Equation 4-10 reduces to

For a rigid porous matrix Bp, the load is equivalent to

Qm 2Bvol=

qd 2fu=

n1

c -------------- pf qd– 0=

pf p0=

n dr u n B 1– p0=


and for a soft porous matrix B1, there is no load since

PO R O U S , P R E S C R I B E D D I S P L A C E M E N T E Q U A T I O N S

For a prescribed displacement u0 at the boundary, set the displacement of the porous matrix in Equation 4-10 as u u0 and assume a sound-hard (impervious) boundary for the fluid pressure in Equation 4-11:

PO R O U S , P R E S C R I B E D VE L O C I T Y E Q U A T I O N S

For a prescribed velocity v0 at the boundary, set the displacement of the porous matrix in Equation 4-10 as

and assume a sound-hard (impervious) boundary condition for the fluid pressure in Equation 4-11

PO R O U S , P R E S C R I B E D A C C E L E R A T I O N E Q U A T I O N S

For a prescribed acceleration a0 at the boundary, set the displacement of the porous matrix in Equation 4-10 as

and assume a sound-hard (impervious) boundary condition for the fluid pressure in Equation 4-11

n dr u n p 1– p0=

n dr u 0=

n1

c -------------- pf qd– 0=

u1i------v0=

n1

c -------------- pf qd– 0=

u1– 2

----------a0=

n1

c -------------- pf qd– 0=


204 | C H A P T E

PO R O U S , R O L L E R E Q U A T I O N S

The roller, or sliding wall boundary, means that the boundary is impervious (sound-hard) to fluid displacements, but it allows tangential displacements of the porous matrix.

The normal displacement of the porous matrix in Equation 4-10 is constrained, but the porous matrix is free to move in the tangential direction

The impervious (sound hard) boundary condition for the fluid pressure in Equation 4-11 is obtained from

PO R O U S , S E P T U M B O U N D A R Y L O A D E Q U A T I O N S

For a prescribed load FA at the boundary, suppose that one side of the septum is fixed to the porous matrix and the other side bears the load.

A septum is a very limp and thin impervious layer with surface density sep. Since the septum can be seen as a boundary mass density, this boundary condition is achieved by setting an effective load FSFAsep

2u on the porous matrix, so the normal stress in Equation 4-10 reduces to

and a sound-hard (impervious) boundary condition is applied for the fluid pressure in Equation 4-11

References for the Elastic Waves and Poroelastic Waves Interfaces

1. M.A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-saturated Porous Solid. I. Low-frequency Range,” J. Acoust. Soc. Am., vol. 28, pp 168–178, 1956.

2. M.A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-saturated Porous Solid. II. Higher Frequency Range,” J. Acoust. Soc. Am., vol. 28, pp 179–191, 1956.

3. M.A. Biot, “Generalized Theory of Acoustic Propagation in Porous Dissipative Media,” J. Acoust. Soc. Am., vol. 34, pp 1254–1264, 1962.

n u 0=

n1

c -------------- pf qd– 0=

n dr u BpfI– FS=

n1

c -------------- pf qd– 0=


4. M.A. Biot, “Mechanics of Deformation and Acoustic Propagation in Porous Media,” J. Appl. Phys., vol. 33, pp 1482–1498, 1962.

5. G. Mavko et al., The Rock Physics Handbook, 2nd ed., Cambridge University Press, 2009.

6. J.M. Carcione, Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media, 2nd ed. Elsevier (Handbook of Geophysical Exploration, vol. 38, Seismic Exploration), 2007.

7. P. Debergue, R. Panneton, and N. Atalla, “Boundary Conditions for the Weak Formulation of the Mixed (u,p) Poroelasticity Problem,” J. Acoust. Soc. Am., vol. 106, pp 2383–2390, 1999.

8. N. Atalla, M.A. Hamdi, and R. Panneton, “Enhanced Weak Integral Formulation for the Mixed (u,p) Poroelastic Equations,” J. Acoust. Soc. Am., vol. 109, pp 3065–3068, 2001.

9. J.F. Allard and N. Atalla, Propagation of Sound in Porous Media, 2nd ed., John Wiley & Sons, 2009.


206 | C H A P T E

Th e A c ou s t i c - S h e l l I n t e r a c t i o n , F r e qu en c y Doma i n I n t e r f a c e

The Acoustic-Shell Interaction, Frequency Domain (acsh) interface ( ), found under the Acoustics>Acoustic-Structure Interaction branch ( ) when adding a physics interface, combines features from the Pressure Acoustics, Frequency Domain and Shell interfaces to connect the acoustics pressure variations in the fluid domain with the structural deformation of a shell boundary. It may for example be used for determining the transmission of sound through a thin elastic structure such as a car hood and analyzing the vibroacoustics of a loudspeaker cone.

Acoustic-structure interaction refers to a multiphysics phenomenon where the acoustic pressure causes a fluid load on the solid surface, and the structural acceleration acts on the fluid domain as a normal acceleration across the fluid-structure boundary.

Special interface conditions are readily defined at the fluid-shell boundary and set up the fluid loads on the shell boundary and the effect of the structural accelerations on the fluid. The physics interface is only available for 3D geometries, and it is capable of modeling the coupled pressure acoustics and shell vibrations in the frequency domain.

When this interface is added, these default nodes are also added to the Model Builder— Pressure Acoustics Model, Sound Hard Boundary (Wall), Free, Exterior Shell, and Initial

Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Acoustic-Shell Interaction,

Frequency Domain to select physics from the context menu.

On the Constituent Physics InterfacesThe Pressure Acoustics, Frequency Domain interface computes the pressure variations when modeling the propagation of acoustic waves in fluids at quiescent background

This interface requires a Structural Mechanics Module license. For theory and interface feature descriptions relating to the Shell interface, see the Structural Mechanics Module User’s Guide.

The interface is only available for 3D geometries, and it is capable of modeling the coupled pressure acoustics and shell vibrations in the frequency domain.


conditions. The physics interface solves the Helmholtz equation and is suited for all linear frequency-domain acoustic simulations with harmonic variations of the pressure field. It includes domain conditions to model losses in a homogenized way, so-called fluid models for porous materials as well as losses in narrow regions. The domain features also include background incident acoustic fields.

The Shell interface is used to model structural shells on 3D faces. Shells are thin flat or curved structures, having significant bending stiffness. The interface uses shell elements of the MITC type, which can be used for analyzing both thin (Kirchhoff theory) and thick (Mindlin theory) shells. Geometric nonlinearity can be taken into account. The material is assumed to be linearly elastic.

The following sections provide information about all nodes specific to this multiphysics interface, but all other acoustics-specific nodes are found under the Pressure Acoustics, Frequency Domain interface.



The default identifier (for the first interface in the model) is acsh.





For transient simulations, use The Acoustic-Solid Interaction, Transient Interface instead.

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T H I C K N E S S

Define the Thickness d by entering a value or expression (SI unit: m) in the field. The default is 0.01 m. Use the Change Thickness node to define a different thickness in parts of the shell or plate. The thickness can be variable if an expression is used.



F O L D - L I N E L I M I T A N G L E

Enter a value for (SI unit: radians). The default is 0.05 radians.

H E I G H T O F E V A L U A T I O N I N S H E L L , [ - 1 , 1 ]

Enter a value for z(dimensionless). The default is 1.


This interface defines these dependent variables (fields), the Pressure p and Displacement field, u, and its components. The name can be changed but the names of fields and dependent variables must be unique within a model.


To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. The Use MITC interpolation check box is selected by default, and this interpolation, which is part of the MITC shell formulation, should normally always be active.


To display this section, click the Show button ( ) and select Discretization. Select Linear, Quadratic (the default), Cubic, or Quartic for the Pressure. Select Linear or Quadratic (the default) for the Displacement field. Specify the Value type when using

splitting of complex variables—Real or Complex (the default).

The links to the nodes described in the Structural Mechanics Module User’s Guide do not work in the PDF, only from the online help.


Domain, Boundary, Edge, Point, and Pair Nodes for the Acoustic-Shell Interaction, Frequency Domain Interface

Because The Acoustic-Shell Interaction, Frequency Domain Interface is a multiphysics interface, almost every node is shared with, and described for, other interfaces. Below are links to the domain, boundary, edge, point, and pair nodes as indicated.

These nodes are described specifically for this interface:

• Exterior Shell

• Initial Values

• Initial Values (Boundary)

• Interior Shell

• Uncoupled Shell


• Domain, Boundary, Edge, Point, and Pair Nodes for the Acoustic-Shell Interaction, Frequency Domain Interface


• Theory for Shell and Plate Interfaces in the Structural Mechanics Module User’s Guide

• Loudspeaker Driver in a Vented Enclosure: model library path Acoustics_Module/Industrial_Models/vented_loudspeaker_enclosure

• Baffled Membrane: model library path Acoustics_Module/

Tutorial_Models/baffled_membrane

The Acoustic-Shell Interaction, Transient Interface also shares the same features as listed below, with an additional domain node: Transient Pressure Acoustics Model as described for the Pressure Acoustics, Transient interface.


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P R E S S U R E A C O U S T I C S , F R E Q U E N C Y D O M A I N M E N U



• Continuity


• Dipole Source


• Impedance






• Monopole Source




• Pressure









S H E L L M E N U

These nodes are described for the Shell interface.

These nodes are described for the Solid Mechanics interface and described in this guide:

Initial Values

The Initial Values node adds initial values for the sound pressure. Add more Initial

Values nodes from the Physics toolbar.



Both the Shell interface and its nodes are described in the Structural Mechanics Module User’s Guide as this interface requires the Structural Mechanics Module. For that reason, these links do not work in the PDF.

Applied Force (Rigid Connector), Applied Moment (Rigid Connector), and Mass and Moment of Inertia (Rigid Connector), Gravity, and Rotating Frame are also described in the Structural Mechanics Module User’s Guide, but for the Solid Mechanics interface.

• Antisymmetry

• Body Load

• Change Thickness

• Edge Load

• Face Load

• No Rotation

• Pinned


• Prescribed Displacement/Rotation


• Point Load

• Symmetry

• Added Mass


• Free

• Pre-Deformation



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Enter a value or expression for the Pressure p (SI unit: Pa) initial value. The default is 0 Pa.

Initial Values (Boundary)

The Initial Values node adds initial values for the displacement field and the displacement of shell normals.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required.


Based on space dimension, enter coordinate values for the Displacement field u (SI unit: m) and the Displacement of shell normals ar (dimensionless).

Exterior Shell

Use the Exterior Shell boundary condition to model any deformable shell boundary, only one side of which is adjacent to the acoustic domain.

The normal acceleration for the acoustic pressure on the boundary equals the acceleration based on the second time derivative of the shell displacement

In addition, the pressure load (force per unit area) on the shell is: Fpn p.

n–10------– p qd–

n u tt=



To choose specific boundaries, select Manual from the Selection list, or select All

boundaries.

Interior Shell

Use the Interior Shell boundary condition to model any deformable shell with both sides adjacent to the acoustic domains.

The normal accelerations for the acoustic pressure on both sides equal to the acceleration based on the second time derivative of the shell displacement

where 1 and 2 subscripts stand for two adjacent domains on different sides of the shell.

In addition, the pressure load (force per unit area) on the shell is: Fp (np)1(np)2.



A Linear Elastic Material node is automatically added to this boundary condition. For the default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required.

n–10------– p qd–

1

n u tt= n–10------– p qd–

2

n u tt=

An Elastic Material subnode is automatically added to this boundary condition. Right-click Interior Shell to add more subnodes if required.


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Uncoupled Shell

Use the Uncoupled Shell boundary condition to model deformable shells that are not adjacent to the acoustic domains.



An Elastic Material subnode is automatically added to this boundary condition. Right-click Uncoupled Shell to add more subnodes if required.


Th e A c ou s t i c - S h e l l I n t e r a c t i o n , T r a n s i e n t I n t e r f a c e

The Acoustic-Shell Interaction, Transient (acshtd) interface ( ), found under the Acoustics>Acoustic-Structure Interaction branch ( ) when adding a physics interface, combines features from the Pressure Acoustics, Transient and Shell interfaces to connect the acoustics pressure variations in the fluid domain with the structural deformation of a shell boundary. It may for example be used for determining the transmission of sound through a thin elastic structure such as a car hood and analyzing the vibroacoustics of loudspeaker cone.

Acoustic-structure interaction refers to a multiphysics phenomenon where the acoustic pressure causes a fluid load on the solid surface, and the structural acceleration affects the fluid domain as a normal acceleration across the fluid-structure boundary.

Special interface conditions are readily defined at the fluid-shell boundary and set up the fluid loads on the shell boundary and the effect of the structural accelerations on the fluid. The interface is only available for 3D geometries, and it is capable of modeling the coupled pressure acoustics and shell vibrations in the time domain.

When this interface is added, these default nodes are also added to the Model Builder— Transient Pressure Acoustics Model, Sound Hard Boundary (Wall), Free, Acoustic-Structure

Boundary, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Acoustic-Shell Interaction, Transient to select physics from the context menu.

On the Constituent Physics InterfacesThe Pressure Acoustics, Transient interface computes the pressure variations when modeling the propagation of acoustic waves in fluids at quiescent background

This interface requires a Structural Mechanics license. For theory and interface feature descriptions relating to the Shell interface, see the Structural Mechanics Module User’s Guide.

The interface is available for 3D geometry only.

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conditions. The physics interface solves the scalar wave equation and is suited for time-dependent simulations with arbitrary time-dependent fields and sources. Domain conditions also include background incident acoustic fields. User-defined sources can be added to, for example, include certain nonlinear effects such as a square pressure dependency of the density variations.

The Shell interface is used to model structural shells on 3D faces. Shells are thin flat or curved structures, having significant bending stiffness. The interface uses shell elements of the MITC type, which can be used for analyzing both thin (Kirchhoff theory) and thick (Mindlin theory) shells. Geometric nonlinearity can be taken into account. The material is assumed to be linearly elastic.

The following sections provide information about all nodes specific to this multiphysics interface, but all other nodes are found under the Pressure Acoustics,

Transient and Shell interfaces, respectively.



The default identifier (for the first interface in the model) is acshtd.

The Acoustic-Shell Interaction, Frequency Domain Interface contains additional functionality that is not applicable for modeling in the time domain.

See The Acoustic-Shell Interaction, Frequency Domain Interface for the rest of the interface settings. The Gaussian Pulse sources are described in Gaussian Pulse Point, Edge, Circular, and Line Sources.



• Domain, Boundary, Edge, Point, and Pair Nodes for the Acoustic-Shell Interaction, Frequency Domain Interface


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Th e P i p e A c ou s t i c s I n t e r f a c e s

In this section:

• The Pipe Acoustics, Frequency Domain Interface

• The Pipe Acoustics, Transient Interface

• For links to all the physics features, go to Edge, Boundary, Point, and Pair Nodes for the Pipe Acoustics Interfaces

The Pipe Acoustics, Frequency Domain Interface

The Pipe Acoustics, Frequency Domain (pafd) interface ( ), found under the Acoustics>Acoustic-Structure Interaction branch ( ) when adding a physics interface, is used to compute the acoustic pressure and velocity variations when modeling the propagation of sound waves in flexible pipe systems. The governing equations are formulated in a general way to include the possibility of a stationary background flow. The physics interface can for example be used to compute the propagation of sound waves in HVAC systems, other large piping systems, or simply in an organ pipe.

In the frequency domain all sources and variations are assumed to be harmonic. The solved equations assume that the propagating waves are plane. The propagation of higher-order modes that exist above their cut-off frequency, dictated by the pipe cross section, is not modeled.

The equations governing the propagation of sound in pipes stem from considering momentum, mass, and energy balances for a control volume of a piece of pipe. The resulting equations are expressed in the cross-sectional averaged variables and reduce the equations to a 1D model with scalar dependent variables. The interface is available in 3D on edges and points, and in 2D on boundaries and points.


Builder—Fluid Properties, Pipe Properties, Closed, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions

These interfaces require both the Pipe Flow Module and the Acoustics Module.


and point conditions. You can also right-click Pipe Acoustics, Frequency Domain to select physics from the context menu.



The default identifier (for the first interface in the model) is pafd.

E D G E O R B O U N D A R Y S E L E C T I O N

The default setting includes All edges or All boundaries in the model. To choose specific edges or boundaries, select Manual from the Selection list.


This section is used to define the dependent variables (fields) for Pressure p (SI unit: Pa) and Tangential velocity u (SI unit: m/s). If required, edit the name, but dependent variables must be unique within a model.


To display this section, click the Show button ( ) and select Discretization. It controls the element types used in the finite element formulation.

Select discretization options from the Pressure and Tangential velocity lists—Linear, Quadratic, Cubic, Quartic, or Quintic. The defaults is quadratic for the pressure and linear for the tangential velocity.

For each Dependent variable in the table under Value types when using splitting of

complex variables, choose either a Complex or Real Value type. Click the cell to select from a drop-down list.


• Edge, Boundary, Point, and Pair Nodes for the Pipe Acoustics Interfaces


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The Pipe Acoustics, Transient Interface

The Pipe Acoustics, Transient (patd) interface ( ), found under the Acoustics>Acoustic-Structure Interaction branch ( ) when adding a physics interface, is used to compute the acoustic pressure and velocity variations when modeling the propagation of sound waves in flexible pipe systems. The governing equations are formulated in a general way to include the possibility of a stationary background flow. The physics interface can for example be used to compute the propagation of sound waves in HVAC systems, other large piping systems, or simply in an organ pipe.

The solved equations assume that the propagating waves are plane. The propagation of higher-order modes that exist above their cut-off frequency, dictated by the pipe cross section, is not modeled.

The equations governing the propagation of sound in pipes stem from considering momentum, mass, and energy balances for a control volume of a piece of pipe. The resulting equations are expressed in the cross-sectional averaged variables and reduce the equations to a 1D model with scalar dependent variables. The physics interface is available in 3D on edges and points, and in 2D on boundaries and points.


Builder—Fluid Properties, Pipe Properties, Closed, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and point conditions. You can also right-click Pipe Acoustics, Transient to select physics from the context menu.



The default identifier (for the first interface in the model) is patd.

The rest of the settings are the same as for The Pipe Acoustics, Frequency Domain Interface.


Edge, Boundary, Point, and Pair Nodes for the Pipe Acoustics Interfaces

The Pipe Acoustics, Transient Interface has these edge, boundary, point, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).


• Edge, Boundary, Point, and Pair Nodes for the Pipe Acoustics Interfaces



• Closed

• End Impedance

• Fluid Properties

• Initial Values

• Pipe Properties

• Pressure

• Velocity

Volume Force is described for the Pipe Flow interface in the Pipe Flow User’s Guide.




The links to the nodes described in the COMSOL Multiphysics Reference Manual do not work in the PDF, only from the on line help in COMSOL Multiphysics.


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Initial Values

The Initial Values node adds initial values for the pressure and tangential velocity that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear solver.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific edges or boundaries or select All edges or All

boundaries as required.


Enter values or expressions for the initial value of the Pressure p (SI unit: Pa) and the Tangential Velocity u (SI unit: m/s).

Fluid Properties

The Fluid Properties node adds the momentum and continuity equations solved by the physics interface, except for volume forces which are added by the Volume Force node. The node also provides an interface for defining the material properties of the fluid.





Edit input variables to the fluid-flow equations if required. For fluid flow, these are typically introduced when a material requiring inputs has been applied.

Volume Force is described for the Pipe Flow interface in the Pipe Flow User’s Guide.


B A C K G R O U N D P R O P E R T I E S

Enter a value or expression for the Background velocity u0 (SI unit: m/s) and Background pressure p0 (SI unit: Pa).

P H Y S I C A L P R O P E R T I E S

Select a Fluid model—Linear elastic (the default).

The default Density (SI unit: kg/m3) uses the value From material. Select User defined to enter a different value or expression.

The default Speed of sound cs (SI unit: m/s) uses the value From material. Select User

defined to enter a different value or expression.

Pipe Properties

The Pipe Properties node is used to define the pipe shape, pipe model, wall drag force, and swirl correction.




Physically sound background property variables for the pressure p0 and velocity u0 can be obtained by solving a Pipe Flow model on the same geometry.


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P I P E S H A P E

Select a pipe shape from the list—Not set (the default), Round, Square, Rectangular, or User defined.

• If Round is selected, enter a value or expression for the Inner diameter di (SI unit: m). The default is 10 cm (0.01 m).

• If Square is selected, enter a value or expression for the Inner width wi (SI unit: m). The default is 5 cm (0.005 m).

• If Rectangular is selected, enter a value or expression for the Inner width wi (SI unit: m; the default is 5 cm) and Inner height hi (SI unit: m; the default is 10 cm).

• If User defined is selected, enter a value or expression for the Cross sectional area A (SI unit: m2; the default is 0.01 m2) and Wetted perimeter Z (SI unit: m; the default is 0.4 m).

P I P E M O D E L

Select a Pipe model—Incompressible cross section (the default), Zero axial stress, Anchored at one end, or Anchored at both ends.

When Zero axial stress, Anchored at one end, or Anchored at both ends is chosen, select an option from the Young’s modulus E (SI unit: Pa) and Wall thickness w lists—Not

set (the default) or User defined. If User defined is selected in either case, enter different values or expressions.

For Anchored at one end or Anchored at both ends also select an option from the Poisson’s ratio v (dimensionless) list—Not set (the default) or User defined. If User

defined is selected, enter a value or expression.

WA L L D R A G F O R C E

Enter a value or expression for w (SI unit: N/m2). The default is 0 N/m2.

S W I R L C O R R E C T I O N

Enter a value or expression for (dimensionless). The default is 1. For most practical applications this factor is 1 as the propagating waves are assumed plane and uniform. This value should typically be changed if a wall drag force is introduces or if a background flow field is used.

Swirl Correction Factor


Closed

Use the Closed node to impose zero velocity. This is the default condition added on all end points.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific points or select All points as required.

Pressure

Use the Pressure node to define the boundary pressure at the pipe ends.


From the Selection list choose the points where you want to define a boundary pressure.

P R E S S U R E

Enter a value or expression for the Pressure p (SI unit: Pa). The default is 0 Pa.



Theory for the Pipe Acoustics Boundary Conditions

In the frequency domain p represents the amplitude an phase (as it is complex valued) of a harmonic pressure source.

In the time domain enter an expression for the pressure p, for example, a forward moving sinusoidal wave of amplitude 1 Pa can be written as 1[Pa]*sin(omega*t-k*x), where omega and k are parameters defining the angular frequency and wave number.


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Velocity

Use the Velocity node to prescribe a velocity at the pipe ends.



VE L O C I T Y

Enter a value or expression for the velocity uin (SI unit: m/s) at the inlet and/or outlet of a pipe. The default is 0 m/s.

End Impedance

Use the End Impedance node to model conditions at the end of a pipe. The condition can either model an infinite pipe and thus represent the characteristic impedance of the pipe system at that point. This results in a zero reflection condition. Alternatively the condition can represent the radiation impedance of an open pipe in either a flanged (in an infinite baffle) or unflanged (a pipe ending in free open space). The end impedance


The velocity uin is defined relative to background flow u0 and thus in the tangential coordinate system. Enable the Show physics symbols from the Graphics and Plot Windows menu on the Preference dialog box in order to visualize the boundary or edge tangent direction. Click the Fluid

Properties node to see the tangents as a red arrows.

• Theory for the Pipe Acoustics Boundary Conditions

• The Preferences Dialog Box in the COMSOL Multiphysics Reference Guide


can also be user-defined and could represent modeled or experimental values for a specific pipe configuration.]



E N D I M P E D A N C E

Select an End impedance.

For The Pipe Acoustics, Transient Interface choose Infinite pipe (low Mach number

limit) (the default) or User defined. The Infinite pipe (low Mach number limit) models and infinite pipe by specifying the characteristic impedance at that point. This condition creates anon reflecting boundary. The expression is valid for small values of the Mach number Ma = u0/c. If User defined is selected, enter an End impedance Zend (SI unit: Pas/m). The default is 0.

For The Pipe Acoustics, Frequency Domain Interface choose from the following—Infinite pipe (low Mach number limit) (the default), Infinite pipe, Flanged pipe, circular, Flanged pipe, rectangular, Unflanged pipe, circular (low ka limit), Unflanged pipe, circular, or User defined.

• If Infinite pipe is selected, enter a Wave number k (SI unit: rad/m). The default expression is pafd.omega*(sqrt(pafd.invc2)). This end impedance models the infinite pipe using the full (nonlinear) dispersion relation. It is valid for all Mach numbers but require the additional input of the wave number k.

• If Flanged pipe, circular is selected, enter an Inner radius a (SI unit: m). The default expression is pafd.dh/2. This end impedance models the radiation impedance of a circular pipe terminated in an infinite baffle. It is an exact analytical result valid for all frequencies and pipe radii. In the low frequency limit it reduces to the classical results:

Note that the wave speed c in the pipe can be different from the speed of sound cs in an open space. It then depends on the elastic properties of the pipe structure. It is defined in Equation 4-16 in the Governing Equations section.

The wave speed can be evaluated as sqrt(1/patd.invc2) or sqrt(1/pafd.invc2) during the analysis and results stage.


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• If Flanged pipe, rectangular is selected, enter an Inner width wi (SI unit: m). The default is 5 cm (0.005 m). Also enter an Inner height hi (SI unit: m). The default is 10cm (0.01 m). This end impedance models the radiation impedance of a pipe of rectangular cross section terminated in an infinite baffle. The model is only valid in the low frequency range where and .

• If Unflanged pipe, circular (low ka limit) or Unflanged pipe, circular is selected, enter an Inner radius a (SI unit: m). The default expression is pafd.dh/2. These two end impedance models prescribe the radiation impedance of an unflanged circular pipe (a pipe ending in free open space). The first model is the classical low frequency approximation valid for . While the second model extends the frequency range to .

• If User defined is selected, enter an End impedance Zend (SI unit: Pas/m). The default expression is pafd.rho*(sqrt(1/pafd.invc2)).

Zend c 12--- ka 2 i 0,8216 ka + =

kwi 1« khi 1«

ka 1«

ka 3,83

For a detailed review of the end impedance models see: Theory for the Pipe Acoustics Boundary Conditions.


Th eo r y f o r t h e P i p e A c ou s t i c s I n t e r f a c e s

The equations governing the propagation of sound in pipes stem from considering momentum, mass, and energy balances for a control volume of a piece of pipe. The resulting equations are expressed in the cross-sectional averaged variables and reduces the equations to a 1D model with scalar dependent variables. The present theory assumes no thermal conduction and thus no losses due to thermal conduction (isentropic sound propagation).

Governing Equations

The continuity equation derived for a control volume is given by

(4-12)

and the corresponding momentum balance equation is

(4-13)

where Z is the inner circumference of the pipe and A = A(x,p,...) is the inner wetted cross-sectional area. u is the area-averaged mean velocity, which is also defined in the tangential direction u = uet, p is the mean pressure along the pipe, w is the wall drag force, and F is a volume force. The gradient is taken in the tangential direction et. The term is a swirl-correction factor relating the mean of the squared total velocity to the square of the mean velocity. Such that

(4-14)

where

The Pipe Acoustics, Transient and the Pipe Acoustics, Frequency Domain interfaces require both the Pipe Flow Module and the Acoustics Module.

A t

---------------- Au + 0=

Au t

-------------------- Au2 + Ap– wZ AF+ +=

u 1A---- u Ad= p 1

A---- p Ad= 1

A---- u

2Ad

u2=

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230 | C H A P T E

and

are the local non-averaged parameters. Again p and u are the are the area-averaged dependent variables.

L I N E A R I Z A T I O N

The governing equations are now linearized, that is, all variables are expanded to first order assuming stationary zero (0th) order values (steady-state background properties). The acoustic variations of the dependent variables are assumed small and on top of the background values. This is done according to the following scheme:

where A0 is often only function of x; however, A0 can be changed by external factors such as heating or structural deformation, thus the time dependency. The 1st order terms represent small perturbations on top of the background values (0th order). They are valid for

Moreover, the perturbations for the fluid density and cross-sectional area are expanded to first order in p0 in a Taylor series such that

where the subscript s refers to constant entropy; that is, the processes are isentropic. The relations for the fluid compressibility and the cross-sectional area-compressibility are

p p x = u u x =

u x t u0 x u1 x t +=

p x t p0 x p1 x t +=

x t 0 x 1 x t +=

A x t A0 x A1 x t +=

1 0« p1 0c02

« u1 c0« A1 A0«

1 0– p p0– p------

s 0

= =

A1 A A0– p p0– Ap-------

s 0

= =


Here, 0 is the fluid compressibility at the given reference pressure p0, the isentropic bulk speed of sound is denoted cs, and 0 is the fluid density at the given reference temperature and reference pressure. A is the effective compressibility of the pipe’s cross-sectional A0 due to changes in the inner fluid pressure. The bulk modulus K is equal to one over the compressibility.

Inserting the above expansions into the governing equations (Equation 4-12 and Equation 4-13) and retaining only 1st-order terms yield the pipe acoustics equations including background flow. These are:

(4-15)

where c is the effective speed of sound in the pipe (it includes the effect due to the elastic properties of the pipe defined through KA). The bulk modulus for the cross-sectional area KA is given by the pipe material properties according to the so-called Korteweg formula (see Ref. 2). For a system with rigid pipe walls cs = c as KA tends to infinity.

Using the fact that the velocity is taken along the tangential direction et the governing equations are rewritten in terms of the scalar values u and p and projected onto the tangent. The 0 subscript is dropped on the density and area and the 1 subscript is also dropped on the dependent variables.

01

K0------- 1

Ks------ 1

0------

p------

s 0

10------ 1

cs2

-----= = = =

A1

A0-------=

Ap-------

s 0

1KA-------=

A01

c2-----p1t

--------- A00 u1u0

0c2-----------p1+

+ 0=

0A0u1t

----------u0

0c2-----------

p1t

---------+

A0u0

2

c2------p1 20A0u0u1+

+

A0 p1 p1+ Ap0 + wZ AF+ + 0=

1

c2----- 0 0 A+ 0

1K0------- 1

KA-------+

1

cs2

-----0KA-------+= = =


232 | C H A P T E

(4-16)

where is the tangential derivative, w is the tangential wall drag force (SI unit: N/m2) and F is a volume force (SI unit: N/m3).

G OV E R N I N G E Q U A T I O N S F O R T H E P I P E A C O U S T I C S , TR A N S I E N T

I N T E R F A C E

Finally, the expression for the time derivative of the pressure in the momentum equation is replaced by spatial derivatives using the continuity equation. This yields the equations solved in the Pipe Acoustics, Transient interface:

(4-17)

G OV E R N I N G E Q U A T I O N S F O R T H E P I P E A C O U S T I C S , F R E Q U E N C Y

D O M A I N I N T E R F A C E

In the frequency domain all variables are assumed to be time harmonic such that

(4-18)

inserting this into the governing Equation 4-17 (and dropping the tilde) yields the equations solved in the Pipe Acoustics, Frequency Domain interface:

A 1

c2-----p

t------ t A u

u0

c2---------p+

et+ 0=

A ut------

u0

c2---------p

t------+

t Au0

2

c2------p 2Au0u+

et+

A tp p+ Atp0 et wZ+ A F et + + 0=

1

c2----- 0 A+ 1

K0------- 1

KA-------+

1

cs2

----- KA-------+= = =

t

A 1

c2-----p

t------ t A u

u0

c2---------p+

et+ 0=

Aut------ t A

u02

c2------p 2Au0u+

et u0t A uu0

c2---------p+

et++


1

c2----- 0 A+ 1

K0------- 1

KA-------+

1

cs2

----- KA-------+= = =

p p x eit=

u u x eit=


(4-19)

where is the angular frequency and f is the frequency.


P R E S S U R E , O P E N , A N D C L O S E D C O N D I T I O N S

The simplest boundary conditions to specify are to prescribe the pressure or the velocity at the pipe ends. These result in the Pressure condition

and the Velocity condition

and can be set independently of each other leaving the other dependent variable free. A special subclass of the velocity condition is the Closed condition where

this corresponds to the sound-hard wall condition in pressure acoustics. It is also assumed here that u0 = 0 at a closed boundary.

E N D I M P E D A N C E C O N D I T I O N

At the end of pipes the relation between the pressure and the velocity can be defined in terms of an end impedance Zend. The End Impedance condition is in the Pipe Acoustics interface given by

(4-20)

iA

c2-----p t A u

u0

c2---------p+

et+ 0=

iAu t Au0

2

c2------p 2Au0u+

et u0t A uu0

c2---------p+

et++


1

c2----- 0 A+ 1

K0------- 1

KA-------+

1

cs2

----- KA-------+= = =

2f=

p pin=

u uin=

u 0=

A uu0

c2------p+

A 1Zend-----------

u0

c2------+

p=


234 | C H A P T E

where Zend = p/u (Si unit: (Pa·s)/m). Different models for the end impedance exist in the Pipe Acoustics interfaces. The variety depend on if the transient or the frequency domain equations are solved.

Transient End-Impedance ModelsIn the transient interface the end impedance can be user-defined or set to mimic an infinite long pipe for low Mach number background flow conditions. In this case it is assumed that the pipe continues with constant cross section A and that there is no external body force F and drag w. Because the acoustic waves are, by design, always normal to the pipe ends. In order to define the relation between the pressure and the velocity (the impedance) the dispersion relation for a plane wave needs to be determined.

In order to do so insert the assumed plane wave form

into the governing Equation 4-17 and solve for the desired relations. After some manipulation this results in

with the dispersion relation

(4-21)

This dispersion relation is nonlinear in k. In the limit where A tends to zero and for small Mach numbers M (= u0/c) the expression is expanded to

Hence, the infinite pipe (low Mach number limit) end impedance relation reads

(4-22)

where the sign in front of c depends on the direction of propagation of the wave.

p Re pei t kx– =

u Re uei t kx– =

1Zend----------- u

p--- 1

c2-----1---

k---- u0– = =

k---- u0 c 1–

u0c

------

21 1

k---A– p0

+=

k---- u0 c 1 1

2--- 1–

u0c

------

2+

1Zend----------- 1

c2-----1--- 1– u0 c 1 1

2--- 1–

u0c

------

2+

=


Frequency Domain End-Impedance ModelsIn the frequency domain many engineering relations exist for the end impedance or radiation impedance of a pipe or waveguide. Most of the relations apply only to a specific geometry or frequency range. The relations available in the Pipe Acoustics, Frequency Domain interface are:

• Infinite pipe (low Mach number limit): This is the same relation as for the transient study and the end impedance is given by Equation 4-22. This can be thought of as the characteristic impedance of the tube.

• Infinite pipe: This relation uses the full dispersion relation given in Equation 4-21 and yields the expression

(4-23)

where the wave number k at the right hand side is a user input. In the frequency domain a good estimate for this quantity is simply /c.

• Flanged pipe, circular: In the case of a circular pipe terminated in an infinite baffle (a flanged pipe) an analytical expression exists for the radiation impedance (see Ref. 1),

(4-24)

where J1 is the Bessel function of order 1, H1 is the Struve function of order 1, a is the pipe radius, and k is the wave number. The Struve function is approximated according to Ref. 3 by

(4-25)

In the low frequency limit (small ka) Equation 4-24 reduces to the classical expression for the radiation impedance

(4-26)

• Flanged pipe, rectangular: In the case of a pipe of rectangular cross-section (with sides wi and hi) terminated in an infinite baffle (a flanged pipe) the radiation impedance can be approximated by

1Zend----------- 1

c2-----1--- 1– u0 c 1–

u0c

------

21 1

k---A– p0

+=

Zend c 12J1 2ka

2ka--------------------------– i

2H1 2ka 2ka

---------------------------+ =

H1 x 2--- J0 x –

16

------ 5– xsin

x------------ 12 36

------–

1 xcos–

x2---------------------+ +

Zend c 12--- ka 2 i 0,8216 ka + =


236 | C H A P T E

(4-27)

see Ref. 4 and Ref. 5.

• Unflanged pipe, circular (low ka limit): In the case of a circular pipe of radius a ending in free air the classical low ka limit for the radiation impedance is given by

(4-28)

see Ref. 1 and Ref. 5.

• Unflanged pipe, circular: A solution for the unflanged pipe exists for the case when , it is presented in Ref. 6 and is based on solving the

Wiener-Hopf integral, it reads

(4-29)

where is an interpolation function found by numerical integration for ka = 0, = 0.6133.

Common for the last four radiation impedance relations is that they do only apply when there is no background flow present u0 = 0 (or at least when it is very small).

Solving Transient Problems

When solving transient acoustic problems where the wave shape is not necessarily harmonic it might be necessary to resolve its spatial variations with a fine mesh, say with a minimal scale dx. Now, in order for the numerical solution of the temporal development of the acoustic field to be good it is necessary to restrict the maximal time steps dt taken by the solver. The condition is known as the CFL condition (Courant–Friedrichs–Lewy condition). For transient acoustic problems it is defined as

Zendc2------ k2 wihi 2 ik wihi 3 2 f

wihi------ +

= kwi 1 khi 1««

f x 2x1 2 sinh 1– x 1– 2x 1– 2 sinh 1– x 23---x3 2 2

3---x 3– 2 2

3--- x x 1–

+ 3 2

–+ + +=

Zend c 14--- ka 2 i 0,6133 ka + = ka 1«

ka 3,83« 1,22=

Zend c1 R+1 R–--------------= R R e2ika

=

R e ka 2 2– 1 16--- ka 4 1

ka--------- 19

12------+

ln+ e0,5772 ka 1==

R kae ka– 1 332------ 1

ka 2--------------+

1 ka 3,83=

C cxd

dt=


where C is the Courant number, and c is the velocity.

For applications where all the shape functions are quadratic the Courant number should be around 0.2. This condition restricts any acoustic disturbances to propagate more than 20% of the mesh size dx during one time step dt. In the Pipe Acoustics interface where a mixed formulation exists, with linear elements for the pressure and quadratic elements for the velocity, the condition might have to be tightened such that C 0.2.

Cut-off Frequency

The Pipe Acoustics interface assumes plane wave propagation. This means that it cannot model the propagation of the higher order modes that can propagate above their cut-off frequency fc. In a rectangular pipe of cross section width wi and height hi the cut-off frequency is

In a pipe of circular cross section (with radius a) the cut-off frequency is

where 'mn is the n’th zero of the differential of the Bessel function J’m(x) or order m. The first few values are '01 = 0, '02 = 3.83, '11 = 1.84, and '21 = 3.05 (see Ref. 1 and Ref. 5 for further details).

Swirl Correction Factor

The swirl correction factor accounts for the ratio of the integrated local square velocity field to the square of the integrated local velocity field (see Equation 4-14). It is defined in terms of the total velocity field (background plus acoustic variations).

In the case of no-background flow (u0 = 0) is 1 in the absence of a wall drag coefficient, as only plane wave modes propagate. If a wall drag force is introduced, to

For an example of a model where the CFL condition is used see Water Hammer: model library path Pipe_Flow_Module/Verification_Models/

water_hammer_verification.

fmnc 1

2---c m

wi------ 2 n

hi----- 2

+=

fmnc 'mnc

2a---------------=


238 | C H A P T E

model some loss mechanism, starts to differ slightly from 1. This can for example be losses introduced to model viscous and thermal effects in narrow pipes.

In the presence of a background u0 the factor can be set different 1 in order to model a non-flat velocity profile inside the tube. The value of b (and the actual shape of the background field) influences the convective momentum transfer balances. The places where enter the governing equations are multiplied with either the Mach number or the Mach number squared, indicating that the effects become important for an increasing background flow.

References for the Pipe Acoustics Interfaces


2. M.S. Ghidaoui, M. Zhao, D.A. McInnis, and D.H. Axworthy, “A Review of Water Hammer Theory and Practice,” Applied Mechanics Reviews, ASME, 2005.

3. R.M. Aarts and A.J.E.M. Janssen, “Approximation of the Struve Function H1 Occurring in Impedance Calculations,” J. Acoust. Soc. Am., vol. 113, pp. 2635–2637, 2003.

4. O.A. Lindemann, “Radiation Impedance of a Rectangular Piston at Very Low Frequencies,” J. Acoust. Soc. Am., vol 44, pp. 1738–1739, 1968.

5. A.D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustics Society of America, 1994.

6. H. Levine and J. Schwinger, “On the Radiation of Sound from an Unflanged Circular Pipe,” Phys. Rev., vol. 73, pp. 383–406, 1948.


5

T h e A e r o a c o u s t i c s B r a n c h

This chapter describes the interfaces found under the Aeroacoustics branch when adding a physics interface.

The aeroacoustic branch has interfaces to solve the fully linearized acoustic equation for several physical conditions, they are: the linearized potential flow equations, the linearized Euler (LE) equations, and the linearized Navier-Stokes (LNS) equations. In this chapter:

• The Linearized Potential Flow, Frequency Domain Interface

• The Linearized Potential Flow, Transient Interface

• The Linearized Potential Flow, Boundary Mode Interface

• The Compressible Potential Flow Interface

• The Linearized Euler, Frequency Domain Interface

• The Linearized Euler, Transient Interface

• The Linearized Navier-Stokes, Frequency Domain Interface

• The Linearized Navier-Stokes, Transient Interface

• Theory Background for the Aeroacoustics Branch

• References for the Aeroacoustics Branch Interfaces

223

224 | C H A P T E

Th e L i n e a r i z e d Po t e n t i a l F l ow , F r e qu en c y Doma i n I n t e r f a c e

The Linearized Potential Flow, Frequency Domain (ae) interface ( ), found under the Acoustics>Aeroacoustics branch ( ) when adding a physics interface, is used to compute the acoustic variations in the velocity potential in the presence of an inviscid and irrotational background mean-flow, that is, a potential flow. The physics interface is used for aeroacoustic simulations that can be described by the linearized compressible potential flow equations.

