theory_ale_cfd_sph

RADIOSS THEORY MANUAL 10.0 version – January 2009

ALE, CFD, SPH

Altair Engineering, Inc., World Headquarters: 1820 E. Big Beaver Rd., Troy MI 48083-2031 USA

Phone: +1.248.614.2400 • Fax: +1.248.614.2411 • www.altair.com • [email protected]

RADIOSS THEORY Version 10.0 CONTENTS

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CONTENTS 1.0 ALE FORMULATION 3

1.1 REFERENTIAL DOMAIN 3 1.2 CONSERVATION OF MOMENTUM 4

1.2.1 MOMENTUM TRANSPORT FORCE 5 1.3 CONSERVATION OF MASS 5 1.4 CONSERVATION OF INTERNAL ENERGY 7 1.5 REZONED QUANTITIES 7 1.6 ALE MATERIALS 7 1.7 NUMERICAL INTEGRATION 8 1.8 IMPROVED INTEGRATION METHOD 9 1.9 MOMENTUM TRANSPORT FORCE 9

1.9.1 UPWINDING TECHNIQUE 10 1.10 STABILITY 10 1.11 ALE KINEMATIC CONDITIONS 10

1.11.1 BOUNDARY CONDITIONS 10 1.11.2 ALE LINKS 11

1.12 AUTOMATIC GRID COMPUTATION 12 1.12.1 /DONEA - J. DONEA GRID FORMULATION 12 1.12.2 /DISP - AVERAGE DISPLACEMENT FORMULATION 12 1.12.3 /SPRING - NONLINEAR SPRING FORMULATION 12

1.13 TYPE 1 INTERFACE - FLUID-STRUCTURE INTERACTION 13 1.14 ALE RIGID WALL 14 1.15 EXAMPLE 15

2.0 COMPUTATIONAL AERO-ACOUSTIC 17 2.1 MATHEMATICAL FORMULATION 18

2.1.1 COMPRESSIBLE NAVIER STOKES 18 2.1.2 TRANSIENT ANALYSIS, EXPLICIT FORMULATION 20 2.1.3 LARGE EDDY SIMULATION TURBULENCE MODELING 20 2.1.4 BOUNDARY CONDITIONS 21 2.1.5 FLUID STRUCTURE COUPLING 22

2.2 MODELING METHODOLOGY 23 2.2.1 MESH DEFINITION 23 2.2.2 POST PROCESSING 23

2.3 APPLICATION AND VALIDATION 24 2.4 CONCLUSION 24

3.0 SMOOTH PARTICLE HYDRODYNAMICS 26 3.1 SPH APPROXIMATION OF A FUNCTION 26 3.2 CORRECTED SPH APPROXIMATION OF A FUNCTION 27 3.3 SPH INTEGRATION OF CONTINUUM EQUATIONS 28 3.4 ARTIFICIAL VISCOSITY 29 3.5 STABILITY: TIME STEP CONTROL 30

3.5.1 CELL TIME STEP 30 3.5.2 NODAL TIME STEP 30

3.6 CONSERVATIVE SMOOTHING OF VELOCITIES 31 3.7 SPH CELL DISTRIBUTION 31

3.7.1 HEXAGONAL COMPACT NET 31 3.7.2 CUBIC NET 32

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Chapter

ARBITRARY LAGRANGIAN EULERIAN FORMULATION

RADIOSS THEORY Version 10.0 ALE FORMULATION

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1.0 ALE FORMULATION ALE or Arbitrary Lagrangian Eulerian formulation is used to model the interaction between fluids and solids; in particular, the fluid loading on structures. It can also be used to model fluid like behavior, as seen in plastic deformation of materials.

ALE derives its name from a combination of two different finite element modelling techniques.

• Lagrangian Formulation - where the observer follows material points.

• Eulerian Formulation - where the observer looks at fixed points in space.

• Arbitrary Lagrangian Eulerian Formulation - where the observer follows moving points in space.

1.1 Referential Domain At any location in space x and time t, there is one material point, identified by its space coordinates x at time t=0, and one grid point identified by its coordinates ⎩ at time t=0. Figure 1.1.1 provides a pictorial representation and defines the velocities in each formulation.

Figure 1.1.1 ALE Formulation

The derivative of any physical quantity can be computed either following the material point or following the grid point. They can then be related to each other.

Given that F is a function f of space and time representing a physical property:

The spatial domain is given by f(x,t).

The material domain is given by f*(X,t).

The mixed domain is given by f**(ξ , t).


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Therefore:

( ) ( )t

jkjXX

j

jXX x

txftxVtf

tx

xf

tf

tf

∂∂

+∂∂

=∂∂

×∂∂

+∂∂

=∂

∂ ∗ ,, EQ. 1.1.0.1

Also:

( ) ( )t

jjjX x

txfWVt

ft

f∂

∂−+

∂∂

=∂

∂ ∗∗∗ ,ξ EQ. 1.1.0.2

This relates to acceleration by:

( ) tjjX vx

wvvtdt

vd rvr

r

∂∂

−+∂∂

== ξγ EQ. 1.1.0.3

where v = material velocity

w = grid velocity

1.2 Conservation of Momentum Conservation of momentum, expressed in terms of a finite element formulation, is given by:

∫ =⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂

∂−

∂∂

ΦV

ij

ijiI dVb

xtv 0ρ

σρ EQ. 1.2.0.4

where F I = the weight functions

ρ = Material density

v = Velocity

σij = Stress Matrix

bi = Body acceleration vector

V = Volume

This can be rewritten in a form similar to the explicit Lagrangian formulation with the addition of a new nodal force ftrm , accounting for transport of momentum:

{ } { } { } { } { }trmhgrbodext FFFFFvt

M +++−=∂∂ int EQ. 1.2.0.5

where { } trmtrm fF ∑=

The transport of momentum force is calculated by:

( ) ( )∫ ∂∂

−Φ+=V j

ijjII

trmiI dV

xvvwf ρη1 EQ. 1.2.0.6

where i,j = direction index

I = connectivity

Iη = Upwind factor


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1.2.1 Momentum transport force Momentum transport forces are computed using the relation:

( ) ( ) VxvvwF

j

ijjII

trmiI ∂

∂−Φ+= ρη1 EQ. 1.2.1.1

The upwinding technique is introduced to add numerical diffusion to the scheme, which otherwise is generally under diffuse and thus unstable.

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−

∂Φ∂

= jjj

II wv

xsignηη EQ. 1.2.1.2

10 ≤≤η Upwind coefficient given in input.

Full upwind 1=η (default value) is generally used.

Development of less diffusive flux calculation is currently under investigation.

