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Time Integration
Suri Bala
Time Integration
Main objective is to find unknown displacements by numerically integrating the equations of motion which is a second-order linear/nonlinear ODE
For a single degree of freedom with no damping:
)(tfkxxm ext=+&& )()(int tftfxm ext=+&&Linear Non-Linear
Numerical Solutions
Direct and Indirect Integration Techniques
Direct» No transformations
» Examples► Explicit
► Implicit
Indirect» Transformation
» Examples► Mode Superposition
Numerical Solution - Explicit
Among many explicit methods, the central-difference technique is the most popular and is used in LS-DYNA
Time
Numerical Solution - Explicit
Discretization in Time
F(t)
Time
Numerical Solution - Explicit
Current Time
F(t)
Timent1−nt 1+nt
Numerical Solution - Explicit
Current Acceleration
F(t)
Timent1−nt 1+nt
)(1 nnn FPMa −=−
Numerical Solution - Explicit
Mid-step parameteres
F(t)
Timent1−nt 1+nt
21−n
t21+n
t
)(1 nnn FPMa −=−
Numerical Solution - Explicit
Mid-Step Velocity
F(t)
Timent1−nt 1+nt
21−n
t21+n
t
nnn tavv Δ+= −+ 21
21
Numerical Solution - Explicit
Unknown displacement
F(t)
Timent1−nt 1+nt
21−n
t21+n
t
211 ++ Δ+= nnn tvdd
Explicit – Timestep
Choosing an incremental timestep is based on the highest natural frequency of the system
max
2ω
Explicit – Timestep
Choosing an incremental timestep is based on the highest natural frequency of the system
lc2
max =ω
max
2ω
Explicit – Timestep
Choosing an incremental timestep is based on the highest natural frequency of the system
lc2
max =ω
max
2ω
Characteristic Length, lc
Element based
Computed Every Cycle
Beam» Length between two nodes
Excluded elements» Discrete beams and springs
Edge or Diagonal Length ?
),,,max(
),,,min(
4321
4321
lllll
lllll
c
c
=
=
1l
4l
3l
2l
),max(
),min(
21
21
ddl
ddl
c
c
=
=
1d4d
Why Edge and Diagonal Length Fail
Edge or diagonal length based method fails for collapsed or nearcollapsed elements
0
0
=
>
A
lc
Shell Element Characteristic Length
),,,max( 4321 llllAlc =
1l
4l
3l
2l1d
4d
Default Shell Element Characteristic Length
))1(,,,max()1(
4321 llllAlc β
β−
+=
1l
4l
3l
2l1d
4d1l
2l
3l
0=β 1=β
Shell Element Characteristic Length - Options
),max()1(
21 ddAlc
β+=ISDO = 1
⎥⎦
⎤⎢⎣
⎡+
−+
= )2010,,,min(,))1(,,,max(
)1(4321
4321ellll
llllAMAXlc ββ
βISDO = 2
Solid Element
maxAreaVolumelc =
Springs – Timestep
Choosing an incremental timestep is based on the highest natural frequency of the system
2121
max)(
mmmmk +
=ω
max
2ω
Wave/Sound Speed, c
ρEC trussbeamliuhughesrod =/__/
ρυυυ
)21)(1()1(
−+−
=ECsolid
ρυ )1( 2−=
ECshell
Global Timestep
),.......,min(* 1 nttTSSFACt ΔΔ=Δ
Time Integration Loop
Start
Time Integration Loop
Apply boundaryloads
Start
Time Integration Loop
Apply boundaryloads
Start
Process elements
Time Integration Loop
Apply boundaryloads
Start
Process elements
Process penalty contacts
Time Integration Loop
Apply boundaryloads
Start
Process elements
Process penalty contactsUpdate accelerations and
Kinematic constraints
Time Integration Loop
Apply boundaryloads
Start
Process elements
Process penalty contactsUpdate accelerations and
Kinematic constraints
Kinematic contacts and
Rigidwalls
Time Integration Loop
Apply boundaryloads
Start
Process elements
Process penalty contactsUpdate accelerations and
Kinematic constraints
Kinematic contacts and
Rigidwalls
Write Databases
Time Integration Loop
Apply boundaryloads
Start
Process elements
Process penalty contactsUpdate accelerations and
Kinematic constraints
Kinematic contacts and
Rigidwalls
Write Databases
Update velocities
Time Integration Loop
Apply boundaryloads
Start
Process elements
Process penalty contactsUpdate accelerations and
Kinematic constraints
Kinematic contacts and
Rigidwalls
Write Databases
Update velocities
Update displacements and new geometry
Time Integration Loop
Apply boundaryloads