The equations are formulated in the frequency domain and assume harmonic variation of all sources and fields. The physics interface is limited to flows with a Mach number M<1, partly due to limitations in potential flow and partly due to the acoustic boundary settings needed for supersonic flow. The coupling between the acoustic field and the background flow does not include any predefined flow-induced noise.

When this interface is added, these default nodes are also added to the Model Builder— Aeroacoustics Model, Sound Hard Boundary (Wall), and Initial Values. For axisymmetric models an Axial Symmetry node is also added.

Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Linearized Potential Flow,




The default identifier (for the first interface in the model) is ae.

The Linearized Potential Flow, Frequency Domain (ae) interface was previously called Aeroacoustics, Frequency Domain (ae). The interface identifier is the same, however.

R 5 : T H E A E R O A C O U S T I C S B R A N C H



L I N E A R I Z E D PO T E N T I A L F L O W E Q U A T I O N S E T T I N G S


The settings are the same as Sound Pressure Level Settings for the Pressure Acoustics, Frequency Domain interface.


The settings are the same as Typical Wave Speed for the Pressure Acoustics, Frequency Domain interface.


This interface defines one dependent variable (field), the Velocity potential phi. The name can be changed but the names of fields and dependent variables must be unique within a model.


To display this section, click the Show button ( ) and select Discretization. Set the element order for the Velocity potential to Linear, Quadratic (the default), Cubic, or

For 1D axisymmetric models, the Circumferential wave number m (dimensionless) is 0 by default. The Out-of-plane wave number kz (SI unit: rad/m) is 0 rad/m by default. Enter different values or expressions as required.

For 2D models, the Out-of-plane wave number kz (SI unit: rad/m) is 0 rad/m by default.

For 2D axisymmetric models the Circumferential wave number m (dimensionless) is 0 by default.

Enter different values or expressions as required.

T H E L I N E A R I Z E D P O T E N T I A L F L O W , F R E Q U E N C Y D O M A I N I N T E R F A C E | 225

226 | C H A P T E

Quartic. Specify the Value type when using splitting of complex variables—Real or Complex (the default).

Domain, Boundary, Edge, Point, and Pair Nodes for the Linearized Potential Flow, Frequency Domain Interface

The Linearized Potential Flow, Frequency Domain Interface has these domain, boundary, edge, point and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users)..


• Domain, Boundary, Edge, Point, and Pair Nodes for the Linearized Potential Flow, Frequency Domain Interface


• Flow Duct: model library path Acoustics_Module/Industrial_Models/flow_duct

• Doppler Shift: model library path Acoustics_Module/Tutorial_Models/doppler_shift


• Continuity

• Initial Values

• Impedance and Pair Impedance


• Linearized Potential Flow Model

• Mass Flow Circular Source

• Mass Flow Edge Source

• Mass Flow Point Source

• Normal Mass Flow

• Normal Velocity





• Velocity Potential

• Vortex Sheet


Linearized Potential Flow Model

The Linearized Potential Flow Model node adds the equations for frequency domain aeroacoustics modeling. You here need to enter the material properties as well as the background mean flow information.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains where the aeroacoustics model is valid and to compute the velocity potential, or select All domains as required.

L I N E A R I Z E D PO T E N T I A L F L O W M O D E L

The default values for the Density (SI unit: kg/m3) and the Mean flow speed of sound cmf (SI unit: m/s) are taken From material. Select User defined to enter other values or

For axisymmetric models, COMSOL Multiphysics takes the axial symmetry (at r = 0) into account and automatically adds an Axial

Symmetry node to the model that is valid on the axial symmetry edges/points only.


• Destination Selection




The Linearized Potential Flow Model was previously called Aeroacoustics

Model. However, the tag (aem) is still the same.


228 | C H A P T E

expressions. This could for example bee to select the values taken from a simulation run using The Compressible Potential Flow Interface.

Also enter values or expressions for the Mean flow velocity V (SI unit: m/s). The defaults are 0 m/s. It is here important to know that the velocity field needs to be a solution to a compressible potential flow simulation. It has to be an irrotational and inviscid flow, for example, a constant flow field V is of this type. Any other type of flow yields non-physical solutions for this formulation of the governing equations.

Initial Values

The Initial Values node adds initial values for the velocity potential. For The Linearized Potential Flow, Transient Interface it adds initial values for the velocity potential, first time derivative. Add more Initial Values nodes from the Physics toolbar.




Enter a value or expression for the initial value of the Velocity potential phi (SI unit: m2/s). The default is 0 m2/s.

For The Linearized Potential Flow, Transient Interface enter a Velocity potential, first

time derivative, (SI unit: m2/s2). The default is 0 m2/s2.

Sound Hard Boundary (Wall)

Use the Sound Hard Boundary (Wall) condition to model rigid boundary surfaces or walls. It prescribes a vanishing normal component of the particle velocity at the boundary. Multiplied by the density, it can equivalently be expressed as a no-flow condition:

The sound-hard boundary condition is available for all analysis types. The equation above applies to the time domain calculations in The Linearized Potential Flow,

t

n– V

cmf2

--------t

V +

– 0=


Transient Interface; to obtain the corresponding condition for frequency domain, simply replace t by i.

The Linearized Potential Flow, Boundary Mode Interface the no-flow or wall condition, known as sound hard, sets the normal acceleration—and thus also the normal velocity—to zero at the edge.





Velocity Potential

Use the Velocity Potential node when coupling two Linearized Potential Flow, Frequency

Domain interfaces together because sometimes be necessary to set the velocity potential: 0.


From the Selection list, choose the boundaries to define. When using the pair node, this list cannot be edited. It shows the boundaries in the selected pairs.



VE L O C I T Y PO T E N T I A L

Enter a Velocity potential 0 (SI unit: m2/s). The default is 0 m2/s.

n– V

cmf2

----------- i V + – 0=

Use Pairs to couple two Linearized Potential Flow, Frequency Domain interfaces together as it can sometimes be necessary to set the velocity potential 0.


230 | C H A P T E




Normal Mass Flow

Use the Normal Mass Flow node to set the inward mass flow boundary condition.

For The Linearized Potential Flow, Frequency Domain Interface, the natural boundary condition for the total wave has the meaning of a mass flow through the boundary surface:

For The Linearized Potential Flow, Boundary Mode Interface, the natural edge condition for the total wave has the meaning of normal mass flow.



N O R M A L M A S S F L O W

Enter an Inward mass flow mn (SI unit: kg/(m2·s)). The default is 0 kg/(m2·s).

Plane Wave Radiation

The Plane Wave Radiation is a class of non-reflecting boundary conditions, which assume that there is an outgoing plane wave, and optionally also an incoming exciting wave.

For transient analysis the boundary condition is

n– V

cmf2

-----------t

V +

– mn=

n– V

cmf2

----------- i V Vn+ + – mn=


while the corresponding time-harmonic equation reads

Specify an Incident Velocity Potential (incoming plane wave)

by supplying its amplitude, 0, and propagation wave direction vector, ek. The vector nk is the normalized wave direction vector of unit length.



Incident Velocity Potential

Right-click the Plane Wave Radiation node to add an Incident Velocity Potential subnode, which is then used to add a velocity potential and wave direction.

n– V

cmf2

-----------t

V + –

kn t

– n V

cmf2

--------t

kn t

n V– – =

t 0kk n nk n V

cmf2

--------kk t0nk V kn t

0– n V

cmf2

-------- kn t0 n V

+–

kk1

cmf V nk+------------------------------= kn

1cmf V n+---------------------------=

n– V

cmf2

----------- i V+ –

ikn– n V

cmf2

-------- i iknn V– – =

ikkn nk nV

cmf2

-------- ikknk V ikn– nV

cmf2

-------- iknn V +–

0ei– k r

kk

cmf V nk+------------------------------= kn

cmf V n+---------------------------= k k nk= nk

ekek--------=

0e ik r–

This boundary condition is most relevant for ports, because many waveguide structures are only interesting in the plane-wave region.


232 | C H A P T E



I N C I D E N T VE L O C I T Y PO T E N T I A L

Enter a Velocity potential 0 (SI unit: m2/s) and Wave direction ek (SI unit: m). The default for the velocity potential is 0 m2/s, and for the wave direction, the default is the inward normal direction of the boundary.

Sound Soft Boundary

The Sound Soft Boundary creates a boundary condition for a sound soft boundary, where the acoustic pressure vanishes and p = 0.


From the Selection list, choose the boundaries to define. If Sound Soft Boundary is selected from the Pairs menu, this list cannot be edited. It shows the boundaries in the selected pairs.




To display this section, click the Show button ( ) and select Advanced Physics Options. Select the Use weak constraints check box to replace the standard constraints with a weak implementation.

Normal Velocity

Use the Normal Velocity node in time-harmonic analysis to specify the velocity component normal to the boundary:

This boundary condition is an appropriate approximation for a liquid-gas interface and in some cases for external waveguide ports.

n V

cmf2

----------- i V + – vn

1i------V vn+

=


Here vn denotes the outward normal velocity at the boundary surface, which is specified in the vn field.



N O R M A L VE L O C I T Y

Enter a Normal velocity vn (SI unit: m/s). The default is 0 m/s.

Impedance and Pair Impedance

Use the Impedance or Pair Impedance node in time-harmonic analysis to define the input impedance of an external domain or at the boundary between parts in an assembly as the ratio of pressure to normal velocity, Zip(n · v) at the boundary. The associated impedance boundary condition is

For the Pair Impedance node the above equation applies on each side of the boundary.


From the Selection list, choose the boundaries to define. For the Pair Impedance node, this list is not editable and shows the boundaries in the selected pairs.



I M P E D A N C E / P A I R I M P E D A N C E

Enter an input Impedance Zi (SI unit: Pa·s/m). The default value is 1Pa·s/m for the boundary condition and 0 Pa·s/m for the pair condition.

Vortex Sheet

Use the Vortex Sheet boundary condition to model a shear layer that separates a stream from the free velocity field. Because the velocity potential is discontinuous over this boundary, use a slit boundary condition or a pair in an assembly. Vortex sheets are only applicable on interior boundaries.

n–

cmf2

----------- i V + V– p

Z---- 1

i------ V p

Z----+

=


234 | C H A P T E

The equations defining the vortex sheet boundary condition are

where w denotes the outward normal displacement (SI unit: m) of the boundary surface, which this boundary condition adds as ae.w or aepf.w, using the default name for the physics interface. The subscripts “up” and “down” refer to the two sides of the boundary.


From the Selection list, choose the boundaries to define. If selected from the Pairs submenu, this list is not editable and shows the boundaries in the selected pairs.



Interior Sound Hard Boundary (Wall)

For The Linearized Potential Flow, Frequency Domain Interface and The Linearized Potential Flow, Transient Interface, use the Interior Sound Hard Boundary (Wall) condition to model interior rigid boundary surfaces, or walls. It prescribes a vanishing normal component of the particle velocity at the boundary. Multiplied by the density, it can equivalently be expressed as a no-flow condition:

Here, the subscripts “up” and “down” refer to the two sides of the boundary.

The sound-hard boundary condition is available for all analysis types. The equation above applies to the time domain calculations in the Linearized Potential Flow, Transient interface; to obtain the corresponding condition for the frequency domain, simply replace t with i.



n V

cmf2

----------- i V + –

i

i V + w i= i up down=

pup pdown= wup wdown–=

n– V

cmf2

--------t

V +

–

i

0= i up down=


Continuity

Continuity is available as an option at interfaces between parts in a pair.

For the interfaces under the Aeroacoustics branch, this condition gives continuity in the velocity potential as well as continuity in the mass flow. It corresponds to a situation where the boundary has no direct effect on the acoustic velocity potential field (subscripts 1 and 2 in the equation refers to the two sides of the pair):


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface.




These settings are the same as for Velocity Potential.

Axial Symmetry

n V

cs2

----- i V + –

1

V

cs2

----- i V + –

2

– 0=




The Axial symmetry feature is a default node added for all axisymmetric models. The boundary condition is active on all boundaries on the symmetry axis.


236 | C H A P T E


The boundaries section shows on which boundaries the feature is active. All boundaries on the symmetry axis are automatically selected.

Mass Flow Line Source on Axis


From the Selection list, choose the boundaries on the symmetry axis to define.

M A S S F L O W L I N E S O U R C E O N A X I S

Enter a Mass flow rate m' (SI unit: kg/(m·s)). The default is 0 kg/(m·s).

Mass Flow Edge Source

For 3D models, use a Mass Flow Edge Source to specify the mass flow rate on an edge:



M A S S F L O W E D G E S O U R C E

Enter a Mass flow rate m' (SI unit: kg/(m2·s)). The default is 0 kg/(m2·s).

Mass Flow Point Source

Add a Mass Flow Point Source node to specify the mass flow rate on a point:

For 2D axisymmetric models, use a Mass Flow Line Source on Axis node to add a line source along the symmetry axis.

V

cs2

------- i V + – m'=

V

cmf2

----------- i V + – m'=




M A S S F L O W PO I N T S O U R C E

Enter a Mass flow rate m' (SI unit: kg/s for 3D and 2D axisymmetric models; kg/(m·s) for 2D models). The default is 0 for all dimensions.

Mass Flow Circular Source



M A S S F L O W C I R C U L A R S O U R C E

Enter a Mass flow rate m' (SI unit: kg/(m·s). The default is 0 kg/(m·s).

V

cs2

------- i V + – m'=

Doppler Shift: model library path Acoustics_Module/Tutorial_Models/doppler_shift

For 2D axisymmetric models, use a Mass Flow Circular Source node to add a circular source located at rr0.

V

cs2

------- i V + – m'=


238 | C H A P T E

Th e L i n e a r i z e d Po t e n t i a l F l ow , T r a n s i e n t I n t e r f a c e

The Linearized Potential Flow, Transient (aetd) interface ( ), found under the Acoustics>Aeroacoustics branch ( ) when adding a physics interface, is used to compute the acoustic variations in the velocity potential in the presence of an inviscid and irrotational background mean-flow, that is, a potential flow. The physics interface is used for aeroacoustic simulations that can be described by the linearized compressible potential flow equations.

The equations are formulated in the time domain. The physics interface is limited to flows with a Mach number M<1, partly due to limitations in potential flow and partly due to the acoustic boundary settings needed for supersonic flow. The coupling between the acoustic field and the background flow does not include any predefined flow-induced noise.

When this interface is added, these default nodes are also added to the Model Builder— Linearized Potential Flow Model, Sound Hard Boundary (Wall), and Initial Values. For axisymmetric models an Axial Symmetry node is also added.


Transient to select physics from the context menu.



The Linearized Potential Flow, Transient (aetd) interface was previously called Aeroacoustics, Transient (aetd). The interface identifier is the same, however.


The default identifier (for the first interface in the model) is aetd.

Domain, Boundary, Edge, Point, and Pair Nodes for the Linearized Potential Flow, Transient Interface

The Linearized Potential Flow, Transient Interface has these domain, boundary, edge, point, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

The remainder of the settings are shared with The Linearized Potential Flow, Frequency Domain Interface.


• Domain, Boundary, Edge, Point, and Pair Nodes for the Linearized Potential Flow, Transient Interface



• Continuity

• Initial Values



• Mass Flow Point Source

• Mass Flow Circular Source

• Mass Flow Edge Source









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240 | C H A P T E



Th e L i n e a r i z e d Po t e n t i a l F l ow , Bounda r y Mode I n t e r f a c e

The Linearized Potential Flow, Boundary Mode (aebm) interface ( ), found under the Acoustics>Aeroacoustics branch ( ) when adding a physics interface, is used to compute eigenmodes and out-of-plane wave numbers for the linearized compressible potential flow equations. This study is used, for example, when specifying sources at inlets or analyzing transverse acoustic modes in ducts.

The interface solves an eigenvalue equation on boundaries, searching for the out-of-plane wave numbers at a given frequency.

When this interface is added, these default nodes are also added to the Model Builder— Linearized Potential Flow Model, Sound Hard Boundary (Wall), and Initial Values. For 2D axisymmetric models an Axial Symmetry node is also added.


Boundary Mode to select physics from the context menu.



The Linearized Potential Flow, Boundary Mode (aebm) interface was previously called Boundary Mode Aeroacoustics (aebm). The interface identifier is the same, however.

This interface is limited to flows with a Mach number M < 1, partly due to limitations in the potential flow formulation and partly due to the acoustic boundary settings needed for supersonic flow.

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The default identifier (for the first interface in the model) is aebm.


The default setting is to include All boundaries in the model. To choose specific boundaries, select Manual from the Selection list.

L I N E A R I Z E D PO T E N T I A L F L O W E Q U A T I O N S E T T I N G S


The settings are the same as Sound Pressure Level Settings for the Pressure Acoustics,

Frequency Domain interface.


This interface defines one dependent variable (field), the Velocity potential phi. The name can be changed but the names of fields and dependent variables must be unique within a model.


To display this section, click the Show button ( ) and select Discretization. Select Quadratic (the default), Linear, Cubic, or Quartic for the Velocity potential. Specify the Value type when using splitting of complex variables—Real or Complex (the default).

For 2D axisymmetric models, the Circumferential wave number m (dimensionless) is 0 by default. It is an integer entering the axisymmetric expression for the velocity potential:

r z r ei kzz m+ –

=


• Boundary, Edge, Point, and Pair Nodes for the Linearized Potential Flow, Boundary Mode Interface


Flow Duct: model library path Acoustics_Module/Industrial_Models/flow_duct


Boundary, Edge, Point, and Pair Nodes for the Linearized Potential Flow, Boundary Mode Interface

The Linearized Potential Flow, Boundary Mode Interface has these boundary, edge, point, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).


• Continuity

• Initial Values






For the Linearized Potential Flow, Boundary Mode interface, apply the features to boundaries instead of domains for 3D models.





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Th e Comp r e s s i b l e Po t e n t i a l F l ow I n t e r f a c e

The Compressible Potential Flow (cpf) interface ( ), found under the Acoustics>Aeroacoustics branch ( ) when adding a physics interface, is used to compute for the velocity potential in a compressible potential flow field assuming an ideal barotropic, irrotational fluid at constant entropy, that is, fluid is also inviscid. The physics interface is used for modeling the background flow to be used as input to The Linearized Potential Flow, Frequency Domain Interface or The Linearized Potential Flow, Transient Interface.

When this interface is added, these default nodes are also added to the Model Builder— Compressible Potential Flow, Slip Velocity, and Initial Values. For axisymmetric models an Axial Symmetry node is also added.

Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Compressible Potential Flow

to select physics from the context menu.



The default identifier (for the first interface in the model) is cpf.

The potential flow formulation for steady compressible flow is in general not suited for modeling shocks. In the region after a shock the flow is typically rotational, hence it is only suited for problems with a Mach number M < 1.



The default setting is to include All domains in the model where the compressible potential flow model is valid and to compute the density and the mean flow velocity potential. To choose specific domains, select Manual from the Selection list.

R E F E R E N C E V A L U E S

Edit or enter the values as required:

• Reference pressure pref (SI unit: Pa). The default is 1 atm.

• Reference density ref (SI unit: kg/m3). The default is 1.2 kg/m3.

• Reference velocity ref (SI unit: m/s). The default is 0 m/s.

• Reference force potential ref (SI unit: J/kg). The default is 0 J/kg.


This interface defines two dependent variables (field), the Mean flow velocity potential Phi and the Density, rho. The name can be changed but the names of fields and dependent variables must be unique within a model.


To display this section, click the Show button ( ) and select Discretization. Select Quadratic (the default), Linear, Cubic, or Quartic for the Mean flow velocity potential and Density. Specify the Value type when using splitting of complex variables—Real or Complex (the default).


• Domain, Boundary, and Pair Nodes for the Compressible Potential Flow Interface


Flow Duct: model library path Acoustics_Module/Industrial_Models/flow_duct

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Domain, Boundary, and Pair Nodes for the Compressible Potential Flow Interface

The Compressible Potential Flow Interface has these domain, boundary, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

• Compressible Potential Flow Model

• Initial Values

• Mass Flow

• Normal Flow


• Slip Velocity

Compressible Potential Flow Model

The Compressible Potential Flow Model node adds equations for time-dependent or stationary modeling of compressible potential flow.



• Using Symmetries







C O M P R E S S I B L E P O T E N T I A L F L O W M O D E L

Enter a Ratio of specific heats (dimensionless). The default is 1.4. Also enter a Force

potential (SI unit: J/kg). The default is 0 J/kg.

Initial Values

The Initial Values node adds initial values for the mean flow velocity potential and density variables. Add more Initial Values nodes from the Physics toolbar.




Enter a value or expression for the initial values Mean flow velocity potential Phi (SI unit: m2/s). The default is 0 m2/s. Enter a Density rho (SI unit: kg/m3). The default is cpf.rhoref.

Normal Flow

The Normal Flow node implies that the flow is normal to the boundary and thus that the tangential velocity is zero. This corresponds to a constant velocity potential along the boundary. Because the velocity potential is determined only up to a constant, imposing this condition fixes the arbitrary constant to zero where

Setting the Normal Flow condition on two or more disjoint boundaries can result in the wrong features unless symmetry implies that the velocity potential is equal on the boundaries in question.


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Mass Flow

The Mass Flow node specifies the mass flow through the boundary. The mass flow is given by the product of two variables: the normal velocity, vn, and the density at the boundary, bnd:





M A S S F L O W

Enter the Normal Velocity vn (SI unit: m/s) and Fluid density at the boundary bnd (SI unit: kg/m3). The defaults are cpf.vref and cpf.rhoref, respectively.

Slip Velocity

The Slip Velocity node is the natural condition at a boundary impervious to the flow, meaning that the velocity normal to the boundary is zero. By multiplying with the density, this condition can be alternatively be expressed as a vanishing mass flow through the boundary:


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the

n vnbnd=

n 0=


interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required.


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Th e L i n e a r i z e d Eu l e r , F r e qu en c y Doma i n I n t e r f a c e

The Linearized Euler, Frequency Domain (lef) interface ( ), found under the Acoustics>Aeroacoustics branch ( ) when adding a physics interface, is used to compute the acoustic variations in density, velocity, and pressure in the presence of a stationary background mean-flow that is well approximated by an ideal gas flow. The physics interface is used for aeroacoustic simulations that can be described by the linearized Euler equations.

The equations defined by the Linearized Euler, Frequency Domain interface are the linearized continuity, momentum (Euler), and energy equations. The interface solves for the acoustic variations in the density , velocity field u, and pressure p. The equations are formulated in the frequency domain and assume harmonic variation of all sources and fields. The background mean flow can be any stationary gas flow that is well approximated by an ideal gas. The coupling between the acoustic field and the background flow does not include any predefined flow induced noise. Even though the equations do not include any loss mechanisms, only acoustic modes exist in the frequency domain as the driving frequency is predefined and real valued.

When this interface is added, these default nodes are also added to the Model Builder— Linearized Euler Model, Rigid Wall, and Initial Values. For axisymmetric models an Axial

Symmetry node is also added.

Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Linearized Euler, Frequency

Domain to select physics from the context menu.



The default identifier (for the first interface in the model) is lef.





See Sound Pressure Level Settings for the Pressure Acoustics, Frequency Domain interface. Only Use reference pressure for air or User-defined reference pressure are available selections.


See Typical Wave Speed for the Pressure Acoustics, Frequency Domain interface.


This interface defines these dependent variables (fields), the Density rho, Velocity field u and its components, and Pressure p. The name can be changed but the names of fields and dependent variables must be unique within a model.

S T A B I L I Z A T I O N

To display this section, click the Show button ( ) and select Stabilization. The Streamline diffusion check box is selected by default and the Stabilization parameter SD (dimensionless) default is 5·10-3. Click to clear the check box as required.

The stabilization scheme implements the streamline upwind Petrov-Galerkin (SUPG) formulation of the weak form equations used for the finite element method. The stabilization parameter SD can be tuned depending on the problem solved, the nature of the background mean flow, and on the computational mesh. The implementation follows the one discussed in Ref. 10 and Ref. 14.


To display this section, click the Show button ( ) and select Discretization. Select Linear (the default for all dependent variables), Quadratic, Cubic, or Quartic for the Element order for density, Element order for velocity, and Element order for pressure.

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Specify the Value types when using splitting of complex variables—Real or Complex (the default).

Domain, Boundary, and Pair Nodes for the Linearized Euler, Frequency Domain Interface

The Linearized Euler, Frequency Domain Interface has these domain, boundary, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

Linearized Euler Model

Use the Linearized Euler Model to set up the governing equations, define the background mean flow, the ideal gas fluid properties, and select gradient term


• Domain, Boundary, and Pair Nodes for the Linearized Euler, Frequency Domain Interface



• Domain Sources

• Impedance

• Incident Acoustic Fields

• Initial Values

• Interior Wall

• Linearized Euler Model

• Rigid Wall

• Prescribed Acoustic Fields

• Moving Wall

• Symmetry

See Continuity as described for The Linearized Potential Flow, Frequency Domain Interface.


suppression stabilization, if needed. The governing equations solved are (in the time domain):

(5-1)

where , u, and p are the acoustic perturbations to the density, velocity, and pressure, respectively. In the frequency domain the time derivatives of the dependent variables are replaced by multiplication with i. The variables with a zero subscript are the background mean flow values, is the ratio of specific heats. The right-hand-side source terms Sc, Sm and Se are zero. They can be defined in the Domain Sources node. Details about the physics and references are found in the Theory Background for the Aeroacoustics Branch section.


In order to model the influence of the background mean flow in the propagation of the acoustic waves in the fluid the background mean flow temperature T0, pressure p0, and velocity field u0 need to be defined. As the flow is assumed to be an ideal gas the background density 0 is readily defined as

where Rs is the specific gas constant or it can be defined as a material input and is then defined as 0 = 0(p0,T0).

All the background flow parameters can be functions of space. They can be either analytical expressions (user defined) or they can be picked up from a flow simulation performed using the CFD Module. By default they are set to the quiescent background conditions of air.

Enter User defined values for the Background mean flow temperature T0 (SI unit: K), Background mean flow pressure p0 (SI unit: Pa), and Background mean flow velocity u0 (SI unit: m/s). The defaults are 293.15 K, 1 atm, and 0 m/s, respectively.

Note that the Background mean flow density needs to be defined or entered in the Fluid

Properties section below.

t------ u0 0u+ + Sc=

ut------ uu0

T p0------I+

0------ u0 u0 p 0

1– – u0 u0 I– u+ ++ Sm=

pt------ pu0 p0u+ 1 – u p0 1 – u0 p–++ Se=

0p0

RsT0-------------=


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Select an option for the Background mean flow density 0 (SI unit: kg/m3), either From

material, Ideal gas (the default selected), or User defined (default value 1.2 kg/m3).

Define the remaining fluid properties necessary for defining an ideal gas. Select theGas constant type:

• Specific gas constant (the default selected) and select Rs From material (the default selected) or User defined (default value 287.058 J/(kg·K)).

• Mean molar mass and select Mn From material (the default selected) or User defined (default value 28.97 g/mol). This option calculates Rs = R/Mn, where R is the gas constant.

Also select whether to specify the ratio of specific heats (dimensionless) or the heat capacity at constant pressure Cp (SI unit: J/(kgK)). Select Specify Cp or :

• Ratio of specific heats (the default selected) and select From material (the default selected) or User defined (default value 1.4).

• Heat capacity at constant pressure and select Cp From material (the default selected) or User defined (default value 1005.4 J/(kgK)). This option calculates Cp /(Cp - Rs).

G R A D I E N T TE R M S U P P R E S S I O N S T A B I L I Z A T I O N

When the linearized Euler (LE) equations are solved in the time domain (or in the frequency domain with an iterative solver), linear physical instability waves can develop, the so-called Kelvin-Helmholtz instabilities. They are instabilities that grow exponentially because no losses exist in the LE equations (no viscous dissipation and no heat conduction). Further more, they are limited by non-linearities in the full Navier-Stokes flow equations. It has been shown that the growth of these instabilities can be limited, while the acoustic solution is retained, by canceling terms involving gradients of the mean flow quantities. This is known as gradient terms suppression (GTS) stabilization.

More details in Ref. 9, Ref. 10, Ref. 11, and the Theory Background for the Aeroacoustics Branch section.

Select the following check boxes to activate the applicable gradient term suppression (GTS), which is a form of physical stabilization where certain terms involving gradients of the background mean flow properties are removed:

• Suppression of mean flow density gradients

This option sets the following terms in the governing equations to zero:


• Suppression of mean flow velocity gradients


• Suppression of mean flow pressure gradients


Rigid Wall

The Rigid Wall condition is used to model a rigid wall, corresponding to the sound hard wall condition in Pressure Acoustics. In the case of lossless flows this reduces to the slip condition:

where n is the surface normal.





Initial Values

The Initial Values node adds initial values for the density, the velocity field, and pressure. Add more Initial Values nodes from the Physics toolbar.

p 01– 0=

0------ u0 u0 u0 u0 I– u+ 0=

1 – u0 p 0=

1 – u p0 0=

u n 0=


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Enter a value or expression for the initial values of the Density rho (SI unit: kg/m3) (the default is 0 kg/m3), the Velocity field u (SI unit: m/s) (the default is 0 m/s), and the Pressure p (SI unit: Pa) (the default is 0 Pa).

Domain Sources

Add a Domain Sources node to define the mass source, and momentum and energy source types. This domain feature adds the right-hand-side to the governing Equation 5-1. This condition can be used to create any user defined source by entering expressions into the field. The sources can for example be point-like Gaussian sources to model a single vortex or two interacting vortices. Note that specifying the source term to the continuity equation often also requires specifying source term to the energy equation. For example, for an isotropic fluid pressure and density are related by

, where c0 is the local speed of sound.



D O M A I N S O U R C E S

Enter a Mass source Sc (SI unit: kg/(m3s)). The default is 0 kg/(m3s).

Select a Momentum source type—Acceleration (the default) or Volume force. If Acceleration is chosen, enter vector expressions or values for the Acceleration source Sm (SI unit: m/s2). The defaults are 0 m/s2. If Volume force is chosen, enter vector expressions or values for the Volume force source Fm (SI unit: N/m3). The defaults are 0 N/m3.

Select an Energy source type—Pressure rate of change (the default) or Heat source. If Pressure rate of change is chosen, enter an expression or value for the Pressure rate of

change source Se (SI unit: Pa/s). The default 0 Pa/s. If Heat source is chosen, enter vector expressions or values for the Heat source Qe (SI unit: W/m3). The default is 0 W/m3.

p c02=


Incident Acoustic Fields

The Incident Acoustic Fields makes it possible to define an incident field in a domain. This condition can be used at an inlet, where a perfectly matched layer is also present, to set up a model with an incident field that also lets any reflected waves leave the computational domain.



I N C I D E N T A C O U S T I C F I E L D S


• Incident acoustic density i (SI unit: kg/m3). The default is 0 kg/m3.

• Incident acoustic velocity ui (SI unit: m/s). The defaults are 0 m/s.

• Incident acoustic pressure pi (SI unit: Pa). The default is 0 Pa.

Prescribed Acoustic Fields

The Prescribed Acoustic Fields condition allows the user to prescribe one or more of the dependent variables at a boundary. When specifying (constraining) the dependent variables for the linearized Euler equations, it is necessary to not under-constrain the system. Typically, this requires defining both the density and pressure, defining the velocity, or defining all three at the same time.



Several predefined variables exist for the incident and scattered field: the incident and scattered fields, the entropy, and the intensity. They are located in the plot menu group Incident and scattered fields, when post processing.

For systems that can be assumed isentropic, where the acoustic changes in the entropy are zero, enter an expression for the pressure p_user and the density p_user/lef.c0^2. This corresponds to the classical relation

.p c02=


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P R E S C R I B E D V A L U E S F O R A C O U S T I C V A R I A B L E S

By default no check boxes are selected. Click to select the following check boxes as required.

• Prescribed density p (SI unit: kg/m3). The default is 0 kg/m3.

• Prescribed velocity (SI unit: m/s). If this check box is selected, choose one or more of these additional check boxes—Prescribed in x direction upx, Prescribed in y

direction upy, or Prescribed in z direction upz (for 3D models). The default is 0 m/s.

• Prescribed pressure pp (SI unit: Pa). The default is 0 Pa.



Symmetry

The Symmetry node adds a symmetry condition at a boundary, it acts the same as the Rigid Wall condition.





Impedance

Use the Impedance condition to specify an acoustic impedance on a boundary. The impedance can be any expression and can, for example, be a function of the frequency (freq). The condition can be used to, for example, model a porous lining or specifying an outlet impedance. The condition is based on Myers’ condition, see Ref. 1 and About the Impedance Boundary Condition located in the Theory Background for the Aeroacoustics Branch section. The condition is a so-called low frequency approximation; the viscous boundary layer of the background flow is assumed infinitely thin at the impedance wall.




I M P E D A N C E

Enter the value of the Impedance Zn (SI unit: Pa·s/m). The default value is 0 Pa·s/m. Click to select the Wall impedance at solid boundary check box if the impedance is at a wall with no normal velocity component (slip condition). This will simplify the equations solved for.



Moving Wall

The Moving Wall condition is used to model a vibrating wall with harmonic variations. The condition is a variant of Myers’ condition used for the Impedance. See Ref. 1 and About the Moving Wall Boundary Condition located in the Theory Background for the Aeroacoustics Branch section for further details.



M O V I N G WA L L

Select a Displacement—Inward normal displacement vn (SI unit: m) (the default), Displacement field vw (SI unit: m), Inward normal velocity un (SI unit: m/s), or Velocity

field uw(SI unit: m/s). Then enter values or expressions based on the selection.



Interior Wall

The Interior Wall boundary condition is used to model a wall condition on an interior boundary. It is similar to the Rigid Wall boundary condition available on exterior boundaries except that it applies on both sides (up and down) of an internal boundary. It allows discontinuities (in density, velocity, and pressure) across the boundary. Use the Interior Wall boundary condition to avoid meshing thin structures by applying this slip-like conditions on interior curves and surfaces instead.


From the Selection list, choose the boundaries on which to apply the condition. The Interior Wall condition can only be applied on interior boundaries.


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Th e L i n e a r i z e d Eu l e r , T r a n s i e n t I n t e r f a c e

The Linearized Euler, Transient (let) interface ( ), found under the Acoustics>Aeroacoustics branch ( ) when adding a physics interface, is used to compute the acoustic variations in density, velocity, and pressure in the presence of a stationary background mean-flow that is well approximated by an ideal gas flow. The physics interface is used for aeroacoustic simulations that can be described by the linearized Euler equations.

The equations defined by the Linearized Euler, Transient interface are the linearized continuity, momentum (Euler), and energy equations. The interface solves for the acoustic variations in the density , velocity field u, and pressure p. The equations are formulated in the time domain. The background mean flow can be any stationary gas flow that is well approximated by an ideal gas. The coupling between the acoustic field and the background flow does not include any predefined flow induced noise. As the equations do not include any loss mechanisms, non-acoustic modes and instabilities can be modeled in the time domain.

When this interface is added, these default nodes are also added to the Model Builder— Linearized Euler Model, Rigid Wall, and Initial Values. For axisymmetric models an Axial

Symmetry node is also added.

Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Linearized Euler to select physics from the context menu.




The default identifier (for the first interface in the model) is let.

Domain, Boundary, and Pair Nodes for the Linearized Euler, Transient Interface

The Linearized Euler, Transient Interface has these nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

This physics interface shares some of its node settings with The Linearized Euler, Frequency Domain Interface.

Initial Values

The Initial Values node adds initial values for the density, the velocity field, pressure, and the first time derivatives for each variable. Add more Initial Values nodes from the Physics toolbar.

The Stabilization, Discretization, and Dependent Variables settings are the same as for The Linearized Euler, Frequency Domain Interface.


• Domain, Boundary, and Pair Nodes for the Linearized Euler, Transient Interface



• Domain Sources

• Initial Values

• Interior Wall

• Linearized Euler Model

• Moving Wall

• Rigid Wall

• Prescribed Acoustic Fields

• Symmetry

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Enter a value or expression for the initial values of the:

• Density rho (SI unit: kg/m3). The default is 0 kg/m3.

• Density, first time derivative rhot (SI unit: kg/(m3s)). The default is 0 kg/(m3s).

• Velocity field u (SI unit: m/s). The default is 0 m/s).

• Velocity field, first time derivative ut (SI unit: m/s2).

• Pressure p (SI unit: Pa). The default is 0 Pa.

• Pressure, first time derivative pt (SI unit: Pa/s). The default is 0 Pa/s.

Moving Wall

The Moving Wall condition can in the time domain only be defined in terms of the normal displacement. It is used to define a vibrating or moving wall. The displacement can be any time dependent expression. The condition can, for example, be used to model an actuator creating sound in a gas flow meter.



M OV I N G WA L L

Enter a Normal displacement vn (SI unit: m). The default is 0 m.




Th e L i n e a r i z e d Na v i e r - S t o k e s , F r e qu en c y Doma i n I n t e r f a c e

The Linearized Navier-Stokes, Frequency Domain (lnsf) interface ( ), found under the Acoustics>Aeroacoustics branch ( ) when adding a physics interface, is used to compute the acoustic variations in pressure, velocity, and temperature in the presence of any stationary isothermal or non-isothermal background mean-flow. The physics interface is used for aeroacoustic simulations that can be described by the linearized Navier-Stokes equations.

The equations are formulated in the frequency domain and assume harmonic variation of all sources and fields. The equations include viscous losses and thermal conduction as well as the heat generated by viscous dissipation, if relevant. The coupling between the acoustic field and the background flow does not include any predefined flow induced noise.

The equations defined by the Linearized Navier-Stokes, Frequency Domain interface are the linearized continuity, momentum (Navier-Stokes), and energy equations. The interface solves for the acoustic variations in the pressure p, velocity field u, and temperature T. The equations are formulated in the frequency domain for any fluid including losses due to viscosity and thermal conduction. The background mean flow can be any stationary flow.

When this interface is added, these default nodes are also added to the Model Builder— Linearized Navier-Stokes Model, No Slip, Isothermal, and Initial Values. For axisymmetric models, an Axial Symmetry node is also added.

Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Linearized Navier-Stokes,


Select the Enable technology preview functionality check box from the Preferences>Model Builder menu to enable preview of this interface and other new technology included in this release of COMSOL Multiphysics. See The Preferences Dialog Box in the COMSOL Multiphysics Reference Manual.

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The default identifier (for the first interface in the model) is lnsf.




See Sound Pressure Level Settings for the Pressure Acoustics, Frequency Domain interface.

TY P I C A L WA V E S P E E D

See Typical Wave Speed for the Pressure Acoustics, Frequency Domain interface.


This interface defines these dependent variables (fields), the Density rho, Velocity field u and its components, and Pressure p. The name can be changed but the names of fields and dependent variables must be unique within a model.

S T A B I L I Z A T I O N

To display this section, click the Show button ( ) and select Stabilization. The Streamline diffusion check box is not selected by default. Click to select the check box and then edit the Stabilization parameter SD (dimensionless) as required. The default is 5·10-3.

The stabilization scheme implements the streamline upwind Petrov-Galerkin (SUPG) formulation of the weak form equations used for the finite element method. The stabilization parameter SD can be tuned depending on the problem solved, the nature of the background mean flow, and on the computational mesh. The implementation follows the one discussed in Ref. 14. It is only implemented for the convective terms, not the diffusive terms.



To display this section, click the Show button ( ) and select Discretization. Select, Linear, Quadratic, Cubic, or Quartic for the Element order for pressure (default: Linear), Element order for velocity (default: Quadratic), and Element order for temperature

(default: Quadratic). Specify the Value types when using splitting of complex variables—Real or Complex (the default).

Domain, Boundary, and Pair Nodes for the Linearized Navier-Stokes, Frequency Domain and Transient Interfaces

The Linearized Navier-Stokes, Frequency Domain Interface has these domain, boundary, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).


• Domain, Boundary, and Pair Nodes for the Linearized Navier-Stokes, Frequency Domain and Transient Interfaces



• Adiabatic

• Domain Sources

• First Order Material Parameters

• Heat Flux

• Initial Values

• Isothermal

• Linearized Navier-Stokes Model

• Prescribed Pressure

• Prescribed Temperature


• Pressure (Adiabatic)

• Normal Stress

• No Slip

• Slip

• Stress


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Linearized Navier-Stokes Model

The Linearized Navier-Stokes Model sets up the governing equations, defines the background mean flow, fluid properties, and the compressibility and thermal expansion properties of the fluid. The governing equations solved are the continuity, momentum, and energy equations:

(5-2)

where p, u, and T are the acoustic perturbations to the pressure, velocity, and temperature, respectively. In the frequency domain, the time derivatives of the dependent variables are replaced by multiplication with i. The stress tensor is and is the viscous dissipation function. The right-hand-side source terms M, F, and Q are initially zero; they can be defined using the Domain Sources feature. The variables with a zero subscript are the background mean flow values. The material parameters are defined below. Details about the physics and references are found in the Theory Background for the Aeroacoustics Branch section.

The constitutive equations are the stress tensor and the linearized equation of state, while the Fourier heat conduction law is readily included in the above energy equation,

(5-3)

The linearized viscous dissipation function is defined as

• Continuity is described for The Linearized Potential Flow, Frequency Domain Interface.

• Symmetry is described for The Linearized Euler, Frequency Domain Interface

pt------ 0u u0+ + M=

0ut------ u u0 u0 u+ + u0 u0+ F+=

0CpTt------- u T0 u0 T+ + Cp u0 T0+

– 0T0pt------ u p0 u0 p+ + 0T u0 p0– kT Q+ +=

pI– u u T+ B23---–

u I+ +=

0 Tp 0T– =


(5-4)




In order to model the influence of the background mean flow on the propagation of the acoustic waves in the fluid, the background mean flow temperature T0, pressure p0, and velocity field u0 need to be defined. The density is defined in the Fluid

Properties section below, and is per default taken from the material. It is thus a function of the model inputs, that is, the background pressure and temperature. Enter User

defined values for the:

• Background mean flow temperature T0 (SI unit K). The default is 293.15 K.

