FEMCI Workshop – May 8, 2003 [amk]

An Explicit-Implicit Analysis Scheme in a General-Purpose FEA

Environment

Abed M. KhaskiaMallett Technology, Inc.

Laurel, Maryland(301) 725-0060

FEMCI Workshop – May 8, 2003 [amk]

Outline

• Introduction

• System of Equations [Equations of Motion]

• Explicit Solution Scheme

• Implicit Scheme

• Mixed Explicit – Implicit

• Illustrative Case

FEMCI Workshop – May 8, 2003 [amk]

Introduction - Applications

• Vibration analysis

• Impact analysis. Crashworthiness, Drop test

• Rotating elements and machinery

• Earthquake analysis

• Explosives

• Metal Forming/stamping/rolling

• Random Vibration

FEMCI Workshop – May 8, 2003 [amk]

Applications

Car CrashProjectile Impact

FEMCI Workshop – May 8, 2003 [amk]

Applications

Metal FormingJet Engine Fan Containment

FEMCI Workshop – May 8, 2003 [amk]

Applications

Pipe Whip Problem

FEMCI Workshop – May 8, 2003 [amk]

Introduction - Challenges

• Large systems

• Material and geometrical behavior

• Unknown material properties

• Loading and system boundary conditions

• Multiphysics and multiple domains

• Available testing and verifications issues

• Changing technologies in numerical analysis

FEMCI Workshop – May 8, 2003 [amk]

System of Equations

• General Equations of Motion

( )tFuKuCuM =++ &&&

• Solutions:• Implicit

• Explicit

• Mixed dictated by physics and numerical behavior

FEMCI Workshop – May 8, 2003 [amk]

Implicit Scheme

( )tFuKuCuM =++ &&&

( )

ttt

ut

ut

utt

u

ttt

ut

utt

ut

utt

u

tFtt

utt

utt

uM

∆∆+

+−+=∆+

∆∆+

+−+∆+=∆+

=∆+

+∆+

+∆+

] )1[(

2 ] )2/1[(

K C

&&&&&&

&&&&&

&&&

δδ

αα

Resulting Equations

Integration Scheme,

Example, Newmark

Effectively Resulting intttt FKu ∆+

−

∆+ = 1

FEMCI Workshop – May 8, 2003 [amk]

Implicit Scheme

Nonlinear Case – Requires Newton-Raphson Iterations and Satisfying Equilibrium

FuK T δδ =

FEMCI Workshop – May 8, 2003 [amk]

Explicit Scheme

( )( ) contetcn

T

extt

FFdBF

FFMa i

++Ω∫Σ=

−=

Ω

−

int

1nt

σtttttt tavv 2/2/ ∆+= ∆−∆+

2/2/ ttttttt tvuu ∆+∆+∆+ ∆+=ttott uxx ∆+∆+ +=

max

2ω

=∆≤∆ crittt

lc=ω 2

max

ρEc=c

lt=∆

• Critical Time Step

• No matrix conversion

• Computations of internal and external force vectors

FEMCI Workshop – May 8, 2003 [amk]

Comparisons/Issues

• Stability

• Time step size

• Nonlinear effects

• Computations

• Convergence

• Mass matrices

FEMCI Workshop – May 8, 2003 [amk]

Mixed Scheme / Explicit - Implicit

• Solution Steps

Explicit SolutionDeformation, Acceleration

Velocity, Stress, Strain

Deformed Shape, Stresses and Strains

Implicit /Static Equilibrium

Solution time, Loads, BCs and Final State

Element Types, Material Models

Equivalency

Solution Accuracy

( )( ) contetcn

T

extt

FFdBF

FFMa i

++Ω∫Σ=

−=

Ω

−

int

1nt

σ FuTK δδ =

FEMCI Workshop – May 8, 2003 [amk]

Mixed Scheme / Explicit - Implicit

• Solution Process on Material Model

Typical Stress - Strain Data

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.02 0.04 0.06 0.08 0.10

Strain

Stre

ss G

Pa

ImplicitExplicit

FEMCI Workshop – May 8, 2003 [amk]

Explicit to ImplicitCase Study – Cup Stamping

• System Description and FE Model• Force is applied at blank holder

• Sinusoidal velocity is applied at punch

• Mass and stiffness damping

• Friction between components

• Mass Scaling

FEMCI Workshop – May 8, 2003 [amk]

Explicit to ImplicitCase Study – Cup Stamping

• Modeling Challenge

• Mass scaling to speed solution

• Solution accuracy and verifications

• Damping

• Friction effects

• Element deformation and proper shape

• Time point and process to go from explicit to implicit

• Preventing Rigid body motion in implicit solution

• Convergence of the nonlinear implicit solution

FEMCI Workshop – May 8, 2003 [amk]

Explicit to ImplicitCase Study – Cup Stamping

• Results

• Animation of process

• Quality of solution – Hourglass energy check

• Force applied by punch and blank velocity

• Fluctuation in stress and strain data

• Deformed shape and plastic strains at end of explicit

• Spring back shape after implicit switch

FEMCI Workshop – May 8, 2003 [amk]

Explicit to ImplicitCase Study – Cup Stamping

• Stamping Process

FEMCI Workshop – May 8, 2003 [amk]

Explicit to ImplicitCase Study – Cup Stamping

• Blank Velocity

FEMCI Workshop – May 8, 2003 [amk]

Explicit to ImplicitCase Study – Cup Stamping

• Stress and Strains

FEMCI Workshop – May 8, 2003 [amk]

Explicit to ImplicitCase Study – Cup Stamping

• Deformed shape and plastic strains at end of explicit

FEMCI Workshop – May 8, 2003 [amk]

Explicit to ImplicitCase Study – Cup Stamping

• Implicit FE Model

FEMCI Workshop – May 8, 2003 [amk]

Explicit to ImplicitCase Study – Cup Stamping

• Spring back shape

FEMCI Workshop – May 8, 2003 [amk]

Explicit to ImplicitCase Study – Cup Stamping

Conclusions

• Experience shows that explicit/implicit is less than 25% of implicit CPU for same application

• Implicit only is easier to validate and hence provides more confidence

• Rigid body constraints in implicit part and time location for switch

• Mass scaling and speed of process introduce simplifications

• Certain aspects of the explicit/implicit process could be automated

• Elements selections and compatibilities among them

• Data management is important as time scale has two different meanings

• The process is very promising for nonlinear applications as solvers will switch automatically between the two schemes based on solution behavior

FEMCI Workshop – May 8, 2003 [amk]

Acknowledgments/References

Certain figures and images are courtesy of ANSYS, Inc. and Livermore Software Technology Corporation.