Comparison of Implicit and Explicit Procedures

Comparison of implicit and explicit procedures Abaqus/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 ys yses are expens ve ve n Abaqus/Standard because each iteration requires a large set of linear e uations to be solved.

Comparison of implicit and explicit procedures Whereas 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 the same analy nalysi siss in Abaqu baqus/ s/S Stand tandaard requi equirres many many iter teration tionss. 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.

Comparison of implicit and explicit procedures Whereas 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 the same analy nalysi siss in Abaqu baqus/ s/S Stand tandaard requi equirres many many iter teration tionss. 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.

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods

Comparison of Implicit and Explicit Methods STATIC

QUASI’ STATIC

‘

DYNAMIC

PUNCH

BLANK DIE

Structural Problems

Metal Forming

F =0

IMPLICIT METHOD

F

0

Impact Problems

=

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:

{u t + Δt } = [K ]−1 {Fta+ Δt } ¾Unconditionally

stable when [K] is linear ¾Large time steps can be taken Nonlinear problems: ¾Solution obtained using a series of linear approximations ¾Requires inversion of nonlinear stiffness matrix [K] ¾Small iterative time steps are required to achieve convergence ¾Convergence is not guaranteed for highly nonlinear problems

Comparison of Implicit and Explicit Methods Explicit 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:

{a t } = [M ]−1

ext

Ft

− Ftint

 F int = Σ⎛  ⎜ ∫ BT σ n d Ω + F hg  ⎞⎟ + F contact 

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

{vt + Δt / 2 } = {vt −Δt / 2 } + {at }Δt t  ut + Δt  = ut  + vt + Δt / 2 Δt t + Δt / 2 where

Δtt+Δt/2=.5(Δtt+ Δtt+ Δt) and Δtt- Δt/2=.5(Δtt- Δtt+ Δt)

Comparison of Implicit and Explicit Methods Explicit Time Integration: The geometry is updated by adding the displacement increments to o

{ xt + Δt } = { xo } + {ut + Δt } • Nonlinear problems: ¾ Lumped mass matrix required for simple inversion ¾ Equations become uncoupled and can be solved for directly (explicitly) ¾ No inversion of stiffness matrix is required. All nonlinearities (including contact) are included in the internal force vector. ¾ Major computational expense is in calculating the internal forces. ¾ No conver ence checks are needed ¾ Very small time steps are required to maintain stability limit

Stability Limit Implicit Time Integration:

Explicit Time Integration:

For linear problems, the time s ep can e ar rar y arge (always stable)

Only stable if time step size s sma er an cr ca me step size

, step size may become small due to convergence difficulties

Δt ≤ Δt  =

2 ω max

Where wmax = largest natural circular frequency Due to this very small time step size, explicit is useful

Critical Time Step Size Critical time step size of a rod - Natural fre uenc : max= 2

ω

c l 

with c=

E

(wave propagation velocity)

ρ

Critical time step: Δt=

l 

- Courant-Friedrichs-Lev -criterion - ∆t is the time needed of the wave to propagate through the rod of length l   : depends on element length and material properties (sonic speed).

ABAQUS/EXPLICIT Time Step Size ABAQUS/EXPLICIT checks all elements when calculating the required time step. The characteristic length l  and the wave propagation velocity de endent on element t e: Beam elements:

l = length of  the element 

c=

E ρ

Shell elements: l= c=

A 2A  , for  triangular   shells: l= max L L L L max L L

L4

E 2

ρ( 1 - ν )

L1

3

A

L2

L

c  are

ABAQUS/EXPLICIT Time Step Size – The concept of a stable time increment is explained .

One-dimensional problem – The stable time increment is the minimum time that a dilatational . • A dilatational wave consists of volume expansion and contraction.

ABAQUS/EXPLICIT Time Step Size – Thus, the stable time increment can be expressed as

Δt=

c

– Decreasing L and/or increasing c will reduce the size of the stable time increment. • . • Increasing material stiffness increases c. • Decreasing material compressibility increases c. • Decreasing material density increases c. – ABAQUS/Explicit monitors the finite element model throu hout the ana sis to deter ine a stab e ti e incre ent.

Summary

Summary Implicit Time Integration (used by ANSYS) -

•Finite Element method used •Average acceleration calculated • •Always stable – but small time steps needed to capture transient response

•Non-linear materials can be used to solve static problems •Can solve non-linear (transient) problems… •…but only for linear material properties •Best for static or ‘quasi’ static problems

Summary

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 stability •Can solve non-linear problems for non-linear materials •Best for d namic roblems

Overview of the Ex licit D namics Procedure • Stress wave propagation • This stress wave ro a ation example illustrates how the explicit dynamics solution procedure works without iterating or solving sets of linear equations. • We consider the ro a ation of a stress wave along a rod modeled with three elements. We study the state of the rod as we increment through time. • Mass is lumped at the nodes.

Initial configuration of a rod with a concentrated load, P, a e ree en

Overview of the Ex licit D namics Procedure

u&&1 =

 P 

⇒ u&1 = u&&1dt ⇒ ε &el 1 =

− u&1

⇒ d ε el 1 = ε &el 1dt 

1

⇒ ε el 1 = ε 0 + d ε el 1 ⇒ σ el 1 = E ε el 1

Configuration at the end of Increment 1

Overview of the Ex licit D namics Procedure

u&&1 = u&& =

−

el 1

 M 1  F el 1 2

⇒ u&1 = u&1

o

⇒ u& = u&& dt 

+ u&&1dt 

ε &el 1 =

& 2 − &1 l 

⇒ d ε el 1 = ε &el 1dt  ⇒ ε el 1 = ε 1 + d ε el 1 ⇒ σ el 1 = ε el 1

Configuration of the rod at the beginning of Increment 2

Configuration of the rod at the beginning of Increment 3

Explicit Dynamics method . && . = U& (t n + (1/2) ) = U&& (t n ) =

U (t n +1 ) − U (t n  )

,

U& (t n + (1/ 2) ) − U& (t n − (1/ 2) )

=

U (t n +1 ) − 2U (t n ) + U (t n −  1 ) 2

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

.U&& (t n ) = F (t n ) − K .U (t n )

&

n + (1/ 2)

= &

n − (1/ 2)

&&

n

U (t n +1 ) = U (t n ) + Δt U& (t n  + (1/ 2) ),

0

=

0

& U& (t 0− (1/ 2) ) = U  0

,