Post on 03-Jun-2018
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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.
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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.
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
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Comparison of Implicit and Explicit Methods
STATIC QUASI STATIC DYNAMIC
PUNCH
DIE
BLANK
Structural Problems Metal Forming Impact Problems
F = 0 F 0
=
IMPLICIT METHOD
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Comparison of Implicit and Explicit Methods
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
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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)
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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
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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).
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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
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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.
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Summary
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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
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Summary
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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
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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.
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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
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&&
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
=
=& &
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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 . , ,