1.3 Conservation of Mass The finite element formulation of the Lagrangian form of the mass conservation equation is given by:

( ) XX dtdVV

dtd /ρρ

−= EQ. 1.3.0.2

When transformed into the ALE formulation it gives:

( ) 0=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅−−∂∂

tK

Kt

iii x

vx

vwt

ρρρξ EQ. 1.3.0.3

Applying a Galerkin variation form for the solution of equation 1.3.0.3:

( ) 0=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅−−∂∂

∫V

tK

Kt

iii x

vx

vwt

ρρρψ ξ EQ. 1.3.0.4

where ψ = Weighting function

Using a finite volume formulation:


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where 1=ψ

ρ = constant density over control volume V

Therefore:

∫ ∫ =∂∂

+∂∂

V V K

K dVxvdV

t0ρρ

EQ. 1.3.0.5

Using the divergence theorem leads to:

( )∫ ∫ =⋅+∂∂

V Sjj dSnvdV

t0ρρ

EQ. 1.3.0.6

Further expansion gives:

( )∫∫ −=S fluxmass

jjjV

dSnvwdVdtd

44 344 21ρρ EQ. 1.3.0.7

This formula is still valid if density p is not assumed uniform over volume V.

The mass flux across a surface is shown in Figure 1.3.1.

Figure 1.3.1 Mass Flux

The density, ρi, is given computed:

( ){ } ( ){ }iJiIi signsign φηρφηρρ ++−= 1211

21

EQ. 1.3.0.8

where 10 ≤≤η is the upwind coefficient given on the input card.

If 0=η , there is no upwind. Therefore: 2

JIi

ρρρ +=

If 1=η , there is full upwind.

The smaller the upwind factor, the faster the solution; however, the solution is more stable with a large upwind factor.


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For a free surface: IJ ρρ =

1.4 Conservation of Internal Energy Conservation of internal energy is used to model temperature dependent material behavior. It also allows an energy balance evaluation. However, internal energy is only calculated if it is turned on, to reduce computation time in problem s not involving heat transfer.

The conservation of energy is given by:

( ) ( ) 0=∂∂

++⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅−−∂

∂

K

K

iii x

vexevw

te ρρρρ

EQ. 1.4.0.9

where e = Internal energy in Joules (Nm)

p = Fluid pressure

Applying a Galerkin variation form for the solution gives:

( ) ( ) 0=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

++⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅−−∂

∂∫V k

k

iii x

vexevw

te ρρρρψ EQ. 1.4.0.10

Making the following assumptions:

1=ψ

=eρ constant over control volume V

Equation 1.4.0.10. reduces to:

( )∫ ∫ =∂∂

++∂

∂

V V k

k dVxvedV

te 0ρρρ

EQ. 1.4.0.11

Applying the divergence theorem gives:

( )( ) 0=∂∂

+⋅+∂

∂∫∫ ∫

k

k

VV Sjj x

vdSnvedVte ρρρ

EQ. 1.4.0.12

Hence:

( ) dVxvdSnvweedV

dtd

k

k

VSjjj

V ∂∂

−−= ∫∫∫ ρρρ EQ. 1.4.0.13

This formula is still valid if ⟩ e is not assumed uniform over volume V.

1.5 Rezoned Quantities The deviatoric stress tensor and the equivalent plastic strain must be rezoned and recalculated after every time step due to the ability of one element to contain a different amount of material.

1.6 ALE Materials The following materials may be used with the ALE formulation.


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ALE Materials Law Number Description

2 Elasto-plastic /MAT/PLAS_JOHN

3 Elasto-plastic-Hydrodynamic /MAT/HYDPLA

4 Johnson Cook /MAT/HYD_JCOOK

6 Hydrodynamic Viscous /MAT/HYD_VISC

10 & 21 Rock Concrete Foam /MAT/LAW10 or /MAT/DPRAG

22 & 23 Elasto-plastic with Damage /MAT/DAMA or /MAT/LAW23

20 Bimaterial /MAT/BIMAT

37 Hydrodynamic - Bi-phase liquid gas /MAT/BIPHAS

11 Boundary - Stagnation conditions in flow calculations /MAT/BOUND

16 Gray model - Multiphase Gray E.O.S + Johnson's shear law /MAT/GRAY

18 Thermal conductivity, purely thermal material /MAT/THERM

For the rest, refer to the next version of the theory manual.

1.7 Numerical Integration The numerical integration techniques used are the same as those used for any other analysis type.

The flow chart of calculations can be seen in Figure 1.7.1.


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Figure 1.7.1 Flow Chart1

1. Simplified Flow Chart

1.8 Improved integration method This method can only be used with the CFD version of RADIOSS, and only available in Eulerian formulation. An eight Gauss point integration scheme is used to determine the shape functions. The shape functions are condensed to one point. This gives an eight point integration scheme with constant stress.

1.9 Momentum Transport Force This scheme is only used with the ALE formulation (Arbitrary Lagrangian Eulerian) and in the CFD version of RADIOSS. The force is calculated using the relation:

( ) ( ) VXvvwF

j

ijjII

Ii

trm

∂∂

−Φ+= ρη1 . EQ. 1.9.0.14

where w = grid velocity

v = material velocity

V = element volume

η = upwind coefficient (user defined, default = 1 for full upwind)

When a Lagrangian formulation is used, the values of wj and vj are equal. Thus, EQ. 1.9.0.14 is equal to zero.


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1.9.1 UPWINDING TECHNIQUE An upwinding technique is introduced to add numerical diffusion to the scheme, otherwise it is generally under diffusive and thus unstable. The upwind coefficient used in EQ. 1.9.0.14 is calculated by:

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂Φ∂

= jjj

II wv

Xsignηη EQ. 1.9.1.1

Development of a less diffusive flux calculation is currently under investigation.

∫ ∂Φ∂

=V j

Iij

Ii dV

XF σ EQ. 1.9.1.2

This option is activated with the flag INTEG (only in the CFD version).

1.10 Stability The Courant condition (neglecting viscosity effects) is used to determine the stability of an ALE process. The maximum time step is calculated by:

wvclkt−+

Δ≤Δ EQ. 1.10.0.3

where k = coefficient

Δl = Smallest characteristic length of an element

c = Material speed of sound

v = Material velocity

w = Grid velocity

The speed of sound is determined by:

ρμ

ρρ

ρ 341

+∂∂

=c EQ. 1.10.0.4

where ρ = Density

μ = Dynamic viscosity

p = Pressure

The relative velocity between the material and grid motion (v-w) is computed by:

( )∑∑= =

−=−3

1 1

21i

N

I

Ii

Ii wv

Nwv EQ. 1.10.0.5

where N = Number of nodes of the considered element (usually N=8)

1.11 ALE Kinematic Conditions

1.11.1 Boundary Conditions Boundaries with Lagrangian materials are declared automatically Lagrangian.

Nodes can be declared Lagrangian.


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Constraints can be applied separately or simultaneously on:

• Material velocity

• Grid velocity These constraints can be applied in one or several directions of a skew reference frame.

When the flag is set to 1, boundary condition is activated w.r.t. global reference frame or skew reference frame.

i = d.o.f with respect to global reference frame or skew reference frame

VELOCITY: Vi = 0

ACCELERATION: Π i = 0

/The boundary conditions can be changed during engine runs with /BCS or /BCSR engine options.