• Background mean flow pressure p0 (SI unit: Pa). The default is 1 atm.

• Background mean flow velocity u0 (SI unit: m/s). The defaults are 0 m/s.


The defaults for the following are taken From material. Select User defined to edit the default values:

• Background mean flow density 0 (p0, T0) (SI unit: kg/m3). The default is 0 kg/m3.

• Dynamic viscosity (SI unit: Pas). The default is 0 Pas.

• Bulk viscosity B (SI unit: Pas). The default is 0 Pas.

• Thermal conductivity k (SI unit: W/(mK)). The default is 0 W/(mK).

• Heat capacity at constant pressure Cp (SI unit: J/(kgK)). The default is 0 J/(kgK).

u: u0 u0: u +=

u u u T+ B23---–

u I+=

u0 u0 u0 T+ B23---–

u0 I+=

When the background mean flow velocity u0 is set to zero, the linearized Navier-Stokes equations reduce to the equations solved in The Thermoacoustics, Frequency Domain Interface.


268 | C H A P T E

T H E R M A L E X P A N S I O N A N D C O M P R E S S I B I L I T Y

Select an option from the Coefficient of thermal expansion 0 list—From material (the default) or User defined. If User defined is chosen, enter a value for 0 (SI unit: 1/K). The default is 0 K-1.

Select an option from the Isothermal compressibility T list— From isentropic

compressibility (the default), From speed of sound, or User defined.

• If From isentropic compressibility is chosen, the values for the Isentropic

compressibility 0 (SI unit: 1/Pa) and Ratio of specific heats (dimensionless) are taken From material. Or select User defined for one or both of the options and enter a different value or expression.

• If From speed of sound is chosen, the values for the Speed of sound c (SI unit: m/s) and Ratio of specific heats (dimensionless) are taken From material. Or select User

defined for one or both of the options and enter a different value or expression.

• If User defined is chosen, enter a value for T (SI unit: 1/Pa). The default is 0 (1/Pa).

V I S C O U S D I S S I P A T I O N F U N C T I O N

Select the Include viscous dissipation function check box to if you wish to include the heat source generated by the viscous losses. The viscous dissipation function is defined in Equation 5-4.

No Slip

The No Slip node is the default mechanical boundary condition. It sets up a no slip condition for the flow on a hard wall:



u 0=




Isothermal

The Isothermal node is the default thermal boundary condition for the temperature. It sets up an isothermal condition:

This is a good approximation on solid walls, as heat conduction is typically much higher in solids than in fluids.





Initial Values

The Initial Values node adds initial values for the pressure, the velocity field, and temperature. Add more Initial Values nodes from the Physics toolbar.



Note that mechanical and thermal boundary conditions contribute with each other such that a condition can be set on the velocity and temperature simultaneously. Thermal type condition override each other the same is true for mechanical type conditions.

T 0=


270 | C H A P T E


Enter a value or expression for the initial values of the Pressure p (SI unit: Pa) (the default is 0 Pa), the Velocity field u (SI unit: m/s) (the default is 0 m/s), and the Temperature T (SI unit: K) (the default is 0 K).

First Order Material Parameters

To display this node in the context menu, click the Show button ( ) and select Advanced Physics Options.

Use the First Order Material Parameters node to include subtle (acoustic) variations in the material parameters due to the acoustic variations of the dependent variables. The selected material parameters will vary according to a linearization about their background values (at pressure p0 and temperature T0). The viscosity will, for example, be replaced by:



F I R S T O R D E R M A T E R I A L P A R A M E T E R S

By default no check boxes are selected. Click to select the following check boxes as required.

• Derivatives of dynamic viscosity / p (SI unit: s) and / T (SI unit: kg/(ms)).

• Derivatives of bulk viscosity B/ p (SI unit: s) and B/ T (SI unit: kg/(ms)).

• Derivatives of heat capacity at constant pressure Cp/ p (SI unit: m3kgK)) and Cp/ T (SI unit: m2s2K2)).

• Derivatives of thermal conduction k/ p (SI unit: m2sK)) and k/ T (SI unit: mkg /s2K2)).

• Derivatives of coefficient of thermal expansion 0/ p (SI unit: ms2kgK)) and 0/ T (SI unit: 1/K2).

pp------

T0

TT-------

p0

+ +


V I S C O U S D I S S I P A T I O N F U N C T I O N

Select the Include viscous dissipation function check box to include the effects of the redefined material parameters on the viscous dissipation function.

Domain Sources

Add a Domain Sources node to define the mass source, volume force source, and heat source, the right-hand-side of Equation 5-2. These can, for example, model varying thermal sources in a combustion chamber or a pulsating laser.



D O M A I N S O U R C E S

Enter values or expressions for the following:

• Mass source M (SI unit: kg/(m3s)). The default is 0 kg/(m3s).

• Volume force source F (SI unit: N/m3). The defaults are 0 N/m3.

• Heat source Q (SI unit: W/m3). The default is 0 W/m3.

Pressure (Adiabatic)

Add a Pressure (Adiabatic) node to give a pressure boundary condition with adiabatic conditions for the temperature. The condition is given by

where pp is the desired pressure at the boundary. This is a good approximation when prescribing a pressure at an inlet or outlet.



P R E S S U R E ( A D I A B A T I C )

Enter a value or expression for the Pressure pp (SI unit: Pa). The default is 0 Pa.



p pp= n ppn= n T 0=


272 | C H A P T E

Slip

Add a Slip node to define slip for the velocity defined by

This condition can be used at boundaries where it is not necessary to model the losses in the viscous boundary layer. Used together with the Adiabatic condition, no acoustic boundary layer is modeled.





Prescribed Velocity

Add a Prescribed Velocity node to define the velocity at a boundary.



P R E S C R I B E D VE L O C I T Y

Enter a Velocity up (SI unit: m/s). The defaults are 0 m/s.



Prescribed Pressure

Add a Prescribed Pressure node to prescribe the pressure at a boundary (using a constraint).



P R E S C R I B E D P R E S S U R E

Enter a Pressure pp (SI unit: Pa). The default is 0 Pa.

n u 0=




Normal Stress

Add a Normal Stress node to define the normal stress vector n at a boundary:



N O R M A L S T R E S S

Enter a Normal stress n (SI unit: N/m2). The default is 0 N/m2.

Stress

Add a Stress node to define the stress p at a boundary:



S T R E S S

Select Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the stress, and enter scalar or vector values or expressions for the Stress p (SI unit: N/m2). The default is 0 N/m2.

Adiabatic

Add an Adiabatic node to set up an adiabatic condition for the temperature, that is, the natural condition for the temperature



n n=

n n p=

n T 0=


274 | C H A P T E

Prescribed Temperature

Add a Prescribed Temperature node to define the temperature variation Tp at a boundary.



P R E S C R I B E D TE M P E R A T U R E

Enter a value or expression for the Temperature Tp (SI unit: K). The default is 0 K.



Heat Flux

Add a Heat Flux node to define the inward normal heat flux qn at a boundary:



H E A T F L U X

Enter a value or expression for the Inward normal heat flux qn (SI unit: W/m2). The default is 0 W/m2.

n kT qn=


Th e L i n e a r i z e d Na v i e r - S t o k e s , T r a n s i e n t I n t e r f a c e

The Linearized Navier-Stokes, Transient (lnst) interface ( ), found under the Acoustics>Aeroacoustics branch ( ) when adding a physics interface, is used to compute the acoustic variations in pressure, velocity, and temperature in the presence of any stationary isothermal or non-isothermal background mean-flow. The physics interface is used for aeroacoustic simulations that can be described by the linearized Navier-Stokes equations.

The equations are formulated in the time domain and include viscous losses and thermal conduction as well as the heat generated by viscous dissipation, if relevant. The coupling between the acoustic field and the background flow does not include any predefined flow induced noise.

The equations defined by the Linearized Navier-Stokes, Transient interface are the linearized continuity, momentum (Navier-Stokes), and energy equations. The interface solves for the acoustic variations in the pressure, velocity, and temperature. The equations are formulated in the time domain for any fluid including losses due to viscosity and thermal conduction. The background mean flow can be any stationary flow.

When this interface is added, these default nodes are also added to the Model Builder— Linearized Navier-Stokes Model, No Slip, Isothermal, and Initial Values. For axisymmetric models, an Axial Symmetry node is also added. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Linearized Navier-Stokes, Transient to select physics from the context menu.

Select the Enable technology preview functionality check box from the Preferences>Model Builder menu to enable preview of this interface and other new technology included in this release of COMSOL Multiphysics. See The Preferences Dialog Box in the COMSOL Multiphysics Reference Manual.

T H E L I N E A R I Z E D N A V I E R - S T O K E S , TR A N S I E N T I N T E R F A C E | 275

276 | C H A P T E



The default identifier (for the first interface in the model) is lnst.

The Stabilization, Discretization, and Dependent Variables settings are the same as for The Linearized Navier-Stokes, Frequency Domain Interface.


• Domain, Boundary, and Pair Nodes for the Linearized Navier-Stokes, Frequency Domain and Transient Interfaces



Th eo r y Ba c k g r ound f o r t h e A e r o a c ou s t i c s B r an c h

The scientific field of aeroacoustics deals with the interaction between a background mean flow and an acoustic field propagating in this flow. In general, this concerns both the, very complex, description of the creation of sound by turbulence in the background flow, that is, flow induced noise, but also the influence the background mean flow has on the propagation of an externally created sound field, that is, flow borne noise or flow borne sound. The computational aeroacoustics (CAA) capabilities of the aeroacoustics interfaces in COMSOL Multiphysics only cover the flow borne noise/sound situation.

Aeroacoustic simulations would ideally involve solving the fully compressible continuity, momentum (Navier-Stokes equations), and energy equations in the time domain. The acoustic pressure waves would then form a subset of the fluid solution. This approach is often impractical for real-world computational aeroacoustics (CAA) applications due to the required computational time and memory resources. Instead, for solving many practical engineering problems, a decoupled two-step approach is used: first solve for the fluid flow, then the acoustic perturbations of the flow.

For the solving the acoustic problem, the governing equations are linearized around the background mean flow and only solved for the acoustic perturbation. Acoustic variables are assumed to be small and perturbation theory can be used, for example, the total pressure

is the sum of the background mean pressure p0 and the acoustic pressure variations p.

This section presents the basic mathematical framework for the aeroacoustic equations solved in the aeroacoustic interfaces, starting with the general governing equations for fluid flow, that is, conservation equations, constitutive equations, and equations of state. Then the linearized potential flow equations, the equations for the compressional potential flow, the linearized Euler equations, and finally the linearized Navier-Stokes equations are presented.

In this section the theory background for:

• General Governing Equations

ptot p0 p+=

T H E O R Y B A C K G R O U N D F O R T H E A E R O A C O U S T I C S B R A N C H | 277

278 | C H A P T E

• Linearized Potential Flow

• Compressible Potential Flow

• Linearized Euler

• Linearized Navier-Stokes

General Governing Equations

The equations governing the physics in any fluid are the foundation for deriving the linearized aeroacoustic equations and in general any acoustics equations. The governing equations for the motion of a compressible fluid are the continuity equation (mass conservation), the Navier-Stokes equation (momentum conservation), and the general heat transfer equation (energy conservation). In order to close the system of equations, constitutive equations are needed, along with the equation of state, and thermodynamic relations. See, for example, Ref. 3, Ref. 4, Ref. 5, Ref. 6, or Ref. 7 for details and further reading.

C O N S E R V A T I O N E Q U A T I O N S

The conservation equations are for mass, momentum, and energy:

(5-5)

where u is the velocity field, is the density, T is the temperature, s is the specific entropy, is the stress tensor, q is the local heat flux, is the viscous dissipation function, and M, F, and Q are source terms. The operator D/Dt is the material derivative (or advection operator) defined as

T H E R M O D Y N A M I C R E L A T I O N S

Some thermodynamic relations are necessary when reformulating the energy equations in terms of other sets of thermodynamic variables, like (p, T) or (, p). They are the density differential, the specific energy relation, a relation due to Helmholtz, and the fundamental entropy relation:

t------ u + M=

DuDt-------- u

t------ u u+ F+= =

TDsDt------- T s

t----- u s+ q Q+ +–= =

DDt-------

t----- u +=


(5-6)

where u is the specific internal energy, 0 is the coefficient of thermal expansion (isobaric), and T is the isothermal compressibility. See, for example, Ref. 5 and Ref. 6 for details. They are defined together with the specific heat at constant pressure Cp and specific heat at constant volume Cv as

(5-7)

Using the above thermodynamic relations, the entropy differential can be expressed as (used for the linearized Navier-Stokes equations)

(5-8)

while for an ideal gas it can be given as (used for the linearized Euler equations)

(5-9)

C O N S T I T U T I V E E Q U A T I O N S

The constitutive equations are the equations of state (density expressed in terms of any set of thermodynamic variables), the Stokes expression for the stress tensor, and the Fourier heat conduction law

(5-10)

where k is the thermal conduction, is the dynamic viscosity, and B is the bulk viscosity. This then also defines the viscous dissipation function

d

------- 0dT– Tdp+=

du uT-------

u------

T+=

u------

T

1

2------ p

0T------T–

=

ds 1T----du p

T----d 1– +=

01---

T-------

p–= T

1---

p------

T=

CvuT-------

= Cp

uT-------

p=

Tds CpdT 0Tdp–=

ds Cvdp

------- Cpdp

-------–=

pI– + pI– u u T+ B23---–

u I+ += =

q kT–=


280 | C H A P T E

(5-11)

P E R T U R B A T I O N T H E O R Y

In the following, the governing equations are linearized and expanded to first order in the small parameters around the average stationary background solution. For details about perturbation theory see Ref. 3, Ref. 4, and Ref. 8. The small parameter variables (1’st order) represent the acoustic variations on top of the stationary background mean (or average) flow (0’th order solution). Note that, when solving the equations, the value of the acoustic field variables can also represent non-acoustic waves like thermal waves (entropy waves) and vorticity waves. In the time domain, these can be linear instabilities and can actually represent the onset of turbulence.

The dependent variables and sources are expanded according to

where A is any of the dependent variables or sources. In the frequency domain, the first order variables are assumed to be harmonic and expanded into Fourier components, such that

The first order variation to material parameters, that are not treated as dependent variables, like, for example, the density in the linearized Navier-Stokes interface, are linearized using a Taylor expansion

The above expansions are inserted into the governing equations and the linearized acoustic equations are derived retaining only first order linear terms.

Linearized Potential Flow

The equations presented here are the linearized potential flow equations. This restricts the applications of the interface to systems where the background flow is well described by a compressible potential flow, that is, a flow that is inviscid, barotropic, and irrotational. The sound sources also need to be external to the flow or at least they need to be represented by simple well defined sources. Application areas typically include modeling of how jet engine noise is influenced by the mean flow.

u:=

A A0 x A1 x t +=

A A0 x A1 x eit+=

0– p p0– 0p---------

TT T0–

0T---------

p+=


The basic dependent variable is the velocity potential conventionally defined by the relationship

where v v(r, t) is the particle velocity associated with the acoustic wave motion. The total particle velocity is given by

(5-12)

where V denotes the local mean velocity for the fluid motion. The dynamic equations for this mean-flow field are described in the next subsection. For now, just assume V to be a given irrotational background velocity field; hence, also the mean-flow velocity can be defined in terms of a potential field , by V.

The linearized equation for the velocity potential , governing acoustic waves in a background flow with mean background velocity V, mean background density , and mean speed of sound cmf, is

(5-13)

In deriving this equation, all variables appearing in the full nonlinear fluid-dynamics equations were first split in time-independent and acoustic parts, in the manner of Equation 5-12. Then, linearizing the resulting equations in the acoustic perturbation and eliminating all acoustic variables except the velocity potential gives Equation 5-13. Thus, the density in this equation is the time-independent part. The corresponding acoustic part is ar, tpr, tcmf

2 where p is the acoustic pressure, given by

Hence, once Equation 5-13 has been solved for the velocity potential, the acoustic pressure can easily be calculated.

When transformed to the frequency domain, the wave Equation 5-13 reads

while the acoustic pressure is

v =

vtot r t, V x v r t +=

cmf2

--------–t

t

V +

cmf2

--------t

V +

V–+ 0=

p r t t

V +

–=

cmf2

--------– i i V +

cmf2

-------- i V + V–+ 0=


282 | C H A P T E

Typical boundary conditions include:

• Sound-hard boundaries or walls

• Sound-soft boundaries


• Radiation boundary conditions

F R E Q U E N C Y D O M A I N E Q U A T I O N S

In the frequency domain the velocity potential is assumed to be a harmonic wave of the form

The governing frequency domain—or time-harmonic—equation is

In 2D, where

the out-of-plane wave number kz enters the equations when the operators are expanded:

The default value of the out-of-plane wave number is 0, that is, no wave propagation perpendicular to the 2D plane. In a mode analysis the equations are solved for kz.

For 2D axisymmetric models

the circumferential wave number m similarly appears in the equation as a parameter:

p r i V + –=

r t r eit=

cmf2

--------– i i V +

cmf2

-------- i V + V–+ 0=

r t x y ei t kz z–

=

i–

cmf2

----------- i V ikzVz–+ V

cmf2

----------- i V ikzVz–+ – +

+ kz2 ikzVz

cmf2

----------- i V ikzVz–+ + 0=

r t r z ei t m– =


T I M E - D E P E N D E N T E Q U A T I O N

In the time domain, the interface solves for the velocity potential with an arbitrary transient dependency. The following equation governs the acoustic waves in a mean potential flow:

(5-14)

Here (SI unit: kg/m3) is the density, V (SI unit: m/s) denotes the mean velocity, and cmf (SI unit: m/s) refers to the speed of sound. The software solves the equation for the velocity potential , with SI unit m2/s. The validity of this equation relies on the assumption that , V, and cmf are approximately constant in time, while they can be functions of the spatial coordinates.

B O U N D A R Y M O D E A N A L Y S I S

The boundary mode analysis type in 3D uses the eigenvalue solver to solve the equation

(5-15)

for the eigenmodes, , and eigenvalues, ikz, on a bounded two-dimensional domain, , given well-posed edge conditions on . In this equation, is the velocity

i–

cs2

------- i V + V

cs2

------- i V + –

mr2-----

2++ 0=

The background velocity field V cannot have a circumferential component because the flow is irrotational.

cmf2

--------–t

t

V +

cmf2

--------t

V +

V–+ 0=

The background velocity field V cannot have a circumferential component because the flow is irrotational.

i–

cmf2

----------- i Vt ikzVn–+ Vt

cmf2

----------- i Vt ikzVn–+ – +

+ kz2 ikzVn

cmf2

----------- i V ikzVn–+ + 0=


284 | C H A P T E

potential, is the density, cmf is the speed of sound, is the angular frequency, and kz is the out-of-plane wave number or propagation constant. Furthermore, Vt denotes the mean velocity in the tangential plane while Vn is the mean-velocity component in the normal direction.

Compressible Potential Flow

Consider a compressible and inviscid fluid in some domain . The motion and state of the fluid is described by its velocity V, density , pressure p, and total energy per unit volume e. Its dynamics is governed by the Euler equations, expressing the conservation of mass, momentum, and energy:

(5-16)

Here a volume force f has been included on the right-hand side of the momentum equation, whereas a possible heat-source term on the right-hand side of the energy equation (the last one) has been set to zero.

To close this system of five equations with six unknowns, an equation of state is required. Here, this is taken to be the equation for an ideal barotropic fluid,

where cp cV is the ratio between the specific heats at constant pressure and constant volume, while p0 and 0 are reference quantities for the pressure and the density, respectively, valid at some point in space. An alternative form of the ideal-fluid state equation is

Although the out-of-plane wave number is called kz, the two-dimensional surface on which Equation 5-15 is defined does not necessarily have to be normal to the z-axis for 3D geometries.

t V + 0=

t

VV V+

p+ f=

et----- e p+ V + 0=

p p00------ =

p 1– e=


The assumption that the fluid is barotropic means that pp. Taking the total time derivative and using the chain rule, leads to the relation

where, using the equation of state,

defines the speed of sound in the ideal fluid.

Assuming the flow to be irrotational, there exists a velocity potential field, , such that V. If, in addition, the volume force is assumed to be given by f, where is referred to as the force potential, the second of Equation 5-16 can be integrated to yield the Bernoulli equation

In this equation, two additional reference quantities have entered: the velocity, v0, and the force potential, 0, both valid at the same reference point as p0 and 0. Note, in particular, that neither the pressure, p, nor the energy per unit volume, e, appears in this equation.

T I M E D E P E N D E N T S T U D Y

Collecting the results, the equations governing the compressible, inviscid, irrotational flow of an ideal fluid are

where is the specific-heat ratio cp/cV and denotes the force potential, that is, the potential energy per unit mass measured in J/kg. In this equation, subscript 0 signifies reference quantities that apply at a specific point or surface. Thus, p0 is a reference pressure, 0 is a reference density, v0 is a reference velocity, and 0 is a reference force potential.

tddp

ddp

tdd c2

tdd=

c p---=

t------- 1

2--- 2 p0

1–

1– 0

----------------------- + +

+12---v0

2 p0 1– 0

----------------------- 0+ +=

t 1

2--- 2 p0

1–

1– 0

----------------------- + +

+v0

2

2--------

p0 1– 0

----------------------- 0+ +=

t + 0= c p

---= cp cV


286 | C H A P T E

S T A T I O N A R Y S T U D Y

In a stationary study, the same equation is used, but all time derivatives are set to zero, such that:

Linearized Euler

The linearized Euler equations are derived from Euler’s equations, that is Equation 5-5 with no thermal conduction and no viscous losses. The fluid in the linearized Euler interface is assumed to be an ideal gas. This is the common approach in literature. A review of the linearized Euler equations is found in, for example, Ref. 12 and Ref. 13.

G OV E R N I N G E Q U A T I O N S

A linearization of the governing equations yield after some manipulation

(5-17)

Here, the subscript “1” has been dropped on the acoustic perturbation variables. The time derivatives are replaced with multiplication by i in the frequency domain. Some ideal gas relations of interest are the equation of state, specific heat capacity and specific gas constant relations, and the compressibility and coefficients of thermal expansion:

2

2--------------

1–-----------

1– p0

0

------------------- + + v0

2

2-----

p0 1– 0

----------------------- 0+ +=

0=

t------ u0 0u+ + Sc=

ut------ uu0

T p0------I+

0------ u0 u0 p 0

1– – u0 u0 I– u+ ++ Sm=

pt------ pu0 p0u+ 1 – u p0 1 – u0 p–++ Se=


(5-18)

It also follows from the governing equations and the thermodynamic relations that the acoustic variations in the specific entropy s and in the temperature T are given by

(5-19)

I N S T A B I L I T I E S

When the linearized Euler (LE) equations are solved in the time domain (or in the frequency domain with an iterative solver), linear physical instability waves can develop, the so-called Kelvin-Helmholtz instabilities. They are instabilities that grow exponentially because no losses exist in the LE equations (no viscous dissipation and no heat conduction). Furthermore, they are limited by non-linearities in the full Navier-Stokes flow equations. It has been shown that in certain cases the growth of these instabilities can be limited, while the acoustic solution is retained, by canceling terms involving gradients of the mean flow quantities. This is known as gradient terms suppression (GTS) stabilization. See more details in Ref. 9, Ref. 10, and Ref. 11.

T H E E N E R G Y C O R O L L A R Y

Expressions for the energy flux, that is, the acoustic intensity vector, are often referred to as Myers’ energy corollary, see Ref. 15 and Ref. 16. The instantaneous intensity vector Ii is defined for both transient and frequency domain models as

(5-20)

c02 p0

0--------- RsT0= =

CpCv-------= Cp Cv Rs+=

01

T0------= T

1p0------=

The specific heat capacity at constant volume and constant pressure are often labeled with a lower case c, here we use upper case C’s but they are the specific quantities.

sCvp0------p

Cp0-------–=

T T0pp0------

0------–

=

Ii 0u u0+ p0------ u0 u+ 0u0Ts+=


288 | C H A P T E

The (time averaged) intensity vector I is given in the frequency domain by

(5-21)

A B O U T T H E I M P E D A N C E B O U N D A R Y C O N D I T I O N

The Linearized Euler, Frequency Domain Interface and The Linearized Euler, Transient Interface have an Impedance physics feature and its theory is included here.

In the frequency domain Myers’ equation (Ref. 1) gives an expression for the normal velocity at a boundary with a normal impedance condition. It is a so-called low-frequency approximation condition in the limit of very thin flow boundary layers (compared to the wavelength). Such conditions are used, for example, for porous lining conditions in ducts (Ref. 2). The condition is given by:

(5-22)

where the surface normal n here points out of the domain and is the surface normal impedance.

If the flow is parallel to the impedance boundary condition u0·n = 0, for example, slip flow over a mechanical impedance boundary condition (the same is true for the moving wall boundary condition described below), one can use a formulation with the tangential derivative for the second term on the right-hand side:

where A is an arbitrary scalar.

The last term on the right-hand side of Equation 5-22 can be reformulated as follows:

Again these terms reduce significantly for the case where nu0. If the boundary does not have curvature (planar boundary) then it is equal to zero. If the boundary is planar and the impedance condition is used inside the flow, for example at an outflow condition, then it reduces to the normal gradient of the velocity normal to the surface.

A B O U T T H E M O V I N G WA L L B O U N D A R Y C O N D I T I O N

The Linearized Euler, Frequency Domain Interface and The Linearized Euler, Transient Interface have a Moving Wall physics feature and its theory is included here.

I12---Re 0u u0+ p

0------ u0 u+ 0u0Ts+

=

u n pZn------ 1

i------u0 p

Zn------ p

iZn-------------n n u0 –+=

u0 A u0 tA u0 n n A + u0 tA= =

n n u0 n n u0 u0 n n –=


Myers’ equation (Ref. 1) gives the expressions used for a boundary condition at a moving wall. In the frequency domain it is given by

(5-23)

The inward normal displacement is given by

where un is the inward normal velocity is.

In the time domain, the condition on the normal velocity s given by

(5-24)

where the inward normal displacement is given by vn nv (SI unit: m).

Linearized Navier-Stokes

The linearized Navier-Stokes equations are derived by linearizing the full set of fluid flow equations given in General Governing Equations. After some manipulation, the continuity, momentum, and energy equations and are:

(5-25)

where p, u, and T are the acoustic perturbations to the pressure, velocity, and temperature, respectively. In the frequency domain, the time derivatives of the dependent variables are replaced by multiplication with i. The stress tensor is and is the viscous dissipation function. The variables with a zero subscript are the background mean flow values and the subscript “1” is dropped on the acoustic variables.

The constitutive equations are the stress tensor and the linearized equation of state, while the Fourier heat conduction law is readily included in the above energy equation,

u n ivn– u0 vn vnn n u0 +–=

vn n vn– i 1– un= = un n ub–=

u nvnt

---------– u0 vn vnn n u0 +–=

pt------ 0u u0+ + M=

0ut------ u u0 u0 u+ + u0 u0+ F+=

0CpTt------- u T0 u0 T+ + Cp u0 T0+

– 0T0pt------ u p0 u0 p+ + 0T u0 p0– kT Q+ +=


290 | C H A P T E

(5-26)

The linearized viscous dissipation function is defined as

pI– u u T+ B23---–

u I+ +=

0 Tp 0T– =

u: u0 u0: u +=

u u u T+ B23---–

u I+=

u0 u0 u0 T+ B23---–

u0 I+=


Re f e r e n c e s f o r t h e A e r o a c ou s t i c s B r an c h I n t e r f a c e s

1. M. K. Myers, “On the basic boundary condition in the presence of flow”, J. of Sound and Vibration, vol. 71, pp. 429, 1980.

2. W. Eversman, “The boundary condition at an impedance wall in a non-uniform duct with potential mean flow”, J. of Sound and Vibration, vol. 246, pp. 63, 2001.


4. H. Bruus, Theoretical Microfluidics, Oxford University Press, 2010.

5. G.K. Bachelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000.

6. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Course on Theoretical Physics volume 6, Butterworth-Heinemann, 2003.

7. B. Lautrup, Physics of Continuous Matter, Exotic and Every Day Phenomena in the Macroscopic World, 2nd ed., CRC Press, 2011.

8. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, Springer, 1999.

9. C. Bogey, C. Bailly, and D. Juvé, “Computation of Flow Using Source Terms in Linearized Euler’s Equations”, AIAA Journal, vol. 40, pp. 235, 2002.

10. P. P. Rao and P. J. Morris, “Use of Finite Element Methods in Frequency Domain Aeroacoustics”, AIAA Journal, vol. 44, pp. 1643, 2006.

11. A. Agarwal, P. J. Morris, and R. Mani, “Calculation of Sound Propagation in Nonuniform Flows: Suppression of Instability Waves”, AIAA Journal, vol. 42, pp.80, 2004.

12. C. Bailly and D. Juvé, “Numerical Solution of Acoustic Propagation Problems Using Linearized Euler Equations”, AIAA Journal, vol. 38, pp. 22, 2000.

13. C. W. Tam, “Computational Aeroacoustics: Issues and Methods”, AIAA Journal, vol. 33, 1995.

R E F E R E N C E S F O R T H E A E R O A C O U S T I C S B R A N C H I N T E R F A C E S | 291

292 | C H A P T E

14. G. Hauke and T. J. R. Hughers, “A comparative study of different sets of variables for solving compressible and incompressible flows”, Comput. Methods Appl. Mech. Engrg., vol. 153, pp. 1–44, 1998.

15. M. K. Myers, “An Exact Energy Corollary for Homentropic Flows”, J. Sound. Vib., vol. 109, pp. 277–284, 1986.

16. M. K. Myers, “Transport of energy disturbances in arbitrary steady flows”, J. Fluid Mech., vol. 226, pp. 383–400, 1991.


6

T h e T h e r m o a c o u s t i c s B r a n c h

This chapter describes the multiphysics interfaces that combine pressure acoustics and thermoacoustics. The interfaces are found under the Thermoacoustics branch ( ) when adding a physics interface.

• The Thermoacoustics, Frequency Domain Interface

• The Thermoacoustic-Solid Interaction, Frequency Domain Interface

• The Thermoacoustic-Shell Interaction, Frequency Domain Interface

• Theory Background for the Thermoacoustics Branch

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264 | C H A P T E

Th e Th e rmoa c ou s t i c s , F r e qu en c y Doma i n I n t e r f a c e

The Thermoacoustics, Frequency Domain (ta) interface ( ), found under the Thermoacoustics branch ( ) when adding a physics interface, is used to compute the acoustic variations of pressure, velocity, and temperature. The physics interface is required to accurately model acoustics in geometries of small dimensions. Near walls, viscosity and thermal conduction become important because it creates a viscous and thermal boundary layer where losses are significant. For this reason, it is necessary to include thermal conduction effects and viscous losses explicitly in the governing equations. The physics interface includes features from the Pressure Acoustics, Frequency Domain interface with predefined couplings to thermoacoustics. It is, for example, used when modeling the response of small transducers like microphones and receivers. Other applications include analyzing feedback in hearing aids and in mobile devices, or studying the damped vibrations of MEMS structures.

The physics interface solves the equations in the frequency domain assuming all fields and sources to be harmonic. Linear acoustics is assumed.

The equations defined by the Thermoacoustics, Frequency Domain interface are the linearized Navier-Stokes equations in quiescent background conditions solving the continuity, momentum, and energy equations. The physics interface solves for the acoustic pressure p, the velocity variation u (particle velocity), and the acoustic temperature variations T. It is available for 3D, 2D, and 1D Cartesian geometries as well as for 2D and 1D axisymmetric geometries.

When this interface is added, these default nodes are also added to the Model Builder— Thermoacoustics Model, Sound Hard Wall, Isothermal, Pressure Acoustics Model, Acoustic-Thermoacoustic Boundary, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Thermoacoustics to select physics from the context menu.

On the Contributing Physics InterfaceThe Pressure Acoustics, Frequency Domain interface is used to compute the pressure variations when modeling the propagation of acoustic waves in fluids at quiescent background conditions. The physics interface solves the Helmholtz equation and is suited for all linear frequency-domain acoustic simulations with harmonic variations of the pressure field. It includes domain conditions to model losses in a homogenized

R 6 : T H E T H E R M O A C O U S T I C S B R A N C H

way, so-called fluid models for porous materials as well as losses in narrow regions. The domain features also include background incident acoustic fields.

On Using the Physics InterfaceThe Thermoacoustics, Frequency Domain interface is necessary when modeling acoustics accurately in geometries with small dimensions. Near walls viscosity and thermal conduction become important because it creates a viscous and a thermal boundary layer (also called the viscous and thermal penetration depth) where losses are significant. For this reason it is necessary to include thermal conduction effects and viscous losses explicitly in the governing equations. For this reason, the thermoacoustic interface solves the full linearized Navier-Stokes (momentum), continuity, and energy equations. It solves for the propagation of compressible linear waves in a general viscous and thermally conduction fluid. Thermoacoustics is also known as thermoviscous acoustics or as viscothermal acoustics.

Due to the detailed description necessary when modeling thermoacoustics, the model simultaneously solves for the acoustic pressure p, the velocity vector u, and the acoustic temperature variations T.

The length scale at which the thermoacoustic description is necessary is given by the thickness of the, above-mentioned, viscous boundary layer, which is

and the thickness of the thermal boundary layer

The thickness of both boundary layers depends on the frequency f and they decreases with increasing frequency. The ratio of the two length scales is related to the non-dimensional Prandtl number Pr, by

which define the relative importance of the thermal and viscous effects for a given material. In air at 20oC and 1 atm the viscous boundary layer thickness is 0.22 mm at

v

f0------------=

tk

f0Cp-------------------=

vt-----

Cp

k----------- Pr= =

T H E T H E R M O A C O U S T I C S , F R E Q U E N C Y D O M A I N I N T E R F A C E | 265

266 | C H A P T E

100 Hz while it is only 55 m in water under the same conditions. The Prandtl number is 0.7 in air and 7 in water.

The physical quantities commonly used in the Thermoacoustics interfaces are defined in Table 6-1 below.


The interface identifier is used primarily as a scope prefix for variables defined by the physics interface. Refer to such interface variables in expressions using the pattern <identifier>.<variable_name>. In order to distinguish between variables

Evaluate the value of the viscous and thermal boundary layer thickness as well as the Prandtl number in post processing. They are defined by the variables ta.d_visc, ta.d_therm, and ta.Pr, respectively.

TABLE 6-1: THERMOACOUSTICS, FREQUENCY DOMAIN PHYSICAL QUANTITIES

QUANTITY SYMBOL SI UNIT ABBREVIATION

Pressure p pascal Pa

Temperature T kelvin K

Particle velocity u = u, v, w meter/second m/s

Dynamic viscosity pascal-second Pa·s

Bulk viscosity B pascal-second Pa·s

Thermal conductivity k watt/meter-kelvin W/(m·K)

Heat capacity at constant pressure

Cp joule/meter3-kelvin J/(m3·K)

Isothermal compressibility T 1/pascal 1/Pa

Coefficient of thermal expansion (isobaric)

0 1/kelvin 1/K

Ratio of specific heats (dimensionless) 1

Frequency f hertz Hz

Wave number k 1/meter 1/m

Equilibrium pressure p0 pascal Pa

Equilibrium density 0 kilogram/meter3 kg/m3

Equilibrium temperature T0 kelvin K

Speed of sound c meter/second m/s

Acoustic impedance Z pascal-second/meter Pa·s/m


belonging to different physics interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter.

The default identifier (for the first interface in the model) is ta.






Enter a value or expression for the Typical wave speed for perfectly matched layers cref (SI unit: m/s).


This interface defines these dependent variables (fields), the Pressure p, the Velocity

field, u and its components, and the Temperature variation, T. The name can be changed but the names of fields and dependent variables must be unique within a model.


To display this section, click the Show button ( ) and select Discretization. From the Discretization of fluids list select the element order for the velocity components, the temperature, and the pressure: P2+P1 (the default) or P3+P2.

• P2+P1 means second-order elements for the velocity components and the temperature. The pressure field has second-order elements in pressure acoustic domains and linear elements in thermoacoustic domains.

• P3+P2 means third-order elements for the velocity components and the temperature. The pressure field has third-order elements in the pressure acoustic domains and second-order elements in the thermoacoustic domains. This can add


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additional accuracy but it also adds additional degrees of freedom compared to P2+P1 elements.

Specify the Value type when using splitting of complex variables—Real or Complex (the default).

Domain, Boundary, Edge, Point, and Pair Nodes for the Thermoacoustics, Frequency Domain Interface

The Thermoacoustics, Frequency Domain Interface has these domain, boundary, edge, point, and pair nodes available from the Physics ribbon toolbar (Windows users),

For both options, the velocity components and the temperature share the same element order as they vary similarly over the same length scale in the acoustic boundary layer. Therefore, both require the same spatial accuracy.


• Domain, Boundary, Edge, Point, and Pair Nodes for the Thermoacoustics, Frequency Domain Interface


• Uniform Layer Waveguide: model library path Acoustics_Module/Verification_Models/uniform_layer_waveguide

• Generic 711 Coupler: An Occluded Ear-Canal Simulator: model library path Acoustics_Module/Industrial_Models/generic_711_coupler

As the Thermoacoustic interface solves for both pressure, velocity, and temperature, models can easily become large and contain many dofs. See Solver Suggestions for Large Thermoacoustic Models for suggestions on how to solve large thermoacoustic models


Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

Some of the following nodes are available from the Mechanical and Thermal submenus (listed in alphabetical order):


• Acoustic-Thermoacoustic Boundary

• Adiabatic

• Heat Source

• Initial Values

• Isothermal

• Normal Impedance

• Normal Stress

• No Stress


• Slip

• Sound Hard Wall

• Stress

• Symmetry

• Temperature Variation

• Thermoacoustics Model

• Velocity

• Wall

The Continuity node with this interface is available as a pair boundary condition. This gives continuity in pressure, temperature variation, velocity and in the flux on a pair boundary between thermoacoustic domains. For a pair boundary between pressure acoustic domains see Continuity as described for the Pressure Acoustics, Frequency Domain interface.

For more information about pairs, see Identity and Contact Pairs in the COMSOL Multiphysics Reference Manual.



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Thermoacoustics Model

Use the Thermoacoustics Model node to define the model inputs (the background equilibrium temperature and pressure) and the material properties of the fluid (dynamic viscosity, bulk viscosity, thermal conductivity, heat capacity at constant



• Dipole Source


• Impedance






• Monopole Source




• Pressure




The Periodic Condition is best used by adding one feature for each physics in the model, for example, one feature for Pressure Acoustics and one for Thermoacoustics.



• Destination Selection




pressure, and equilibrium density) necessary to model the propagation of acoustic compressible waves in a thermoacoustic context. Extended inputs are available for the coefficient of thermal expansion and the compressibility, which enables modeling of any constitutive relation for the fluid.




This section contains field variables that appear as model inputs, if the current settings include such model inputs. From the Equilibrium temperature T0 (SI unit: K) list, select an existing temperature variable (from another physics interface) if available, or select User defined to define a different value or expression. The default is User defined and set to 293.15 K (that is 20oC). From the Equilibrium pressure p0 (SI unit: Pa) list, select an existing pressure variable (from another physics interface) if available, or select User

defined to define a different value or expression. The default is User defined and set to 1 atm.

T H E R M O A C O U S T I C S M O D E L

Define the material parameters of the fluid by selecting an Equilibrium density—Ideal

gas (the default), From material, or User defined.

• If Ideal gas is selected, also select the Gas constant type—select Specific gas constant

Rs (SI unit: J/(kg·K) (the default) or Mean molar mass Mn (SI unit: kg/mol). Both take the values From material by default, or select User defined to enter different values. This model

• If From material is selected the equilibrium density, and its dependence on the equilibrium pressure p0 and temperature T0 is, is taken from the defined material.

• If User defined is selected, enter a value or expression for the Equilibrium density 0(p0, T0) (SI unit: kg/m3). The default is ta.p0/(287[J/kg/K]*ta.T0).

The other thermoacoustic model parameters defaults use values From material. If User

defined is selected, enter another value or expression for:

• Dynamic viscosity (SI unit: Pa·s). The default is 0 Pa·s.

• Bulk viscosity B (SI unit: Pa·s). The default is 0 Pa·s.


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• Thermal conductivity k (SI unit: W/(m·K)). The default is 0 W/(m·K).

• Heat capacity at constant pressure Cp (SI unit: J/(kg·K)). The default is 0 J/(kg·K).

T H E R M A L E X P A N S I O N A N D C O M P R E S S I B I L I T Y

One of the main characteristics of an acoustic wave is that it is a compressional wave. In the detailed thermoacoustic description this property is closely related to the constitutive relation between the density, the pressure, and also the temperature. This results in the important (linear) relation for the acoustic density variation:

where is the density variation, p is the acoustic pressure, T is the acoustic temperature variations, T is the (isothermal) compressibility of the fluid, and 0 the coefficient of thermal expansion. If this constitutive relation is not correct then no waves propagate or possibly they propagate at an erroneous speed of sound. When the From equilibrium

density option (the default) is selected for the coefficient of thermal expansion and the compressibility, both values are derived from the equilibrium density 0(p0,T0) using their defining relations

If the equilibrium density 0 is a user-defined constant value or the material model does not define both a pressure and temperature dependence for 0 the coefficient of thermal expansion and the compressibility needs to be set manually, or it evaluates to 0.

The Thermal Expansion and Compressibility section displays if From material or User

defined is selected as the Equilibrium density under Thermoacoustics Model.

0 Tp 0T– =

T10------

0p---------

T= 0

10------–

0T---------

p=

For most materials, selected form the material library, it is necessary to set the coefficient of thermal expansion and the compressibility using one of the non-default option.

If the material is air the From equilibrium density option works well as the equilibrium density 0 = 0(p0,T0) is a function of both pressure and temperature.

For water the coefficient of thermal expansion is well defined as 0 = 0(T0), while the compressibility can easily be defined using the From speed of sound option.


