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.