1.11.2 ALE Links An ALE link is identical to a rigid link. The slave node sets' grid velocity can be controlled by two master nodes, M1 and M2.

There are three options to choose from:

Option 0:

Velocity is linearly interpolated with respect to order of input.

1)( 1221 +

−+=N

IWWWW MMMNI EQ. 1.11.2.1

Option 1:

Velocity is set to maximum absolute velocity of master nodes.

1MNI WW = if 21 MM WW > EQ. 1.11.2.2

Option 2:

Velocity is set to minimum absolute velocity of master nodes.


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1MNI WW = if 21 MM WW < EQ. 1.11.2.3

The input data is specified at each restart run.

1.12 Automatic Grid Computation There are three different grid velocity formulations that can be used in an ALE simulation. New keywords define the type of method used. The different formulations are:

• 0 - J. Donea Grid Formulation: use keyword /DONEA

(NWALE =0 for version <4.1)

• 1 - Average Displacement Formulation: use keyword /DISP


• 2 - Nonlinear Spring Formulation: use keyword /SPRING


1.12.1 /DONEA - J. Donea Grid Formulation This formulation [8], [72] computes grid velocity using:

( ) ( ) ( ) ( ) ( )( )∑∑∑ −

Δ+Δ−=Δ+

J IJ

IJ

JIJ

JJI tL

tututLtN

ttWN

ttW α2

12/12/ EQ. 1.12.1.1

where, γγ +≤≤− 11vw

N = Number of nodes connected to node I

IJL = Distance between node I and node J

γα , = adimensional factors given in input

Therefore, the grid displacement is given by:

( ) ( ) ( ) tttwtuttu ΔΔ++=Δ+ 2/ EQ. 1.12.1.2

1.12.2 /DISP - Average Displacement Formulation The average displacement formulation calculates average velocity to determine average displacement.

( ) ( )∑=Δ+J

j twN

ttu 1 EQ. 1.12.2.1

1.12.3 /SPRING - Nonlinear Spring Formulation Each grid node is connected to neighboring grid nodes through a non-linear viscous spring, similar to that shown in Figure 1.12.1.


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Figure 1.12.1 Spring Force Graph

The input parameters required are:

0TΔ = typical time step (Must be greater than the time step of the current run.)

10 << γ = Nonlinearity factor

η = Damping coefficient

v= Shear factor (stiffness ratio between diagonal springs and springs along connectivities)

This formulation is the best of the three, but it is the most computationally expensive.

1.13 Type 1 interface - Fluid-Structure Interaction Type 1 interface is used to model fluid-structure interactions, as shown in Figure 1.13.1

Figure 1.13.1 Fluid-Structure Interaction

This interface allows Lagrangian elements (structure) to interact with ALE (Arbitrary Lagrangian Eulerian) elements, which model a viscous fluid. Full slip conditions are applied at the boundary between the two domains.

The acceleration of the Lagrange node is computed by:

al

all mm

FF++

=rr

rγ EQ. 1.13.0.1


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The acceleration of the ALE node is computed by:

a

aa m

Fr

r=γ EQ. 1.13.0.2

The grid velocity of the ALE node is equal to the material velocity of the Lagrange node:

la vw rr= EQ. 1.13.0.3

The normal material velocities of Lagrange and ALE nodes are equal. Therefore:

nvnv larrrr

⋅=⋅ EQ. 1.13.0.4

.

1.14 ALE Rigid Wall An ALE rigid wall has similar properties to other types of rigid walls. There are two different types:

1. Shaped Charged : use keyword /DFS/WALL_SHAP

2. Penetration: use keyword /DFS/WALL_PEN

(For further explanation of each rigid wall, contact Altair Development France).

Impacting nodes can either have a sliding contact or be tied to the rigid surface contact point. The wall can also be moving.

An example of an object impacting an ALE rigid wall can be seen in Figure 1.14.1.

Figure 1.14.1 ALE Rigid Wall Impact

A gap is required for the wall, Figure 1.14.1. When a slave node distance to the rigid wall is within the gap:

• Nodes are forced onto the rigid wall and set as Lagrangian.

• Zero volume elements are emptied into neighboring elements and deleted. In addition to the information explained above, an ALE rigid wall definition requires the number of nodes impacting simultaneously to be defined, along with the order of slave node impact.


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1.15 Example A typical application of the ALE method is using high velocity impacts. Below, a cylinder, moving at 227 m/s, impacts with a rigid wall. The material is copper, with a yield stress of 400 MPa. The initial diameter is 6.4 mm and initial length is 32.4 mm. The simulation was performed using two different methods: ALE and standard Lagrangian. The results can be seen in Figure 1.15.1.

Figure 1.15.1 Cylinder Impact Deformation

It can be seen that the cylinder mesh using ALE remains regular, unlike the Lagrange method, where large element deformation creates very small and skewed elements. This reduces the time step, leading to more time step cycles. However, each ALE cycle takes longer than a Lagrangian.

RADIOSS THEORY Version 10.0 COMPUTATIONAL AERO-ACOUSTIC

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Chapter

COMPUTATIONAL AERO-ACOUSTIC


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2.0 COMPUTATIONAL AERO-ACOUSTIC This chapter presents a resumed state-of-the-art of simulation in aero-acoustic domain as presented in [88]. Aero-Acoustics is the engineering field dealing with noise generated generally (but not necessarily) by a turbulent fluid flow interacting with a vibrating structure. This field differs from the pure acoustic domain where the object is the propagation of acoustic pressure waves, including reflections, diffractions and absorptions, in a medium at rest. Aero-Acoustic questions arise in many industrial design problems and are heavily represented in the noise nuisances related to the transportation industry.

A classification of Aero-Acoustic problems can be made using the following three categories:

• External wind noise transmitted to the inside through a structure: In the automotive industry, a pillar, side mirror and windshield wipers noise are typical problems of this category.

• Internal flow noise transmitted to the outside through a structure: Examples of this class of problems are exhaust, HVAC and Intakes noises.

• Rotating machines noise: Axial and centrifugal fans are noisy components that bring with them many interesting Aero-Acoustic problems.

Most of the Aero-Acoustic R&D works are performed experimentally but this method has some critical pitfalls. Although it is relatively simple to setup a microphone, measure a noise level and derive a spectrum at any given location in space, the correct analysis of an Aero Acoustic problem involves the use of advanced experimental techniques and is complex to use. The Aero Acoustic engineering community seeks more and more the help of CAE tools as they become available. Those tools complement the experimentations and allow a thorough visualization and understanding of the pressure and velocity fields as well as the structural vibrations. Furthermore, parametric studies can be carried out with little added cost since a numerical model modification is often straightforward and the CPU time is becoming cheaper and cheaper.