Select an option from the Coefficient of thermal expansion 0 list—From equilibrium

density (the default), From material, or User defined. If User defined is chosen, enter a value for 0 (SI unit: 1/K = K-1). The default is 0 K-1.

Select an option from the Isothermal compressibility T lists—From equilibrium density (the default), From speed of sound, From isentropic compressibility, or User defined.

• If User defined is chosen, enter a value for T (SI unit: 1/Pa). The default is 0 (1/Pa).

• If From speed of sound is chosen, the values for the Speed of sound c (SI unit: m/s) and Ratio of specific heats (dimensionless) are taken From material. Or select User

defined for one or both of the options and enter a different value or expression.

• If From isentropic compressibility is chosen, the values for the Isentropic

compressibility 0 (SI unit: 1/Pa) and Ratio of specific heats (dimensionless) are taken From material. Or select User defined for one or both of the options and enter a different value or expression.

See the Theory Background for the Thermoacoustics Branch section for a detailed description of the governing equations and the constitutive relations.

Visualize the dissipated energy due to viscosity and thermal conduction in post processing. Three post processing variables exist:

• The viscous power dissipation density ta.diss_visc.

• The thermal power dissipation density ta.diss_therm.

• The total thermo-viscous power dissipation density ta.diss_tot.


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Initial Values

The Initial Values node adds initial values for the sound pressure, velocity field, and temperature variation. Add more Initial Values nodes from the Physics toolbar.




Enter values or expressions for the Pressure p (SI unit: Pa) (the default is 0 Pa), Velocity

field u (SI unit: m/s) (the defaults are 0 m/s), and Temperature variation T (SI unit: K) (the default is 0 K).

Heat Source

Use the Heat Source node to define the heat source for the thermoacoustic model. This adds a domain heat source Q to the right-hand side of the energy equation.



In certain cases it can be interesting to not include thermal conduction in the model and treat all processes as adiabatic (isentropic). This is, for example, relevant for fluids where the thermal boundary layer is much thinner than the viscous. Not solving for the temperature field T also saves some degrees of freedom (DOF).

This is achieved by setting the Isothermal compressibility to User defined and here enter the adiabatic value 0 = ·T. Then, in the solver sequence under Solver Configuration > Solver 1 > Dependent Variables select Define

by study step to User defined and under > Temperature variation (mod1.T) click to clear the Solver for this field box.

See also Solver Suggestions for Large Thermoacoustic Models for suggestions on how to set up the solver for large problems.


H E A T S O U R C E

Enter a value for the Heat source Q (SI unit: W/m3). The default is 0 W/m3.

Sound Hard Wall

Use the Sound Hard Wall node to model a physical wall. The boundary condition corresponds to the no-slip condition in the thermoacoustic domain (Wall condition) and the Sound Hard Boundary (Wall) condition in the pressure acoustic domain.



Isothermal

Use the Isothermal node to model a wall that is assumed to be a good thermal conductor and backed by a large heat reservoir kept at constant temperature. This implies that the harmonic temperature variations vanish: T = 0.






Acoustic-Thermoacoustic Boundary

Use the Acoustic-Thermoacoustic Boundary node to couple the thermoacoustic domain to a pressure acoustic domain. As it is only necessary to solve the full thermally conduction and viscous model near walls in the boundary layer region, it makes sense


276 | C H A P T E

to switch to the classical pressure acoustics outside this region. This saves a lot of memory and solution time due to the reduced number of degrees of freedom. The coupling is done normally to the interface:



Pressure (Adiabatic)

Use the Pressure (Adiabatic) node to specify a prescribed pressure pbnd, that acts as a pressure source at the boundary, typically an inlet or outlet. In the frequency domain pbnd is the amplitude of a harmonic pressure source. The adiabatic condition states that no heat flows into or out of the boundary:

This condition is in general not physically correct on a solid wall because solids are generally better thermal conductors than air.



P R E S S U R E

Enter the value of the Pressure pbnd (SI unit: Pa) at the boundary. The default is 0 Pa.

pI– u u T+ 23

------- B– u I–+

n ptn–=

n10------ pt q– –

– n iu–=

p pbnd=

pI– u u T+ 23

------- B– u I–+

n pbndn–=

n– k T– 0=


Wall

Use the Wall node to model a sound-hard wall where the no-slip condition applies. All velocity components are zero and u 0.




See Isothermal for the settings.

Symmetry

The Symmetry node for The Thermoacoustics, Frequency Domain Interface adds a boundary condition that represents symmetry. This corresponds to the Sound Hard Boundary (Wall) condition in pressure acoustics domains. In the thermoacoustic domains it corresponds to the Slip condition for the mechanical degrees of freedom and the Adiabatic condition for the temperature variation.





Velocity

Use the Velocity node to define the prescribed velocities u0 on the boundary: u = u0. This condition is useful, for example, when modeling a vibrating wall.






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VE L O C I T Y



Slip

Use the Slip node to prescribe a no-penetration condition specifying zero normal velocity on the boundary





Stress

Use the Stress node to define the coordinates of the stress vector on the boundary:



To define a prescribed velocity for each space direction (x and y, plus z for 3D), select one or more of the Prescribed in x direction, Prescribed in y

direction, and Prescribed in z direction check boxes. Then enter a value or expression for the prescribed velocities u0, v0, or w0 (SI unit: m/s).

To define a prescribed velocity for each space direction (r and z), select one or both of the Prescribed in r direction and Prescribed in z direction check boxes. Then enter a value or expression for the prescribed velocities u0, or w0 (SI unit: m/s).

n u 0=

pI– u u T+ 23

------- B– u I–+ n =




S T R E S S

Enter the Stress (SI unit: N/m2) coordinates for each space direction (x, y, and z or r and z for 2D axisymmetric models). The defaults are 0 N/m2.

No Stress

Use the No Stress node to set the total surface stress equal to zero:



Normal Stress

Use the Normal Stress node to define the inward normal stress, n, on the boundary:



N O R M A L S T R E S S

Enter a value or expression for the Inward normal stress n (SI unit: N/m2). The default is 0 N/m2.

Normal Impedance

Use the Normal Impedance node to specify a normal impedance Z0 on a boundary. This feature is useful outside the viscous boundary layer, as this condition mimics the behavior of a corresponding Pressure Acoustics Model with a normal impedance condition. The boundary condition reads:

pI– u u T+ 23

------- B– u I–+ n 0=

pI– u u T+ 23

------- B– u I–+ n – nn=


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N O R M A L I M P E D A N C E

Enter a value or expression for the Normal impedance Z0 (SI unit: Pas/m). The default is 0 Pas/m.

Adiabatic

Use the Adiabatic node to define a situation with no heat flow into or out of the boundary:



Temperature Variation

Use the Temperature Variation node to define the temperature variation on the boundary Tbnd. In the frequency domain this is the amplitude of a harmonic temperature variation:



TE M P E R A T U R E V A R I A T I O N

Enter a value or expression for the Temperature variation Tbnd (SI unit: K). The default is 0 K.



pI– u u T+ 23

------- B– u I–+ n Z0 u n n–=

n– k T– 0=

T Tbndeit=


Th e Th e rmoa c ou s t i c - S o l i d I n t e r a c t i o n , F r e qu en c y Doma i n I n t e r f a c e

The Thermoacoustic-Solid Interaction, Frequency Domain (tas) interface ( ), found under the Thermoacoustics branch ( ) when adding a physics interface, combines the Pressure Acoustics, Frequency Domain, Thermoacoustics, Frequency Domain, and Solid Mechanics interfaces. The physics interface solves for and has predefined couplings between the displacement field in the solid and the acoustic variations in the fluid domains.


When this interface is added, these default nodes are also added to the Model Builder— Thermoacoustics Model, Sound Hard Wall, Isothermal, Free, Pressure Acoustics Model, Linear Elastic Material, Continuity, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Thermoacoustic-Solid Interaction, Frequency Domain to select physics from the context menu.

On the Constituent Physics InterfacesThe Pressure Acoustics, Frequency Domain interface is used to compute the pressure variations when modeling the propagation of acoustic waves in fluids at quiescent background conditions. The physics interface solves the Helmholtz equation and is suited for all linear frequency-domain acoustic simulations with harmonic variations of the pressure field. It includes domain conditions to model losses in a homogenized way, so-called fluid models for porous materials as well as losses in narrow regions. The domain features also include background incident acoustic fields.

The Thermoacoustics, Frequency Domain interface is used to compute the acoustic variations of pressure, velocity, and temperature. The physics interface is required to accurately model acoustics in geometries of small dimensions. Near walls, viscosity and thermal conduction become important because it creates a viscous and a thermal boundary layer where losses are significant. For this reason, it is necessary to include thermal conduction effects and viscous losses explicitly in the governing equations.

T H E T H E R M O A C O U S T I C - S O L I D I N T E R A C T I O N , F R E Q U E N C Y D O M A I N I N T E R F A C E | 281

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The Solid Mechanics interface is intended for general structural analysis of 3D, 2D, or axisymmetric bodies. In 2D, plane stress or plane strain assumptions can be used. This interface is based on solving Navier's equations, and results such as displacements, stresses, and strains are computed.

It is available for 3D, 2D, and 2D axisymmetric geometries.



The default identifier (for the first interface in the model) is tas.








Enter the coordinates for the Reference point for moment computation xref (SI unit: m). All moments are then computed relative to this reference point.


This interface defines these dependent variables (fields), the Pressure p, the Velocity

field ufluid and its components, the Temperature variation, T, and the Displacement field

usolid and its components. The name can be changed but the names of fields and dependent variables must be unique within a model.



To display this section, click the Show button ( ) and select Discretization. From the Discretization of fluids list select the element order for the velocity components, the temperature, and the pressure: P2+P1 (the default) or P3+P2.


• P3+P2 means third-order elements for the velocity components and the temperature. The pressure field has third-order elements in the pressure acoustic domains and second-order elements in the thermoacoustic domains. This can add additional accuracy but it also adds additional degrees of freedom compared to P2+P1 elements.

From the Displacement field list select the element order for the displacement field in the solid: Linear, Quadratic (the default), Cubic, Quartic, or (in 2D) Quintic. Specify the Value type when using splitting of complex variables—Real or Complex (the default).

Domain, Boundary, Edge, Point, and Pair Nodes for the Thermoacoustic-Solid Interaction, Frequency Domain Interface

The Thermoacoustic-Solid Interaction, Frequency Domain Interface has these domain, boundary, edge, point, and pair nodes available from the Physics ribbon

For both options, the velocity components and the temperature share the same element order as they vary similarly over the same length scale in the acoustic boundary layer. Therefore, both require the same spatial accuracy.


• Domain, Boundary, Edge, Point, and Pair Nodes for the Thermoacoustic-Solid Interaction, Frequency Domain Interface





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toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).

These nodes are described in this section:

• Initial Values

• Continuity

• Symmetry

T H E R M O A C O U S T I C S , F R E Q U E N C Y D O M A I N M E N U

These nodes are described for the Thermoacoustics, Frequency Domain interface (listed in alphabetical order):


Destination Selection in the COMSOL Multiphysics Reference Manual

• Adiabatic

• Heat Source

• Isothermal


• No Stress

• Normal Stress


• Slip

• Sound Hard Wall

• Stress



• Velocity

• Wall




S O L I D M E C H A N I C S M E N U




• Dipole Source


• Impedance






• Monopole Source




• Pressure





• Added Mass

• Antisymmetry

• Body Load

• Boundary Load

• Edge Load


• Free

• Gravity


• Point Load




• Roller

• Rotating Frame


• Symmetry



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Initial Values

The Initial Values node adds initial values for the sound pressure, velocity field, temperature variation, and displacement field. Add more Initial Values nodes from the Physics toolbar.




Enter values or expressions for the Pressure p (SI unit: Pa), Velocity field ufluid (SI unit: m/s), Temperature variation T (SI unit: K), and Displacement field usolid (SI unit: m). The defaults are 0.

Continuity

Use the Continuity node condition in The Thermoacoustic-Solid Interaction, Frequency Domain Interface to model continuity in a general sense. This boundary condition is automatically added between pressure acoustics and thermoacoustic domains, between pressure acoustic and solid domains, and also between thermoacoustic and solid domains.

The finals condition simply states that the fluid velocity should be equal to the structure velocity at the thermoacoustic-structure interface:


• Acoustic-Structure Boundary

ufluid iusolid=


where usolid is the deformation field (SI unit: m) and ufluid is the fluid velocity (SI unit: m/s). Multiplication by corresponds to a time derivative in frequency domain.

Symmetry

The Symmetry node in The Thermoacoustic-Solid Interaction, Frequency Domain Interface adds a boundary condition that represents symmetry. This corresponds to the Sound Hard Boundary (Wall) condition in pressure acoustics domains. In the thermoacoustic domains it corresponds to the Slip condition for the mechanical degrees of freedom and the Adiabatic condition for the temperature variation. In solid domains is corresponds to the Symmetry condition (described for the Solid Mechanics interface).






i

The Continuity pair condition is a multiphysics feature combining the features from Thermoacoustics, Pressure Acoustics and Solid Mechanics. See Continuity on Interior Boundaries and Identity and Contact Pairs as described in the COMSOL Multiphysics Reference Manual.


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Th e Th e rmoa c ou s t i c - S h e l l I n t e r a c t i o n , F r e qu en c y Doma i n I n t e r f a c e

The Thermoacoustic-Shell Interaction, Frequency Domain (tash) interface ( ), found under the Acoustics>Thermoacoustics branch ( ) when adding a physics interface, combines the Pressure Acoustics, Frequency Domain, Thermoacoustics, Frequency Domain, and Shell interfaces. The physics interface solves for and has a predefined coupling between the displacement field of the shell and the acoustic variations in the fluid domains. It can be used, for example, for modeling the vibrating response of micromirrors in MEMS applications.


Three types of shells are available—exterior shells (on exterior boundaries), interior shells (on interior boundaries), and uncoupled shells (on boundaries that are not connected to any acoustic domain).

When this interface is added, these default nodes are also added to the Model Builder—Thermoacoustics Model, Pressure Acoustics Model, Sound Hard Wall, Isothermal, Acoustic-Thermoacoustic Boundary, Exterior Shell (including a Linear Elastic Material default node), Free, and two Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions and sources. You can also right-click Thermoacoustic-Shell Interaction, Frequency Domain to select physics from the context menu.

This interface requires a Structural Mechanics Module license. For theory and interface feature descriptions relating to the Shell interface, see the Structural Mechanics Module User’s Guide.

The interface is only available for 3D geometries, and it is capable of modeling the coupled thermoacoustics and shell vibrations in the frequency domain.


On the Constituent Physics InterfacesThe Pressure Acoustics, Frequency Domain interface is used to compute the pressure variations when modeling the propagation of acoustic waves in fluids at quiescent background conditions. The physics interface solves the Helmholtz equation and is suited for all linear frequency-domain acoustic simulations with harmonic variations of the pressure field. It includes domain conditions to model losses in a homogenized way, so-called fluid models for porous materials as well as losses in narrow regions. The domain features also include background incident acoustic fields.

The Thermoacoustics, Frequency Domain interface is used to compute the acoustic variations of pressure, velocity, and temperature. The physics interface is required to accurately model acoustics in geometries of small dimensions. Near walls, viscosity and thermal conduction become important because it creates a viscous and a thermal boundary layer where losses are significant. For this reason, it is necessary to include thermal conduction effects and viscous losses explicitly in the governing equations.

The Shell interface is used to model structural shells on 3D boundaries. Shells are thin flat or curved structures, having significant bending stiffness. The physics interface uses shell elements of the MITC type, which can be used for analyzing both thin (Kirchhoff theory) and thick (Mindlin theory) shells. Geometric nonlinearity can be taken into account. The material is assumed to be linearly elastic.



The default identifier (for the first interface in the model) is tash.





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T H I C K N E S S

Enter a value for the thickness d (SI unit: m). The default is 0.01 m. Select an Offset

definition—No offset (the default), Relative offset, or Physical offset.

• If Relative offset is selected, enter a value for zrel_offset (dimensionless). The default is 0.

• If Physical offset is selected, enter a value for zoffset (SI unit: m). The default is 0 m.


Enter the coordinates for the Reference point for moment computation xref (SI unit: m). All moments are then computed relative to this reference point.

F O L D - L I N E L I M I T A N G L E

Enter a value for (SI unit: radians). The default is 0.05 radians.

H E I G H T O F E V A L U A T I O N I N S H E L L , [ - 1 , 1 ]

Enter a value for z(dimensionless). The default is 1.


This interface defines these dependent variables (fields), the Pressure p, Velocity field

ufluid (and its components), the Temperature variation T, Displacement field, ushell (and its components), and the Displacement of shell normals ar (and the components). The names can be changed but the names of fields and dependent variables must be unique within a model.


To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed. The Use MITC interpolation check box is selected by default, and this interpolation, which is part of the MITC shell formulation, should normally always be active.


To display this section, click the Show button ( ) and select Discretization.

From the Discretization of fluids list select the element order for the velocity components, the temperature, and the pressure: P2+P1 (the default) or P3+P2.



• P3+P2 means third-order elements for the velocity components and the temperature. The pressure field has third-order elements in the pressure acoustic domains and second-order elements in the thermoacoustic domains. This can add additional accuracy but it also adds additional degrees of freedom compared to P2+P1 elements.

From the Displacement field list select the element order for the displacement field of the shell: Linear or Quadratic (the default). Specify the Value type when using splitting

of complex variables—Real or Complex (the default).

Domain, Boundary, Edge, Point, and Pair Nodes for the Thermoacoustic-Shell Interaction, Frequency Domain Interface

The Thermoacoustic-Shell Interaction, Frequency Domain Interface has these domain, boundary, edge, point, and pair nodes available from the Physics ribbon

For both options, the velocity components and the temperature share the same element order as they vary similarly over the same length scale in the acoustic boundary layer. Therefore, both require the same amount spatial accuracy.


• Domain, Boundary, Edge, Point, and Pair Nodes for the Thermoacoustic-Shell Interaction, Frequency Domain Interface



• Theory for Shell and Plate Interfaces in the Structural Mechanics Module User’s Guide

The links to the nodes described in the Structural Mechanics Module User’s Guide do not work in the PDF, only from the on line help in COMSOL Multiphysics.


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The Continuity node with this interface is available as a pair boundary condition. This gives continuity in pressure, temperature variation, velocity and in the flux on a pair boundary between thermoacoustic domains. For a pair boundary between pressure acoustic domains see Continuity as described for the Pressure Acoustics, Frequency Domain interface.

For more information, see Identity and Contact Pairs in the COMSOL Multiphysics Reference Manual.

• Exterior Shell

• Initial Values

• Initial Values (Boundary)

• Interior Shell

• Uncoupled Shell




T H E R M O A C O U S T I C S , F R E Q U E N C Y D O M A I N M E N U

These nodes are described for the Thermoacoustics, Frequency Domain interface (listed in alphabetical order):



• Dipole Source


• Impedance







• Monopole Source




• Pressure






• Adiabatic

• Heat Source

• Isothermal


• Normal Stress

• No Stress


• Slip

• Sound Hard Wall

• Stress

• Symmetry



• Wall


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S H E L L M E N U

These nodes are described for the Shell interface.

These nodes are described for the Solid Mechanics interface and described in this guide:

Initial Values

The Initial Values node adds initial values for the pressure, velocity field (and its components), and the temperature variation. Add more Initial Values nodes from the Physics toolbar.

Both the Shell interface and its nodes are described in the Structural Mechanics Module User’s Guide as this interface requires the Structural Mechanics Module. For that reason, these links do not work in the PDF.

• Antisymmetry

• Body Load


• Damping

• Edge Load

• Face Load

• Initial Stress and Strain


• No Rotation

• Phase

• Pinned


• Prescribed Displacement/Rotation


• Point Load

• Rigid Connector

• Symmetry

• Thermal Expansion

Applied Force (Rigid Connector), Applied Moment (Rigid Connector), and Mass and Moment of Inertia (Rigid Connector) are described for the Solid Mechanics interface in the Structural Mechanics Module User’s Guide.

• Added Mass


• Free

• Pre-Deformation






Enter a value or expression for the Pressure p (SI unit: Pa), Velocity field ufluid (and its components) (SI unit: m/s), the Temperature variation T (SI unit: K).

Initial Values (Boundary)

The Initial Values node adds initial values for the displacement field (and its components) and the displacement of shell normals (and the components).


The default setting is to include All boundaries in the model. To choose specific boundaries, select Manual from the Selection list.


Based on space dimension, enter coordinate values for the Displacement field ushell (SI unit: m) and Displacement of shell normals ar (dimensionless).

Exterior Shell

Use the Exterior Shell boundary condition to model any deformable shell boundary, only one side of which is adjacent to an acoustic domain. See Exterior Shell for the Acoustic-Shell Interaction, Frequency Domain interface for a description of a shell exterior to a pressure acoustic domain.

The condition for a shell exterior to a thermoacoustic domain simply states that the fluid velocity should be equal to the structure velocity at the thermoacoustic-shell interface:

ufluid iushell=


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where ushell is the deformation field (SI unit: m) and ufluid is the fluid velocity (SI unit: m/s). Multiplication by corresponds to a time derivative in the frequency domain.


From the Selection list, choose Manual to select the boundaries to define or select All

boundaries.

Interior Shell

Use the Interior Shell boundary condition to model any deformable shell with both sides adjacent to the acoustic domains. See Interior Shell for the Acoustic-Shell Interaction, Frequency Domain interface for a description of a shell between two a pressure acoustic domains.

The condition for a shell between two thermoacoustic domains simply states that the fluid velocity should be equal to the structure velocity at the thermoacoustic-shell interface:

where ushell is the deformation field (SI unit: m) and ufluid is the fluid velocity (SI unit: m/s). Multiplication by corresponds to a time derivative in the frequency domain.

When the shell is positioned between a pressure acoustic domain and a thermoacoustic domain, the same constraint is set on the thermoacoustic side. On the pressure acoustic side of the shell, the normal acceleration for the acoustic pressure on the boundary equals the acceleration based on the second time derivative of the shell displacement

In addition, the pressure load (force per unit area) on the shell is: Fpn p,

where p is the acoustic pressure in the pressure acoustics domain.

i

A Linear Elastic Material node is automatically added to this boundary condition. Right-click Exterior Shell to add more subnodes if required.

ufluid iushell=

i

n–10------– p q–

n u tt=




Uncoupled Shell

Use the Uncoupled Shell boundary condition to model deformable shells that are not adjacent to the acoustic domains.



A Linear Elastic Material node is automatically added to this boundary condition. Right-click Interior Shell to add more subnodes if required.

A Linear Elastic Material node is automatically added to this boundary condition. Right-click Uncoupled Shell to add more subnodes if required.


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Th eo r y Ba c k g r ound f o r t h e Th e rmoa c ou s t i c s B r an c h

The Thermoacoustics, Frequency Domain Interface is designed for the analysis of acoustics in viscous and thermally conducting, compressible Newtonian fluids. The interface solves the linearized Navier-Stokes equation, the continuity equation, and the energy equation. This corresponds to a small parameter expansion of the dependent variables. The interface solves for the acoustic pressure variations p, the fluid velocity variations u, and the acoustic temperature variations T.

The Thermoacoustic interfaces are available for 3D, 2D, and 1D Cartesian geometries as well as for 2D and 1D axisymmetric geometries. The interface solves problems in the frequency domain, that is, Frequency Domain, Frequency-Domain Modal, and Eigenfrequency type analysis. In 2D and 1D axisymmetric systems a Mode Analysis study is also available.

In this section:

• The Viscous and Thermal Boundary Layers

• General Linearized Compressible Flow Equations

• Formulation for Eigenfrequency Studies

• Formulation for Mode Analysis

• Solver Suggestions for Large Thermoacoustic Models

• References for the Thermoacoustics, Frequency Domain Interface

The Viscous and Thermal Boundary Layers

In general, a tangential harmonic oscillation of amplitude u0 and frequency f applied to a wall at z 0 creates a viscous wave of the form

This theory also applies to The Thermoacoustic-Solid Interaction, Frequency Domain Interface and The Thermoacoustic-Shell Interaction, Frequency Domain Interface, although the Mode Analysis study is not available for these interfaces.


where, f is the frequency, 0 is the static density, and is the dynamic viscosity. The viscous shear waves are therefore dispersive with wavelength

and highly damped since their amplitude decays exponentially with distance from the boundary (see Ref. 3). In fact, in just one wavelength, the amplitude decreases to about 1500 of its value at the boundary. Therefore, the viscous boundary layer thickness can for most purposes be considered to be less than Lv. The length scale v is the so-called viscous penetration depth or viscous boundary layer thickness.

Similarly, a harmonically oscillating temperature with amplitude T0 and frequency f at z 0 creates a thermal wave of the form

where Cp is the heat capacity at constant pressure and k is the thermal conductivity. The wavelength is here

and a decay behavior similar to the viscous waves. The length scale t is here the thermal penetration depth.

The ratio of viscous wavelength to thermal wavelength is a non-dimensional number related to the Prandtl number Pr, as

In air, this ratio is roughly 0.8, while in water, it is closer to 2.7. Thus, at least in these important cases, the viscous and thermal boundary layers are of the same order of magnitude. Therefore, if one effect is important for a particular geometry, so is probably the other.

u z u0ef0

----------- 1 i+ z–

=

Lv 2 0f------------ 2v= =

T z T0e

f0Cp

k------------------ 1 i+ z–

=

Lt 2 k0fCp------------------- 2t= =

LvLt------

Cp

k----------- Pr= =

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General Linearized Compressible Flow Equations

In general, the motion of a viscous compressible Newtonian fluid, including the energy equation, is governed by the set of equations listed in Equation 6-1:

(6-1)

where the dependent variables are pressure p, velocity u, temperature T, and density . The first equation is the momentum equation (the Navier-Stokes equation), the second is the continuity equation, and the third is the energy equation formulated using the Fourier heat law. The last equation defines the total stress tensor and the viscous stress tensor - this is a constitutive relation. See, for example, Ref. 1 to 6 for further details. The material time derivatives are in the following expanded according to

where A is a dummy variable.

To close the system above, an equation of state must also be added to the ones displayed. The equation of state relates local values of pressure, density and temperature and is therefore an algebraic equation or an ODE, rather than a PDE. A common form of state equations is to know the density as function of pressure and temperature, p, TIn the following, it is assumed that the state equation has this form.

The basic properties of the fluid are the dynamic viscosity and thermal conductivity k. The coefficient B is the bulk (or second) viscosity and describes losses due to compressibility (expansion and contraction of the fluid), where describes losses due to shear friction. The bulk viscosity can in some cases be used to model an empirically observed deviation from Stokes’ assumption but is usually negligible compared to unless the motion is really irrotational, see Ref. 3 and Ref. 4. These three properties are taken to be constant or at most weakly temperature-dependent. The specific heat

td

du F+=

tdd u + 0=

Cp tddT 0T

tddp

– k– T – B u Q+ +=

pI– + pI– u u T+ 23

------- B– u I–+= =

d dt

dA x t dt

--------------------- At------- u A+=


at constant pressure Cp and the (isobaric) coefficient of volumetric thermal expansion 0

are both possibly functions of pressure and temperature.

In the energy equation

is the viscous dissipation function, that is, the scalar contraction of the viscous stress tensor with the rate of train tensor S. Both tensors are seen as functions of a velocity vector. If the mean velocity is zero, this term vanishes in the following linearization since it is homogenous of second order in the velocity gradients. Otherwise, it acts as an oscillating source/sink. In the right-hand sides of Equation 6-1, F and Q are a volume force and a heat source, respectively.

For small harmonic oscillations about a steady state solution, the dependent variables and sources can be assumed to take on the following form:

Assuming zero mean flow u0 = 0 and after inserting into the governing Equation 6-1, the steady state equations can be subtracted from the system, which is subsequently linearized to first order by ignoring terms quadratic in the primed variables. Dropping the primes for readability yields the thermoacoustic equations:

(6-2)

01---

T

–p

=

u :S u =

u u0 u'eit+=

p p0 p'eit+=

T T0 T'eit+=

0 'eit+=

F F0 F'eit+=

Q Q0 Q'eit+=

i0u pI– u u T+ 23

------- B– u I–+

F+=

i 0 u + 0=

i0CpT k– T – ipT00 Q+ +=


302 | C H A P T E

where the unprimed variables are now the acoustic deviation from the steady state. The density is expressed in terms of the pressure and the temperature using a first order Taylor expansion about the steady quiescent values

The two thermodynamic quantities (the coefficients terms in square brackets) define the isobaric coefficient of thermal expansion 0 and the isothermal compressibility T, according to the following relations

(6-3)

where K0 is the isentropic bulk modulus, KT the isothermal bulk modulus, Cv is the heat capacity at constant volume, c is the speed of sound, and is the ratio of specific heats (the adiabatic index). The isothermal compressibility T is related to the isentropic (or adiabatic) compressibility 0 and the coefficient of thermal expansion 0 via the thermodynamic relations

(6-4)

it is derived using the Maxwell relations, see, for example, Ref. 5 and Ref. 6.

Using the state equation, both the equilibrium density and the density variation can be eliminated from the system of equations. Moreover, the volume fore F is set equal to zero. For clarity, the equilibrium density is retained, though:

(6-5)

p 10------

0p---------

TT 10------

0T---------

p+=

T10------

0p---------

T

1KT------- 1

0------

c2----- 0= = = =

CpCv-------

K0KT-------= =

010------–

0T---------

p=

0 T0

2T00Cp-------------–= T 0=

i0u pI– u u T+ 23

------- B– u I–+

=

i 0 u + 0=

i0CpT k– T – ipT00 Q+ +=

0 Tp 0T– =


This set of equations describes the behavior of a general compressible fluid under small harmonic oscillations. This is the system of equations implemented in the Thermoacoustic interface.

I D E A L G A S

For an ideal gas, the equation of state p RT, where R is the specific gas constant, leads to

and the density

Inserting these expressions and dividing the continuity equation by the reference density, the system of equations takes on the following simplified form:

This is the system of equations implemented in the Thermoacoustic, Frequency Domain interface when ideal gas law is selected.

I S E N T R O P I C ( A D I A B A T I C ) C A S E

If the process is assumed to be adiabatic and reversible, that is isentropic, the thermal conductivity is effectively zero. Then also the temperature can be eliminated, giving for an ideal gas:

(6-6)

Defining the speed of sound c in analogy with the standard assumptions for linear acoustics (term in front of the pressure in the continuity equation), it is found that

T1p0------= 0

1T0------=

0pp0------ T

T0------–

=

i0u pI– u u T+ 23

------- B– u I–+

=

i pp0------ T

T0------–

u+ 0=

i0CpT k– T – ip Q+ +=

i0u pI– u u T+ 23

------- B– u I–+

=

i 1p0------ 1

0CpT0--------------------–

p u+ 0=


304 | C H A P T E

or

In the case with a general fluid, the corresponding relation is using Equation 6-3 and Equation 6-4:

where K0 is the adiabatic bulk modulus, KTthe isothermal bulk modulus, and 0the coefficient of thermal expansion.

I S O T H E R M A L C A S E

If, on the other hand, the thermal conductivity is high, or the thermoacoustic waves propagate in a narrow space between highly conductive walls, the temperature can be assumed to be constant (isothermal assumption) and the system of equations for an ideal gas becomes:

which, again comparing to standard assumptions, gives

or equivalently

Therefore, thermal conductivity and/or conducting walls decrease the apparent speed of sound in narrow domains.

10c2------------ 1

p0------ 1

0CpT0--------------------–

1p0------ 1 R

Cp-------–

1p0---------= = =

cp00---------=

10c2------------ 1

K0------- 1

0------

0p---------

T

T00Cp------------- 1

0------

0T---------

p

2–

1KT-------

T002

0Cp-------------–= = =

i0u pI– u u T+ 23

------- B– u I–+

=

i 1p0------ p u+ 0=

10c2------------ 1

p0------=

cp00------=


T H E H E L M H O L T Z E Q U A T I O N

If the thermodynamic processes in the system are assumed to be adiabatic and viscosity can be neglected Equation 6-6 reduces to

Now, taking the divergence of the momentum equations and inserting the expression for the divergence of the velocity, taken from the continuity equations, yields the Helmholtz equation

Formulation for Eigenfrequency Studies

When performing an eigenfrequency study the governing equations (Equation 6-5) are on the form:

(6-7)

where the eigenvalue is = i. It is important to note that there is a difference between regular pressure acoustics and thermoacoustics in terms of what modes can exist and which modes are found during an eigenfrequency study. In pressure acoustics only the pure acoustic modes exist, here the equations and assumptions made ensure this. In thermoacoustics on the other hand the equations are formulated for all small signal components that can exist. This means that other non-acoustic modes also exist they are thermal and vorticity modes.

VO R T I C I T Y A N D T H E R M A L M O D E S

When solving an eigenfrequency problem in thermoacoustics it is important to take a close look at the obtained eigenfrequencies and assess if they are acoustic or not. The nature of the solution is of the form

i0u pI– p= =

i 10c2------------p u+ 0=

2p 2

c2------p+ 0=

0– u pI– u u T+ 23

------- B– u I–+

=

– 0 u + 0=

0CpT pT00– – k– T – Q+=

0 Tp 0T– =


306 | C H A P T E

where is the eigenvalue. Typically, eigenvalues exist near the positive real axis, where . These are exponentially decaying non-acoustic (non-oscillating) modes that

stem from the thermal equation or the deviatoric part of the momentum equation (the non-pressure and non-volume part of the stress tensor) also called the vorticity modes. The acoustic eigenvalues on the other hand lie close to the imaginary axis and are oscillating and slightly damped.

O T H E R S P U R I O U S M O D E S

Note that other spurious and non-acoustic modes can also exist when for example a PML layer is used to model an open boundary. These modes stem from non-physical phenomena and the scaling inside the PML layer. In all cases it is a good idea to have an a priori knowledge of the location/type of the eigenvalues, maybe from solving an lossless pressure acoustics model, and also to look at the modes in terms of, for example, the pressure field.

Formulation for Mode Analysis

The Mode Analysis study type is available for thermoacoustics in 2D and 1D axisymmetric models. This type of study is used to determine the form of the propagating acoustic modes in waveguide structures. The analyzed 2D and 1D axisymmetric geometries can be thought of as the cross sections of a waveguide. The spatial dependency in the (out-of-plane) axial z-direction along the waveguide is assumed to be of the form of a traveling wave with wave number kz. The dependent variables in 2D are rewritten as

and in 1D axisymmetric as

p x t p x e t– p x eit= i+= – i+=

0

p p x y e ikzz–=

u u x y eikzz–

=

T T x y e ikzz–=

p p r eikzz–

=

u u r eikzz–

=

T T r e ikzz–=


Using this form of the dependent variables, differentiation with respect to z reduces to a multiplication with . The propagating modes are determined by solving an eigenvalue problem in the variable .

The expression for the pressure can now be written retaining the harmonic time dependency, as

(6-8)

where x is the in plane coordinate(s). The axial wave number is split into a real and an imaginary part. The imaginary part of the wave number describes how fast the propagating modes decay along the waveguide, it is often referred to as the attenuation coefficient. The real part is related to the phase speed cph of the propagating mode by cph = /. In thermoacoustics the obtained wave numbers always have an imaginary part as the modeled system always includes losses. The relation between the angular frequency and the axial wave number kz is called the dispersion relation.

Solver Suggestions for Large Thermoacoustic Models

Solving thermoacoustic problems can easily involve solving for many degrees of freedom (dofs) as the model solves for the acoustic variations in both pressure, velocity field (2 or 3 components), and temperature. First of all, it is important to restrict the use of the thermoacoustic model to domains and regions of the model where it is necessary. Secondly, care should be taken when meshing the computational domain. If these two things have been carefully considered, the solver can be changed from its default Fully Coupled setting to use a Segregated Solver or even use an iterative solver.

S E G R E G A T E D D I R E C T S O L V E R

The temperature is only loosely coupled to the pressure and velocity field. This enables the use of a segregated solver that iteratively solves the pressure and velocity in one step and then the temperature in another step. This approach will save some dofs and allow solution of a larger model with the direct solver.

ikz–

ikz–=

kz i+=

p p x eikzz–

eit p x ezei t z– = =

• Mode Analysis Study

• Mode Analysis in the COMSOL Multiphysics Reference Manual


308 | C H A P T E

To set this solver up, first right click on the relevant study node (for example, Study 1) and select Show Default Solver. Expand the Solver nodes and go to Stationary Solver 1, right click and select Segregated. Now right click on the Segregated 1 node and add an extra Segregated Step. In the first of the two segregated step nodes, remove the Temperature variation (comp1.T) and add it to the second segregated step.

Note, that if you solve a coupled thermoacoustic and structure model, then the dependent variables for the displacement field must be in the same group as the velocity and pressure.

I T E R A T I V E S O L V E R W I T H D O M A I N D E C O M P O S I T I O N

A more advanced approach is to again use the segregated solver but now add an iterative solver to handle the velocity and pressure degrees of freedom. The best method here is to use the Domain Decomposition solver with, for example, the following settings:

Solver: Multiplicative Schwarz, Number of iterations: 1, Number of subdomains: 2, Maximum number of DOFs per subdomain: 10000, Maximum number of nodes per

subdomain: 1, Additional overlap: 2, Overlap method: Matrix based, Hierarchy generation

method: Coarse mesh and lower order, and Mesh coarsening factor: 8. The coarse and domain solvers should be the PARDISO solver.

This type of approach should allow you to solve large pure thermoacoustic models using a minimum of RAM. Possibly increase the value of the Maximum number of DOFs

per subdomain option to use the amount of RAM at your disposition.

S O L V I N G I N T H E A D I A B A T I C C A S E

In certain cases, it can be interesting not to include thermal conduction in the model and treat all processes as adiabatic (isentropic). This is, for example, relevant for fluids where the thermal boundary layer is much thinner than the viscous. Not solving for the temperature field T also saves some degrees of freedom (dofs).

This is achieved by setting the Isothermal compressibility to User defined and here enter the adiabatic value 0 = ·T. Then, in the solver sequence under Solver Configuration

> Solver 1 > Dependent Variables select Define by study step to User defined and under > Temperature variation (mod1.T) click to clear the Solver for this field box.

References for the Thermoacoustics, Frequency Domain Interface

1. W.M. Beltman, P.J.M. van der Hoogt, R.M.E.J. Spiering, and H. Tijdeman, “Implementation and Experimental Validation of a New Viscothermal Acoustic Finite


Element for Acousto-Elastic Problems”, J. Sound and Vibration, vol. 216, pp. 159–185, 1998.

2. M. Malinen, M. Lyly, et al., “A Finite Element Method for the Modeling of Thermo-Viscous Effects in Acoustics”, Proc. ECCOMAS 2004, Jyväskylä, 2004.


4. H. Bruus, Theoretical Microfluidics, Oxford University Press, 2010.

5. G.K. Bachelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000.

6. B. Lautrup, Physics of Continuous Matter, Exotic and Every Day Phenomena in the Macroscopic World, 2nd ed., CRC Press, 2011.


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7

T h e S t r u c t u r a l M e c h a n i c s B r a n c h

This chapter describes the enhanced Solid Mechanics interface, which is found under the Structural Mechanics branch ( ) when adding a physics interface and included with the Acoustics Module. This version of the interface simplifies multiphysics modeling for the acoustics-structure interaction.

In this chapter:

• The Solid Mechanics Interface


• Calculating Reaction Forces

• Geometric Nonlinearity, Frames, and the ALE Method

• Springs and Dampers

• Damping and Loss

See also The Piezoelectric Devices Interface described in another chapter.

307

308 | C H A P T E

Th e S o l i d Me chan i c s I n t e r f a c e

The Solid Mechanics (solid) interface ( ), found under the Structural Mechanics branch ( ) when adding a physics interface, is intended for general structural analysis of 3D, 2D, or axisymmetric bodies. In 2D, plane stress or plane strain assumptions can be used. The Solid Mechanics interface is based on solving Navier’s equations, and results such as displacements, stresses, and strains are computed.

With the Nonlinear Structural Materials Module or the Geomechanics Module, the interface is extended with, for example, material models for plasticity, hyperelasticity, creep, and concrete.

The Linear Elastic Material is the default material, which adds a linear elastic equation for the displacements and has a settings window to define the elastic material properties.


Builder— Linear Elastic Material, Free (a boundary condition where boundaries are free, with no loads or constraints), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, solid mechanics material models, boundary conditions, and loads. You can also right-click Solid Mechanics to select physics from the context menu.



The default identifier (for the first interface in the model) is solid.


The default setting is to include All domains in the model to define the displacements and the equations that describe the solid mechanics. To choose specific domains, select Manual from the Selection list.

R 7 : T H E S T R U C T U R A L M E C H A N I C S B R A N C H

2 D A P P R O X I M A T I O N

T H I C K N E S S

S T R U C T U R A L TR A N S I E N T B E H A V I O R

From the Structural transient behavior list, select Include inertial terms (the default) or Quasi-static. Use Quasi-static to treat the elastic behavior as quasi-static (with no mass effects; that is, no second-order time derivatives). Selecting this option gives a more efficient solution for problems where the variation in time is slow when compared to the natural frequencies of the system. The default solver for the time stepping is changed from Generalized alpha to BDF when Quasi-static is selected.


Enter the coordinates for the Reference point for moment computation xref (SI unit: m; variable refpnt). The resulting moments (applied or as reactions) are then computed relative to this reference point. During the results and analysis stage, the coordinates can be changed in the Parameters section in the result nodes.


The typical wave speed cref is a parameter for the perfectly matched layers (PMLs) if used in a solid wave propagation model. The default value is solid.cp, the

From the 2D approximation list select Plane stress or Plane strain (the default). For more information see the theory section.