Start
Process elements
Process penalty contactsUpdate accelerations and
Kinematic constraints
Kinematic contacts and
Rigidwalls
Write Databases
Update velocities
Update displacements and new geometry Update time and check for
termination
Increasing CFL based timestep
Stems from the desire to improve job turnaround with negligible effects on accuracy
Two method exists» Mass Scaling
» Stiffness scaling
Mass Scaling» Sound speed is slower in denser materials thereby allows larger timestep
Stiffness» Sound speed is slower in softer materials thereby allows larger timestep
Mass-Scaling, DT2MS > 0 in *CONTROL_TIMESTEP
Mass Scaling, DT2MS < 0 in *CONTROL_TIMESTEP
Limiting mass-scaling to first cycle
Mass-scaling is performed at every cycle by default
MS1ST in *CONTROL_TIMESTEP allows to limit the mass-scaling routing to be executed at cycle 1 which allows the timestep to drop thereafter
Selective Mass-Scaling
Available from 971
Allows a larger mass-scaled timestep with negligible reduction in accuracy
Automotive Applications» Detailed steering wheel
» Any subsystem
» Localized study
Stiffness Scaling
Alternative way of increasing the computed timestep
Alter (reduce) the elastic stiffness, E, to decrease the sound speed thereby increasing the resulting timestep
Options» Manually by updating the parameter in the *MAT keyword
► Can be used for ALL element types
» Automatic by specifying the desired timestep, TSMIN, in *CONTROL_TIMESTEP keyword
► Applies for only shell elements using limited elastic-plastic material laws
100 smallest timesteps in D3HSP
Element Id
Tim
este
p
NOT RECOMMENDED
DESIRED
Explicit – Advantages/Disadvantages
+ Ideal for Highly Non-Linear short duration transient events+ Low Memory requirements+ Mature Contact treatments+ Inexpensive Timestep Calculations
- Limited by Courant stability limitNeed to ignore geometric details
- Long duration events not feasible
Numerical Solution - Implicit
Unknowns are embedded in a system of linear/non-linear equations» Stiffness matrix is formed» Need Efficient Linear/Non-Linear solver
0~0
~1
1~usxx nn Δ+=+
1
~
1~~
11
~~~~)( ++
++ =−⎟⎠
⎞⎜⎝⎛=Δ n
i
ni
nn
iitQxFxPuK
j
Numerical Solution - Implicit
Advantages/Disadvantageous+ Unconditionally stable for any load/time step
+ Geometric details can be included. A huge benefit for certain automotive structures+ Ideal for Long and Short Duration events
- High Memory Requirements- Very expensive Time Step calculations- Contact Inexperience
Choosing Solution Type
Explicit» Short duration
» High Strain-rate
» Intertia dominated
Implicit» Static
» Quasi-static
» Zero to low strain-rates
Combination of both ?
IMFLAG in *CONTROL_GENERAL
Time Integration�����Suri BalaTime IntegrationNumerical SolutionsNumerical Solution - ExplicitNumerical Solution - ExplicitNumerical Solution - ExplicitNumerical Solution - ExplicitNumerical Solution - ExplicitNumerical Solution - ExplicitNumerical Solution - ExplicitExplicit – TimestepExplicit – TimestepExplicit – TimestepCharacteristic Length, lcEdge or Diagonal Length ?Why Edge and Diagonal Length FailShell Element Characteristic LengthDefault Shell Element Characteristic LengthShell Element Characteristic Length - OptionsSolid ElementSprings – TimestepWave/Sound Speed, cGlobal TimestepTime Integration LoopTime Integration LoopTime Integration LoopTime Integration LoopTime Integration LoopTime Integration LoopTime Integration LoopTime Integration LoopTime Integration LoopTime Integration LoopIncreasing CFL based timestep Mass-Scaling, DT2MS > 0 in *CONTROL_TIMESTEPMass Scaling, DT2MS < 0 in *CONTROL_TIMESTEPLimiting mass-scaling to first cycleSelective Mass-ScalingStiffness Scaling100 smallest timesteps in D3HSPExplicit – Advantages/DisadvantagesNumerical Solution - ImplicitNumerical Solution - ImplicitChoosing Solution TypeIMFLAG in *CONTROL_GENERAL