CFD codes are available since over several years, able to predict with a reasonable precision steady state flows (drag and lift) and slow transient flows like heating and defrosting. Highly transient flows involved in the Aero-Acoustic phenomena have not been treated since they were not in the bulk of the needs and they required way too much CPU to be industrially feasible. Acoustic Propagation numerical tools have also been industrially available since quite a few years. These tools operate in the frequency domain and are able to propagate a given boundary condition signal in a fluid at rest, including the noise reflections, diffractions, transmissions and attenuations thanks to the various geometrical obstacles and different materials.

Attempts have been made to combine existing CFD and Acoustic propagation tools to predict Aero-Acoustic problems. Most methodologies are based on the Lighthill &Curle method, developed in the mid 50’s and Ffowcs Williams & Hawkings contributions made in the late 60 ’s [67], [68], [69], [70]. The ideas underlying these methods are to decouple the flow pressure field and the acoustic pressure field. The fluid flow can then be computed by a standard CFD code and the noise derived from the curvature and turbulent intensities of the flow. A propagation tool is then used to compute the noise on a sub grid of the CFD computational domain loosing therefore quite some local information and high frequency content. First attempts were made with incompressible steady state CFD simulations and were not able to deliver valuable result in many cases. A good example of these limitations is highlighted by the study of the noise generated by a simple ‘side mirror’ shape written by R. Siegert [71]. Recent developments of this family of techniques require the use of transient simulations and filtering to avoid loosing to much information on the coarser acoustic mesh. Reasonable success has been met in specific areas involving low frequencies (up to a couple hundred Hz) and considerable CPU time is needed.

An alternative methodology is to incorporate in a single numerical tool, right from the beginning, the ingredients that are necessary to perform direct Aero-Acoustic numerical simulation. They are:


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• Compressible Navier Stokes: To be able to propagate pressure waves and therefore take into account in a single simulation the flow and the noise, including all possible cavity modes.

• Fluid structure coupling: To be able to treat the problems involving a turbulent flow on one side of the structure and the noise radiation on the other side.

• Small time step: To be able to deal accurately with frequencies going up to several thousand Hertz.

• Transient turbulence modeling: Unlike the Reynolds Averaged Navier Stokes (RANS) methods that makes the assumption that the flow is a combination of a steady state and turbulent fluctuations. Aero-Acoustic noise is directly linked to the small scale turbulence fluctuations and strongly time dependant.

• Acoustic boundaries with prescribed impedance: This is a critical point of a good Aero-Acoustic simulation. Boundaries need to be able to perform tasks such as giving a free field impedance to an inlet with fixed velocity, prescribing a specific impedance at the outlet of a duct to make sure long wavelength stay trapped inside, treat exterior air impedance effect on a vibrating structure and be used to model absorbing materials (carpet, foams …) that are used to coat many components.

These ingredients have been implemented in a single numerical code. The outcome is RADIOSS solver which is different from the existing CFD codes in its capabilities and particularly well suited to short time transient analysis.

2.1 Mathematical Formulation The objective of the development described in this document is underlined by the search for fully suited numerical technologies in order to model all physical phenomena involved in noise source generation. This criteria will be used to perform all the major numerical choices described here below without any trade-off tight to other applications.

2.1.1 Compressible Navier Stokes Correct prediction of Aero-Acoustic phenomena must obviously include a solution of the 3D Navier Stokes equations since noise sources lie in a turbulent viscous fluid flow. The equations have been written with the Arbitrary Lagrange Euler (ALE) formulation [8].This means that an arbitrarily moving frame is used in order to be able to have a fluid grid that can undergo deformation. The deformation can range from small vibrations (an exhaust pipe) to large deformations (a door slamming). The conservation equations are written for mass, momentum and energy:

Mass ( )( ) 0.. =∇+∇−+∂∂ uwu ρρρ

t EQ. 2.1.1.1

Momentum ( )( ) 0. =∇+∇−∇−+∂∂ p

tτρρ uwuu

EQ. 2.1.1.2

Energy ( )( ) ( ) 0.. =∇++∇−+∂

∂ uwu peete ρρρ

EQ. 2.1.1.3

Where u is the fluid velocity, w the grid velocity, ρ the density, p the pressure e the energy and τ the stress tensor.

NOTE: In case of w=u, the system degenerates into the Lagrangian formulation, meanwhile w=0 describes the classical Eulerian formulation. Lagrangian formulation will be used for the structures when needed. An ALE fluid mesh is attached to the structure mesh, therefore able to undergo deformations tight to the structural vibrations.


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Acoustic phenomena are driven by pressure waves propagating in the materials. Those waves velocity, the sound speed velocity c can be expressed by the following equations:

ρ∂∂

=Pc

EQ. 2.1.1.4

Another useful relation defining c as a function of the material characteristics is:

ρKc = EQ. 2.1.1.5

where K is the bulk modulus of the material.

In the usual engineering fields, the sound speed is about 340 m/s for air and 1500 m/s for water.

Most of the CAA research in the industry is aimed at flow motions of relatively small Mach numbers.

Therefore, CFD codes have widely used the incompressibility hypothesis which allows a fair simplification of the Navier Stokes equations. An incompressible flow is a flow for in which:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

<<∇+∂∂

=zw

yv

xu

tDtD ρρρρ .u EQ. 2.1.1.6

If this condition is fulfilled the mass conservation equation can be rewritten.

0=∇u EQ. 2.1.1.7

It has to be noted that an incompressible flow does not necessarily implies that ρ remains constant throughout the flow but that ρ remains constant along a streamline since one can substitute EQ.2.1.1.6 in EQ.2.1.1.1. Practically, this condition can be reasonably assumed for Mach numbers lower than 0.3 and in the case of transient flows, if the time rate of velocity change is long compared to the time for a sound wave to traverse the flow field. Thus, the incompressibility hypothesis yields an instantaneous transmission of pressure waves in the fluid domain. Incompressible approach will not be able to simulate the propagation of acoustic waves nor the cavity modes tight to composition of reflected waves on a given geometry. It then becomes necessary to apply one of the following techniques:

• Solve the compressible Navier Stokes equations to capture correctly the propagative nature of the noise waves.

• Use additional equations to derive the noise sources from the flow field.

Accordingly to the base criteria of always to the best solution for CAA, it has been decided to follow the first path that requires no specific assumptions and is physically and mathematically much better grounded. By solving the fully compressible Navier Stokes equations, it becomes possible to compute in a single simulation the flow and the sound pressure wave’s propagation. Since there is a large gap between the flow pressure variations (in the order of hundreds of Pa in most cases) and the acoustic pressure variations (in the order of 1 Pa ~94 dB), it has become critical to use 64 bits double precision arithmetic. Using 32 bits would yield flawed results because the acoustic pressure levels would not be handled properly in many cases.

The spatial integration of momentum equation is performed with a Finite Element integration using Streamline Upwind Petrov Galerkin (SUPG) scheme [72], which was shown accurate enough to capture flow instabilities. Advection of state variables is achieved via a simple finite volume technique as in Donea's original paper [8].