When modeling using plane stress, the Solid Mechanics interface solves for the out-of-plane strain displacement derivative, , in addition to the displacement field u.

wZ

-------

For 2D models, enter a value or expression for the Thickness d (SI unit: m). The default value of 1 m is suitable for plane strain models, where it represents a a unit-depth slice, for example. For plane stress models, enter the actual thickness, which should be small compared to the size of the plate for the plane stress assumption to be valid.

Use a Change Thickness node to change thickness in parts of the geometry if necessary.

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pressure-wave speed. To use another wave speed, enter a value or expression in the Typical wave speed for perfectly matched layers field.


The interface uses the global spatial components of the Displacement field u as dependent variables. You can change both the field name and the individual component names. If a new field name coincides with the name of another displacement field, the two fields (and the interfaces which define them) share degrees of freedom and dependent variable component names. You can use this behavior to connect a Solid Mechanics interface to a Shell directly attached to the boundaries of the solid domain, or to another Solid Mechanics interface sharing a common boundary.

A new field name must not coincide with the name of a field of another type, or with a component name belonging to some other field. Component names must be unique within a model except when two interfaces share a common field name.


To display this section, click the Show button ( ) and select Discretization. Select a Displacement field—Linear, Quadratic (the default), Cubic, Quartic, or Quintic. Specify the Value type when using splitting of complex variables—Real or Complex (the default). The Frame type in the Solid Mechanics interface is always Material.

Domain, Boundary, Edge, Point, and Pair Nodes for Solid Mechanics

The Solid Mechanics Interface has these domain, boundary, edge, point, and pair nodes and subnodes, listed in alphabetical order, are available from the Physics ribbon


• Domain, Boundary, Edge, Point, and Pair Nodes for Solid Mechanics


• Stresses in a Pulley: model library path COMSOL_Multiphysics/

Structural_Mechanics/stresses_in_pulley

• Eigenvalue Analysis of a Crankshaft: model library path COMSOL_Multiphysics/Structural_Mechanics/crankshaft



Note that you can add force loads acting on all levels of the geometry for the physics interface. Add a:

• Body Load to domains (to model gravity effects, for example).

• Boundary Load to boundaries (a pressure acting on a boundary, for example).

• Edge Load to edges in 3D (a force distributed along an edge, for example).

• Point Load to points (concentrated forces at points).

In general, to add a node, go to the Physics toolbar, no matter what operating system you are using. However, to add subnodes, right-click the parent node.

If there are subsequent constraints specified on the same geometrical entity, the last one takes precedence.

For information about the Perfectly Matched Layers feature, see Infinite Element Domains and Perfectly Matched Layers in the COMSOL Multiphysics Reference Manual.

For 2D axisymmetric models, COMSOL Multiphysics takes the axial symmetry boundaries (at r = 0) into account and automatically adds an Axial Symmetry node to the model that is valid on the axial symmetry boundaries only.


• Harmonic Perturbation, Prestressed Analysis, and Small-Signal Analysis




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Linear Elastic Material

The Linear Elastic Material node adds the equations for a linear elastic solid and an interface for defining the elastic material properties. Right-click to add a Damping subnode.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically set up and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains to define a linear elastic solid and compute the displacements, stresses, and strains, or select All domains as required.

• Added Mass

• Antisymmetry

• Body Load

• Boundary Load


• Damping

• Edge Load


• Free

• Gravity


• Initial Values


• Low-Reflecting Boundary


• Phase

• Point Load

• Pre-Deformation




• Roller

• Rotating Frame


• Symmetry


Also right-click to add an Initial Stress and Strain subnode.



Define model inputs, for example, the temperature field of the material uses a temperature-dependent material property. If no model inputs are required, this section is empty.


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes (except boundary coordinate systems). The coordinate system is used for interpreting directions of orthotropic and anisotropic material data and when stresses or strains are presented in a local system.

L I N E A R E L A S T I C M A T E R I A L

Define the Solid model and the linear elastic material properties.

Solid ModelSelect a linear elastic Solid model—Isotropic (the default), Orthotropic, or Anisotropic. Select:

• Isotropic for a linear elastic material that has the same properties in all directions.

• Orthotropic for a linear elastic material that has different material properties in orthogonal directions, so that its stiffness depends on the properties Ei, ij, and Gij.

• Anisotropic for a linear elastic material that has different material properties in different directions, and the stiffness comes from the symmetric elasticity matrix, D.

DensityThe default Density (SI unit: kg/m3) uses values From material. If User defined is selected, enter another value or expression.

Specification of Elastic Properties for Isotropic MaterialsFor an Isotropic Solid model, from the Specify list select a pair of elastic properties for an isotropic material—Young’s modulus and Poisson’s ratio, Young’s modulus and shear

modulus, Bulk modulus and shear modulus, Lamé parameters, or Pressure-wave and


• Orthotropic Material

• Anisotropic Material


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shear-wave speeds. For each pair of properties, select from the applicable list to use the value From material or enter a User defined value or expression.

The individual property parameters are:

• Young’s modulus (elastic modulus) E (SI unit: Pa). The default is 0 Pa.

• Poisson’s ratio (dimensionless). The default is 0.

• Shear modulus G (SI unit: N/m2. The default is 0 N/m2.

• Bulk modulus K (SI unit: N/m2). The default is 0 N/m2.

• Lamé parameter (SI unit: N/m2) and Lamé parameter (SI unit: N/m2). The defaults are 0 N/m2.

• Pressure-wave speed (longitudinal wave speed) cp (SI unit: m/s). The default is 0 m/s.

• Shear-wave speed (transverse wave speed) cs (SI unit: m/s). The default is 0 m/s.

Specification of Elastic Properties for Orthotropic MaterialsWhen Orthotropic is selected from the Solid model list, the material properties vary in orthogonal directions only. The Material data ordering can be specified in either Standard or Voigt notation. When User defined is selected in 3D, enter three values in the fields for Young’s modulus E, Poisson’s ratio , and the Shear modulus G. This defines the relationship between engineering shear strain and shear stress. It is applicable only to an orthotropic material and follows the equation

Each of these pairs define the elastic properties and it is possible to convert from one set of properties to another (see Table 7-6).

This is the wave speed for a solid continuum. In plane stress, for example, the actual speed with which a longitudinal wave travels is lower than the value given.

ijijGij--------=

ij is defined differently depending on the application field. It is easy to transform among definitions, but check which one the material uses.


Specification of Elastic Properties for Anisotropic MaterialsWhen Anisotropic is selected from the Solid model list, the material properties vary in all directions, and the stiffness comes from the symmetric Elasticity matrix, D (SI unit: Pa). The Material data ordering can be specified in either Standard or Voigt notation. When User defined is selected, a 6-by-6 symmetric matrix is displayed.

G E O M E T R I C N O N L I N E A R I T Y

In this section there is always one check box, either Force linear strains or, if you have opened an older model, Include geometric nonlinearity.

Force Linear StrainsIf a study step is geometrically nonlinear, the default behavior is to use a large strain formulation in all domains. There are, however, some cases when the use of a small strain formulation for a certain domain is needed. In those cases, select the Force linear

strains check box. When selected, a small strain formulation is always used, independently of the setting in the study step. The check box is not selected by default to conserve the properties of the model.

Change Thickness

The Include geometric nonlinearity check box is displayed in this section only if the model was created in a version prior to 4.2a, and geometric nonlinearity was originally used for the selected domains. Once the check box is cleared on this settings window, it is permanently removed and the study step assumes control over the selection of geometric nonlinearity.

When selected on this window, it automatically selected the Include

geometric nonlinearity check box on the study settings window. This check box is still available.

• Geometric Nonlinearity for the Piezoelectric Devices Interface

• Studies and Solvers in the COMSOL Multiphysics Reference Manual

The Change Thickness node is available in 2D.


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Use the Change Thickness node to model domains with a thickness other than the overall thickness defined in the physics interface’s Thickness section.


From the Selection list, choose the domains to use a different thickness.

C H A N G E T H I C K N E S S

Enter a value for the Thickness d (SI unit: m). This value replaces the overall thickness for the domains selected above.

Damping

Right-click the Linear Elastic Material node to add a Damping subnode, which is used in Time Dependent, Eigenfrequency, and Frequency Domain studies to model damped problems. The node adds Rayleigh damping by default.


For a default node, the setting inherits the selection from the parent node. Or select Manual from the Selection list to choose specific domains or select All domains as required.

D A M P I N G S E T T I N G S

Select a Damping type—Rayleigh damping (the default), Isotropic loss factor, or Anisotropic loss factor. If Orthotropic is selected as the Linear Elastic Material Solid

model, Orthotropic loss factor is also available.

Rayleigh DampingEnter the Mass damping parameter dM (SI unit: 1/s) and the Stiffness damping

parameter dK (SI unit: s). The default values are 0 (no damping).

In this damping model, the damping parameter is expressed in terms of the mass m and the stiffness k as

The time-stepping algorithms also add numerical damping, which is independent of any explicit damping added.

For the generalized alpha time-stepping algorithm it is possible to control the amount of numerical damping added.

dMm dKk+=


That is, Rayleigh damping is proportional to a linear combination of the stiffness and mass; there is no direct physical interpretation of the mass damping parameter dM and the stiffness damping parameter dM.

About Loss Factor DampingThe loss factor is a measure of the inherent damping in a material when it is dynamically loaded. It is typically defined as the ratio of energy dissipated in unit volume per radian of oscillation to the maximum strain energy per unit volume. Loss factor damping is sometimes referred to as material or structural damping.

The use of loss factor damping traditionally refers to a scalar-valued loss factor s. But there is no reasonthat s must be scalar. Because the loss factor is a value deduced from true complex-valued material data, it can be represented by a matrix of the same dimensions as the anisotropic stiffness matrix. Especially for orthotropic material, there should be a set of loss factors of all normal and shear elasticity modulus components. The following loss-elasticity combinations are available:

• Isotropic Loss Factor Damping

• Anisotropic Loss Factor Damping

• Orthotropic Loss Factor Damping

Isotropic Loss Factor DampingAn isotropic material is described by the different isotropic material constants. Is likely to only have isotropic loss, described by the isotropic loss factor s. When Isotropic loss

factor is selected as the Damping type, from the Isotropic structural loss factor list, the default s (dimensionless) uses values From material. If User defined is selected, enter another value or expression. The default value is 0.

Anisotropic Loss Factor DampingA symmetric anisotropic material is described by a symmetric 6-by-6 elasticity matrix D, and the loss can be isotropic or symmetric anisotropic. The loss is described by the isotropic loss factor s or by a symmetric anisotropic 6-by-6 loss factor matrix D.

Loss factor damping applies to frequency domain studies (that is, frequency response and damped eigenfrequency studies).


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When Anisotropic loss factor is selected as Damping type from the Loss factor for

elasticity matrix D list, the default D or DVo (dimensionless) uses values From

material. If User defined is selected, choose:

• Isotropic (the default) to enter a single scalar loss factor.

• Symmetric to enter the components of D in the upper-triangular part of a symmetric 6-by-6 matrix.

Orthotropic Loss Factor Damping

An orthotropic material is described by three normal Young’s modulus components (Ex, Ey, and Ez) and three shear modulus components (Gxy, Gyz, and Gxz. The loss can be isotropic (described by the isotropic loss factor s) or orthotropic (described by three plus three orthotropic loss factors corresponding to the elastic moduli components for an orthotropic material).

When Orthotropic loss factor is selected as the Damping type from the Loss factor for

orthotropic Young’s modulus list E (dimensionless), the default uses values From

material. If User defined is selected, enter another value or expression (defaults are 0).

From the Loss factor for orthotropic shear modulus list, the default G or GVo (dimensionless) use values From material. If User defined is selected, enter other values or expressions. The defaults are 0.

The values for the shear modulus loss factors are ordered in two ways, consistent with the selection of either Standard (XX, YY, ZZ, XY, YZ, XZ) or Voigt (XX, YY, ZZ, YZ, XZ,

XY) notation in the corresponding Linear Elastic Model. The default values are 0. If the values are taken from the material, these loss factors are found in the Orthotropic or Orthotropic, Voigt notation property group for the material.

The values for the loss factors are ordered in two ways, consistent with the selection of either Standard (XX, YY, ZZ, XY, YZ, XZ) or Voigt (XX, YY,

ZZ, YZ, XZ, XY) notation in the corresponding Linear Elastic Model. The default values are 0. If the values are taken from the material, these loss factors are found in the Anisotropic or Anisotropic, Voigt notation property group for the material.

This option is available when Orthotropic is selected as the Linear Elastic Material Solid model.


Initial Values

The Initial Values node adds initial values for the displacement field and structural velocity field that can serve as an initial condition for a transient simulation or as an initial guess for a nonlinear analysis. Add additional Initial Values nodes from the Physics toolbar.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically set up and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or select All domains as required.


Enter values or expressions for the initial values of the Displacement field u (SI unit: m) (the displacement components u, v, and w in 3D) (the default is 0 m), and the Structural velocity field ut (SI unit: m/s) (the default is 0 m/s)).

Body Load

Add a Body Load to domains for modeling gravity or centrifugal loads, for example. Right-click to add a Phase for harmonic loads in frequency-domain computations or to add a Harmonic Perturbation subnode.





F O R C E

Select a Load type—Load defined as force per unit volume (the default) or Total force. For 2D models, Load defined as force per unit area is also an option.

Then enter values or expressions for the components in the matrix based on the selection and the space dimension:

• Body load FV (SI unit: N/m3)


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• Total force Ftot (SI unit: N). For total force, COMSOL Multiphysics divides the total force by the volume of the domains where the load is active.

• For 2D models: Load FA (SI unit: N/m2). The body load as force per unit volume is then the value of F divided by the thickness.

Free

The Free node is the default boundary condition. It means that there are no constraints and no loads acting on the boundary.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically set up and is the same as for the physics interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific boundaries or select All boundaries as required.



Boundary Load

Add a Boundary Load to boundaries for a pressure acting on a boundary, for example. Right-click to add a Phase for harmonic loads in frequency-domain computations or to add a Harmonic Perturbation subnode.








F O R C E

Select a Load type—Load defined as force per unit area (the default), Pressure, or Total

force. For 2D models, Load defined as force per unit length is also an option.

Then enter values or expressions for the components in the matrix based on the selection and the space dimension:

• Load FA (SI unit: N/m2). The body load as force per unit volume is then the value of F divided by the thickness.

• For 2D models: Load FL (SI unit: N/m).

• Total force Ftot (SI unit: N). For total force, COMSOL Multiphysics then divides the total force by the area of the surfaces where the load is active.

• Pressure p (SI unit: Pa), which can represent a pressure or another external pressure. The pressure is positive when directed toward the solid.

Fixed Constraint

The Fixed Constraint node adds a condition that makes the geometric entity fixed (fully constrained); that is, the displacements are zero in all directions. For domains, this condition is selected from the More submenu.

D O M A I N , B O U N D A R Y, E D G E , O R PO I N T S E L E C T I O N

From the Selection list, choose, the geometric entity (domains, boundaries, edges, or points) to define.


If this node is selected from the Pairs menu, choose the pair use. An identity pair has to be created first.



After selecting a Load type, the Load list normally only contains User

defined. When combining the Solid Mechanics interface with another physics interface, it is also possible to choose a predefined load from this list.


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Prescribed Displacement

The Prescribed Displacement node adds a condition where the displacements are prescribed in one or more directions to the geometric entity (domain, boundary, edge, or point). For domains, this condition is selected from the More submenu.

If a displacement is prescribed in one direction, this leaves the solid free to deform in the other directions.

You can also define more general displacements as a linear combination of the displacements in each direction.

D O M A I N , B O U N D A R Y , E D G E , O R PO I N T S E L E C T I O N

From the Selection list, choose the geometric entity (domains, boundaries, edges, or points) to define.




The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. If you choose another, local coordinate system, the displacement components change accordingly.

• Using Weak Constraints to Evaluate Reaction Forces

• Boundary Conditions in the COMSOL Multiphysics Reference Manual

• If a prescribed displacement is not activated in any direction, this is the same as a Free constraint.

• If a zero displacement is applied in all directions, this is the same as a Fixed Constraint.


P R E S C R I B E D D I S P L A C E M E N T

Define the prescribed displacements using a Standard notation (the default) or a General

notation.

Standard NotationTo define the displacements individually, click the Standard notation button.

Select one or all of the Prescribed in x direction, Prescribed in y direction, and for 3D models, Prescribed in z direction check boxes. Then enter a value or expression for u0, v0, and for 3D models, w0 (SI unit: m). For 2D axisymmetric models, select one or both of the Prescribed in r direction and Prescribed in z direction check boxes. Then enter a value or expression for u0 and w0 (SI unit: m).

General NotationClick the General notation to specify the displacements using a general notation that includes any linear combination of displacement components. For example, for 2D models, use the relationship

For H matrix H (dimensionless) select Isotropic, Diagonal, Symmetric, or Anisotropic and then enter values as required in the field or matrix. Enter values or expressions for the R vector R (SI unit: m)

For example, to achieve the condition u = v, use the settings

which force the domain to move only diagonally in the xy-plane.


See Fixed Constraint for these settings.

Roller

The Roller node adds a roller constraint as the boundary condition; that is, the displacement is zero in the direction perpendicular (normal) to the boundary, but the boundary is free to move in the tangential direction. See Fixed Constraint for all the settings.

H uv

R=

H 1 1–0 0

= R 00

=


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Periodic Condition

The Periodic Condition node adds a periodic boundary condition. This periodicity makes uix0uix1 for a displacement ui. You can control the direction that the periodic condition applies to. If the source and destination boundaries are rotated with respect to each other, this transformation is automatically performed, so that corresponding displacement components are connected.


From the Selection list, choose the boundaries to define. The software automatically identifies the boundaries as either source boundaries or destination boundaries.

P E R I O D I C I T Y S E T T I N G S

Select a Type of periodicity—Continuity (the default), Antiperiodicity, Floquet periodicity, Cyclic symmetry, or User defined. If User defined is selected, select the Periodic in u, Periodic in v (for 3D and 2D models), and Periodic in w (for 3D and 2D axisymmetric models) check boxes as required. Then for each selection, choose the Type of

periodicity—Continuity (the default) or Antiperiodicity.

• If Floquet periodicity is selected, enter a k-vector for Floquet periodicity kF (SI unit: rad/m) for the X, Y, and Z coordinates (3D models), or the R and Z coordinates (2D axisymmetric models), or X and Y coordinates (2D models).


defined. If User defined is selected, enter a value for S (SI unit: rad; default value: 0). For any selection, also enter a Azimuthal node number m (dimensionless; default value: 0).

This works fine for cases like opposing parallel boundaries. In other cases right-click the Periodic Condition node to add a Destination Selection subnode to control the destination. By default it contains the selection that COMSOL Multiphysics has identified.

In cases where the periodic boundary is split into several boundaries within the geometry, it might be necessary to apply separate periodic conditions to each pair of geometry boundaries.




Edge Load

Add an Edge Load to 3D models for a force distributed along an edge, for example. ight-click to add a Phase for harmonic loads in frequency-domain computations or to add a Harmonic Perturbation subnode.





F O R C E

Select a Load type—Load defined as force per unit area (the default) or Total force. Then enter values or expressions for the components in the matrix based on the selection:

• Load FL (SI unit: N/m). When combining the Solid Mechanics interface with, for example, film damping, it is also possible to choose a predefined load from this list.

• Total force Ftot (SI unit: N). COMSOL Multiphysics then divides the total force by the volume where the load is active.

Point Load

Add a Point Load to points for concentrated forces at points. Right-click to add a Phase for harmonic loads in frequency-domain computations or to add a Harmonic Perturbation subnode.



• Cyclic Symmetry and Floquet Periodic Conditions





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F O R C E

Enter values or expressions for the components of the Point load Fp (SI unit: N).

Initial Stress and Strain

A solid mechanics model can include the Initial Stress and Strain subnode, which is the stress-strain state in the structure before applying any constraint or load. Initial strain can, for example, describe moisture-induced swelling, and initial stress can describe stresses from heating. Think of initial stress and strain as different ways to express the same thing. Right-click to add this subnode to a Linear Elastic Material.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains to define, or select All domains as required.


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. The given initial stresses and strains are interpreted in this system.

I N I T I A L S T R E S S A N D S T R A I N

Enter values or expressions for the Initial stress S0 (SI unit: N/m2) and Initial strain 0 (dimensionless). The default values are 0, which is no initial stress or strain. For both, enter the diagonal and off-diagonal components (based on space dimension):

• For a 3D Initial stress model, diagonal components S0x, S0y, and S0z and off-diagonal components S0xy, S0yz, and S0xz, for example.

• For a 3D Initial strain model, diagonal components 0x, 0y, and 0z and off-diagonal components 0xy, 0yz, and 0xz, for example.


Phase

Add a Phase node to a Body Load, Boundary Load, Edge Load, or Point Load. For modeling the frequency response the physics interface splits the harmonic load into two parameters:

• The amplitude, F, which is specified in the feature node for the load.

• The phase (FPh).

Together these define a harmonic load, for which the amplitude and phase shift can vary with the excitation frequency, f:

D O M A I N , B O U N D A R Y, E D G E , O R PO I N T S E L E C T I O N

For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains, boundaries, edges, or points to define, or select All domains, All boundaries, All edges, or All points, as required.

P H A S E

Enter the components of Load phase in radians (for a pressure the load phase is a scalar value). Add [deg] to a phase value to specify it using degrees.

Prescribed Velocity

The Prescribed Velocity node adds a boundary or domain condition where the velocity is prescribed in one or more directions. The prescribed velocity condition is applicable for Time Dependent and Frequency Domain studies. With this boundary or domain condition it is possible to prescribe a velocity in one direction, leaving the solid free in the other directions. For domains, this condition is selected from the More submenu.

The Prescribed Velocity node is a constraint and overrides any other constraint on the same selection.

Ffreq F f 2f FPh f + cos=

Typically the load magnitude is a real scalar value. If the load specified in the parent feature contains a phase (using a complex-valued expression), the software adds the phase from the Phase node to the phase already included in the load.


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D O M A I N O R B O U N D A R Y S E L E C T I O N

From the Selection list, choose the geometric entity (domains or boundaries) to define.


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. Coordinate systems with directions that change with time should not be used. If you choose another, local coordinate system, the velocity components change accordingly.

P R E S C R I B E D VE L O C I T Y

Select one or all of the Prescribed in x direction, Prescribed in y direction, and for 3D models, Prescribed in z direction check boxes. Then enter a value or expression for vx, vy, and for 3D models, vz (SI unit: m/s). For 2D axisymmetric models, select one or both of the Prescribed in r direction and Prescribed in z direction check boxes. Then enter a value or expression for vr and vz (SI unit: m/s).

Prescribed Acceleration

The Prescribed Acceleration node adds a boundary or domain condition, where the acceleration is prescribed in one or more directions. The prescribed acceleration condition is applicable for Time Dependent and Frequency Domain studies. With this boundary condition, it is possible to prescribe a acceleration in one direction, leaving the solid free in the other directions. For domains, this condition is selected from the More submenu.

The Prescribed Acceleration node is a constraint and overrides any other constraint on the same selection.

D O M A I N O R B O U N D A R Y S E L E C T I O N

From the Selection list, choose the geometric entity (domains or boundaries) to define.


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. Coordinate systems with directions that change with time should not be used. If you choose another, local coordinate system, the acceleration components change accordingly.


P R E S C R I B E D A C C E L E R A T I O N

Select one or all of the Prescribed in x direction, Prescribed in y direction, and for 3D models, Prescribed in z direction check boxes. Then enter a value or expression for ax, ay, and for 3D models, az (SI unit: m/s2). For 2D axisymmetric models, select one or both of the Prescribed in r direction and Prescribed in z direction check boxes. Then enter a value or expression for ar and az (SI unit: m/s2).

Symmetry

The Symmetry node adds a boundary condition that represents symmetry in the geometry and in the loads. A symmetry condition is free in the plane and fixed in the out-of-plane direction.








Using Weak Constraints to Evaluate Reaction Forces


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Antisymmetry

The Antisymmetry node adds a boundary condition for an antisymmetry boundary, which must exist in both the geometry and in the loads. An antisymmetry condition is fixed in the plane and free in the out-of-plane direction.






See Symmetry for these settings.

Spring Foundation

The Spring Foundation node has elastic and damping boundary conditions for domains, boundaries, edges, and points. To select this node for the domains, it is selected from the More submenu. Also right-click to add a Pre-Deformation subnode.

The Spring Foundation and Thin Elastic Layer nodes are similar, with the difference that a Spring Foundation connects the structural part on which it is acting to a fixed “ground,” while a Thin Elastic Layer acts between two parts, either on an interior boundary or on a pair boundary.




The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. The spring and damping constants are given with respect to the selected coordinate system.

In a geometrically nonlinear analysis, large rotations must not occur at the antisymmetry plane because this causes artificial straining.


S P R I N G

Select the Spring type and its associated spring constant of force using Table 7-1 as a guide. The default option is the spring type for the type of geometric entity and space dimension, and there are different combinations available based on this.

L O S S F A C T O R D A M P I N G

Enter values or expressions in the table for each coordinate based on space dimension for the Loss factor for spring k. The loss factors act on the corresponding components of the spring stiffness. All defaults are 0.

TABLE 7-1: SPRING TYPES FOR THE SPRING FOUNDATION FEATURE

SPRING TYPE VARIABLE SI UNITS GEOMETRIC ENTITY LEVEL

SPACE DIMENSION

Spring constant per unit volume

kV N/(mm3) domains 3D, 2D, and 2D axisymmetric

Total spring constant ktot N/m domains, edges 3D, 2D, and 2D axisymmetric

Spring constant per unit area

kA N/(mm)2 domains, boundaries

3D, 2D

Spring constant per unit length

kL N/(mm) edges, boundaries (2D)

3D, 2D

Spring constant kP N/m points 3D, 2D, and 2D axisymmetric

Force per volume as function of extension

FV N/m3 domains 3D, 2D, and 2D axisymmetric

Total force as function of extension

Ftot N domains, boundaries, edges

3D, 2D, and 2D axisymmetric

Force per area as function of extension

FA N/m2 domains, boundaries

3D, 2D

Force per length as function of extension

FL N/m edges 3D

Force as function of extension

FP N points 3D, 2D, and 2D axisymmetric


332 | C H A P T E

V I S C O U S D A M P I N G

Select the Damping type using Table 7-2 as a guide. The default option is the default damping type for the type of geometric entity and space dimension, and there are different combinations available based on this.

Thin Elastic Layer

The Thin Elastic Layer node has elastic and damping boundary conditions for boundaries and acts between two parts, either on an interior boundary or on a pair boundary. Also right-click to add a Pre-Deformation subnode.

The Thin Elastic Layer and Spring Foundation nodes are similar, with the difference that a Spring Foundation connects the structural part on which it is acting to a fixed “ground”.




If this node is selected from the Pairs menu, choose the pair to define. An identity pair has to be created first. Ctrl-click to deselect. A default Free node is added when a Thin Elastic Layer pair node is added.

TABLE 7-2: DAMPING TYPES FOR THE SPRING FOUNDATION FEATURE

DAMPING TYPE VARIABLE SI UNITS GEOMETRIC ENTITY LEVEL

SPACE DIMENSION

Viscous force per unit volume

dV Ns/(mm3) domains, boundaries (2D)

3D, 2D

Viscous force per unit area

dA Ns/(mm2) domains, boundaries


Total viscous force dtot Ns/m domains, boundaries, edges, points


Viscous force per unit length

dL Ns/(mm) edges 3D


• About Spring Foundations and Thin Elastic Layers



The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. The spring and damping constants are given with respect to the selected coordinate system.

S P R I N G

Select the Spring type and its associated spring constant of force using Table 7-3 as a guide. The default option is the spring type for the space dimension.


Enter values or expressions in the table for each coordinate based on space dimension for the Loss factor for spring k. The loss factors act on the corresponding components of the spring stiffness. All defaults are 0.

TABLE 7-3: SPRING TYPES FOR THE THIN ELASTIC LAYER FEATURE

SPRING TYPE VARIABLE SI UNITS SPACE DIMENSION

Total spring constant ktot N/m 3D, 2D, and 2D axisymmetric

Spring constant per unit area kA N/(mm)2 3D, 2D, and 2D axisymmetric

Spring constant per unit length kL N/(mm) 2D

Total force as function of extension Ftot N 3D, 2D, and 2D axisymmetric

Force per area as function of extension

FA N/m2 3D, 2D, and 2D axisymmetric

Force per length as function of extension

FL N/m 2D


334 | C H A P T E


Select the Damping type using Table 7-4 as a guide. The default option is the default damping type for the space dimension.

Pre-Deformation

Right-click the Spring Foundation or Thin Elastic Layer nodes to add a Pre-Deformation subnode and define the coordinates. By including a pre-deformation, you can model cases where the unstressed state of the spring is in another configuration than the one modeled.



S P R I N G P R E - D E F O R M A T I O N

Based on space dimension, enter the coordinates for the Spring Pre-Deformation u0 (SI unit: m). The defaults are 0 m.

Added Mass

The Added Mass node is available on domains, boundaries, and edges and can be used to supply inertia, which is not part of the material itself. Such inertia does not need to

TABLE 7-4: DAMPING TYPES FOR THE THIN ELASTIC LAYER FEATURE

DAMPING TYPE VARIABLE SI UNITS SPACE DIMENSION

Viscous force per unit area dA Ns/(mm2) 3D, 2D, and 2D axisymmetric

Total viscous force dtot Ns/m 3D, 2D, and 2D axisymmetric

Viscous force per unit length dL Ns/(mm) 2D




be isotropic, in the sense that the inertial effects are not the same in all directions. To select this node for the domains, it is selected from the More submenu .

D O M A I N , B O U N D A R Y, O R E D G E S E L E C T I O N

From the Selection list, choose the geometric entity (domains, boundaries, or edges) to define.


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. The added mass values are given with respect to the selected coordinate directions.

A D D E D M A S S

Select a Mass type using Table 7-5 as a guide. The default option is the type for the geometric entity. Then enter values or expressions into the table for each coordinate based on space dimension. All defaults are 0.

Low-Reflecting Boundary

Use the Low-Reflecting Boundary node to let waves pass out from the model without reflection in time-dependent analysis. As a default, it takes material data from the

To include the added mass as a static self weight, separate load physics need to be added for the domains, boundaries, or edges. The Added Mass node only contributes to the inertia in the dynamic sense.

TABLE 7-5: AVAILABLE MASS TYPES BASED ON GEOMETRIC ENTITY

MASS TYPE VARIABLE SI UNITS GEOMETRIC ENTITY LEVEL

Mass per unit volume pV kg/m3 domains

Mass per unit area pA kg/m2 domains, boundaries

Mass per unit length pL kg/m edges

Total mass m kg domains, boundaries, edges

About Added Mass


336 | C H A P T E

domain in an attempt to create a perfect impedance match for both pressure waves and shear waves. It can be sensitive to the direction of the incoming wave.




The Boundary System is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes.

D A M P I N G

Select a Damping type—P and S waves (the default) or User defined. If User defined is selected, enter values or expressions for the Mechanical impedance di (SI unit: Pas/m). The defaults for all values are 0.5*solid.rho*(solid.cp+solid.cs).

Gravity

When you add a Gravity node, gravity forces will be applied to all features in the physics interface that has a density, mass, or mass distribution. The gravity acts in a fixed spatial direction. The load intensity is ggeg where g as a default is the acceleration of gravity. The action of gravity can also be seen as a linearly accelerated frame of reference.


From the Selection list, choose the domains to which the gravity load should be applied.


The Global coordinate system is selected by default. The Coordinate system list contains any additional coordinate systems that the model includes. It can be used when prescribing the direction of the gravitational forces.

About the Low-Reflecting Boundary Condition


G R AV I T Y

Enter the components of the Gravity g (SI unit: m/s2). The default value is g_const which is the physical constant having the value 9.8066 m/s2.

Rotating Frame

Centrifugal, Coriolis, and Euler forces are “fictitious” forces that need to be introduced in a rotating frame of reference, since it is not an inertial system. Use a Rotating Frame node to add the effect of these forces.


From the Selection list, choose the domains to which the loads are to be applied.

R O T A T I N G F R A M E

Select an Axis of rotation—x-axis, y-axis, z-axis, or User defined. If User defined is chosen, enter a Rotation axis base point rbp (SI unit: m) and Rotation axis direction eax (dimensionless).

Select a Rotational direction—Counterclockwise or Clockwise. The rotational direction does not make any difference for a centrifugal force,

Select a Rotational frequency—Angular velocity, RPM, or Revolutions per time. Then enter a value as required for Angular velocity magnitude (SI unit: rad/s), RPM, or Revolutions per time (SI unit: 1/s).

For 2D models, the default is that the gravity acts in the negative y direction.

For 3D and 2D axisymmetric models, the default is that the gravity acts in the negative z direction.

Theory for Rotating Frame and Gravity


338 | C H A P T E

For 3D and 2D models, use the Centrifugal force, Coriolis force, or Euler force check boxes to determine which effects of a rotating frame that are to be incorporated in the analysis. Only Centrifugal force is selected by default.

For 2D axisymmetric models, the only effect from a rotating frame is the centrifugal force, which is then always included.

The Spin softening check box is selected by default. When including spin-softening effects, an extra contribution to centrifugal force from deformation is taken into account. The Spin softening check box is only available if Centrifugal force or Euler force

is selected.



Th eo r y f o r t h e S o l i d Me chan i c s I n t e r f a c e

The Solid Mechanics Interface theory is described in this section:

Material and Spatial Coordinates

The Solid Mechanics interface, through its equations, describes the motion and deformation of solid objects in a 2- or 3-dimensional space. In COMSOL Multiphysics terminology, this physical space is known as the spatial frame and positions in the physical space are identified by lowercase spatial coordinate variables x, y, and z (or r,

, and z in axisymmetric models).

Continuum mechanics theory also makes use of a second set of coordinates, known as material (or reference) coordinates. These are normally denoted by uppercase variables X, Y, and Z (or R, , and Z) and are used to label material particles. Any material particle is uniquely identified by its position in some given initial or reference configuration. As long as the solid stays in this configuration, material and spatial coordinates of every particle coincide and displacements are zero by definition.

• Material and Spatial Coordinates

• Coordinate Systems

• Lagrangian Formulation

• About Linear Elastic Materials

• Strain-Displacement Relationship

• Stress-Strain Relationship

• Plane Strain and Plane Stress Cases

• Axial Symmetry

• Loads

• Pressure Loads

• Equation Implementation

• Setting up Equations for Different Studies

• Damping Models

• Initial Stresses and Strains


• About Added Mass

• Geometric Nonlinearity Theory for the Solid Mechanics Interface

• About the Low-Reflecting Boundary Condition

• Cyclic Symmetry and Floquet Periodic Conditions

• Theory for Rotating Frame and Gravity

T H E O R Y F O R T H E S O L I D M E C H A N I C S I N T E R F A C E | 339

340 | C H A P T E

When the solid objects deform due to external or internal forces and constraints, each material particle keeps its material coordinates X (bold font is used to denote coordinate vectors), while its spatial coordinates change with time and applied forces such that it follows a path

(7-1)

in space. Because the material coordinates are constant, the current spatial position is uniquely determined by the displacement vector u, pointing from the reference position to the current position. The global Cartesian components of this displacement vector in the spatial frame, by default called u, v, and w, are the primary dependent variables in the Solid Mechanics interface.

By default, the Solid Mechanics interface uses the calculated displacement and Equation 7-1 to define the difference between spatial coordinates x and material coordinates X. This means the material coordinates relate to the original geometry, while the spatial coordinates are solution dependent.

Material coordinate variables X, Y, and Z must be used in coordinate-dependent expressions that refer to positions in the original geometry, for example, for material properties that are supposed to follow the material during deformation. On the other hand, quantities that have a coordinate dependence in physical space, for example, a spatially varying electromagnetic field acting as a force on the solid, must be described using spatial coordinate variables x, y, and z. Any use of the spatial variables is a source of nonlinearity if a geometrically nonlinear study is performed.

Coordinate Systems

Force vectors, stress and strain tensors, as well as various material tensors are represented by their components in a specified coordinate system. By default, material properties use the canonical system in the material frame. This is the system whose basis vectors coincide with the X, Y, and Z axes. When the solid deforms, these vectors rotate with the material.

Loads and constraints, on the other hand, are applied in spatial directions, by default in the canonical spatial coordinate system. This system has basis vectors in the x, y, and z directions, which are forever fixed in space. Both the material and spatial default coordinate system are referred to as the global coordinate system in the physics interface.

x x X t X u X t += =


Vector and tensor quantities defined in the global coordinate system on either frame use the frame’s coordinate variable names as indices in the tensor component variable names. For example, SXY is the material frame XY-plane shear stress, also known as a second Piola-Kirchhoff stress, while sxy is the corresponding spatial frame stress, or Cauchy stress. There are also a few mixed tensors, most notably the deformation gradient FdxY, which has one spatial and one material index because it is used in converting quantities between the material and spatial frames.

It is possible to define any number of user coordinate systems on the material and spatial frames. Most types of coordinate systems are specified only as a rotation of the basis with respect to the canonical basis in an underlying frame. This means that they can be used both in contexts requiring a material system and in contexts requiring a spatial one.

The coordinate system can be selected separately for each added material model, load, and constraint. This is convenient if, for example, an anisotropic material with different orientation in different domains is required. The currently selected coordinate system is known as the local coordinate system.

Lagrangian Formulation

The formulation used for structural analysis in COMSOL Multiphysics for both small and finite deformations is total Lagrangian. This means that the computed stress and deformation state is always referred to the material configuration, rather than to current position in space.

Likewise, material properties are always given for material particles and with tensor components referring to a coordinate system based on the material frame. This has the obvious advantage that spatially varying material properties can be evaluated just once for the initial material configuration and do not change as the solid deforms and rotates.

The gradient of the displacement, which occurs frequently in the following theory, is always computed with respect to material coordinates. In 3D:


342 | C H A P T E

The displacement is considered as a function of the material coordinates (X, Y, Z), but it is not explicitly a function of the spatial coordinates (x, y, z). It is thus only possible to compute derivatives with respect to the material coordinates.

About Linear Elastic Materials

The total strain tensor is written in terms of the displacement gradient

or in components as

(7-2)

The Hooke’s law relates the stress tensor to the strain tensor and temperature:

(7-3)

where is the 4th order elasticity tensor, “:” stands for the double-dot tensor product (or double contraction), s0 and 0 are initial stresses and strains, thTTref) is the thermal strain, and is the coefficient of thermal expansion.

The elastic strain energy density is

(7-4)

or using the tensor components:

u

Xu

Yu

Zu

Xv

Yv

Zv

Xw

Yw

Zw

=

12--- u u T

+ =

mn12---

xnum

xmun+

=

s s0 C 0– th– +=

C

Ws12--- 0– th– C 0– th– =

Ws12---C

ijmnij ij

0– ij

th– mn mn

0– mn

th–

i j m n =


TE N S O R V S . M A T R I X F O R M U L A T I O N S

Because of the symmetry, the strain tensor can be written as the following matrix:

Similar representation applies to the stress and the thermal expansion tensors:

Due to the symmetry, the elasticity tensor can be completely represented by a symmetric 6-by-6 matrix as:

which is the elasticity matrix.

I S O T R O P I C M A T E R I A L A N D E L A S T I C M O D U L I

In this case, the elasticity matrix becomes

x xy xz

xy y yz

xz yz z

sx sxy sxz

sxy sy syz

sxz syz sz

x xy xz

xy y yz

xz yz z

D

D11 D12 D13 D14 D15 D16

D12 D22 D23 D24 D25 D26

D13 D23 D33 D34 D35 D36

D14 D24 D34 D44 D45 D46

D15 D25 D35 D45 D55 D56

D16 D26 D36 D46 D56 D66

C1111

C1122

C1133

C1112

C1123

C1113

C1122

C2222

C2233

C2212

C2223

C2213

C1133

C2233

C3333

C3312

C3323

C3313

C1112

C2212

C3312

C1212

C1223

C1213

C1123

C2223

C3323

C1223

C2323

C2313

C1113

C2213

C3313

C1213

C2313

C1313

= =

D E1 + 1 2–

---------------------------------------

1 – 0 0 0 1 – 0 0 0 1 – 0 0 0

0 0 0 1 2–2

---------------- 0 0

0 0 0 0 1 2–2

---------------- 0

0 0 0 0 0 1 2–2

----------------

=


344 | C H A P T E

and the thermal expansion is the diagonal tensor:

Different pairs of elastic moduli can be used, and as long as two moduli are defined, the others can be computed according to Table 7-6.

According to Table 7-6, the elasticity matrix D for isotropic materials is written in terms of Lamé parameters and ,

or in terms of the bulk modulus K and shear modulus G:

TABLE 7-6: EXPRESSIONS FOR THE ELASTIC MODULI.

DESCRIPTION VARIABLE DE DKG D

Young’s modulus E

Poisson’s ratio

Bulk modulus K

Shear modulus G

Lamé parameter

Lamé parameter

G

Pressure-wave speed

cp

Shear-wave speed cs

0 00 00 0

9KG3K G+------------------- 3 2+

+--------------------

12--- 1 3G

3K G+-------------------–

2 + ---------------------

E3 1 2– ------------------------ 2

3-------+

E2 1 + ---------------------

E1 + 1 2–

--------------------------------------- K 2G3

--------–

E2 1 + ---------------------

K 4G 3+

--------------------------

G

D

2+ 0 0 0 2+ 0 0 0 2+ 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

=


O R T H O T R O P I C A N D A N I S O T R O P I C M A T E R I A L S

There are two different ways to represent orthotropic or anisotropic data. The Standard (XX, YY, ZZ, XY, YZ, XZ) material data ordering converts the indices as:

thus, the Hooke’s law is presented in the form involving the elasticity matrix D and the following vectors:

COMSOL Multiphysics uses the complete tensor representation internally to perform the coordinate system transformations correctly.

