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