20-jan-2009 20

2.1.2 Transient analysis, explicit formulation Sound pressure waves and acoustic phenomena are transient per fundamental nature. Therefore, a transient solution of the full compressible Navier-Stokes equations has been developed, allowing the propagation of pressure waves as well as transient fluid flow simulation.

For time integration, two techniques are available, implicit and explicit. Implicit methods are unconditionally stable; therefore allowing the use of an arbitrary large time step; meanwhile the explicit formulation is conditionally stable. A Courant Friedrichs Levy condition [56] has to be verified at each time step of the simulation:

( )⎟⎟⎠

⎞⎜⎜⎝

⎛

−+≤

wuclMindt EQ. 2.1.1.8

where l is the characteristic element size, c the sound speed, u the flow velocity and w the grid velocity. Typical values of dt for CAA applications are in 1 to 5 μ s range.

The choice of one formulation or the other will actually be dictated by the minimization of the CPU time for the considered application. Shall the goal be to simulate the defrosting of a windshield lasting about 5 minutes, with low flow velocities, the several second time step allowed by the implicit scheme will be much more efficient. In the CAA case however, the frequencies of interest are going up to 5 to 10 kHz and the Nyquist criteria and anti aliasing filtering imposes a sampling for the time domain recording of 20 to 40 kHz. Therefore, the maximum time step that can be used in implicit will be 5x10-5s.The explicit time step being in the range of 2x10-6s, one has to compare the cost of one implicit time step to 25 explicit time steps. A survey yield a ratio of about two orders of magnitude between the CPU cost of both schemes in favor of the explicit and therefore, the explicit scheme CPU efficiency has been considered superior for CAA applications.

2.1.3 Large Eddy Simulation Turbulence modeling Another critical aspect of a correct CAA modeling is to take into account properly the noise induced by turbulent structures. Unfortunately, the turbulent structures that are simultaneously active at any given time range from the full size of the problem to the microscopic Kolmogorov size. The ideal solution would be to use DNS but this is unfortunately out of reach of today computers. Consequently, a turbulence model has to be used. The choice among the turbulence models will be performed by evaluating their interest for the CAA simulation. Today, there are two major families of turbulence models that are available for implementation in an industrial oriented CFD code.

The Reynolds Average Navier-Stokes (RANS) turbulence models family relies on the assumption that the flow can be separated between a steady state flow and a turbulent fluctuation. The steady state flow can be interpreted as the spatial average of the flow field. Transient problems featuring vortex shedding cannot be treated this way and therefore Unsteady Reynolds Average Navier-Stokes (URANS) have been developed in which the averaging is performed on a considerably smaller time scale. Although RANS and URANS models are easy and cheap to implement, the basic assumption that there is a combination of a steady state flow and turbulent fluctuations is not very well suited to the CAA analysis where the key to the noise generation is the transient turbulent behavior. The Large Eddy Simulation or LES family is based on a fully transient flow simulation. Large turbulent scales which depend heavily on the boundary conditions are solved directly by the computational grid and the small scales are assumed to be more or less non problem dependant, are solved by a model called a Sub Grid Scale model (SGS).The Navier-Stokes equations are filtered in space and the spatial high frequency terms (beyond the filter cutoff frequency) are modeled by the SGS .The SGS main role is to model the viscous dissipation of the energy within the small scale eddies and absorbs energy from the large scales to this effect. The filter used most often is actually the grid itself and the spatial cutoff is about 6 element size. The drawback that prevents the LES to be used in many cases is the need for small elements for the base assumption to be valid (independence of the modeled scales from the boundary conditions) and the transient assumption which requires the use of a CFL condition on the time step to guarantee a correct simulation of the larger scales and yields large CPU time when combined with classical implicit resolution techniques.


20-jan-2009 21

In our case, since it has already been chosen to solve the transient Navier Stokes equations with an explicit scheme yielding a very small time step, the CFL condition does not imply any major constraint. Mesh size criteria have been developed in order to ensure a good CAA behavior and LES accuracy (also see Modeling Methodology in the next chapter).

In noise generation zones where most of the turbulence is generated and small vortices are very active, the mesh criteria is:

⎟⎟⎠

⎞⎜⎜⎝

⎛<

fuh 1.0 EQ. 2.1.1.9

Where h is the element size, u the flow velocity and f the highest frequency of interest. The last equation yields the expression of local Reynolds number:

⎟⎟⎠

⎞⎜⎜⎝

⎛<=

νν .1.0.Re

2

fuhu

EQ. 2.1.1.10

Usually, u is in the 20 m/s range, f is 2500 Hz and the air viscosity ν =1.5x10-5m2/s. This yields a local Reynolds number of 1000. A reinforced mesh criteria applied close to the walls is to have a y+

lower than 100. Although, these numbers might be considered as large by academic standards, it has been verified for various industrial applications that the numerical predictions reasonably match the experimental data [73]. In the SGS model, the sub grid scale level energy dissipation SGSε is proportional to the resolved scales stress tensor ijS and the

modeled small scales stress tensor ijτ through a turbulent viscosity coefficient SGSν .This turbulent viscosity is a

function of the filter width l (mesh size) and of the resolved scales (via the stress tensor).Cs is known as the Smagorinsky constant and has been experimentally evaluated to be 0.18 (this constant can be tuned for specific applications and 0.1 is used by default in RADIOSS).

ijijSGSSGS SS ..2νε −= EQ. 2.1.1.11

( ) ( )SlCSSGS2=ν EQ. 2.1.1.12

It has to be noted that an original pressure damping factor has been developed in order to better take into account the absorption of the acoustic waves on the boundary layers.

The elements neighboring the walls have a specific treatment depending upon the element size. A logarithmic velocity profile is assumed in the first layer of elements and the corresponding modified viscosity is derived.

2.1.4 Boundary Conditions Non-reflective boundary conditions are critical for CFD simulations and it is obviously going to be even more critical for CAA analysis. Acoustic boundary conditions treatment can be classified in three different categories:

• Non reflective fluid boundaries in the open field. Under this class lie the external wind noise problems. Not only outlet and sides need to be treated but unless and infinite impedance or supersonic conditions are assumed, the inlet shall be able to let acoustic waves go out of the computational domain.

• Boundaries including geometrical details. Exhaust tailpipe for instance have a cutoff frequency related to its diameter and shape.

• Exterior air boundaries. In order to correctly treat problems like an exhaust noise, the exterior air impedance has to be applied to the structure in order to get the correct vibrations and noise. It has to be noted that in the case of a duct surrounded by other structures (e.g. an exhaust line under a car), the acoustic impedance of the exterior air is vastly different from the free field conditions and non trivial to compute.