Beside the Standard (XX, YY, ZZ, XY, YZ, XZ) Material data ordering, the elasticity coefficients are entered following the Voigt notation. In the Voigt (XX, YY, ZZ, YZ, XZ,

XY) Material data ordering, the sorting of indices is:

D

K 4G3

--------+ K 2G3

--------– K 2G3

--------– 0 0 0

K 2G3

--------– K 4G3

--------+ K 2G3

--------– 0 0 0

K 2G3

--------– K 2G3

--------– K 4G3

--------+ 0 0 0

0 0 0 G 0 00 0 0 0 G 00 0 0 0 0 G

=

112233

12 2123 3213 31

123456

xyz

xyyzxz

sx

sy

sz

sxy

syz

sxz

sx

sy

sz

sxy

syz

sxz 0

D

x

y

z

2xy

2yz

2xz

x

y

z

2xy

2yz

2xz 0

x

y

z

2xy

2yz

2xz

––

+=


346 | C H A P T E

thus the last three rows and columns in the elasticity matrix D are swapped.

O R T H O T R O P I C M A T E R I A L

The elasticity matrix for orthotropic material in the Standard (XX, YY, ZZ, XY, YZ, XZ)

Material data ordering has the following structure:

where the components are as follows:

where

112233

23 3213 3112 21

123456

xyz

yzxzxy

D

D11 D12 D13 0 0 0

D12 D22 D23 0 0 0

D13 D23 D33 0 0 0

0 0 0 D44 0 0

0 0 0 0 D55 0

0 0 0 0 0 D66

=

D11Ex

2 Ezyz2 Ey–

Ddenom----------------------------------------,= D12

ExEy Ezyzxz Eyxy+ Ddenom

-----------------------------------------------------------------–=

D13ExEyEy xyyz xz+

Ddenom----------------------------------------------------------,–= D22

Ey2 Ezxz

2 Ex– Ddenom

----------------------------------------=

D23EyEz Eyxyxz Exyz+

Ddenom-----------------------------------------------------------------,–= D33

EyEz Eyxy2 Ex–

Ddenom-----------------------------------------------=

D44 Gxy= , D55 Gyz= , and D66 Gxz=

Ddenom EyEzxz2 ExEy– 2xyyzxzEyEz ExEzyz

2 Ey2xy

2+ + +=


The values of Ex, Ey, Ez, xy, yz, xz, Gxy, Gyz, and Gxz are supplied in designated fields in the physics interface. COMSOL deduces the remaining components—yx, zx, and zy—using the fact that the matrices D and D1 are symmetric. The compliance matrix has the following form:

The thermal expansion matrix is diagonal:

The elasticity matrix in the Voigt (XX, YY, ZZ, YZ, XZ, XY) Material data ordering changes the sorting of the last three elements in the elasticity matrix:

A N I S O T R O P I C M A T E R I A L

In the general case of fully anisotropic material, provide explicitly 21 components of the symmetric elasticity matrix D, in either Standard (XX, YY, ZZ, XY, YZ, XZ) or Voigt (XX, YY, ZZ, YZ, XZ, XY) Material data ordering, and 6 components of the symmetric thermal expansion matrix.

E N T R O P Y A N D T H E R M O E L A S T I C I T Y

The free energy for the linear thermoelastic material can be written as

D 1–

1Ex------

yx

Ey--------–

zx

Ez--------– 0 0 0

xyEx--------–

1Ey------

zyEz--------– 0 0 0

xzEx--------–

yzEy--------–

1Ez------ 0 0 0

0 0 0 1Gxy--------- 0 0

0 0 0 0 1Gyz--------- 0

0 0 0 0 0 1Gxz---------

=

x 0 0

0 y 0

0 0 z

D44 Gyz= , D55 Gxz= , and D66 Gxy=


348 | C H A P T E

where WsT is given by Equation 7-4. Hence, the stress can be found as

and the entropy per unit volume can be calculated as

where T0 is a reference temperature, the volumetric heat capacity CP can be assumed independent of the temperature (Dulong-Petit law), and

For an isotropic material, it simplifies into

The heat balance equation can be written as

where k are the thermal conductivity matrix, and

where is the strain-rate tensor and the tensor represents all possible inelastic stresses (for example, a viscous stress).

Using the tensor components, the heat balance can be rewritten as:

(7-5)

In many cases, the second term can be neglected in the left-hand side of Equation 7-5 because all Tmn are small. The resulting approximation is often called uncoupled thermoelasticity.

F f0 T Ws T +=

s

F

T W

TC 0– th– = = =

TF

– Cp T T0 log Selast+=

Selast s=

Selast sx sy sz+ + =

Cp tT T

t Selast+ k T Qh+=

Qh ·=

·

Cp tT Tmn t

smn

m n+ k T Qh+=


Strain-Displacement Relationship

The strain conditions at a point are completely defined by the deformation components—u, v, and w in 3D—and their derivatives. The precise relation between strain and deformation depends on the relative magnitude of the displacement.

S M A L L D I S P L A C E M E N T S

Under the assumption of small displacements, the normal strain components and the shear strain components are related to the deformation as follows:

(7-6)

To express the shear strain, use either the tensor form, xy, yz, xz, or the engineering form, xy, yz, xz.

The symmetric strain tensor consists of both normal and shear strain components:

The strain-displacement relationships for the axial symmetry case for small displacements are

L A R G E D E F O R M A T I O N S

As a start, consider a certain physical particle, initially located at the coordinate X. During deformation, this particle follows a path

For simplicity, assume that undeformed and deformed positions are measured in the same coordinate system. Using the displacement u, it is then possible to write

x xu

=

y yv

=

z zw

=

xyxy2

-------=12---

yu

xv

+ =

yzyz2

-------=12---

zv

yw

+ =

xzxz

2-------=

12---

zu

xw

+ .=

x xy xz

xy y yz

xz yz z

=

r ru,=

ur---,= z z

w, and= rz zu

rw

+=

x x X t =


350 | C H A P T E

When studying how an infinitesimal line element dX is mapped to the corresponding deformed line element dx, the deformation gradient, F, defined by

is used.

The deformation gradient contains the complete information about the local straining and rotation of the material. It is a positive definite matrix, as long as material cannot be annihilated. The ratio between current and original volume (or mass density) is

A deformation state where J = 1 is often called incompressible. From the deformation gradient, it is possible to define the right Cauchy-Green tensor as

As can be shown by simple insertion, a finite rigid body rotation causes nonzero values of the engineering strain defined by Equation 7-6. This is not in correspondence with the intuitive concept of strain, and it is certainly not useful in a constitutive law. There are several alternative strain definitions in use that do have the desired properties. The Green-Lagrange strains, , is defined as

Using the displacements, they be also written as

(7-7)

The Green-Lagrange strains are defined with reference to an undeformed geometry. Hence, they represent a Lagrangian description.

The deformation state characterized by finite (or large displacements) but small to moderate strains is sometimes referred to as geometric nonlinearity or nonlinear

x X u+=

dx xX

-------dX F dX= =

dVdV0-----------

0------ det F J= = =

C FTF=

12--- C I– 1

2--- FTF I– = =

ij12---

uiXj

--------ujXi

--------ukXi

---------ukXj

---------+ + =


geometry. This typically occurs when the main part of the deformations presents a finite rigid body rotation

S T R A I N R A T E A N D S P I N

The spatial velocity gradient is defined in components as

where is the spatial velocity field. It can be shown that L can be computed in terms of the deformation gradient as

where the material time derivative is used.

The velocity gradient can be decomposed into symmetric and skew-symmetric parts

where

is called the rate of strain tensor, and

is called the spin tensor. Both tensors are defined on the spatial frame.

It can be shown that the material time derivative of the Green-Lagrange strain tensor can be related to the rate of strain tensor as

The spin tensor Lw(x,t) accounts for an instantaneous local rigid-body rotation about an axis passing through the point x.

Components of both Ld and Lw are available as results and analysis variables under the Solid Mechanics interface.

Lkl xl vk r t =

vk r t

Ltd

dFF1–

=

L Ld Lw+=

Ld12--- L LT

+ =

Lw12--- L LT

– =

tdd FTLdF=


352 | C H A P T E

Stress-Strain Relationship

The symmetric stress tensor describes stress in a material:

This tensor consists of three normal stresses (x, y, z) and six (or, if symmetry is used, three) shear stresses (xy, yz, xz).

For large deformations there are more than one stress measure:

• Cauchy stress (the components are denoted sx, … in COMSOL Multiphysics) defined as force/deformed area in fixed directions not following the body. Symmetric tensor.

• First Piola-Kirchhoff stress P (the components are denoted Px, … in COMSOL Multiphysics). This is an unsymmetric two-point tensor.

• Second Piola-Kirchhoff stress S (the components are denoted Sx, … in COMSOL Multiphysics). This is a symmetric tensor, for small strains same as Cauchy stress tensor but in directions following the body.

The stresses relate to each other as

Plane Strain and Plane Stress Cases

For a general anisotropic linear elastic material in case of plane stress, COMSOL Multiphysics solves three equations si30 for i3 with i = 1, 2, 3, and uses the solution instead of Equation 7-2 for these three strain components. Thus, three components i3 are treated as extra degrees of freedom. For isotropy and orthotropy, only with an extra degree of freedom, 33,is used since all out of plane shear components of both

x xy xz

yx y yz

zx zy z

= xy yx= xz zx= yz zy=

S F 1– P=

J 1– PFT J 1– FSFT= =


stress and strain are zero. The remaining three strain components are computed as in 3D case according to Equation 7-2.

In case of plane strain, set i3 for i1, 2, 3. The out-of-plane stress components si3 are results and analysis variables.

Axial Symmetry

The axially symmetric geometry uses a cylindrical coordinate system. Such a coordinate system is orthogonal but curvilinear, and one can choose between a covariant basis e1, e2, e3 and a contravariant basis e1, e2, e3.

The metric tensor is

in the coordinate system given by e1, e2, e3, and

in e1, e2, e3.

The metric tensor plays the role of a unit tensor for a curvilinear coordinate system.

For any vector or tensor A, the metric tensor can be used for conversion between covariant, contravariant, and mixed components:

For an isotropic material, only the normal out-of-plane component 33 needs to be solved for.

gij 1 0 0

0 r2 00 0 1

=

gij

1 0 0

0 r 2– 00 0 1

=

Aij Aimgmj

m=

Aij Anmgnigmj

m n=


354 | C H A P T E

In both covariant and contravariant basis, the base vector in the azimuthal direction has a nonunit length. To cope with this issue, the so called physical basis vectors of unit length are introduced. These are

The corresponding components for any vector or tensor are called physical.

For any tensor, the physical components are defined as

where no summation is done over repeated indices.

M I X E D C O M P O N E N T S A N D P R I N C I P A L I N V A R I A N T S

The mixed strain components are given by

The principal invariants are

D I S P L A C E M E N T S A N D A X I A L S Y M M E T R Y A S S U M P T I O N S

The axial symmetry implementation in COMSOL Multiphysics assumes independence of the angle, and also that the torsional component of the displacement is identically zero. The physical components of the radial and axial displacement, u and w, are used as dependent variables for the axially symmetric geometry.

For the linear elastic material, the stress components in coordinate system are

er e1 e1= e

1r---e

2re2= = ez e3= e3

= =

Aijphys gii gjjA

ij=

Aji gimAmj

m=

I1 A trace Aii Ai

i

i= A11 A22

1

r2----- A33+ += =

I2 A 12--- I1 A 2 Aj

iAij

i j–

=

I3 A det Aii =

sij s0ij

C+ijkl

kl kl– 0kl– =


where TTref.

For anisotropic and orthotropic materials, the 4th-order elasticity tensor is defined from D matrix according to:

The user input D matrix always contains the physical components of the elasticity tensor

and the corresponding tensor components are computed internally according to:

For an isotropic material:

where and are the first and second Lamé elastic parameters.

Loads

Specify loads as

• Distributed loads. The load is a distributed force in a volume, on a face, or along an edge.

sr

ssz

sr

sz

srz

sr

ssz

sr

sz

srz 0

D

r

z

2r

2z

2rz

r

z

2r

2z

2rz 0

r

z

2r

2z

2rz

––

+=

Cijklphys

Cijkl Cijkl

phys

gii gjj gkk gll

-----------------------------------------------=

Cijkl gijgkl gikgjl gilgjk

+ +=


356 | C H A P T E

• Total force. The specification of the load is as the total force. The software then divides this value with the area or the volume where the force acts.

• Pressure (boundaries only).

Table 7-7 shows how to define distributed loads on different geometric entity levels; the entries show the SI unit in parentheses.

For 2D models choose how to specify the distributed boundary load as a load defined as force per unit area or a load defined as force per unit length acting on boundaries.

In the same way, choose between defining the load as force per unit volume or force per unit area for body loads acting in a domain. Also define a total force (SI unit: N) as required.

For 2D and axisymmetric models, the boundary loads apply on edges (boundaries).

For 2D axisymmetric models, the boundary loads apply on edges (boundaries).

For 3D solids, the boundary loads apply on faces (boundaries).

TABLE 7-7: DISTRIBUTED LOADS

GEOMETRIC ENTITY

POINT EDGE FACE DOMAIN

2D force (N) force/area (N/m2) or force/length (N/m)

Not available

force/volume (N/m3) or force/area (N/m2)

Axial symmetry

total force along the circumferential (N)

force/area (N/m2) Not available

force/volume (N/m3)

3D force (N) force/length (N/m) force/area (N/m2)

force/volume (N/m3)


Pressure Loads

A pressure load is directed inward along the normal of boundary on which it is acting. This load type acts as a source of nonlinearity, since its direction depends on the current direction of the boundary normal. In a linearized context, for example in the frequency domain, the pressure is equivalent to a specified normal stress.

Equation Implementation

The COMSOL Multiphysics implementation of the equations in the Solid Mechanics interface is based on the principle of virtual work.

The principle of virtual work states that the sum of virtual work from internal strains is equal to work from external loads.

The total stored energy, W, for a linear material from external and internal strains and loads equals:

The principle of virtual work states that W0 which leads to

For general cases, if the problem is linear in all other respects, the solution can be made more efficient by forcing the solver to treat the problem as linear. See Stationary Solver in the COMSOL Multiphysics Reference Manual.

W – s u FV dv+V=

u FS sdS u FL ld

L Ut Fp

p+ + +

– tests utest FV utest utt– dv+V

utest FS sdS utest FL ld

L Utest

t Fp p+ + +


358 | C H A P T E

Setting up Equations for Different Studies

The Solid Mechanics interface supports Stationary (static), Eigenfrequency, Time Dependent (transient), requency Domain, and Modal solver study types.

S T A T I O N A R Y S T U D I E S

COMSOL Multiphysics uses an implementation based on the stress and strain variables. The normal and shear strain variables depend on the displacement derivatives.

Using the tensor strain, stress, and displacement variables, the principle of virtual work is expressed as:

T I M E D E P E N D E N T S T U D I E S

(7-8)

where the terms proportional to dM and dK appear if the Rayleigh damping is used

F R E Q U E N C Y - D O M A I N S T U D I E S

In the frequency domain the frequency response is studied when applying harmonic loads. Harmonic loads are specified using two components:

• The amplitude value, Fx

• The phase, FxPh

W – tests utest FV dv+V=


L Utest

t Fp p+ + +

– test s dMst+ utest FV utest utt dMutest ut–– dv+V


L Utest

t Fp p+ + +

For more information about the equation form in case of geometric nonlinearity see Geometric Nonlinearity Theory for the Solid Mechanics Interface.


To derive the equations for the linear response from harmonic excitation loads

assume a harmonic response with the same angular frequency as the excitation load

Also describe this relationship using complex notation

where

E I G E N F R E Q U E N C Y S T U D I E S

The eigenfrequency equations are derived by assuming a harmonic displacement field, similar as for the frequency response formulation. The difference is that this study type

Fxfreq Fx f t FxPh f 180----------+

cos=

Ffreq

Fxfreq

Fyfreq

Fzfreq

=

u uamp t u+ cos=

uuvw

=

u Re uampejuejt Re ue

jt where u uampe

ju= = =

u Re uejt

=

Fxfreq Re Fx ejFxPh f

180----------

ejt

Re Fx˜ ejt = =

Fx˜ Fx f e

jFxPh f 180----------

=

F˜

Fx˜

Fy˜

Fz˜

=


360 | C H A P T E

uses a new variable j explicitly expressed in the eigenvalue jThe eigenfrequency f is then derived from j as

Damped eigenfrequencies can be studied by adding viscous damping terms to the equation. In addition to the eigenfrequency the quality factor, Q, and decay factor, for the model can be examined:

Damping Models

The Solid Mechanics interface offers two predefined damping models: Rayleigh damping and loss factor damping.

R A Y L E I G H D A M P I N G

To model damping effects within the material, COMSOL Multiphysics uses Rayleigh damping, where two damping coefficients are specified.

The weak contribution due to the alpha-damping is always accounted for as shown in Equation 7-2. The contribution from the beta-damping that shown in Equation 7-8 corresponds to the case of small strains. In case of geometric nonlinearity, it becomes

where P is the first Piola-Kirchhoff stress tensor.

To further clarify the use of the Rayleigh damping, consider a system with a single degree of freedom. The equation of motion for such a system with viscous damping is

f Im j 2

-------------------=

Q Im 2Re -------------------=

Re =

dMutest– Pt vdV

Geometric Nonlinearity Theory for the Solid Mechanics Interface


In the Rayleigh damping model the damping coefficient c can be expressed in terms of the mass m and the stiffness k as

The Rayleigh damping proportional to mass and stiffness is added to the static weak term.

A complication with the Rayleigh damping model is to obtain good values for the damping parameters. A more physical damping measure is the relative damping, the ratio between actual and critical damping, often expressed as a percentage of the critical damping. Commonly used values of relative damping can be found in the literature.

It is possible to transform relative damping to Rayleigh damping parameters. The relative damping, , for a specified pair of Rayleigh parameters, dM and dK, at a frequency, f, is

Using this relationship at two frequencies, f1 and f2, with different relative damping, 1 and 2, results in an equation system that can be solved for dM and dK:

md2u

dt2---------- cdu

dt------- ku+ + f t =

c dMm dKk+=

12---

dM2f----------- dK2f+ =

14f1------------ f1

14f2------------ f2

dM

dK

1

2

=


362 | C H A P T E

Using the same relative damping, 1 = 2, does not result in a constant damping factor inside the interval f1 f f2. It can be shown that the damping factor is lower inside the interval, as Figure 7-1 shows.

Figure 7-1: An example of Rayleigh damping.


Loss factor damping (sometimes referred to as material or structural damping) can be applied in the frequency domain.

In COMSOL Multiphysics, the loss information appears as a multiplier of the elastic stress in the stress-strain relationship:

where s is the loss factor, and j is the imaginary unit.

Choose between these loss damping types:

• Isotropic loss damping s

• Orthotropic loss damping with components of E, the loss factor for an orthotropic Young’s modulus, and G, the loss factor for an orthotropic shear modulus.

• Anisotropic loss damping with an isotropic or symmetric loss damping D for the elasticity matrix.

If modeling the damping in the structural analysis via the loss factor, use the following definition for the elastic part of the entropy:

Rel

ativ

e da

mpi

ng

f

Rayleigh damping

Specified damping

f2f1

s s0 1 js+ C 0– – +=

Selast s js C – =


This is because the entropy is a function of state and thus independent of the strain rate, while the damping represents the rate-dependent effects in the material (for example, viscous or viscoelastic effects). The internal work of such inelastic forces averaged over the time period 2 can be computed as:

(7-9)

Equation 7-9 can be used as a heat source for modeling of the heat generation in vibrating structures, when coupled with the frequency-domain analysis for the stresses and strains.

Initial Stresses and Strains

Initial stress refers to the stress before the system applies any loads, displacements, or initial strains. The initial strain is the one before the system has applied any loads, displacements, or initial stresses.

Both the initial stress and strains are tensor variables defined via components on the local coordinate system for each domain. Input these as the following matrices:

In case of nearly incompressible material (mixed formulation), the components of the total initial stress (that is, without volumetric-deviatoric split) are still input. The initial pressure in the equation for the pressure help variable pw is computed as

The initial stresses and strains are available with the Linear Elastic Material.

A X I A L S Y M M E T R Y

User inputs the physical components of 0 and s0:

Qh12---sReal Conj C =

0x 0xy 0xz

0xy 0y 0yz

0xz 0yz 0z

s0x s0xy s0xz

s0xy s0y s0yz

s0xz s0yz s0z

p013---I1 s0 –=

0r 0r 0rz

0r 0 0z

0rz 0z 0z

s0r s0r s0rz

s0r s0 s0z

s0rz s0z s0z


364 | C H A P T E

O T H E R PO S S I B L E U S E S O F I N I T I A L S T R A I N S A N D S T R E S S E S

Many inelastic effects in solids and structure (creep, plasticity, damping, viscoelasticity, poroelasticity, and so on) are additive contributions to either the total strain or total stress. Then the initial value input fields can be used for coupling the elastic equations (solid physics) to the constitutive equations (usually General Form PDEs) modeling such extra effects.

About Spring Foundations and Thin Elastic Layers

In this section, the equations for the spring type physics nodes are developed using boundaries, but the generalizations to geometrical objects of other dimensions are obvious.

S P R I N G F O U N D A T I O N

A spring gives a force that depends on the displacement and acts in the opposite direction (in the case of a force that is proportional to the displacement, this is called Hooke’s law). In a suitable coordinate system, a spring condition can be represented as

where fs is a force/unit area, u is the displacement, and K is a diagonal stiffness matrix. u0 is an optional pre-deformation. If the spring properties are not constant, it is, in general, easier to directly describe the force as a function of the displacement, so that

In the same way, a viscous damping can be described as a force proportional to the velocity

Structural (“loss factor)” damping is only relevant for frequency domain analysis and is defined as

where is the loss factor and i is the imaginary unit (in this case, a constant or a diagonal matrix). If the elastic part of the spring definition is given as a force versus displacement relation, the stiffness K is taken as the stiffness at the linearization point at which the frequency response analysis is performed. Since the loss factor force is proportional to the elastic force, the equation can be written as

fs K u u0– –=

fs f u u0– =

fv Du·–=

fl iK u u0– –=


The contribution to the virtual work is

T H I N E L A S T I C L A Y E R B E T W E E N TW O P A R T S

A spring or damper can also act between two boundaries of an identity pair. The spring force then depends on the difference in displacement between the surfaces.

The uppercase indices refer to “source” and “destination.” When a force versus displacement description is used,

The viscous and structural damping forces have analogous properties,

so that

The virtual work expression is formulated on the destination side of the pair as

Here the displacements from the source side are obtained using the src2dst operator of the identity pair.

T H I N E L A S T I C L A Y E R O N I N T E R I O R B O U N D A R I E S

On an interior boundary, the Thin Elastic Layer decouples the displacements between two sides if the boundary. The two boundaries are then connected by elastic and viscous forces with equal size but opposite directions, proportional to the relative displacements and velocities. The spring force can be written as

fsl fs fl+ 1 i+ fs= =

W uT fsl fv+ AdA=

fsD fsS– K uD uS u0–– –= =

fsD fsS– f u u0– u uD uS–

= ==

fvD fvS– D u· D u· S– –= =

flD flS– iK uD uS u0–– –= =

fslD fsD flD+ 1 i+ fsD= =

W uD uS– T fslD fvD+ ADdAD

=


366 | C H A P T E

or

The viscous force is

and the structural damping force is

The subscripts u and d denote the “up” and “down” sides of the interior boundary.

The virtual work expression is formulated as

About Added Mass

The Added Mass node can be used for supplying inertia that is not part of the material itself. Such inertia does not need to be isotropic, in the sense that the inertial effects are not the same in all directions. This is, for example, the case when a structure immersed in a fluid vibrates. The fluid is added to the inertia for acceleration in the direction normal to the boundary, but not tangential to it.

Other uses for added mass are when sheets or strips of a material that is heavy, but having a comparatively low stiffness, are added to a structure. The data for the base material can then be kept unaltered, while the added material is represented purely as added mass.

The value of an added mass can also be negative. You can use such a negative value for adjusting the mass when a part imported from a CAD system does not get exactly the correct total mass due so simplifications of the geometry.

Added mass is an extra mass distribution that can be anisotropic. It can exist on domains, boundaries, and edges. The inertial forces from added mass can be written as

fsd fsu– K ud uu u0–– –= = fsd fsu– f u u0– u ud uu–

= ==

fvd fvu– D u· d u· u– –= =

fld flu– iK ud uu u0–– –= =

fsld fsd fld+ 1 i+ fsd= =

W ud us– T fsld fvd+ ADdAD

=

fm Mt2

2

u–=


where M is a diagonal mass distribution matrix. The contribution to the virtual work is

for added mass on a boundary, and similarly for objects of other dimensions.


Geometric nonlinearity formulation is suitable for any material, and it is always used for hyperelastic materials and for large strain plasticity.

For other materials, it can be activated via the solver setting. Note however that even together with the geometric nonlinearity, the validity of any linear material model is usually limited to the situation of possibly large displacements but small to moderate strains. A typical example of use is to model large rigid body rotations. The implementation is similar to that for the geometrically linear elastic material, but with the strain tensor replaced with the Green-Lagrange strain tensor, and the stress tensor replaced with the second Piola-Kirchhoff stress tensor, defined as:

where TTref, and i represents all possible inelastic strains (such as plastic or creep strains). In components, it is written as:

where the elasticity tensor is defined from the D matrix (user input). The 2nd Piola-Kirchhoff stress is a symmetric tensor.

The strain energy function is computed as

which is a variable defined in the physics interface. Other stress variables are defined as follows.

W uTfm AdA=

The Hyperelastic Material node is available with the Nonlinear Structural Materials Module.

S S0C I– 0

– i– +=

Sij Sij0

C+ ijkl kl kl– kl0

– kli

– =

Cijkl

Ws12---S I– 0 i

–– =


368 | C H A P T E

The first Piola-Kirchhoff stress P is calculated from the second Piola-Kirchhoff stress as P FS. The first Piola-Kirchhoff stress relates forces in the present configuration with areas in the reference configuration, and it is sometimes called the nominal stress.

Using the 1st Piola-Kirchhoff stress tensor, the equation of motion can be written in the following form:

(7-10)

where the density corresponds to the material density in the initial undeformed state, the volume force vector FV has components in the actual configuration but given with respect to the undeformed volume, and the tensor divergence operator is computed with respect to the coordinates on the material frame. Equation 7-10 is the strong form that corresponds to the weak form equations solved in case of geometric nonlinearity within the Solid Mechanics interface (and many related multiphysics interfaces) in COMSOL Multiphysics. Using vector and tensor components, the equation can be written as

The components of 1st Piola-Kirchhoff stress tensor are non symmetric in the general case, thus

because the component indexes correspond to different frames. Such tensors are called two-point tensors.

The boundary load vector FA in case of geometric nonlinearity can be related to the 1st Piola-Kirchhoff stress tensor via the following formula:

0t2

2

u FV X P–=

0t2

2

ux FVx XPxX

YPxY

ZPxZ+ +

–=

0t2

2

uy FVy XPyX

YPyY

ZPyZ+ +

–=

0t2

2

uz FVz XPzX

YPzY

ZPzZ+ +

–=

PiJ PIj

FA P n0=


where the normal n0 corresponds to the undeformed surface element. Such a force vector is often referred to as the nominal traction. In components, it can be written as

The Cauchy stress, s,can be calculated as

The Cauchy stress is a true stress that relates forces in the present configuration (spatial frame) to areas in the present configuration, and it is a symmetric tensor. Equation 7-10 can be rewritten in terms of the Cauchy stress as

where the density corresponds to the density in the actual deformed state, the volume force vector fV has components in the actual configuration (spatial frame) given with respect to the deformed volume, and the divergence operator is computed with respect to the spatial coordinates.

The pressure is computed as

which corresponds to the volumetric part of the Cauchy stress. The deviatoric part is defined as

The second invariant of the deviatoric stress

is used for the computation of von Mises (effective) stress

FAx PxXnX PxYnY PxZnZ+ +=

FAy PyXnX PyYnY PyZnZ+ +=

FAz PzXnX PzYnY PzZnZ+ +=

s J 1– PFT J 1– FSFT= =

t2

2

u fV x s–=

p 13---– trace s =

sd s pI+=

J2 s 12---sd:sd=

smises 3J2 s =


370 | C H A P T E

N E A R L Y I N C O M P R E S S I B L E M A T E R I A L S

Nearly incompressible materials can cause numerical problems if only displacements are used in the interpolating functions. Small errors in the evaluation of volumetric strain, due to the finite resolution of the discrete model, are exaggerated by the high bulk modulus. This leads to an unstable representation of stresses, and in general to underestimation of the deformation because spurious volumetric stresses might balance also applied shear and bending loads.

In such cases a mixed formulation can be used that represents the pressure as a dependent variable in addition to the displacement components. This formulation removes the effect of volumetric strain from the original stress tensor and replaces it with an interpolated pressure, pw. A separate equation constrains the interpolated pressure to make it equal (in a finite-element sense) to the original pressure calculated from the strains.

For an isotropic linear elastic material, the second Piola-Kirchhoff stress tensor S, computed directly from the strains, is replaced by a modified version:

where I is the unit tensor and the pressure p is calculated from the stress tensor

The auxiliary dependent variable pw is set equal to p using the equation

(7-11)

where is the bulk modulus.

The modified stress tensor is used then in calculations of the energy variation.

For orthotropic and anisotropic materials, the auxiliary pressure equation is scaled to make the stiffness matrix symmetric. Note, however, that the stiffness matrix in this formulation is not positive definite and even contains a zero block on the diagonal in

s s p pw– I+=

p 13---trace s –=

pw

------- p---– 0=

s


the incompressible limit. This limits the possible choices of direct and iterative linear solver.

About the Low-Reflecting Boundary Condition

The low-reflecting boundary condition is mainly intended for letting waves pass out from the model domain without reflection in time-dependent analysis. It is also available in the frequency domain, but then adding a perfectly matched layer (PML) is usually a better option.

As a default, the low-reflecting boundary condition takes the material data from the adjacent domain in an attempt to create a perfect impedance match for both pressure waves and shear waves, so that

where n and t are the unit normal and tangential vectors at the boundary, respectively, and cp and cs are the speeds of the pressure and shear waves in the material. This approach works best when the wave direction in close to the normal at the wall.

In the general case, you can use

where the mechanical impedance di is a diagonal matrix available as the user input, and by default it is set to

In case of linear elastic materials without geometric nonlinearity (and also for hyperelastic materials), the stress tensor s in the above equations is replaced by the 2nd Piola-Kirchhoff stress tensor S, and the pressure p with:

pp13---trace S –=

Infinite Element Domains and Perfectly Matched Layers in the COMSOL Multiphysics Reference Manual

n cp tu n n– cs t

u t t–=

n di cp cs t

u–=


372 | C H A P T E

Cyclic Symmetry and Floquet Periodic Conditions

These boundary conditions are based on the Floquet theory which can be applied to the problem of small-amplitude vibrations of spatially periodic structures.

If the problem is to determine the frequency response to a small-amplitude time-periodic excitation that also possesses spatial periodicity, the theory states that the solution can be sought in the form of a product of two functions. One follows the periodicity of the structure, while the other one follows the periodicity of the excitation. The problem can be solved on a unit cell of periodicity by applying the corresponding periodicity conditions to each of the two components in the product.

The problem can be modeled using the full solution without applying the above described multiplicative decomposition. For such a solution, the Floquet periodicity conditions at the corresponding boundaries of the periodicity cell are expressed as

where u is a vector of dependent variables, and vector kF represents the spatial periodicity of the excitation.

The cyclic symmetry boundary condition presents a special but important case of Floquet periodicity, for which the unit periodicity cell is a sector of a structure that possesses rotational symmetry. The frequency response problem can be solved then in one sector of periodicity by applying the periodicity condition. The situation is often referred to as dynamic cyclic symmetry.

For an eigenfrequency study, all the eigenmodes of the full problem can be found by performing the analysis on one sector of symmetry only and imposing the cyclic symmetry of the eigenmodes with an angle of periodicity , where the cyclic

di cp cs+

2----------------I=

More information about modeling using low-reflecting boundary conditions can be found in M. Cohen and P.C. Jennings, “Silent Boundary Methods for Transient Analysis,” Computational Methods for Transient Analysis, vol 1 (editors T. Belytschko and T.J.R. Hughes), Nort-Holland, 1983.

udestination ikF rdestination rsource– – exp usource=

m=


symmetry mode number m can vary from 0 to N, with N being the total number of sectors so that 2N.

The Floquet periodicity conditions at the sides of the sector of symmetry can be expressed as

where the u represents the displacement vectors with the components given in the default Cartesian coordinates. Multiplication by the rotation matrix given by

makes the corresponding displacement components in the cylindrical coordinate system differ by the factor only. For scalar dependent variables, a similar condition applies, for which the rotation matrix is replaced by a unit matrix.

The angle represents either the periodicity of the eigenmode for an eigenfrequency analysis or the periodicity of the excitation signal in case of a frequency-response analysis. In the latter case, the excitation is typically given as a load vector

when modeled using the Cartesian coordinates; parameter m is often referred to as the azimuthal wave-number.


You can add Rotating Frame and Gravity nodes for creating the loads caused by gravity or accelerated frames. Doing so will give load contributions from all nodes in the physics interface which have a density or mass, such as Linear Elastic Material, Rigid Domain, Added Mass, Point Mass, or Mass and Moment Inertia.

udestination e i– RusourceT

=

R

cos sin– 0 sin cos 0

0 0 1

=

i– exp

F F0 im Y X atan– exp–=

More information about cyclic symmetry conditions can be found in B. Lalanne and M. Touratier, “Aeroelastic Vibrations and Stability in Cyclic Symmetric Domains,” The International Journal of Rotating Machinery, vol. 6, no. 6, pp 445–452, 2000.


374 | C H A P T E

In the following, the mass density should be considered as generalized. It can be represent either mass per unit volume, mass per unit area, mass per unit length, or even mass, depending on the dimensionality of the object giving the contribution.

R O T A T I N G F R A M E

Centrifugal, Coriolis and Euler forces are fictitious forces that need to be introduced in a rotating frame of reference, since it is not an inertial system.They can be added as loads.

Alternatively, the total acceleration in the rotating frame can be augmented to include the frame acceleration effects:

G R A V I T Y

The gravity acts in a fixed spatial direction eg. The intensity is

where g is the acceleration of gravity. The action of gravity can also be presented as a linearly accelerated frame of reference. Thus, it can be accounted for as a contribution into the total acceleration via the frame acceleration term given by:

C E N T R I F U G A L F O R C E

A centrifugal force acts radially outwards from the axis of rotation defined by the axial direction vector eax. The rotation is represented by the angular velocity vector:

where is the angular velocity. On vector form, the acceleration contribution and the loads are:

atot u·· af+=

af acen acor aeul+ +=

g geg=

af g–=

eax=

acen rp =

Fcen acen–=


where rp is the rotation position vector that contains the coordinates with respect to any point on the axis of rotation. The point is given by its radius vector in the global coordinate system rbp.

S P I N - S O F T E N I N G E F F E C T

The structural displacement can be accounted for when computing the rotation position, so that

This results in a contribution from the extra acceleration terms caused by the deformation into the stiffness matrix of the equation system. The effect is often called spin-softening.

C O R I O L I S F O R C E

For a Coriolis force to appear, the object studied must have a velocity relative to the rotating frame. The acceleration contribution and the load are:

This will give a damping contribution since it is proportional to the velocity.

E U L E R F O R C E

The Euler force occurs when the rate of rotation is not constant in time. The force acts in the plane of rotation perpendicular to the centrifugal force. The acceleration contribution and the load are

rp X u rpb–+=

acor 2t

u=

Fcor acor–=

aeul t rp=

Feul aeul–=


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Ca l c u l a t i n g R e a c t i o n Fo r c e s

There are different ways to evaluate reaction forces and these are discussed in this section.

• Using Predefined Variables to Evaluate Reaction Forces

• Using Weak Constraints to Evaluate Reaction Forces

• Using Surface Traction to Evaluate Reaction Forces

• Using Surface Traction to Evaluate Reaction Forces

The following sections describe the merits and costs of these methods.

Using Predefined Variables to Evaluate Reaction Forces

The results analysis capabilities include easy access to the reaction forces and moments. They are available as predefined variables. The reaction force variables are available only at the nodes, and not as a continuous field, so they are not suitable for graphic presentation.

Reaction forces are computed as the sum of the nodal values over the selected volume, face, or edge. Reaction moments are calculated as the sum of the moment from the reaction forces with respect to a reference point, and any explicit reaction moments (if there are rotational degrees of freedom).

Specify the default coordinates of the Reference Point for Moment Computation at the top level of the physics interface main node’s settings window. After editing the reference point coordinates, you need to right-click the Study node and select Update

Solution for the change to take effect on the reaction moment calculation. During

To compute the sum of the reaction forces over a region, use Volume

Integration, Surface Integration, or Line Integration under Derived Values. The integration method discovers that the reaction forces are discrete values and applies a summation instead of an integration.


postprocessing, you can modify the coordinates of the reference point in the Parameters section of a result feature..

Using Weak Constraints to Evaluate Reaction Forces

Select the Use weak constraints check box to get accurate distributed reactions. Extra variables that correspond to the reaction traction distribution are automatically added to the solution components.

With weak constraints activated, COMSOL Multiphysics adds the reaction forces to the solution components. The variables are denoted X_lm, where X is the name of the constrained degree of freedom (as, for example, u_lm and v_lm). The extension lm stands for Lagrange multipliers. It is only possible to evaluate reaction forces on constrained boundaries in the directions of the constraints.

Reaction forces are not available for eigenfrequency analysis or when weak constraints are used.

If reaction forces are summed independently for two adjacent boundaries, the total sum is not the same as if the reaction forces were summed for both boundaries in one operation. The values of the nodes at the common edge always contain contributions from the elements at both sides of the edge.

Derived Values and Tables in the COMSOL Multiphysics Reference Manual

To compute the total reaction force on a boundary, integrate one of the variables X_lm using Volume Integration, Surface Integration, or Line

Integration under Derived Values.

If the constraint is defined in a local coordinate system, the degrees of freedom for the weak constraint variables are defined along the directions of that system.

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Since the reaction force variables are added to the solution components, the number of DOFs for the model increases slightly, depending on the mesh size for the boundaries in question. Boundaries that are adjacent to each other must have the same constraint settings. The reason for this is that adjacent boundaries share a common node.

Using weak constraints affects the structure of the equation system to be solved, and is not suitable for all types of equation solvers.

Using Surface Traction to Evaluate Reaction Forces

As an alternative method, you can obtain values of the reaction forces on constrained boundaries by using boundary integration of the relevant components of the surface traction vector.

Two different types of surface traction results can be computed in COMSOL Multiphysics:

The first type, contained in the variables interface.Tax, is computed from the stresses. It is always available. Since the surface traction vector is based on computed stress results, this method is less accurate for computing reactions than the other methods.

The second type, contained in the variables interface.Tracx, is computed using a method similar to the weak constraints, but without introducing the Lagrange multipliers as extra degrees of freedom. The accuracy is high, but there is an extra


• Derived Values and Tables

• Symmetric and Nonsymmetric Constraints

For 2D models, multiply the surface traction by the cross section thickness before integrating to calculate the total reaction force.


computational cost. These traction variables are computed only if the check box Compute boundary fluxes in the Discretization section of the physics interface is selected.

Evaluating Surface Traction Forces on Internal Boundaries

As opposed to the other methods for reaction force computation, the boundary flux based tractions are computed not only on external boundaries, but also on internal boundaries. On internal boundaries, there are then two traction fields: One acting from each of the domains sharing the boundary. These internal traction fields are contained in the variables interface.iTracux and interface.iTracdx. The letters u and d in the variable names indicate the up and down side of the boundary respectively. If you need the value of the total force acting on an internal section through your model, these variable can be integrated. The interface.iTracux andinterface.iTracdx variables are only available if the Compute boundary fluxes check box is selected in the Discretization section of the physics interface, and there are internal boundaries in your model.

In case of geometric nonlinearity, the two types of traction variables are interpreted differently. The interface.Tax variables are based on Cauchy stress, and contains a force per current area. If you integrate them you must use the spatial frame. The interface.Tracx variables are based on First Piola-Kirchhoff stresses and contains a force per undeformed area. An integration must then be done on the material frame.

Computing Accurate Fluxes in the COMSOL Multiphysics Reference Manual

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Geome t r i c Non l i n e a r i t y , F r ame s , a nd t h e A LE Me t hod

Consider the bending of a beam in the general case of a large deformation (see Figure 7-2). In this case the deformation of the beam introduces an effect known as geometric nonlinearity into the equations of solid mechanics.

Figure 7-2 shows that as the beam deforms, the shape, orientation, and position of the element at its tip changes significantly. Each edge of the infinitesimal cube undergoes both a change in length and a rotation that depends on position. Additionally the three edges of the cube are no longer orthogonal in the deformed configuration (although typically for practical strains the effect of the non-orthogonality can be neglected in comparison to the rotation).

From a simulation perspective it is desirable to solve the equations of solid mechanics on a fixed domain, rather than on a domain that changes continuously with the deformation. In COMSOL Multiphysics this is achieved by defining a displacement field for every point in the solid, usually with components u, v, and w. At a given coordinate (X, Y, Z) in the reference configuration (on the left of Figure 7-2) the value of u describes the displacement of the point relative to its original position. Taking derivatives of the displacement with respect to X, Y, and Z enables the definition of a strain tensor, known as the Green-Lagrange strain (or material strain). This strain is defined in the reference or Lagrangian frame, with X, Y, and Z representing the coordinates in this frame. The displacement is considered as a function of the material coordinates (X, Y, Z), but it is not explicitly a function of the spatial coordinates (x, y, z). It is thus only possible to compute derivatives with respect to the material coordinates.

