Implicit and Explicit

8/11/2019 Implicit and Explicit

1/36

Comparison of implicit and explicit proceduresAbaqus/Standard is more efficient for solving smooth nonlinear

problems; on the other hand, Abaqus/Explicit is the clear choice

for a wave propagation analysis. There are, however, certain

static or quasi-static problems that can be simulated well with

either ro ram. T icall these are roblems that usuall would

be solved with Abaqus/Standard but may have difficulty

converging because of contact or material complexities, resulting

n a arge num er o era ons. uc ana yses are expens ve n

Abaqus/Standard because each iteration requires a large set of

linear e uations to be solved.


2/36

Comparison of implicit and explicit proceduresWhereas Abaqus/Standard must iterate to determine the

solution to a nonlinear problem, Abaqus/Explicit determines the

solution without iterating by explicitly advancing the kinematic

state from the previous increment. Even though a given analysis

ma re uire a lar e number o time increments usin the ex licit

method, the analysis can be more efficient in Abaqus/Explicit if

the same analysis in Abaqus/Standard requires many iterations.

Another advantage of Abaqus/Explicit is that it requires much

less disk s ace and memor than Aba us/Standard for the same

simulation. For problems in which the computational cost of the

two programs may be comparable, the substantial disk space and

memory sav ngs o aqus x c ma e a rac ve.


3/36

Comparison of Implicit and Explicit Methods


4/36



5/36



6/36



7/36



8/36



9/36



10/36



11/36



12/36


13/36



14/36



15/36



16/36



17/36



18/36



19/36


STATIC QUASI STATIC DYNAMIC

PUNCH

DIE

BLANK

Structural Problems Metal Forming Impact Problems

F = 0 F 0

=

IMPLICIT METHOD


20/36


Implicit Time Integration:

Inertia effects ([C] and [M]) are typically not included

Average acceleration - displacements evaluated at time t+Dt:

{ } [ ] { }a tt

1

tt FKu +

+ =

Unconditionally stable when [K] is linear

Large time steps can be takenNonlinear problems:

Solution obtained using a series of linear approximations-

Requires inversion of nonlinear stiffness matrix [K]

Small iterative time steps are required to achieve convergence

Convergence is not guaranteed for highly nonlinear problems


21/36

Comparison of Implicit and Explicit MethodsExplicit Time Integration:

Central difference method used - accelerations evaluated at time t:

ere t s t e app e externa an o y orce vector,

{Ftint} is the internal force vector which is given by:

{ } [ ] inttext

t

1

t FFMa =

contacthgnT FFdBF + += int

Fhg is the hourglass resistance force (see ELEMENTS Chapter)and Fcont is the contact force.

{ } { } { } tttttt tavv += + 2/2/

2/2/ ttttttt tvuu +++ +=

where tt+t/2=.5(tt+ tt+ t) and tt-t/2=.5(tt-tt+ t)


22/36


23/36

Stability LimitExplicit Time Integration:

Only stable if time step size

Implicit Time Integration:

For linear problems, the time

s sma er an cr ca me

step size

2

s ep can e ar rar y arge

(always stable)

t t =

max

,

step size may become small

due to convergence difficulties

Where wmax = largest natural

circular frequency

Due to this very small timestep size, explicit is useful


24/36

Critical Time Step Size

Critical time step size of a rod

- Natural fre uenc :

lc= 2max with

Ec= (wave propagation velocity)

Critical time step:l

t=

- Courant-Friedrichs-Lev -criterion

- t is the time needed of the wave to propagate through therod of length l

:depends on element length and material properties (sonic speed).


25/36

ABAQUS/EXPLICIT Time Step Size

ABAQUS/EXPLICIT checks all elements when calculating therequired time step.

The characteristic length land the wave propagation velocity carede endent on element t e:

Ec=elementtheoflength=lBeam elements:

LLLmaxA2

LLLLmaxA l=shells:triangular for,l=

Shell elements:

)-1(

E2

c=

L1

L43

L2

A


26/36


27/36

ABAQUS/EXPLICIT Time Step Size

Thus, the stable time increment can be expressed as

ct=

Decreasing L and/or increasing c will reduce the size of thestable time increment.

.

Increasing material stiffness increases c.

Decreasing material compressibility increases c.

Decreasing material density increases c. ABAQUS/Explicit monitors the finite element modelthrou hout the ana sis to deter ine a stab e ti e incre ent.


28/36

Summary


29/36

Summary

Implicit Time Integration (used by ANSYS) -

Finite Element method used

Average acceleration calculated Always stable but small time steps needed to capturetransient responseNon-linear materials can be used to solve static problemsCan solve non-linear (transient) problems

but only for linear material propertiesBest for static or quasi static problems


30/36

Summary


31/36

Summary

Explicit Time Integration (used by LS Dyna)

Central Difference method used

Accelerations (and stresses) evaluated

Accelerations -> velocities -> displacements

Small time steps required to maintain stabilityCan solve non-linear problems for non-linear materials

Best for d namic roblems


32/36

Overview of the Ex licit D namics Procedure Stress wave propagation

This stress wave ro a ationexample illustrates how theexplicit dynamics solutionprocedure works withoutiterating or solving sets oflinear equations.

We consider the ro a ation

Initial configuration of a rod

with a concentrated load, P,of a stress wave along a rodmodeled with three elements.We study the state of the rod

a e ree en

as we increment through time. Mass is lumped at the

nodes.


33/36


34/36

Overview of the Ex licit D namics Procedure

&&

==

+=

=

dtuuFu

dtuuuM

u

el

oel

1

1111

11

&&&&&

&&&&&&

111 elel

d +=

== dtdl

elelel 1112

1 &&

Configuration of the rod at the beginning of Increment 22 11 elel

=

Configuration of the rod at the beginning of Increment 3


35/36

&&

Explicit Dynamics method

1(1/2)

( ) ( )( ) ,n nn

U t U t U t ++

=&

. . =

(1/ 2) (1/ 2) 1 1

2

( ) ( ) ( ) 2 ( ) ( )( ) ,n n n n n

n

U t U t U t U t U t U t + + += =& &&&

Errors are of the order O ( (t) 2) for time steps t 0,

& & &&

. ( ) ( ) . ( )n n nU t F t K U t = &&

(1/ 2) (1/ 2)

1 (1/ 2)( ) ( ) ( ),n n n

n n nU t U t t U t +

+ +

=

= + &0 0

0 (1/ 2) 0( )U t U

=

=& &


36/36

Implicit Dynamics method

1 1 1

2

. ( ) (1 ) . ( ) . ( ) ( ) (*)

1

n n n nM U t K U t K U t F t t + + ++ + = + &&

& && &&1 1

1 1

,

2( ) ( ) [(1 ) ( ) ( )],

n n n n n

n n n nU t U t t U t U t

+ +

+ +

=

= + +& & && &&

21/ 3 0 1 / 4 1 2 / 2 = =

U t U=

0 0( )

*

U t U=

& &

&&0 . , ,

Implicit and Explicit

Documents

Transcript of Implicit and Explicit