20-jan-2009 22

The goal of boundary conditions is to replace the influence of the exterior of the computational domain by a condition that will let some frequencies go out and reflect others accordingly to the physical properties of the surrounding materials. Acoustic impedance, defined by the ratio of the acoustic pressure p~ at a given location

over the acoustic velocity nu at the same location is a key notion.

nupZ~

= EQ. 2.1.1.13

The important point to be reminded here is that p~ and nu differs from the traditional fluid dynamics pressure and velocity since they are complex functions, solutions of Helmoltz equation which is describing the propagation of waves in the frequency domain ω :

( ) ( ) ( )ωωω ,,~,~ 2 rFrpkrp rrr=+Δ EQ. 2.1.1.14

With:

ck ω

= EQ. 2.1.1.15

where c is the sound speed. In the acoustic works, items are classically described as a function of the frequency. In particular, impedances describing the boundary conditions acoustic behavior. RADIOSS on the other hand is based on a time domain description of the physical phenomena, it is therefore not possible to use directly those data and it is required to impose time dependant boundary conditions. For the non-reflective boundary conditions, it has been decided to implement the linearized Euler equation by Bayliss and Turkell [74 ]:

( )pplc

tuc

t c

n −+∂

∂=

∂∂

∞2ρρ

EQ. 2.1.1.16

Where n is the normal to the boundary surface and lc a relaxation factor toward the desired pressure ∞p . Practically, the last equation matches the radiation of a monopole situated at a distance 2 lc inside the computational domain.

2.1.5 Fluid Structure Coupling Fluid Structure Interaction (FSI) is an issue in many important sub domains of the CAA. Important applications include and are of course not limited to:

• Duct noise: The noise radiated by a structural system of ducts whose vibrating walls are excited by a turbulent flow inside. Among the many examples, exhausts and Heat Ventilation and Air Conditioning (HVAC).

• Passenger perception of exterior wind noise transmitted by a side glass or plane cabin.

One might suggest that rather than using a complex FSI solution, a decoupled approach could be used. That is performing a CFD (or pure fluid CAA) simulation on one hand and then apply to a structural model as a boundary condition the computed pressure field. Beyond the mere inconvenience tight to the external coupling of two numerical programs (flow and structure), this approach, which is reasonably grounded steady aero elasticity phenomena and for low frequencies applications is not well suited to broad band CAA applications. It is not practical to apply the transient pressure field with a sampling of 20 kHz on a fine mesh and even more important, there is a dependency of the structure vibrations to the surrounding fluid impedances. This impedance can be eventually modeled on the side where there is no flow but is extremely complex on the side exposed to the turbulent airflow. Consequently, in most cases, a decoupled modeling will be flawed from the beginning.


20-jan-2009 23

Accordingly to our goal of modeling all physical phenomena important for CAA, a full FSI has been developed between the well-established structural module of RADIOSS and the CFD/CAA module. Some specific options have been developed in order to make this interface practical among which:

1. ALE Subcycling. In most applications, there are 95% fluid elements and 5% or less structural elements. Unfortunately the sound speed is often one order of magnitude higher in the structure yielding a time step decreased by the same factor. The 5% structure element will impose to the 95% a time step 10 times smaller, which is obviously not desirable. A specific subcycling has been developed in order to compute the fluid elements with a decoupled time step from the structures.

2. Fluid – Structure interfaces have been developed to ease the mesh development. Those interfaces can be used to stitch arbitrary meshes together and can be used in either fixed or sliding mode (fluid-fluid fixed and sliding interfaces have also been developed to take care primarily of fan problems).

2.2 Modeling Methodology The process of defining a numerical model for a given CAA application unfolds in four consecutive phases. The quality of the results is directly tight to the quality of the numerical model developed in the points 1 and 2 below. It is therefore critical to define precisely the questions the model is supposed to answer to before starting any development:

1. Mesh definition: Targeting specific answers to engineering questions and constrained by the available computer resources.

2. Numerical model construction, including material, boundary conditions and desired output.

3. Run monitoring.

4. Post processing of the time domain data. Including 3D visualization, time history and frequency content analysis.

2.2.1 Mesh definition The mesh building process is the art of building models able to solve the CFD and the CAA problem for a given set of boundary conditions and range of frequencies while keeping the model as small as possible to minimize the compute time. To do so, a set of practical rules have been developed. These rules should be used as initial guidelines to get started on a given problem.

However each class of problems has its own requirements and subtleties and a good knowledge of the problem physics through experimental data and/or numerical simulations will be necessary to refine these rules and get the best possible results.

2.2.2 Post processing Post processing of the numerical simulation is very similar to the post processing of a detailed experimental study of the same problem. The analysis will be carried out by using:

• FFT’s of recorded time domain signal to access the frequency domain content at any given location of the computational domain.

• Visualization and analysis of intensities on the structures

• Propagation in the far field (if and when needed) of the pressure signal. Typically, this is required for simulations where the measurement locations are not located in the computational domain. In most of the internal flow problems for instance the limit of the domain is the structure and a boundary elements layer to represent the outside air impedance (exhaust, HVAC …).The noise is often measured at a given distance outside the ducts in still air where there is no reason to have an expensive CAA solution. This propagation can be performed by a simple monopolar approximation that gives satisfactory results in the free conditions or by more sophisticated tools such as BEM methods.

• …


20-jan-2009 24

2.3 Application and Validation As the aim of this manual is not to describe the modeling techniques, the readers are invited to consult the available publications (for example [73], [75], [76], [77] and the proceeding of International RADIOSS User Conference).

2.4 Conclusion Along the whole development process, the CAA orientation has been kept in mind for all the decisions and lead to the choice of the following methods:

• Compressible 3D Navier stokes solver

• Transient explicit time integration

• LES Turbulence

• Acoustic boundary conditions

• Fluid Structure coupling

These ingredients are needed to perform CAA simulations with no particular assumptions on the flow (excepted of course the use of a turbulence model), the fluid structures coupling or the vibrations, making RADIOSS a fairly general code.

Further developments are considered among which we can cite the ability to deal with bent flows. On the real world, in many interesting cases, the object to be studied is not positioned on a flat ground but embedded within a complex geometrical shape (for example a side mirror on a car).The flow, which hits the component, is distorted by this geometry. Simulation of the whole vehicle with a CAA method is not practical beyond a few hundred Hertz because of the huge number of elements needed in the propagation zone. Therefore, a method mixing a steady state simulation of the far field to get proper bent boundary conditions and RADIOSS close to the component and the acoustic sources zones is currently under development to perform CAA analysis of this kind of problem well beyond 1000 Hz.

RADIOSS THEORY Version 10.0 SMOOTH PARTICLES HYDRODYNAMICS

20-jan-2009 25

Chapter

SMOOTH PARTICLE HYDRODYNAMICS


20-jan-2009 26

3.0 SMOOTH PARTICLE HYDRODYNAMICS Smooth Particle Hydrodynamics SPH is a meshless numerical method based on interpolation theory. It allows any function to be expressed in terms of its values at a set of disordered point’s so-called particles. SPH is not based on the particle physics theory. The conservation laws of continuum dynamics, in the form of partial differential equations, are transformed into integral equations through the use of kernel approximation. A comprehensive state-of-the-art of the method is given in [78], [81], and [82]. These techniques were initially developed in astrophysics [79], [80]. During the 1991-1995 periods, SPH has become widely recognized and has been used extensively for fluid and solid mechanics type of applications. SPH method is implemented in RADIOSS in Lagrangian approach whereby the motion of a discrete number of particles is followed in time.