In the Solid Mechanics interface, the Lagrangian frame is equivalent to the material frame. An element at point (X, Y, Z) specified in this frame moves with a single piece of material throughout a solid mechanics simulation. It is often convenient to define material properties in the material frame as these properties move and rotate naturally together with the volume element at the point at which they are defined as the simulation progresses. In Figure 7-2 this point is illustrated by the small cube highlighted at the end of the beam, which is stretched, translated, and rotated as the beam deforms. The three mutually perpendicular faces of the cube in the Lagrange frame are no longer perpendicular in the deformed frame. The deformed frame is


called the Eulerian or (in COMSOL Multiphysics) the spatial frame. Coordinates in this frame are denoted (x, y, z) in COMSOL.

Figure 7-2: An example of the deformation of a beam showing the undeformed state (left) and the deformed state (right) of the beam itself with an element near its tip highlighted (top), of the element (center) and of lines parallel to the x-axis in the undeformed state (bottom).

For example, in the Eulerian system it is easy to define forces per unit area (known as tractions) that act within the solid, and to define a stress tensor that represents all of these forces that act on a volume element. Such forces could be physically measured, for example using an implanted piezoresistor. The stress tensor in the Eulerian frame is called the Cauchy or true stress tensor (in COMSOL this is referred to as the spatial stress tensor). To construct the stress tensor in the Lagrangian frame a tensor transformation must be performed on the Cauchy stress. This produces the second Piola-Kirchhoff (or material) stress, which can be used with the Lagrange or material

It is important to note that, as the solid deforms, the Lagrangian frame becomes a non-orthogonal curvilinear coordinate system (see the lower part of Figure 7-2 to see the deformation of the X-axis). Particular care is therefore required when defining physics in this coordinate system.

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strain to solve the solid mechanics problem in the (fixed) Lagrangian frame. This is how the Solid Mechanics interface works when geometric nonlinearities are enabled.

In the case of solid mechanics, the material and spatial frames are associated directly with the Lagrangian and Eulerian frames, respectively. In a more general case (for example, when tracking the deformation of a fluid, such as a volume of air surrounding a moving structure) tying the Lagrangian frame to the material frame becomes less desirable. Fluid must be able to flow both into and out of the computational domain, without taking the mesh with it. The Arbitrary Lagrangian-Eulerian (ALE) method allows the material frame to be defined with a more general mapping to the spatial or Eulerian frame. In COMSOL, a separate equation is solved to produce this mapping—defined by the mesh smoothing method (Laplacian, Winslow, or hyperelastic) with boundary conditions that determine the limits of deformation (these are usually determined by the physics of the system, whilst the domain level equations are typically being defined for numerical convenience). The ALE method offers significant advantages since the physical equations describing the system can be solved in a moving domain.

Reference for Geometric Nonlinearity

1. A.F. Bower, Applied Mechanics of Solids, CRC Press, Boca Raton, FL (http://www.solidmechanics.org), 2010.

If the strains are small (significantly less than 10 percent), and there are no significant rotations involved with the deformation (significantly less than 10 degrees), geometric nonlinearity can be disabled, resulting in a linear equation system which solves more quickly (Ref. 1). This is often the case for many practical MEMS structures.

Geometric nonlinearity can be enabled or disabled within a given model by changing the Include geometric nonlinearity setting in the relevant solver step.



http://www.solidmechanics.org

http://www.solidmechanics.org

S p r i n g s and Dampe r s

The Spring Foundation and Thin Elastic Layer physics nodes are available with the Solid Mechanics interface and supply elastic and damping boundary conditions for domains, boundaries, edges, and points. The Spring Foundation node is available also in the other structural mechanics interfaces.

The features are completely analogous, with the difference that a Spring Foundation node connects the structural part on which it is acting to a fixed “ground,” while the Thin Elastic Layer acts between two parts, either on an internal boundary or on a pair.

The following types of data are defined by these nodes:

• Spring Data

• Loss Factor Damping

• Viscous Damping

S P R I N G D A T A

The elastic properties can be defined either by a spring constant or by a force as function of displacement. The force as a function of displacement can be more convenient for nonlinear springs. Each spring feature has three displacement variables defined, which can be used to describe the dependency on deformation. These variables are named uspring1_tag, uspring2_tag, and uspring3_tag for the three directions given by the local coordinate system. In the variable names, tag represents the tag of the feature defining the variable The tag could for example be spf1 or tel1 for a Spring Foundation or a Thin Elastic Layer respectively. These variables measure the relative extension of the spring after subtraction of any pre-deformation.


The loss factor damping adds a loss factor to the spring data above, so that the total force exerted by the spring with loss is

where fs is the elastic spring force, and is the loss factor.

Loss factor damping is only applicable in for eigenfrequency and frequency domain analysis. In time-dependent analysis the loss factor is ignored.

fsl 1 i+ fs=

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It is also possible to add viscous damping to the Spring Foundation and Thin Elastic Layer features. The viscous damping adds a force proportional to the velocity (or in the case of Thin Elastic Layer: the relative velocity between the two boundaries). The viscosity constant of the feature can be made dependent on the velocity by using the variables named vdamper1__tag, vdamper2__tag, and vdamper3__tag, which contain the velocities in the three local directions.

The Spring Foundation feature is most commonly used for simulating boundary conditions with a certain flexibility, such as the soil surrounding a construction. An other important use is for stabilizing parts that would otherwise have a rigid body singularity. This is a common problem in contact modeling before an assembly has actually settled. In this case a Spring Foundation acting on the entire domain is useful because it avoids the introduction of local forces.

A Thin Elastic Layer between used as a pair condition can be used to simulate thin layers with material properties which differ significantly from the surrounding domains. Common applications are gaskets and adhesives.

When a Thin Elastic Layer is applied on an internal boundary, it usually simulates a local flexibility, such as a fracture zone in a geological model.


Damp i n g and Lo s s

In this section:

• Overview of Damping and Loss

• Viscoelasticity

• Rayleigh Damping

• Equivalent Viscous Damping


• Explicit Damping

Overview of Damping and Loss

In some cases damping is included implicitly in the material model. This is the case for Viscoelasticity, for which damping operates on the shear components of stress and strain. Damping must be added explicitly as a subnode of the material node for material models that do not include damping, such as linear elastic materials.

Phenomenological damping models are typically invoked to model the intrinsic frictional damping present in most materials (material damping). These models are easiest to understand in the context of a system with a single degree of freedom. The following equation of motion describes the dynamics of such a system with viscous damping:

(7-12)

In this equation u is the displacement of the degree of freedom, m is its mass, c is the damping parameter, and k is the stiffness of the system. The time (t) dependent forcing term is f(t). This equation is often written in the form:

(7-13)

where c2m0 and 02km. In this case is the damping ratio (1 for critical

damping) and 0 is the resonant frequency of the system. In the literature it is more common to give values of than c. can also be readily related to many of the various

md2udt

---------- cdudt------- ku+ + f t =

d2udt

---------- 20dudt------- 0

2u+ +f t m

---------=

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measures of damping employed in different disciplines. These are summarized in Table 7-8.

In the frequency domain, the time dependence of the force and the displacement can be represented by introducing a complex force term and assuming a similar time dependence for the displacement. The equations

and

are written where is the angular frequency and the amplitude terms U and F can in general be complex (the arguments provide information on the relative phase of signals). Usually the real part is taken as implicit and is subsequently dropped. Equation 7-12 takes the following form in the frequency domain:

(7-14)

where the time dependence has canceled out on both sides. Alternatively this equation can be written as:

TABLE 7-8: RELATIONSHIPS BETWEEN MEASURES OF DAMPING

DAMPING PARAMETER

DEFINITION RELATION TO DAMPING RATIO

Damping ratio –

Logarithmic decrement

where t0 is a reference time and is the period of vibration for a decaying, unforced degree of freedom.

Quality factor

where is the bandwidth of the amplitude resonance measured at of its peak.

Loss factor

where Qh is the energy lost per cycle and Wh is the maximum potential energy stored in the cycle. The variables Qh and Wh are available as solid.Qh and solid.Wh.

At the resonant frequency:

c ccritical=

du t0

u t0 + ---------------------- ln=

d 2

1«

Q =

1 2

Q 1 2

1«

12------

Qh

Wh-------- =

2

1«

f t Re Fejt = u t Re Uejt =

2mU– jcU kU+ + F=


(7-15)

There are two basic damping models available—Rayleigh damping and models based on introducing complex quantities into the equation system.

Rayleigh Damping introduces damping in a form based on Equation 7-12. This means that the method can be applied generally in either the time or frequency domain. The parameter c in Equation 7-12 is defined as a fraction of the mass and the stiffness using two parameters, dM and dK, such that

(7-16)

Although this approach seems cumbersome with a one degree of freedom system, when there are many degrees of freedom m, k, and c become matrices and the technique can be generalized. Substituting this relationship into Equation 7-12 and rearranging into the form of Equation 7-13 gives:

Rayleigh damping can therefore be identified as equivalent to a damping factor at resonance of:

(7-17)

Note that Equation 7-17 holds separately for each vibrational mode in the system at its resonant frequency. In the frequency domain it is possible to use frequency dependent values of dM and dK. For example setting dM0 and dK2/0 produces a Equivalent Viscous Damping model at the resonant frequency.

While Rayleigh damping is numerically convenient, the model does not agree with experimental results for the frequency dependence of material damping over an extended range of frequencies. This is because the material damping forces behave more like frictional forces (which are frequency independent) than viscous damping forces (which increase linearly with frequency as implied by Equation 7-14). In the frequency domain it is possible to introduce loss factor damping, which has the desired property of frequency independence.

Loss Factor Dampingintroduces complex material properties to add damping to the model. As a result of this it can only be used in the frequency domain (for

2U– 2j0U 02U+ +

Fm-----=

c dMm dKk+=

d2udt

---------- dM dK02

+ dudt------- 0

2u+ +f t m

---------=

12---

dM

0----------- dK0+ =


388 | C H A P T E

eigenfrequency, frequency domain, or time harmonic studies). In the single degree of freedom case this corresponds to a complex value for the spring constant k. Setting c=0, but modifying the spring constant of the material to take a value k1j where is the loss factor, modifies the form of Equation 7-14 to:

(7-18)

Alternatively writing this in the form of Equation 7-15 gives:

Comparing these equations with Equation 7-14 and Equation 7-15 shows that the loss factor is related to and c by:

Equation 7-18 shows that the loss factor has the desired property of frequency independence. However it is clear that this type of damping cannot be applied in the time domain. In addition to using loss factor damping the material properties can be entered directly as complex values in COMSOL Multiphysics, which results in Explicit Damping

2mU– jkU kU+ + F=

2U– j02U 0

2U+ +Fm-----=

2 0------

k----c= =

are more complex and include coupling and electrical losses in addition to the material terms.

For piezoelectric materials, dK is only used as a multiplier of the structural contribution to the stiffness matrix when building-up the damping matrix as given by Equation 7-16. In the frequency domain studies, you can use the coupling and dielectric loss factors equal to dK to effectively achieve the Rayleigh damping involving the whole stiffness matrix.

• Viscoelasticity


• Equivalent Viscous Damping




Viscoelasticity

If Viscoelasticity is added to a Linear Elastic Material, the viscoelastic branches include damping automatically and no more damping is required. In the frequency domain the damping using a viscoelastic material corresponds to loss factor damping applied to the shear components of the material properties.

Rayleigh Damping

As discussed for a model with a single degree of freedom, the Rayleigh damping model defines the damping parameter c in terms of the mass m and the stiffness k as

where dM and dK are the mass and stiffness damping parameters, respectively. At any resonant frequency, f, this corresponds to a damping factor, given by:

(7-19)

Using this relationship at two resonant frequencies f1 and f2 with different damping factors 1 and 2 results in an equation system

As a result of its non-physical nature, the Rayleigh damping model can only be tuned to give the correct damping at two independent resonant frequencies or to give an approximately frequency independent damping response (which is physically what is usually observed) over a limited range of frequencies.

c dMm dKk+=

12---

dM2f----------- dK2f+ =

14f1------------ f1

14f2------------ f2

dM

dK

1

2

=

Using the same damping factors 1 and 2 at frequencies f1 and f2 does not result in the same damping factor in the interval. It can be shown that the damping parameters have the same damping at the two frequencies and less damping in between (see Figure 7-3).

Care must therefore be taken when specifying the model to ensure the desired behavior is obtained.


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Figure 7-3: An example of Rayleigh damping.

For many applications it is sufficient to leave dM as zero (the default value) and to define damping only using the dK coefficient. Then according to Equation 7-19 linearly increasing damping is obtained. If the damping ratio f0 or loss factor f0 is known at a given frequency f0, the appropriate value for dK is:

This model results in a well-defined, linearly increasing, damping term that has the defined value at the given frequency.

Equivalent Viscous Damping

Although equivalent viscous damping is independent of frequency, it is only possible to use it in a frequency response analysis. Equivalent viscous damping also uses a loss factor as the damping parameter, and can be implemented using the Rayleigh damping feature, by setting the stiffness damping parameter dK, to the loss factor, , divided by the excitation frequency:

Dam

ping

fact

or

f

Rayleigh damping

Specified damping

f2f1

dK f0 2f0 = =

All interfaces under the Structural Mechanics branch use zero default values (that is, no damping) for dM and dK. These default values must be changed to meet the specific modeling situation.

dK

2f---------

----= =


The mass damping factor, dM, should be set to zero.

Loss Factor Damping

Loss factor damping (sometimes referred to as material or structural damping) takes place when viscoelasticity is modeled in the frequency domain. The complex modulus G*() is the frequency-domain representation of the stress relaxation function of viscoelastic material. It is defined as

where G' is the storage modulus, G'' is the loss modulus, and their ratio sG''G' is the loss factor. The term G' defines the amount of stored energy for the applied strain, whereas G'' defines the amount of energy dissipated as heat; G', G'', and s can all be frequency dependent.

In COMSOL Multiphysics the loss information appears as a multiplier of the total strain in the stress-strain relationship:

.

For hyperelastic materials the loss information appears as a multiplier in strain energy density, and thus in the second Piola-Kirchhoff stress, S:

Loss factor damping is available for frequency response analysis and damped eigenfrequency analysis in all interfaces,

Explicit Damping

It is possible to define damping by modeling the dissipative behavior of the material using complex-valued material properties. In COMSOL Multiphysics, you can enter the complex-valued data directly, using i or sqrt(-1) for the imaginary unit.

G G jG+ 1 js+ G= =

D 1 js+ th– 0– 0+=

S 1 js+ WsE

----------=

The Hyperelastic Material node are available with the Nonlinear Structural Materials Module.


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8

T h e P i e z o e l e c t r i c D e v i c e s I n t e r f a c e

This chapter describes the background theory for Piezoelectric Devices interface, which is found under the Structural Mechanics branch ( ) when adding a physics interface.

• The Piezoelectric Devices Interface


• Piezoelectric Damping

389

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Th e P i e z o e l e c t r i c De v i c e s I n t e r f a c e

The Piezoelectric Devices (pzd) interface ( ), found under the Structural Mechanics branch ( ) when adding a physics interface, combines Solid Mechanics and Electrostatics together with the constitutive relationships required to model piezoelectrics. Both the direct and inverse piezoelectric effects can be modeled and the piezoelectric coupling can be formulated using the strain-charge or stress-charge forms.

The piezoelectric coupling can be presented in stress-charge or strain-charge form. All solid mechanics and electrostatics functionality for modeling is also accessible to include surrounding linear elastic solids or air domains. For example, add any solid mechanics material for other solid domain, a dielectric model for air, or a combination.

When this interface is added, these default nodes are also added to the Model Builder— Piezoelectric Material, Free (for the solid mechanics and default boundary conditions), Zero Charge (for the electric potential), and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, loads, constraints, and solid mechanics and electric materials. You can also right-click Piezoelectric Devices to select physics from the context menu.

In 2D and 2D axial symmetry, adding a Piezoelectric Devices interface also adds predefined base-vector coordinate systems for the material’s (in the plane 2D case) XY-, YZ-, ZX-, YX-, XZ-, and XY-planes. These additional coordinate systems are useful for simplifying the material orientation for the piezoelectric material.


The interface identifier is used primarily as a scope prefix for variables defined by the physics interface. Refer to such interface variables in expressions using the pattern

All functionality from the Solid Mechanics and Electric Current interfaces is accessible for modeling the solid and electric properties and non-piezoelectric domains. Only the features unique to this interface are described in this section. For details about the shared features see:

• The Solid Mechanics Interface in this guide

• The Electrostatics Interface in the COMSOL Multiphysics Reference Manual

R 8 : T H E P I E Z O E L E C T R I C D E V I C E S I N T E R F A C E

<identifier>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the identifier string must be unique. Only letters, numbers and underscores (_) are permitted in the Identifier field. The first character must be a letter.

The default identifier (for the first interface in the model) is pzd.



2 D A P P R O X I M A T I O N

T H I C K N E S S

S T R U C T U R A L TR A N S I E N T B E H A V I O R

From the Structural transient behavior list, select Include inertial terms (the default) or Quasi-static. Use Quasi-static to treat the elastic behavior as quasi-static (with no mass effects; that is, no second-order time derivatives). Selecting this option gives a more efficient solution for problems where the variation in time is slow when compared to the natural frequencies of the system. The default solver for the time stepping is changed from generalized alpha to BDF when Quasi-static is selected.

From the 2D approximation list select Plane stress or Plane strain (the default). When modeling using plane stress, the Piezoelectric Devices interface solves for the out-of-plane strain components in addition to the displacement field u.

Enter a value or expression for the Thickness d (SI unit: m). The default value of 1 m is suitable for plane strain models, where it represents a a unit-depth slice, for example. For plane stress models, enter the actual thickness, which should be small compared to the size of the plate for the plane stress assumption to be valid. In rare cases, use a Change Thickness

node to change thickness in parts of the geometry.

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This interface defines these dependent variables (fields): the Displacement field u (and its components) and the Electric potential V. The names can be changed but the names of fields and dependent variables must be unique within a model.


To display this section, click the Show button ( ) and select Advanced Physics Options. Normally these settings do not need to be changed.


To display this section, click the Show button ( ) and select Discretization. Select Linear, Quadratic (the default), Cubic, Quartic, or (in 2D) Quintic for the Displacement

field and Electric potential. Specify the Value type when using splitting of complex

variables—Real or Complex (the default).

Domain, Boundary, Edge, Point, and Pair Nodes for the Piezoelectric Devices Interface

The Piezoelectric Devices Interface has these domain, boundary, edge, point, and pair nodes, listed in alphabetical order, available from the Physics ribbon toolbar (Windows users), Physics context menu (Mac or Linux users), or right-click to access the context menu (all users).


• Domain, Boundary, Edge, Point, and Pair Nodes for the Piezoelectric Devices Interface


In general, to add a node, go to the Physics toolbar, no matter what operating system you are using. However, to add subnodes, right-click the parent node.




• Damping and Loss

• Dielectric Loss

• Electrical Conductivity (Time-Harmonic)

• Electrical Material Model

• Initial Values



• Remanent Electric Displacement

• Added Mass

• Antisymmetry

• Body Load

• Boundary Load

• Edge Load


• Free

• Gravity



• Point Load

• Pre-Deformation




• Roller

• Rotating Frame


• Symmetry



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These nodes are described for the Electrostatics interface in the COMSOL Multiphysics Reference Manual:

Piezoelectric Material

Use the Piezoelectric Material node to define the piezoelectric material properties on stress-charge form using the elasticity matrix and the coupling matrix or on strain-charge form using the compliance matrix and the coupling matrix. The default settings is to use material data defined for the material in the domain. Right-click Piezoelectric Material to add Electrical Conductivity (Time-Harmonic), Initial Stress and Strain, and Damping and Loss subnodes as required.



• Electric Displacement Field

• Electric Potential

• Ground

• Line Charge

• Line Charge (on Axis)

• Line Charge (Out-of-Plane)

• Point Charge

• Point Charge (on Axis)

• Space Charge Density

• Surface Charge Density

• Thin Low Permittivity Gap

• Zero Charge


For entering these matrices, the ordering is different from the standard ordering used in COMSOL Multiphysics. Instead, use the following order (Voigt notation), which is the common convention for piezoelectric materials: xx, yy, zz, yz, xz, zy.



This section has field variables that appear as model inputs, if the current settings include such model inputs. By default, this section is empty.



P I E Z O E L E C T R I C M A T E R I A L P R O P E R T I E S

Select a Constitutive relation—Stress-charge form or Strain-charge form. For each of the following, the default uses values From material. Select User defined to enter other values in the matrix or field as required.

• For Stress-charge form, select an Elasticity matrix (ordering: xx, yy, zz, yz, xz, xy) (cE) (SI unit: 1/Pa).

• For a Strain-charge form, select a Compliance matrix (ordering: xx, yy, zz, yz, xz, xy)

(sE) (SI unit: 1/Pa).

• Select a Coupling matrix (ordering: xx, yy, zz, yz, xz, xy) (d) (SI unit: C/m2 or C/N).

• Select a Relative permittivity (erS or erT) (dimensionless).

• Select a Density (p) (SI unit: kg/m3).


If a study step is geometrically nonlinear, the default behavior is to use a large strain formulation in all domains. There are however some cases when you would still want to use a small strain formulation for a certain domain. In those cases, select the Force

linear strains check box. When selected, a small strain formulation is always used, independently of the setting in the study step.

Electrical Material Model

The Electrical Material Model adds an electric field to domains in a piezoelectric device model that only includes the electric field. Right-click the node to add Electrical Conductivity (Time-Harmonic) and Dielectric Loss subnodes as required.




396 | C H A P T E




This section contains field variables that appear as model inputs, if the current settings include such model inputs. By default, this section is empty.



E L E C T R I C F I E L D

Select a Constitutive relation—Relative permittivity, Polarization, or Remanent

displacement.

• If Relative permittivity is selected, also choose a Relative permittivity (r) (dimensionless). The default uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic and enter values in the matrix or field.

• If Polarization is selected, enter the Polarization P (SI unit: C/m2) coordinates.

• If Remanent displacement is selected, select a Relative permittivity (r). The default uses values From material. If User defined is selected, choose Isotropic, Diagonal, Symmetric, or Anisotropic and enter values in the matrix or field. Then enter the Remanent displacement (Dr) (SI unit: C/m2) coordinates.


If a study step is geometrically nonlinear, the default behavior is to use a large strain formulation in all domains. There are however some cases when you would still want to use a small strain formulation for a certain domain. In those cases, select the Force

linear strains check box. When selected, a small strain formulation is always used, independently of the setting in the study step.


• See The Solid Mechanics Interface for details about this section.


Electrical Conductivity (Time-Harmonic)

Right-click the Piezoelectric Material node or the Electrical Material Model to add an Electrical Conductivity (Time-Harmonic) subnode. This adds ohmic conductivity to the material. For example, if the model has metal electrodes, or if the piezoelectric material might not be a perfect insulator but has some electrical conductivity. Because The Piezoelectric Devices Interface solves for the charge balance equation (that is, electrostatics) this conductivity would lead to a time integral of the ohmic current in the equation. This feature can therefore only operate in a time-harmonic study (as pointed out in the name), and the “equivalent electric displacement” Jij appears in the equation.





C O N D U C T I O N C U R R E N T

Select an Electrical conductivity (SI unit: S/m). Select:

• From material to use the conductivity value from the domain material.

• Linearized resistivity to define the electric resistivity (and conductivity) as a linear function of temperature.

• User defined to enter a value (SI unit: S/m) or expressions for an isotropic or anisotropic conductivity. Select Isotropic, Diagonal, Symmetric, or Anisotropic from the list based on the properties of the conductive media.

If Linearized resistivity is selected, each default setting in the corresponding Reference

temperature (Tref), Resistivity temperature coefficient (), and Reference resistivity (0) lists is From material, which means that the values are taken from the domain material. To specify other values for these properties, select User defined from the corresponding list and then enter a value or expression in the applicable field.


398 | C H A P T E

Damping and Loss

Right-click the Piezoelectric Material node to add a Damping and Loss subnode, which adds damping (Rayleigh damping or loss damping), coupling losses, and dielectric losses to the piezoelectric material.



D A M P I N G S E T T I N G S

Select a Damping type—Rayleigh damping, Loss factor for cE, Loss factor for sE, No

damping, or Isotropic loss factor:

• No damping

• For Rayleigh damping, enter the Mass damping parameter dM and the Stiffness

damping parameter in the dM corresponding fields. The default values are 0, which means no damping.

• For Loss factor for cE, select From material (the default) from the Loss factor for

elasticity matrix cE list to use the value from the material or select User defined to enter values or expressions for the loss factor in the associated fields. Select Symmetric to enter the components of cE in the upper-triangular part of a symmetric 6-by-6 matrix or select Isotropic to enter a single scalar loss factor. The default values are 0.

• For Loss factor for sE, from the Loss factor for compliance matrix sE list, select From

material (the default) to use the value from the material or select User defined to enter values or expressions for the loss factor in the associated fields. Select Symmetric to enter the components of sE in the upper-triangular part of a symmetric 6-by-6 matrix or select Isotropic to enter a single scalar loss factor. The default values are 0.

• For an Isotropic loss factor s, select From material (the default) from the Isotropic

structural loss factor list to take the value from the material or select User defined to enter a value or expression for the isotropic loss factor in the field. The default value is 0.

C O U P L I N G L O S S S E T T I N G S

Select a Coupling loss—No loss, Loss factor for e, or Loss factor for d.


For Loss factor for e and Loss factor for d, select a Loss factor for coupling matrix e or d

from the list. Select User defined to enter values or expressions for the loss factor in the associated fields. Select Symmetric to enter the components of e or d in the upper-triangular part of a symmetric 6-by-6 matrix or select Isotropic to enter a single scalar loss factor. The default values are 0.

D I E L E C T R I C L O S S S E T T I N G S

From the Dielectric loss list, select Loss factor for S, Loss factor for T, or No loss.

For Loss factor for S and Loss factor for T, select a Loss factor for permittivity. Select From material (the default) to use the value from the material or select User defined to enter values or expressions for the loss factor in the associated fields. Select Symmetric to enter the components of eS or eT in the upper-triangular part of a symmetric 6-by-6 matrix, select Isotropic to enter a single scalar loss factor, or select Diagonal. The default values are 0.

Remanent Electric Displacement

Right-click the Piezoelectric Material node to add a Remanent Electric Displacement subnode to include a remanent electric displacement vector Dr (the displacement when no electric field is present).



R E M A N E N T E L E C T R I C D I S P L A C E M E N T

Enter the components of the remanent electric displacement Dr (SI unit: C/m2) in the Remanent displacement fields (the default values are 0 C/m2).

Dielectric Loss

Right-click the Electrical Material Model node to add a Dielectric Loss subnode to include a dielectric loss using a dielectric loss factor.


For a default node, the setting inherits the selection from the parent node, and cannot be edited; that is, the selection is automatically selected and is the same as for the


400 | C H A P T E

interface. When nodes are added from the context menu, you can select Manual from the Selection list to choose specific domains or select All domains as required.

D I E L E C T R I C L O S S S E T T I N G S

The default Dielectric loss factor uses values From material. If User defined is selected, then also select Isotropic, Diagonal, Symmetric, or Anisotropic and enter one or more components in the field or matrix. The default values are 0.

Initial Values

The Initial Values node adds an initial value for the displacement field and the electric potential. Add additional Initial Values nodes from the Physics toolbar.




Enter the initial values as values or expressions for the Displacement field u (SI unit: m) and the Electric potential V (SI unit: V).

Periodic Condition

The Periodic Condition node adds a periodic boundary condition. This periodicity make uix0uix1 for a displacement component ui or similarly for the electric potential. You can control the direction that the periodic condition applies to and if it applies to the electric potential. Right-click the Periodic Condition node to add a Destination

Selection subnode if required. If the source and destination boundaries are rotated with respect to each other, this transformation is automatically performed, so that corresponding displacement components are connected.

This feature works well for cases like opposing parallel boundaries. In other cases use a Destination Selection subnode to control the destination. By default it contains the selection that COMSOL Multiphysics identifies.


PE R I O D I C I T Y S E T T I N G S

Select a Type of periodicity—Continuity (the default), Antiperiodicity, Floquet periodicity, Cyclic symmetry, or User defined.

• If Floquet periodicity is selected, enter a k-vector for Floquet periodicity kF (SI unit: rad/m) for the X, Y, and Z coordinates (3D models), or the R and Z coordinates (2D axisymmetric models), or X and Y coordinates (2D models).


defined. If User defined is selected, enter a value for S (SI unit: rad; default value: 0). For any selection, also enter a Azimuthal mode number m (dimensionless; default value: 0).

• If User defined is selected, select the Periodic in u, Periodic in v (for 3D and 2D models), and Periodic in w (for 3D and 2D axisymmetric models) check boxes as required. For all dimensions the Periodic in V check box is also available. Then for each selection, choose the Type of periodicity—Continuity (the default) or Antiperiodicity.





402 | C H A P T E

Th eo r y f o r t h e P i e z o e l e c t r i c De v i c e s I n t e r f a c e

The Piezoelectric Devices Interface theory is described in this section:

• The Piezoelectric Effect

• Piezoelectric Constitutive Relations


• Piezoelectric Dissipation

• Initial Stress, Strain, and Electric Displacement


• Damping and Losses Theory

• References for the Piezoelectric Devices Interface

The Piezoelectric Effect

The piezoelectric effect manifests itself as a transfer of electric to mechanical energy and vice versa. It is present in many crystalline materials, while some materials such as quartz, Rochelle salt, and lead titanate zirconate ceramics display the phenomenon strongly enough for it to be of practical use.

The direct piezoelectric effect consists of an electric polarization in a fixed direction when the piezoelectric crystal is deformed. The polarization is proportional to the deformation and causes an electric potential difference over the crystal.

The inverse piezoelectric effect, on the other hand, constitutes the opposite of the direct effect. This means that an applied potential difference induces a deformation of the crystal.

P I E Z O E L E C T R I C I T Y C O N V E N T I O N S

The documentation and the Piezoelectric Devices interface use piezoelectricity conventions as much as possible. These conventions differ from those used in other structural mechanics interfaces. For instance, the numbering of the shear components in the stress-strain relation differs, as the following section describes. However, the names of the stress and strain components remain the same as in the other structural mechanics interfaces.


Piezoelectric Constitutive Relations

It is possible to express the relation between the stress, strain, electric field, and electric displacement field in either a stress-charge form or strain-charge form:

S T R E S S - C H A R G E

S T R A I N - C H A R G E

The naming convention differs in piezoelectricity theory compared to structural mechanics theory, but the Piezoelectric Devices interface uses the structural mechanics nomenclature. The strain is named instead of S, and the stress is named instead of T. This makes the names consistent with those used in the other structural mechanics interfaces.

The numbering of the strain and stress components is also different in piezoelectricity theory and structural mechanics theory, and it is quite important to keep track of this aspect in order to provide material data in the correct order. In structural mechanics the following is the most common numbering convention, and it is also the one used as default in the structural mechanics interfaces:

In contrast, textbooks on piezoelectric effects and the IEEE standard on piezoelectric effects use the following numbering convention (also called Voigt notation):

T cES eTE–=

D eS SE+=

S sET dTE+=

D dT TE+=

xx

yy

zx

xy

yz

xz

=

xx

yy

zz

xy

yz

xz

xx

yy

zz

2xy

2yz

2xz

= =

T H E O R Y F O R T H E P I E Z O E L E C T R I C D E V I C E S I N T E R F A C E | 403

404 | C H A P T E

The Piezoelectric Devices interface uses the immediately preceding piezo numbering convention (Voigt notation) to make it easier to work with material data and to avoid mistakes.

The constitutive relation using COMSOL Multiphysics symbols for the different constitutive forms are thus:

S T R E S S - C H A R G E

S T R A I N - C H A R G E

Most material data appears in the strain-charge form, and it can be easily transformed into the stress-charge form. In COMSOL Multiphysics both constitutive forms can be used; simply select one, and the software makes any necessary transformations. The following equations transform strain-charge material data to stress-charge data:

xx

yy

zz

yz

xz

xy

=

xx

yy

zz

yz

xz

xy

xx

yy

zz

2yz

2xz

2xy

= =

cE eTE–=

D e 0rSE+=

sE dTE+=

D d 0rTE+=

cE sE1–

=

e d sE1–

=

S 0rT d sE1– dT

–=


Piezoelectric Material

The Piezoelectric Devices interface also has different materials for easier modeling of piezo components. This means that the material for each domain can be defined as:

• Piezoelectric material (the default material)

• Purely solid as a linear elastic or nonlinear material

• Purely dielectric using an electrical material (to model surrounding air, for example)

The piezoelectric material operates as described above, whereas using the two other materials, structural and electrical problems can be modeled, together or either of them independently.

Piezoelectric Dissipation

In order to define dissipation in the piezoelectric material for a time-harmonic analysis, all material properties in the constitutive relations can be complex-valued matrices where the imaginary part defines the dissipative function of the material.

As described in Damping and Losses Theory complex-valued data can be defined directly in the fields for the material properties, or a real-valued material X and a set of loss factors X can be defined, which together form the complex-valued material data

See also the same references for an explanation of the sign convention. It is also possible to define the electrical conductivity of the piezoelectric material: S or T depending on the constitutive relation. Electrical conductivity does not appear directly in the constitutive equation, but it appears as an additional term in the variational formulation (weak equation).

Initial Stress, Strain, and Electric Displacement

Using the piezoelectrical interfaces initial stress (0), initial strain (0), and initial electric displacement (D0) can be defined for models. In the constitutive relation for piezoelectric material these additions appear in the stress-charge formulation:

X˜

X 1 jX =

The conductivity does not change during transformation between the formulations. S and T are used to get fully-defined materials in each formulation.


406 | C H A P T E

When solving the model, these program does not interpret these fields as a constant initial state, but they operate as additional fields that are continuously evaluated. Thus use these initial field to add, for example, thermal expansion or pyroelectric effects to models.

Geometric Nonlinearity for the Piezoelectric Devices Interface

P I E Z O E L E C T R I C M A T E R I A L S W I T H L A R G E D E F O R M A T I O N S

The linear piezoelectric equations as presented in Piezoelectric Constitutive Relations with engineering strains are valid if the model undergoes only relatively small deformations. As soon as the model contains larger displacements or rotations, these equations produce spurious strains that result in an incorrect solution. To overcome this problem, so-called large deformation piezoelectrical equations are required.

The Piezoelectric Devices interface implements the large deformation piezoelectrical equations according to Yang (Ref. 8). Key items of this formulation are:

• The strains are calculated as the Green-Lagrange strains, ij:

(8-1)

Green-Lagrange strains are defined with reference to an undeformed geometry. Hence, they represent a Lagrangian description. In a small-strain, large rotational analysis, the Green-Lagrange strain corresponds to the engineering strain in directions that follow the deformed body.

• Electrical field variables are calculated in the material directions, and the electric displacement relation is replaced by an expression that produce electric polarization in the material orientation of the solid.

• In the variational formulation, the electrical energy is split into two parts: The polarization energy within the solid and the electric energy of free space occupied by the deformed solid.

The first two items above result in another set of constitutive equations for large deformation piezoelectricity:

cE 0– eTE– 0+=

D e 0– 0rSE D0+ +=

ij12---

uiXj

--------ujXi

--------ukXi

---------ukXj

---------+ + =


where S is the second Piola-Kirchhoff stress; is the Green-Lagrange strain, Em and Pm are the electric field and electric polarization in the material orientation; I is the identity matrix; and cE, e, and rS are the piezoelectric material constants. The expression within parentheses equals the dielectric susceptibility of the solid:

The electric displacement field in the material orientation results from the following relation

where C is the right Cauchy-Green tensor

Fields in the global orientation result from the following transformation rules:

(8-2)

where F is the deformation gradient; J is the determinant of F; and v and V are the volume charge density in spatial and material coordinates respectively. The deformation gradient is defined as the gradient of the present position of a material point xX + u:

D E C O U P L E D M A T E R I A L S W I T H L A R G E D E F O R M A T I O N S

The large deformation formulation described in the previous section applies directly to non-piezoelectric materials if the coupling term is set to zero: e0. In that case, the structural part corresponds to the large deformation formulation described for the solid mechanics interfaces.

S cE eTEm–=

Pm e 0rS 0I– Em+=

0rS 0I– =

Dm Pm 0JC 1– Em+=

C FTF=

E F T– Em=

P J 1– FPm=

D J 1– FDm=

v VJ 1–=

F xX

-------=


408 | C H A P T E

The electrical part separates into two different cases: For solid domains the electric energy consists of polarization energy within the solid and the electric energy of free space occupied by the deformed solid—the same as for the piezoelectric materials. For nonsolid domains this separation does not occur, and the electric displacement in these domains directly results from the electric field—the electric displacement relation:

L A R G E D E F O R M A T I O N A N D D E F O R M E D M E S H

The Piezoelectric Devices interface can be coupled with the Moving Mesh (ALE) interface in a way so that the electrical degrees of freedom are solved in an ALE frame. This feature is intended to be used in applications where a model contains nonsolid domains, such as modeling of electrostatically actuated structures. This functionality is not required for modeling of piezoelectric or other solid materials.

The use of ALE has impacts on the formulation of the electrical large deformation equations. The first impact is that with ALE, the gradient of electric potential directly results in the electric field in the global orientation, and the material electric field results after transformation.

The most visible impact is on the boundary conditions. With ALE any surface charge density or electric displacement is defined per the present deformed boundary area, whereas for the case without ALE they are defined per the undeformed reference area.

Damping and Losses Theory

H Y S T E R E T I C L O S S

The equivalent viscous and loss factor damping are special cases of a more general way of defining damping: hysteretic loss. Generally, and independently of the microscopic origin of the loss, the dissipative behavior of the material can be modeled using complex-valued material properties. For the case of piezoelectric materials, this means that the constitutive equations are written as follows:

For the stress-charge formulation

Dm 0rEm=

On nonsolid domains the global orientation of the fields is not known unless the ALE method is used.


and for the strain-charge formulation

where , , , , and are complex-valued matrices, where the imaginary part defines the dissipative function of the material.

Similarly to the real-valued material data, it is not possible to freely define the complex-valued data. Instead the data must fulfill certain requirement to represent physically proper materials. A key requirement is that the dissipation density is positive; that is, there is no power gain from the passive material. This requirement sets rules for the relative magnitudes for all material parameters. This is important to be aware of, especially when defining the coupling losses.

In COMSOL Multiphysics the complex-valued data can be entered directly, or the concept of loss factors can be used. Similarly to the loss factor damping, the complex data is represented as pairs of a real-valued parameter

and a loss factor

the ratio of the imaginary and real part, and the complex data is then

where the sign depends on the material property used. The loss factors are specific to the material property, and thus they are named according to the property they refer to, for example, cE. For a structural material without coupling, simply use s, the structural loss factor.

Depending on the field, different terminology is in use. For example, the loss tangent tan might be referred to when working with electrical applications. The loss tangent has the same meaning as the loss factor. Often the quality factor Qm is defined for a material. The quality factor Qm and the loss factor i are inversely related: i1 Qm,

cE eT

E–=

D e SE+=

sE dT

E+=

D d TE+=

cE sE S˜ T

˜ d

X˜

X real X˜

=

X imag X˜

real X˜

=

X˜

X 1 jX =


410 | C H A P T E

where i is the loss factor for cE, sE, or the structural loss factor depending on the material.

The Piezoelectric Devices interface uses a formulation that assumes that a positive loss factor corresponds to a positive loss. The complex-valued data is then based on sign rules. For piezoelectric materials, the following equations apply (m and n refer to elements of each matrix):

(8-3)

The losses for the non-coupled material models are more straightforward to define. Using the complex stiffness and permittivity, the following equations describe the lossy material:

(8-4)

Often fully defined complex-valued data is not accessible. In the Piezoelectric Devices interface the loss factors can be defined as full matrices or as scalar isotropic loss factors independently of the material and the other coefficients.

For more information about hysteretic losses, see Ref. 1 to Ref. 4.

T H E L O S S F A C T O R U S I N G D I F F E R E N T D A M P I N G TY P E S

The following damping types use an isotropic loss factor s:

• Loss factor damping

• Equivalent viscous damping

• Isotropic loss

In each case the meaning of the loss factor is the same: the fractional loss of energy per cycle.

cEm n

cEm n 1 jcE

m n+ =

em n

em n 1 jem n

– =

Sm n

Sm n 1 jS

m n– =

sEm n

sEm n 1 jsE

m n– =

dm n

dm n 1 jdm n

– =

Tm n

Tm n 1 jT

m n– =

D˜ m n

1 jm n+ D

m n=

em n

1 jem n

– em n

=


The difference between these damping types is how the loss enters the equations. Using the isotropic loss, s is used to build complex-valued material properties, whereas when using the loss factor damping, s appears in a complex-valued multiplier in the stress-strain relation. In the equivalent viscous damping, s appears in a complex-valued and frequency-dependent expression for dK of the Rayleigh damping model.

E L E C T R I C A L C O N D U C T I V I T Y A N D D I E L E C T R I C L O S S E S

For frequency response and damped eigenfrequency analyses, the electrical conductivity of the piezoelectric and decoupled material (see Ref. 2, Ref. 5, and Ref. 6) can be defined. Depending on the formulation of the electrical equation, the electrical conductivity appears in the variational formulation (the weak equation) either as an effective electric displacement

(the actual displacement variables do not contain any conductivity effects) or in the total current expression

where Jp = eE is the conductivity current and Jd is the electric displacement current.

Both a dielectric loss factor (Equation 8-3 and Equation 8-4) and the electrical conductivity can be defined at the same time. In this case, ensure that the loss factor refers to the alternating current loss tangent, which dominates at high frequency, where the effect of ohmic conductivity vanishes (Ref. 7).

References for the Piezoelectric Devices Interface

1. R. Holland and E. P. EerNisse, Design of Resonant Piezoelectric Devices, Research Monograph No. 56, The M.I.T. Press, 1969.