SPH is a complementary approach with respect to ALE method. When the ALE mesh is too distorted to handle good results (for example in the case of vortex creation), SPH method may allow getting a sufficiently accurate solution.

3.1 SPH approximation of a function

Let ( )∏ xf the integral approximation of a scalar function f in space:

( ) ( ) ( )dyhyxWyfxf ,−= ∫∏Ω

EQ.3.1.0.1

with h the so-called smoothing length and W a kernel approximation such that:

( )∫Ω=−∀ 1,, dyhyxWx EQ. 3.1.0.2

and ( ) ( )yxhyxWx h −=−∀ → δ,lim, 0 (in a suitable sense) EQ. 3.1.0.3

δ denotes the Dirac function.

Let a set of particles i=1, n at positions xi (i=1,n) with mass mi and density iρ . The smoothed approximation of the function f is (summation over neighbouring particles and the particle i itself):

( ) ( ) ( )hyxWxfm

xfni

ii

is

,,1

−= ∑∏= ρ

EQ. 3.1.0.4

The derivatives of the smoothed approximation are obtained by ordinary differentiation.

( ) ( ) ( )hyxWxfmxfni

ii

i ,,1

−∇=∇ ∑= ρ

EQ. 3.1.0.5

The following kernel [83] which is an approximation of Gaussian kernel by cubic splines was chosen (Figure 3.1.1):

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−=⇒≤

32

3 21

32

23,

hr

hr

hhrWhr

π EQ. 3.1.0.6

( )3

3 24

1,2 ⎟⎠⎞

⎜⎝⎛ −=⇒≤≤

hr

hhrWhrh

π EQ. 3.1.0.7


20-jan-2009 27

and ( ) 0,2 =⇒≤ hrWrh EQ. 3.1.0.8

Figure 3.1.1 Kernel based on spline functions

This kernel has compact support, so that for each particle i, only the closest particles contribute to approximations at i (this feature is computationally efficient). The accuracy of approximating EQ. 3.1.0.1 by EQ. 3.1.0.4 depends on the order of the particles.

3.2 Corrected SPH approximation of a function Corrected SPH formulation [85], [86] has been introduced in order to satisfy the so-called consistency conditions:

( )∫Ω

∀=− xhxyW ,1, EQ. 3.2.0.1

( ) ( )∫Ω

∀=−− xhxyWxy ,0, EQ. 3.2.0.2

These equations insure that the integral approximation of a function f coincides with f for constant and linear functions of space.

CSPH is a correction of the kernel functions :

( ) ( ) ( ) ( ) ( )[ ]jjj xxxxhxWhxW −•+= βα 1,,ˆ , with ( ) ( )hxxWhxW jj ,, −= EQ. 3.2.0.3

where the parameters ( )xα and ( )xβ are evaluated by enforcing the consistency condition, now given by the point wise integration as :

( ) xhxWVj

jj ∀=∑ ,1,ˆ EQ. 3.2.0.4

( ) ( ) xhxWxxV jjj

j ∀=−∑ ,0,ˆ EQ. 3.2.0.5

W(r,h)

2h

W(r,h)

2h


20-jan-2009 28

These equations enable the explicit evaluation of the correction parameters ( )xα and ( )xβ as follows:

( ) ( ) ( ) ( ) ( ) ( )hxWxxVhxWxxxxVx jjj

jjjj

jj ,,1

−⎥⎦

⎤⎢⎣

⎡−⊗−= ∑∑

−

β EQ. 3.2.0.6

( ) ( ) ( ) ( )[ ]∑ −•+=

jjjj xxxhxWV

xβ

α1,

1 EQ. 3.2.0.7

Since the evaluation of gradients of corrected kernel (which are used for the SPH integration of continuum equations) becomes very expensive, corrected SPH limited to order 0 consistency has been introduced. Therefore, the kernel correction reduces to the following equations :

( ) ( ) ( )xhxWhxW jj α,,ˆ = EQ. 3.2.0.8

( ) xhxWV jj

j ∀=∑ ,1,ˆ EQ. 3.2.0.9

that is ( ) ( )∑=

jjj hxWV

x,

1α EQ. 3.2.0.10

Note that SPH corrections generally insure a better representation even if the particles are not organized into a hexagonal compact net, especially close to the integration domain frontiers. SPH corrections also allow the smoothing length h to values different to the net size xΔ to be set.

3.3 SPH Integration of continuum equations In order to keep an almost constant number of neighbors contributing at each particle we use smoothing length varying in time and in space. Consider id the smoothing length related to particle i ;

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

2,ˆ ji

jij

ddxxWiW and ( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=∇

2,ˆ ji

jxj

ddxxWgradiW

i if kernel correction,

EQ. 3.3.0.1

or ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

2, ji

jij

ddxxWiW and ( ) ⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=∇

2, ji

jxj

ddxxWgradiW

i without kernel correction.

EQ. 3.3.0.2 At each time step, density is updated for each particle i, according to:

iii

vdtd

⋅∇−= ρρ EQ. 3.3.0.3

with ( ) ( )iWvvm

v jjij

ji

∇⋅−=⋅∇ ∑ ρ EQ. 3.3.0.4

( jm indicates the mass of a particle i, iρ its density, iv its velocity).


20-jan-2009 29

Strain tensor is obtained by the same way when non pure hydrodynamic laws are used or in the other words when law uses deviatoric terms of the strain tensor:

( ) ( ) .3...1,3...1, ==−= ∑ βαρ β

ααβ

α

idxdW

vvm

dxdv j

jij

j

i

EQ. 3.3.0.5

Next the constitutive law is integrated for each particle. Then Forces are computed according to :

( ) ( )[ ] ( ) ( )[ ]2

jWiWmmjWpiWpVV

dtdvm ij

ijjj

iijjijj

ii

i

∇−∇−∇−∇−= ∑∑ π EQ. 3.3.0.6

where ip and jp are pressures at particles i and j, and ijπ is a term for artificial viscosity. The expression is more complex for non pure hydrodynamic laws. Note that the previous equation reduces to the following one when there is no kernel correction:

[ ] ( ) ( ),iWmmiWppVVdtdvm jijj

jijjij

ji

ii ∇−∇+−= ∑∑ π since ( ) ( )iWjW ji −∇=∇ EQ. 3.3.0.7

Then, search distances are updated according to:

( )3

ii

iv

ddtdd ⋅∇

= EQ. 3.3.0.8

in order particles to keep almost a constant number of neighbors into their kernels ( 3dρ is kept constant).

3.4 Artificial viscosity As usual in SPH implementations [83], viscosity is rather an inter-particles pressure than a bulk pressure. It was shown that the use of EQ. 3.4.0.1 and EQ. 3.4.0.2 generates a substantial amount of entropy in regions of strong shear even if there is no compression.

( )2

22

ji

ijijji

ij

qcc

qb

ρρ

αμμπ

+

++

−= EQ. 3.4.0.1

with ( ) ( )

22

ijji

jijiijij

dXX

XXvvd

εμ

+−

−•−= EQ. 3.4.0.2

where iX (resp. jX ) indicates the position of particle I (resp.j) and ci (resp cj) is the sound speed at location i (resp.j), and qa and qb are constants. This leads us to introduce EQ. 3.4.0.3 and EQ. 3.4.0.4, as explained in [82]. The artificial viscosity is decreased in regions where vorticity is high with respect to velocity divergence.

( )2

22

ji

ijijji

ij

qcc

qb

ρρ

αμμπ

+

++

−= EQ. 3.4.0.3


20-jan-2009 30

with ( ) ( ) ( )

k

kkk

kk

ji

ijji

jijiijij

dcvv

vf

ff

dXX

XXvvd

εεμ

′+×∇+⋅∇

⋅∇=

+

+−

−•−= ,

222 EQ. 3.4.0.4

Default values for qa and qb are respectively set to 2 and 1.

3.5 Stability: Time step control The stability conditions of explicit scheme in SPH formulation can be written over cells or on nodes.

3.5.1 Cell time step In case of cell stability computation (when no nodal time step is used), the stable time step is computed as:

( ) ( )ijjii

iii

iii

iisca and

cdqaqbwith

cdtt μμμα

ααmax,,

1min

2=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ⋅⋅+=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++⋅Δ=Δ

EQ. 3.5.1.1

scatΔ is the user defined coefficient (RADIOSS option /DT or /DT/SPHCEL). The value of scatΔ =0.3 is recommended in [83].

3.5.2 Nodal time step In case of nodal time step, stability time step is computed in a more robust way:

i

ii K

mt 2=Δ at particle i EQ. 3.5.2.1

Using the following notations:

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=∇⎟⎟

⎠

⎞⎜⎜⎝

⎛ +−= ′′ 2

ˆ2

ˆ jijxij

jijij

ddxxWgradiWand

ddxxWiW if kernel correction,

EQ. 3.5.2.2

or ( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=∇⎟⎟

⎠

⎞⎜⎜⎝

⎛ +−= ′′ 22

jijxij

jijij

ddxxWgradiWand

ddxxWiW if no kernel correction,

EQ. 3.5.2.3

Recalling that apart from the artificial viscosity terms:

( ) ( )[ ]∑ ∇−∇==j

jjjijiijiji jWpiWpVVFFF , EQ. 3.5.2.4

we write ( ) ( ) ( ) ( )[ ]( )jWpiWpVVuud

duud

dFK ijjiji

jiji

ijij ∇+∇

−≤

−= EQ. 3.5.2.5


20-jan-2009 31

where ui-uj is the relative displacement of particles i and j. Keeping the only first order terms leads to :

( ) ( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡∇

−+∇

−≤ jW

uuddp

iWuud

dpVVK iji

jj

ji

ijiij EQ.3.5.2.6

where ( ) ( ) ( ) ( ) ( ) ( )iWuud

dcVViWuud

dddpVViW

uuddpVV j

ji

iijij

ji

i

i

ijij

ji

iji ∇

−=∇

−⋅=∇

−ρρ

ρ2

EQ.3.5.2.7

that is ( ) ( ) ( ) 222 iWVcmiWuud

dpVV jjiijji

iji ∇=∇

−& EQ.3.5.2.8

Same reasoning leads to :

( ) ( ) ( ) 222 jWVcmjWuud

dpVV iijji

ji

jji ∇=∇

−& EQ.3.5.2.9

So that ( ) ( ) 222222 jWVcmiWVcmK iijjjjiiij ∇+∇≤ && EQ.3.5.2.10

Stiffness around node i is then estimated as :

∑≤j

iji KK EQ.3.5.2.11

3.6 Conservative smoothing of velocities It can be shown that the SPH method is unstable in tension. The instability is shown to result from an effective stress with a negative modulus (imaginary sound speed) being produced by the interaction between the constitutive relation and the kernel function, and is not caused by the numerical time integration algorithm [84]. According to [82], we use special filtering of velocities (so called conservative smoothing, because momentum quantities are not modified):

( ) ( ) ( ) ( )∑

+−

++=

j

jiij

ji

jcsii

iWjWvv

mvsmoothedV

22

ρρα EQ.3.6.0.1

3.7 SPH cell distribution It is recommended to distribute the particles through a hexagonal compact or a cubic net.

3.7.1 Hexagonal compact net A cubic centered faces net realizes a hexagonal compact distribution and this can be useful to build the net (Figure 3.7.1). The nominal value h0 is the distance between any particle and its closest neighbour. The mass of the particle mp may be related to the density of the material ρ and to the size h0 of the hexagonal compact net, with respect to the following equation:


20-jan-2009 32

ρ2

30hmp ≈ EQ.3.7.1.1

since the space can be partitioned into polyhedras surrounding each particle of the net, each one with a volume :

2

30hVp ≈ EQ.3.7.1.2

But, due to discretization error at the frontiers of the domain, mass consistency better corresponds to nVmP

ρ=

where V is the total volume of the domain and n the number of particles distributed in the domain.

Figure 3.7.1 Local view of hexagonal compact net and perspective view of cubic centered faces net

Note that choosing h0 for the smoothing length insures naturally consistency up to order 1 if the previous equation is satisfied.

Weight functions vanish at distance 2h where h is the smoothing length. In an hexagonal compact net with size h0, each particle has exactly 54 neighbors within the distance 2 h0 (Table 3.7.1).

Table 3.7.1 Number of neighbors in a hexagonal compact net.

distance d number of particles at distance d number of particles within distance d

0h 12 12

02h 6 18

03h 24 42

02h 12 54

05h 24 78

3.7.2 Cubic net Let c the side length of each elementary cube into the net. The mass of the particles mp should be related to the density of the material ρ and to the size c of the net, with respect to the following equation:


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ρ3cmp ≈ EQ.3.7.1.3

By experience, a larger number of neighbors must be taken into account w.r.t. the hexagonal compact net, in order to solve the tension instability as explained in following sections. A value of the smoothing length between 1.25c and 1.5c seems to be suitable. In the case of smoothing length h=1.5c, each particle has 98 neighbors within the distance 2h.

Table 3.7.2 Number of neighbors in a cubic net

distance d number of particles at distance d number of particles within distance d

c 6 6

c2 12 18

c3 8 26

2c 6 32

c5 24 56

c6 24 80

2 c2 12 92

3c 6 98

theory_ale_cfd_sph

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