2. T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, 1990.

3. A.V. Mezheritsky, “Elastic, Dielectric, and Piezoelectric Losses in Piezoceramics: How it Works all Together,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 51, no. 6, 2004.

D˜

r0E jJp

------–=

J Jd Jp+=


412 | C H A P T E

4. K. Uchino and S. Hirose, “Loss Mechanisms in Piezoelectrics: How to Measure Different Losses Separately,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 48, no. 1, pp. 307–321, 2001.

5. P.C.Y. Lee, N.H. Liu, and A. Ballato, “Thickness Vibrations of a Piezoelectric Plate with Dissipation,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 51, no. 1, 2004.

6. P.C.Y. Lee and N.H. Liu, “Plane Harmonic Waves in an Infinite Piezoelectric Plate with Dissipation,” Frequency Control Symposium and PDA Exhibition, IEEE International, pp. 162–169, 2002.

7. C.A. Balanis, “Electrical Properties of Matter,” Advanced Engineering Electromagnetics, John Wiley & Sons, 1989.

8. J. Yang, An Introduction to the Theory of Piezoelectricity, Springer Science and Business Media, N.Y., 2005.


P i e z o e l e c t r i c Damp i n g

In this section:

• About Piezoelectric Materials

• Piezoelectric Material Orientation

• Piezoelectric Losses

• References for Piezoelectric Damping

About Piezoelectric Materials

Piezoelectric materials become electrically polarized when strained. From a microscopic perspective, the displacement of atoms within the unit cell (when the solid is deformed) results in electric dipoles within the medium. In certain crystal structures this combines to give an average macroscopic dipole moment or electric polarization. This effect, known as the direct piezoelectric effect, is always accompanied by the converse piezoelectric effect, in which the solid becomes strained when placed in an electric field.

Within a piezoelectric there is a coupling between the strain and the electric field, which is determined by the constitutive relation:

(8-5)

Here, S is the strain, T is the stress, E is the electric field, and D is the electric displacement field. The material parameters sE, d, and T, correspond to the material compliance, the coupling properties and the permittivity. These quantities are tensors of rank 4, 3, and 2 respectively, but, since the tensors are highly symmetric for physical reasons, they can be represented as matrices within an abbreviated subscript notation, which is usually more convenient. In the Piezoelectric Devices interface, the Voigt notation is used, which is standard in the literature for piezoelectricity but which differs from the defaults in the Solid Mechanics interface. Equation 8-5 is known as the strain-charge form of the constitutive relations. The equation can be re-arranged into the stress-charge form, which relates the material stresses to the electric field:

S sET dTE+=

D dT TE+=

P I E Z O E L E C T R I C D A M P I N G | 413

414 | C H A P T E

(8-6)

The material properties, cE, e, and S are related to sE, d, and T. Note that it is possible to use either form of the constitutive relations. In addition to Equation 8-5 or Equation 8-6, the equations of solid mechanics and electrostatics must also be solved within the material.

Piezoelectric Material Orientation

The orientation of a piezoelectric crystal cut is frequently defined by the system introduced by the I.R.E. standard of 1949 (Ref. 8). This standard has undergone a number of subsequent revisions, with the final revision being the IEEE standard of 1989 (Ref. 9). Unfortunately the more recent versions of the standard have not been universally adopted, and significant differences exist between the 1949 and the 1987 standards. The 1987 standard was ultimately withdrawn by the IEEE. COMSOL Multiphysics follows the conventions used in the book by Auld (Ref. 10) and defined by the 1987 standard. While these conventions are often used for many piezoelectric materials, unfortunately practitioners in the quartz industry usually adhere to the older 1947 standard, which results in different definitions of crystal cuts and of material properties.

The stiffness, compliance, coupling, and dielectric material property matrices are defined with the crystal axes aligned with the local coordinate axes. In the absence of a user-defined coordinate system, the local system corresponds to the global X, Y, and Z coordinate axes.

The crystal axes used to define material properties correspond to the 1987 IEEE standard. All piezoelectric material properties are defined using the Voigt form of the abbreviated subscript notation, which is almost universally employed in the literature

T cES eTE–=

D eS SE+=

• The Piezoelectric Devices Interface


The material properties are defined in the material frame, so that if the solid rotates during deformation the material properties rotate with the solid. See Geometric Nonlinearity, Frames, and the ALE Method.


(this differs from the standard notation used for the Solid Mechanics interface material properties). To define a particular crystal cut, a local set of rotated coordinates must be defined; this local system then corresponds to the orientation of the crystal axes within the model.

The crystal cuts are also defined differently within the 1949 and 1987 standards. Both standards use a notation that defines the orientation of a virtual slice (the plate) through the crystal. The crystal axes are denoted X, Y, and Z and the plate, which is usually rectangular, is defined as having sides l, w, and t (length, width, and thickness). Initially the plate is aligned with respect to the crystal axes and then up to three rotations are defined, using a right-handed convention about axes embedded along the l, w, and t sides of the plate. Taking AT cut quartz as an example, the 1987 standard defines the cut as: (YXl) 35.25°. The first two letters in the bracketed expression always refer to the initial orientation of the thickness and the length of the plate. Subsequent bracketed letters then define up to three rotational axes, which move with the plate as it is rotated. Angles of rotation about these axes are specified after the bracketed expression in the order of the letters, using a right-handed convention. For AT cut quartz only one rotation, about the l axis, is required. This is illustrated in Figure 8-2. Note that within the 1949 convention AT cut quartz is denoted as: (YXl) 35.25°, since the X-axis rotated by 180° in this convention and positive angles therefore correspond to the opposite direction of rotation (see Figure 8-1).

For some materials, the crystal X, Y, and Z axes are defined differently between the 1987 IEEE standard and the 1949 I.R.E. standard. Figure 8-1 shows the case of right-handed quartz (which is included in the COMSOL material library as quartz; see Piezoelectric Materials Database in the COMSOL Multiphysics Reference Manual), which has different axes defined within the two standards.

The different axes sets result in different material properties so, for example, the elasticity or stiffness matrix component cE14 of quartz takes the value 18 GPa in the 1987 standard and 18 GPa in the 1949 standard.


416 | C H A P T E

Figure 8-1: Crystallographic axes defined for right-handed quartz in COMSOL and the 1987 IEEE standard (color). The 1949 standard axes are shown for comparison (gray).

Figure 8-1 is reproduced with permission from: IEEE Std 176-1987 - IEEE Standard on Piezoelectricity, reprinted with permission from IEEE, 3 Park Avenue, New York, NY 10016-5997 USA, copyright 1987, by IEEE. This figure must not be reprinted or further distributed without prior written permission from the IEEE.


Figure 8-2: Definition of the AT cut of quartz within the IEEE 1987 standard. The AT cut is defined as: (YXl) 35.25°. The first two bracketed letters specify the initial orientation of the plate, with the thickness direction, t, along the crystal Y axis and the length direction, l, along the X axis. Then up to three rotations about axes that move with the plate are specified by the corresponding bracketed letters and the subsequent angles. In this case only one rotation is required about the l axis, of 35.25° (in a right-handed sense).

When defining material properties it is necessary to consider the orientation of the plate with respect to the global coordinate system in addition to the orientation of the plate with respect to the crystallographic axes. Consider once again the example of AT cut quartz in Figure 8-2. The definition of the appropriate local coordinate system depends on the desired final orientation of the plate in the global coordinate system. One way to set up the plate is to orientate its normal parallel to the Y axis in the global coordinate system. Figure 8-3 shows how to define the local coordinate system in this

Because COMSOL Multiphysics allows user-defined material parameters, it is possible to add a user-defined material defined within the 1949 standard if the use of the 1987 standard is inconvenient. In any case, significant care must be taken when entering material properties and when defining the rotated coordinate system for a given cut. In the literature, the particular standard being employed to define material properties and cuts is rarely cited.


418 | C H A P T E

case. Figure 8-4 shows how to define the local system such that the plate has its normal parallel to the global Z axis.

Usually it is most convenient to define the local coordinates with a Rotated System node, which defines three Euler angles according to the ZXZ convention (rotation about Z, then X, then Z again). Note that these Euler angles define the local (crystal) axes with respect to the global axes—this is distinct from the approach of defining the cut (global) axes with respect to the crystal (local) axes.

Figure 8-3: Defining an AT cut crystal plate within COMSOL, with normal in the global Y-direction. Within the 1987 IEEE standard the AT cut is defined as (YXl) -35.25°. Start with the plate normal or thickness in the Ycr direction (a) and rotate the plate 35.25°

In both cases it is critical to keep track of the orientation of the local system with respect to the global system, which is defined depending on the desired orientation of the plate in the model.

There are also a number of methods to define the local coordinate system with respect to the global system.


about the l axis (b). The global coordinate system rotates with the plate. Finally rotate the entire system so that the global coordinate system is orientated as it appears in COMSOL (c). The local coordinate system should be defined with the Euler angles (ZXZ - 0, 35.25°, 0).(d) shows a coordinate system for this system in COMSOL.

Figure 8-4: Defining an AT cut crystal plate within COMSOL, with normal in the global Z-direction. Within the 1987 IEEE standard the AT cut is defined as (YXl) 35.25°. Begin with the plate normal in the Zcr-direction, so the crystal and global systems are coincident. Rotate the plate so that its thickness points in the Ycr-direction (the starting point for the IEEE definition), the global system rotates with the plate (b). Rotate the plate 35.25° about the l axis (d). Finally rotate the entire system so that the global coordinate system is orientated as it appears in COMSOL (d). The local coordinate system should be


420 | C H A P T E

defined with the Euler angles (ZXZ: 0, -54.75°, 0). (e) shows a coordinate system for this system in COMSOL.

Piezoelectric Losses

Losses in piezoelectric materials can be generated both mechanically and electrically.

In the frequency domain these can be represented by introducing complex material properties in the elasticity and permittivity matrices, respectively. Taking the mechanical case as an example, this introduces a phase lag between the stress and the strain, which corresponds to a Hysteretic Loss. These losses can be added to the Piezoelectric Materialby a Damping and Losssubnode, and are typically defined as a loss factor (see below). For the case of electrical losses, hysteretic electrical losses are usually used to represent high frequency electrical losses that occur as a result of friction impeding the rotation of the microscopic dipoles that produce the material permittivity. Low frequency losses, corresponding to a finite material conductivity, can be added to the model through an Electrical Conductivity (Time Harmonic) node. This feature also operates in the frequency domain. Note that the option to add Rayleigh damping, or explicit damping (which is a particular case of Rayleigh damping in the frequency domain), is also available in the Damping and Loss node for the frequency domain.

In the time domain, material damping can be added using the Rayleigh Damping option in the Damping and Loss node. Electrical damping is currently not available in the time domain.

H Y S T E R E T I C L O S S

In the frequency domain, the dissipative behavior of the material can be modeled using complex-valued material properties, irrespective of the loss mechanism. Such hysteretic losses can be applied to model both electrical and mechanical losses. For the case of piezoelectric materials, this means that the constitutive equations are written as follows:

For the stress-charge formulation




and for the strain-charge formulation

where , , and are complex-valued matrices, where the imaginary part defines the dissipative function of the material.

Both the real and complex parts of the material data must be defined so as to respect the symmetry properties of the material being modeled and with restrictions imposed by the laws of physics.

In COMSOL Multiphysics the complex-valued data can be entered directly, or by means of loss factors. When loss factors are used, the complex data is represented as pairs of a real-valued parameter

and a loss factor

the ratio of the imaginary and real part, and the complex data is then:

where the sign depends on the material property used. The loss factors are specific to the material property, and thus they are named according to the property they refer to, for example, cE. For a structural material without coupling, simply use s, the structural loss factor.

cE eT

E–=

D e SE+=

sE dT

E+=

D d TE+=

cE d

A key requirement is that the dissipation density is positive; that is, there is no power gain from the passive material. This requirement sets rules for the relative magnitudes for all material parameters. This is important when defining the coupling losses.

X˜

X real X˜

=

X imag X˜

real X˜

=

X˜

X 1 jX =


422 | C H A P T E

The Piezoelectric Devices interface defines the loss factors such that a positive loss factor corresponds to a positive loss. The complex-valued data is then based on sign rules. For piezoelectric materials, the following equations apply (m and n refer to elements of each matrix):

(8-7)

The losses for non-piezoelectric materials are easier to define. Again, using the complex stiffness and permittivity, the following equations describe the material:

(8-8)

Often there is no access to fully defined complex-valued data. The Piezoelectric Devices interface defines the loss factors as full matrices or as scalar isotropic loss factors independently of the material and the other coefficients. For more information about hysteretic losses, see Ref. 1 to Ref. 4.

E L E C T R I C A L C O N D U C T I V I T Y ( T I M E H A R M O N I C )

For frequency domain analyses the electrical conductivity of the piezoelectric and decoupled material (see Ref. 2, Ref. 5, and Ref. 6) can be defined. Depending on the formulation of the electrical equation, the electrical conductivity appears in the variational formulation (the weak equation) either as an effective electric displacement

(the actual displacement variables do not contain any conductivity effects) or in the total current expression J = Jd Jp where Jp = eE is the conductivity current and Jd is the electric displacement current.

cEm n

cEm n 1 jcE

m n+ =

em n

em n 1 jem n

– =

Sm n

Sm n 1 jS

m n– =

sEm n

sEm n 1 jsE

m n– =

dm n

dm n 1 jdm n

– =

Tm n

Tm n 1 jT

m n– =

D˜ m n

1 jm n+ D

m n=

em n

1 jem n

– em n

=

D˜

r0E jJp------–=


Both a dielectric loss factor (Equation 8-3 and Equation 8-4) and the electrical conductivity can be defined at the same time. In this case, ensure that the loss factor refers to the alternating current loss tangent, which dominates at high frequencies, where the effect of ohmic conductivity vanishes (Ref. 7).

The use of electrical conductivity in a damped eigenfrequency analysis leads to a nonlinear eigenvalue problem, which must be solved iteratively. To compute the correct eigenfrequency, run the eigenvalue solver once for a single mode. Then set the computed solution to be the linearization point for the eigenvalue solver, defined in the settings window for the Eigenvalue Solver node. Re-run the eigenvalue solver repeatedly until the solution no longer changes. This process must be repeated for each mode separately.

References for Piezoelectric Damping

1. R. Holland and E.P. EerNisse, Design of Resonant Piezoelectric Devices, Research Monograph No. 56, The M.I.T. Press, 1969.

2. T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, 1990.

3. A.V. Mezheritsky, “Elastic, Dielectric, and Piezoelectric Losses in Piezoceramics: How it Works all Together,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 51, no. 6, 2004.

4. K. Uchino and S. Hirose, “Loss Mechanisms in Piezoelectrics: How to Measure Different Losses Separately,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 48, no. 1, pp. 307–321, 2001.

5. P.C.Y. Lee, N.H. Liu, and A. Ballato, “Thickness Vibrations of a Piezoelectric Plate With Dissipation,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 51, no. 1, 2004.

6. P.C.Y. Lee and N.H. Liu, “Plane Harmonic Waves in an Infinite Piezoelectric Plate With Dissipation,” Frequency Control Symposium and PDA Exhibition, pp. 162– 169, IEEE International, 2002.


• Selecting a Stationary, Time-Dependent, or Eigenvalue Solver

• Eigenvalue Solver


424 | C H A P T E

7. C.A. Balanis, “Electrical Properties of Matter,” Advanced Engineering Electromagnetics, John Wiley & Sons, chapter 2, 1989.

8. “Standards on Piezoelectric Crystals, 1949”, Proceedings of the I. R. E.,vol. 37, no.12, pp. 1378–1395, 1949.

9. IEEE Standard on Piezoelectricity, ANSI/IEEE Standard 176-1987, 1987.

10. B.A. Auld, Acoustic Fields and Waves in Solids, Krieger Publishing, 1990.


9

G l o s s a r y

This Glossary of Terms contains finite element modeling terms in an acoustics context. For mathematical terms as well as geometry and CAD terms specific to the COMSOL Multiphysics® software and documentation, see the glossary in the COMSOL Multiphysics Reference Manual. For references to more information about a term, see the index.

425

426 | C H A P T E

G l o s s a r y o f T e rm sacoustic impedance At a specified surface, the complex quotient of acoustic pressure by normal fluid velocity. SI unit: Pa/(m/s) .

acoustic reactance The imaginary part of the acoustic impedance.

acoustic resistance The real part of the acoustic impedance.

acoustic-structure interaction A multiphysics phenomenon where the fluid’s pressure causes a fluid load on the solid domain, and the structural acceleration affects the fluid domain as a normal acceleration across the fluid-solid boundary.

adiabatic bulk modulus One over the compressibility s measured at constant entropy. The adiabatic bulk modulus is denoted Ks and gives a measure of the compressibility of the fluid and is directly related to the speed of sound cs in the fluid. SI unit: Pa.

admittance The reciprocal of impedance.

aeroacoustics The scientific field of study used to couple acoustics and fluid dynamics. In COMSOL Multiphysics the formulation based on the potential field for the particle velocity and Bernoulli’s equation.

anisotropy Variation of material properties with direction.

arbitrary Lagrangian-Eulerian (ALE) method A technique to formulate equations in a mixed kinematical description. An ALE referential coordinate system is typically a mix between the material (Lagrangian) and spatial (Eulerian) coordinate systems.

Bernoulli equation An integrated form of Euler’s momentum equation along a line of flow. The equation gives an expression for an invariant quantity in an inviscid fluid. A decrease in the speed of the fluid translates to an increase in the fluid pressure and/or potential energy.

Ks1s----- 0cs

2= =

R 9 : G L O S S A R Y

bulk modulus One over the compressibility. It gives a measure of the compressibility of the fluid and is related to the speed of sound in the fluid. SI unit: Pa. See also adiabatic bulk modulus.

characteristic impedance The product of the equilibrium density and the speed of sound in a medium. SI unit: Pa/(m/s).

compliance Reciprocal of stiffness.

compliance matrix The inverse of the elasticity matrix. See elasticity matrix.

Cauchy stress The most fundamental stress measure defined as force/deformed area in fixed directions not following the body.

constitutive equations Equations that relate two physical quantities. In thermoacoustics both the stress tensor (relating velocity to stress) and Fourier’s law of heat conduction (relating heat conduction to temperature) are constitutive relations. In structural mechanics this is the equation formulating the stress-strain relationship of a material. Constitutive equations are supplemented by equilibrium equations (mass, momentum, and energy) and an equation of state to provide a full physical description.

creep Time-dependent material nonlinearity that usually occurs in metals at high temperatures in which the effect of the variation of stress and strain with time is of interest.

damping Dissipation of energy in the fluid or a vibrating structure. The damping is typically due to viscous losses or thermal conduction. In acoustics this happens in structures with small geometrical dimensions, for example, small pipes or porous materials. In structures a common assumption is viscous damping where the damping is proportional to the velocity. See also Rayleigh damping.

decibel (dB) Logarithmic unit that indicates the ratio of a physical quantity relative to a reference value.

dipole source An acoustic source that behaves as a translational oscillating sphere.

Doppler effect Change in the observed frequency of a wave caused by a time rate of change in the effective length of the path of travel between the source and the observation point.

G L O S S A R Y O F TE R M S | 427

428 | C H A P T E

effective sound pressure RMS instantaneous sound pressure at a point during a time interval, T, long enough that the measured value is effectively independent of small changes in T. SI unit: Pa = N/m2.

equation of state The thermodynamic relation between three independent thermodynamic variables. Typically in acoustics it is the density = (p,s) given as function of the entropy s and the pressure p.

eigenmode A possible propagating mode of an acoustic wave.

elasticity matrix The matrix D relating strain to stresses:

Eulerian Model described and solved in a coordinate system that is fixed (spatial). See also Lagrangian and arbitrary Lagrangian-Eulerian method.

Green-Lagrange strain Nonlinear strain measure used in large-deformation analysis. In a small strain, large rotation analysis, the Green-Lagrange strain corresponds to the engineering strain, with the strain values interpreted in the original directions. The Green-Lagrange strain is a natural choice when formulating a problem in the undeformed state. The conjugate stress is the second Piola-Kirchhoff stress.

impedance At a specified frequency, the quotient of a dynamic field quantity (such as force, sound, pressure) by a a kinematic field quantity (such as vibration velocity, particle velocity).

instantaneous sound pressure Total instantaneous pressure at a point in a medium minus the static pressure at the same point. SI unit: Pa = N/m2.

irrotational background velocity field A velocity field u that has the property of having rotation everywhere, where the first term is the vorticity of the fluid. In such a fluid the viscous stress does not contribute to the acceleration of the fluid. The mean pressure in this fluid is described by Bernoulli’s equation.

Lagrangian Model described and solved in a coordinate system that moves with the material. See also Eulerian and arbitrary Lagrangian-Eulerian method.

monopole source An acoustic source that behaves as a radially oscillating sphere.

D=

u 0=


particle velocity In a sound field, the velocity caused by a sound wave of a given infinitesimal part of the medium relative to the medium as a whole.

PML (perfectly matched layer) Domain adjoined at a system boundary designed to emulate a non-reflecting boundary condition independently of the shape and frequency of the incident wave front.

principle of virtual work States that the variation in internal strain energy is equal to the work done by external forces.

propagating acoustic modes The acoustic modes or wave shapes that propagate with no significant damping for a given frequency in a duct of a given cross-section.

Rayleigh damping A viscous damping model where the damping is proportional to the mass and stiffness through the mass and stiffness damping parameters.

reference sound pressure See definition in the entry for sound pressure level.

resonance frequency A frequency at which the system has the tendency to oscillate at a greater amplitude than at non-resonance frequencies. At the resonance frequencies the system can easily transfer energy from the actuation to the vibrating structure or acoustic wave.

RMS value Root-mean-square value; for the (complex) sound pressure, p(t), over the time interval T1 < t < T2 defined as

For a harmonic pressure wave, , the time interval is taken to be a complete period, resulting in pRMS p0 /2.

second Piola-Kirchhoff stress Conjugate stress to Green-Lagrange strain used in large deformation analysis.

sound energy Total energy in a given part of a medium minus the energy that would exist at the same part in the absence of sound waves. SI unit: J.

sound-energy flux density See sound intensity.

pRMS1

T2 T1–------------------- Re p t 2 td

T1

T2=

p t p0eit=


430 | C H A P T E

sound intensity Average rate of sound energy transmitted in a specified direction at a point through a unit area normal to this direction. SI unit: W/m2.

sound pressure See effective sound pressure.

sound pressure amplitude Absolute instantaneous sound pressure in any given cycle of a sound wave at some specified time. SI unit: W/m2.

sound power density See sound intensity.

sound pressure level Ten times the logarithm (to the base ten) of the ratio of the time-mean-square pressure of a sound, in a stated frequency band, to the square of a reference sound pressure, pref. For gases, pref = 20 Pa, for other media (unless otherwise specified) pref = 1 Pa. Unit: dB (decibel).

sound source strength Maximum instantaneous rate of volume displacement produced by a source when emitting a harmonic sound wave. SI unit: m3/s.

specific acoustic impedance At a point in a sound field, the quotient of sound pressure by particle velocity. SI unit: Pa/(m/s).

speed of sound The rate of change of particle displacement with distance for a sound wave. SI unit: m/s.

spin tensor The skew-symmetric part of the velocity gradient tensor.

static pressure Pressure that would exist at a point in the absence of a sound wave.

stiffness Ratio of change of force (or torque) to the corresponding change in translational (or rotational) displacement of an elastic element.

thermoacoustics The interaction between thermodynamic and acoustic phenomena, which takes into account the temperature oscillations that accompany the acoustic pressure oscillations. The combination of these oscillations produces thermoacoustic effects. Thermoacoustic phenomena are modeled by solving the full linearized Navier-Stokes equation (momentum equation), the continuity equation, and the energy equation. Thermoacoustics is also known as thermo-viscous or visco-thermal acoustics.


velocity potential When a flow is irrotational the vector field (velocity field) can always be derived from a scalar potential (x) as , where is the velocity potential. See also irrotational background velocity field.

waveguide structures Structures that have the property of guiding sound waves. See also propagating acoustic modes.

u 0=

u =


432 | C H A P T E

I n d e x

A acoustic intensity vector 60

acoustic-piezoelectric interaction, fre-

quency domain (acpz) interface 175

acoustic-piezoelectric interaction, tran-

sient (acpztd) interface 182

acoustic-shell interaction, frequency do-

main (acsh) interface 212

acoustic-shell interaction, transient

(acshtd) interface 221

acoustic-solid interaction, frequency do-

main (acsl) interface 150

acoustic-solid interaction, transient

(actd) interface 172

acoustic-structure boundary (node) 163

acoustic-thermoacoustic boundary

(node) 327

added mass (node) 390

added mass, theory 422

adiabatic (node) 332

adiabatic bulk modulus 122

advanced settings 21

ALE method 468

analysis. see study types.

anisotropic materials

defining 403

elastic properties 371

loss factor damping, and 373, 418

antisymmetry (node)

solid mechanics 386

Arbitrary Lagrangian-Eulerian (ALE)

method 438

artificial boundary conditions 45

attenuation coefficient 359

axial symmetry

initial stress and strain 419

axial symmetry (node) 85, 257

azimuthal wave-number 429

B background pressure field (node) 95

background pressure wave 68

barotropic fluids 306

Bernoulli equation 307

Biot equivalents fluid model 157

Biot theory 204

Biot’s high frequency range 205

body load (node) 375

boundary load (node) 376

boundary loads theory 412

boundary mode acoustics (acbm) inter-

face 117

theory 127

boundary nodes

acbm interface 119

acpr interface 72

acpz and acpztd interfaces 178

acsh and acshtd interfaces 215

acsl and astd interfaces 153

actd interface 113

ae interface 248

aebm interface 265

aetd interface 261

cpf interface 268

lef and let interfaces 274

lnsf and lnst interfaces 287

pafd interface 227

patd interface 227

piezoelectric devices 452

solid mechanics 366

ta interface 320

tas interface 335

tash interface 343

boundary selection 23

boundary-layer absorption 144

I N D E X | 433

434 | I N D E X

boundary-layer absorption fluid model

158

bulk modulus 237

elastic moduli 400

pressure acoustics 40

C canonical systems 396

Cauchy stress 408

Cauchy stress tensor 437

Cauchy-Green tensors 406

CFL condition 242

change thickness (node) 372

circular source (node) 110

circumferential wave number 125, 128

closed (node) 231

complex conjugate operator 63

complex impedance 43

complex modulus 447

complex wave numbers 43

compressible potential flow (cpf) inter-

face 266

compressible potential flow model

(node) 268

consistent stabilization

settings 22

constraint settings 22

continuity (node)

acpr interface 85

ae interface 257

elw interface 200

tas interface 338

converse piezoelectric effect 473

coordinate system selection 23

coordinate systems

solid mechanics theory 396

Courant number 243

crystal cut standards 474

cyclic symmetry, settings 380

cyclic symmetry, theory 428

cylindrical wave radiation (node) 80

D damped eigenfrequency study 447

damping

boundaries 44

equation of motion, and 441

loss factors 373

piezoelectric devices 458, 468

solid mechanics 372

types 470

viscoelastic material, and 445

damping (node) 372

damping and loss (node) 458

damping models 416, 418

defining

anisotropic materials 403

isotropic materials 399

orthotropic materials 402

thermoelastic materials 403

deformation gradient 406

degrees of freedom 41

dielectric loss 471

dielectric loss (node) 459

dielectric loss factor 483

dipole source (node) 74

dipole sources 40, 121

direct piezoelectric effect 462, 473

discretization 21

dispersion curves 57

dispersion relation 359

displacement field, defining 436

displacement gradients 397

dissipation, piezoelectric materials 465

distributed loads, theory 411

documentation 24

domain nodes

acbm interface 119

acpr interface 72




actd interface 113

ae interface 248

aetd interface 261

cpf interface 268




solid mechanics 366

ta interface 320

tas interface 335

tash interface 343

domain selection 23

domain sources (node) 278, 293

double dot operator 64

Dulong-Petit law 404

dynamic cyclic symmetry 428

E edge load (node) 381

edge nodes

acbm interface 119

acpr interface 72




actd interface 113

ae interface 248

aebm interface 265

aetd interface 261

pafd interface 227

patd interface 227


solid mechanics 366

ta interface 320

tas interface 335

tash interface 343

edge selection 23

eigenfrequency study 55

pressure acoustics 126

solid mechanics 415

elastic material properties 369–371

elastic moduli 399

elastic waves (elw) interface 185

theory 203

elasticity matrix 399

electrical conductivity (time-harmonic)

(node) 457

electrical material model (node) 455

elkernel element 134

emailing COMSOL 26

empty study 59

end impedance (node) 232

equation of motion, damping and 441

equation view 21

equivalent fluid model 87, 157

equivalent viscous damping 446

Euler equations 306

Eulerian frame 437

evanescent modes 56

evanescent wave components 47, 131

excitation frequency 446

expanding sections 21

explicit damping 447

exterior shell (node)

acsh interface 218

tash interface 347

F far field variables 49

far-field calculation (node) 97

far-field limits 48, 133

far-field regions 47, 131

far-field variables 48, 134

first order material parameters (node)

292

first Piola-Kirchhoff stress 408

fixed constraint (node) 377

Floquet periodicity, settings 380

I N D E X | 435

436 | I N D E X

Floquet periodicity, theory 428

flow line source on axis (node) 164

fluid models, pressure acoustics 87, 157

fluid properties (node) 228

force linear strains (check box) 371

Fraunhofer diffraction 47, 132

free (node) 376

frequency domain modal study 58

frequency domain study 54

ae interface 304

solid mechanics 414

frequency response study

loss factor damping 447

viscous damping 446

Fresnel numbers 47, 132

G geometric entity selection 23

geometric nonlinearity 406

micromechanics, and 436, 438



global

coordinate systems 396

gradient displacements 397

gravity (node) 392

Green’s function 133

Green-Lagrange strain 436

Green-Lagrange strains 405

H Hankel function 133

harmonic loads 414

harmonic time dependence 61

heat dissipation 447

heat flux (node) 296

heat source (node) 326

Helmholtz-Kirchhoff integral 48, 132

hide (button) 21

Higdon conditions 129

high frequency range 205

hysteretic loss 468, 480

I I.R.E. standard, for material orientation

474

ideal barotropic fluids 306

ideal gas fluid model 88, 157

IEEE standard, for material orientation

474

IEEE standard, piezoelectric materials

463

impedance (node) 280

acpr interface 78

ae interface 255

impedance, complex 43

incident acoustic fields (node) 279

incident pressure field (node) 81

incident velocity potential (node) 253

include geometric nonlinearity (check

box) 371

inconsistent stabilization

settings 22

initial stress and strain 465

theory 419

initial stress and strain (node) 382

initial values (node)

acbm interface 118

acpr interface 75

acpz interface 181

acsh interface 217–218

acsl interface 164

ae interface 250

astd interface 173

cpf interface 269

elw interface 190

lem interface 277

let interface 283


pafd interface 228

patd interface 228


solid mechanics 375

ta interface 326

tas interface 338

tash interface 346–347

insertion loss curves 60

intensity edge source (node) 166

intensity line source on axis (node) 165

intensity point source (node) 169

intensity sources, pressure acoustics

line, on axis 165

intensity variables 60–61

interior impedance (node) 100

interior normal acceleration (node) 99

interior perforated plate (node) 101

interior shell (node)

acsh interface 219

tash interface 348

interior sound hard boundary (wall)

(node)

acpr interface 84

ae interface 256

interior wall (node) 281

Internet resources 24

inverse piezoelectric effect 462

irrotational velocity fields 303

isentropic speed of sound 63

isothermal (node) 291, 327

isotropic materials

defining 399


loss damping, and 418

loss factor damping, and 373

isotropic meshes 41

K knowledge base, COMSOL 26

Korteweg formula 237

L Lagrange elements 42

Lagrangian formulations 397

Lagrangian frame 436

Lamé parameters 400

large acoustics problems 49

large deformations 405

piezoelectric materials 466

limp porous matrix model 142

line source (node) 102

line source on axis (node) 105

line sources

pressure acoustics, intensity, on axis

165

linear elastic attenuation fluid model 137

linear elastic fluid model 136

linear elastic material (node) 368

linear elastic materials 398

linear elastic with attenuation 88, 157

linearized Euler model (node) 274

linearized Euler, frequency domain inter-

face 272

linearized Euler, transient interface 282

linearized Navier-Stokes model (node)

288

linearized Navier-Stokes, frequency do-

main interface 285

linearized Navier-Stokes, transient inter-

face 297

linearized potential flow model (node)

249

linearized potential flow, boundary mode

interface 263

linearized potential flow, frequency do-

main interface 246

linearized potential flow, transient inter-

face 260

liquids and gases materials 52

loads


local coordinate systems 397

logarithmic decrement 442

I N D E X | 437

438 | I N D E X

loss factor damping

modeling 447

solid mechanics and 373


springs, and 439

loss modulus 447

loss tangents 469

low-reflecting boundary (node) 391

low-reflecting boundary, theory 427

M macroscopic empirical porous model 92,

157

mass damping parameter 445

mass flow (node) 270

mass flow circular source (node) 259

mass flow edge source (node) 258

mass flow line source on axis (node) 258

mass flow point source (node) 258

mass flows 252

matched boundary (node) 96

matched boundary conditions 45

material coordinates 395

material frame 436

materials

linear elastic 398

nearly incompressible 426

piezoelectric 463

porous absorbing 43

materials, piezoelectric devices 465

mixed formulations 426

modal reduced order model study 58

mode analysis study 56

boundary mode acoustics 127

theory 358

Model Library 25

Model Library examples

acpr interface 72

acpz interface 178

acsh interface 215

acsl interface 152

actd interface 113

ae interface 248

aebm interface 264

background pressure field 96

Biot equivalent fluid model 144

cpf interface 267

cylindrical wave radiation 81

damping 45

eigenfrequency study 126

far field calculation 99

far field plots 49

flow point source 108

Gaussian pulse point source 116

interior normal acceleration 100

macroscopic empirical porous fluid

model 141

mass flow point source 259

mode analysis study 57, 128

monopole source 74

pipe acoustics, transient 243

plane wave radiation 80

poroelastic waves 188

power point source 108, 171

pressure acoustics, frequency domain

72

radiation boundary condition 46

solid mechanics 366

spherical wave radiation 80

ta interface 320

moment computations 432

monopole point source (node) 106

monopole source (node) 74

monopole sources 40, 121

moving mesh interface, piezoelectric de-

vices and 468

moving wall (node) 281, 284

MPH-files 25

multigrid solvers 49

N narrow ducts 145

narrow region acoustics (node) 94

near-field regions 47, 131

nearly incompressible materials 419, 426

no slip (node) 290

no stress (node) 331

no-flow conditions 250, 256

nominal stress 424

nonlinear geometry 406, 423

nonreflecting boundary conditions 45

normal acceleration (node) 76

normal flow (node) 269

normal impedance (node) 331

normal mass flow (node) 252

normal stress 408

normal stress (node) 295, 331

normal velocity (node) 254

NRBC. see nonreflecting boundary con-

dition. 96

Nyquist criterion 50

O orientation, piezoelectric material 474

orthotropic materials

defining 402


loss damping, and 418

loss factor damping, and 374

out-of-plane wave number 124, 304

override and contribution 21–22

P pair impedance (node)

acpr interface 100

ae interface 255

pair nodes

acpr interface 72




ae interface 248

aebm interface 265

aetd interface 261

cpf interface 268



pafd interface 227

patd interface 227


solid mechanics 366

ta interface 320

tas interface 335

tash interface 343

pair perforated plate (node) 101

pair selection 23

particle velocity 61

perfectly matched layers (node) 46

perforated plate (node) 101

periodic boundary conditions 380

periodic condition (node) 82


solid mechanics 380

periodic conditions, theory 428

phase (node) 383

phase factors, pressure acoustics 125

phase variables 55

piezoelectric coupling 450

piezoelectric crystal cut 474

piezoelectric devices interface 450

theory 462

piezoelectric losses 480

piezoelectric material (node) 454

pipe acoustics, frequency domain (pafd)

interface 224

pipe acoustics, transient (patd) interface

226

theory 235

Pipe Flow Module 33

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pipe properties (node) 229

plane stress and strain 365

plane wave radiation (node)

acpr interface 79

ae interface 252

point load (node) 381

point nodes

acbm interface 119

acpr interface 72




actd interface 113

ae interface 248

aebm interface 265

aetd interface 261

pafd interface 227

patd interface 227


solid mechanics 366

ta interface 320

tas interface 335

tash interface 343

point selection 23

point source (node) 108

Poisson’s ratio 400

poroacoustics (node) 92

poroelastic material (node) 190

poroelastic septum boundary load

(node) 200

poroelastic waves (elw) interface 187

theory 203

porous absorbing materials 43

porous, fixed constraint (node) 194

porous, free (node) 195

porous, prescribed acceleration (node)

198

porous, prescribed displacement (node)

196

porous, prescribed velocity (node) 198

porous, pressure (node) 195

porous, roller (node) 199

power dissipation variables 62

power edge source (node) 167

power line source on axis (node) 166

power point source (node) 170

ppr() operator 98

Prandtl number 317

pre-deformation (node) 390

prescribed acceleration (node) 384

prescribed acoustic fields (node) 279

prescribed displacement (node) 378

prescribed pressure (node) 294

prescribed temperature (node) 296

prescribed velocity (node) 294, 383

pressure (adiabatic) (node) 293

pressure (node) 77

patd interface 231

pipe acoustics 231

pressure acoustics (node) 86

pressure acoustics interfaces 120

pressure acoustics model (node) 155

pressure acoustics model, fluid models

137

pressure acoustics models, theory 136

pressure acoustics theory 120

pressure acoustics, frequency domain

(acpr) interface 68

pressure acoustics, transient (actd) in-

terface 112

pressure loads 413

pressure, adiabatic (node) 328

pressure-wave speeds 400

principle of virtual work 413

propagating acoustic modes 358

propagating modes 56

Q quality factors and losses 469

R radiation boundary conditions 45

rate of strain tensor 407

Rayleigh damping 416, 443, 445

reference coordinates 395

reference point for moment computa-

tion 432

refpnt variable 365

remanent electric displacement (node)

459

resonance frequency 55

resonant frequency 443

results analysis and variables 60

rigid porous matrix model 142

rigid wall (node) 277

RMS 60

roller (node) 379

rotated coordinate system 478

rotating frame (node) 393

S sdiabatic (node) 295

second Piola-Kirchhoff stress 408, 437

settings windows 21

shear modulus expression 400

shear stress 408

shear-wave speeds 400

show (button) 21

skew-symmetric part 407

slip (node) 294, 330

slip velocity (node) 270

solid mechanics

damping 372

prescribed acceleration 384

prescribed displacement 378

prescribed velocity 383

solid mechanics interface 364

theory 395

solving large problems 49

sound hard boundary (wall) (node)

acpr interface 75

ae interface 250

sound hard wall (node) 327

sound soft boundary (node)

acpr interface 76

ae interface 254

source term 62

spatial coordinates 395

spatial frame 437

spatial stress tensor 437

spherical wave radiation (node) 80

spin tensor 407

spring constant 439

spring foundation (node) 386

spring foundation, solid mechanics 439

spring foundation, theory 420

stabilization settings 22

stationary study 53, 308

stiffness damping parameter 445

storage modulus 447

stress

Cachy 408

first Piola-Kirchhoff 408

normal 408

second Piola-Kirchhoff 408

shear 408

stress (node) 295, 330

stress and strain, piezoelectric devices

463

structural damping 447

study types

acbm interface 127

actd interface 127

ae interface 304

eigenfrequency 415

frequency domain, solid mechanics 414

stationary, solid mechanics 414

surface traction and reaction forces 434

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swirl-correction factor 235

symmetry (node) 78, 280

solid mechanics 385

ta interface 329

tas interface 339

symmetry planes, far-field analysis 98

T Taylor series 236

technical support, COMSOL 26

temperature variation (node) 332

theory 120

acbm interface 127

elw interface 203

patd interface 235


pressure acoustics models 136

solid mechanics 395

ta interface 350

tas interface 350

thermal dissipation 62

thermally conducting and viscous fluid

model 140

thermally conducting and viscous fluid

model, defining 90, 159

thermally conducting fluid model 88, 139,

158

thermoacoustics model (node) 322

thermoacoustics, frequency domain (ta)

interface 316

theory 350

thermoacoustic-shell interaction, fre-

quency domain (tash) interface 340

thermoacoustic-solid interaction, fre-

quency domain (tas) interface 333

theory 350

thermoelastic materials, defining 403

thin elastic layer (node) 388

thin elastic layer, solid mechanics 439

thin elastic layer, theory 421

time dependent study 57

pressure acoustics, transient 127

time-dependent modal study 58

time-harmonic study

aeroacoustics 304

total force loads 412

tractions 437

transient pressure acoustics model

(node) 113

transmission loss 60

true stress tensor 437

U uncoupled shell (node)

acsh interface 220

tash interface 349

U-P formulation 205

user community, COMSOL 26

using

coordinate systems 396

predefined variables 432

spatial and material coordinates 395

weak constraints 433

uspring variable 439

V variables

for far fields 49

intensity 60

material and spatial coordinates 396

phase 55

power dissipation 62

predefined 432

refpnt 365

results analysis 60

vdamper variable 440

velocity (node) 232, 329

pipe acoustics 232

velocity potential (node) 251

velocity potential, compressible poten-

tial flow, and 307

viscoelastic material, damping and 445

viscous damping 440, 447

viscous dissipation 62

viscous fluid model 88, 139, 158

Voigt form 474

Voigt notation 374, 401, 463

volume ratio 406

vortex sheet (node) 255

W wall (node) 329

wave numbers 43, 124, 304

wave radiation equations 129

wave speeds 370

waveguide 56

waveguide structures 358

weak constraint settings 22

weak constraints, using 433

web sites, COMSOL 26

wide ducts 144

Y Young’s modulus expression 400

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the acoustics module user’s guide - lost … 3 contents chapter 1: introduction acoustics module...

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