new features in ls-dyna efg method for solids and structures … · adaptive methods for...

7th European LS-DYNA Conference

© 2009 Copyright by DYNAmore GmbH

New Features in LS-DYNA EFG Method for Solids and Structures Analysis

C. T. Wu

LSTC

Summary: In this presentation, an update on LS-DYAN EFG method for solids and structures analysis will be given. Several features were developed in the past two years to solve specific challenging problems as well as to improve the efficiency. This talk will emphasize on three new features including an adaptive Meshfree scheme based on a local Maximum Entropy approximation for metal forging and extrusion analysis, a semi-Lagrangain formulation in foam materials under severe compression, and a discrete meshfree approach in the failure analysis of brittle materials. Several practical examples are included to demonstrate these capabilities.





New Features in LS-DYNA EFG Method for Solids and Structures Analysis

C. T. Wu*, Yong Guo, Jing Xiao Xu and Hong Sheng LuLSTC

May 14th ~ 15th, 2009Salzburg, Austria



Current EFG Formulations for Solids and Structures Analysis

• Metal materials in Forging/Extrusion analysis: Adaptive formulation

• Foam materials: Semi-Lagrangian kernel formulation

• Rubber materials: Lagrangian kernel formulation

• Quasibrittle material fracture: Strong discontinuities formulation

• E.O.S. materials: Eulerian kernel formulation (trial version)

• Meshfree Shell: Lagrangian kernel, adaptivity …

Stabilized Method





37th European LS-DYNA Conference3

Reasons for Adaptivity High accuracy requirement (surface representation, high

gradient …)

Residual stress effects the crash result

Current Numerical LimitationsRH-adaptivity for solids (H-adaptivity is limited to shell

structures).

No failure is allowed if failure energy is important (can not be extended to metal cutting, riveting ..)

Do not apply to rubber-like materials

Adaptive Methods for Manufacturing Simulations


1. Adaptive EFG Method

Adaptive Forging/Extrusion analysisAn explicit/implicit solver coupled with thermal analysis.

Introduce a fast transformation meshfree method and a modified Maximum Entropy approximation to improve the efficiency.

A second-order interpolation scheme for state variable transfer.

Include global/local adaptive refinements.

Available in SMP and MPP.






• Momentum equation

• Continuity equationvx∇−= ρρ&

Particle

Stress point

I

( )t ,xr I

EFG Fast Transformation Method

∑∑∑∈∈∈

∈∈

≡⋅=

ΩΩI

I[m]

IΩJ

JI[m]J

ΩΩI

[n]I

hΩ )(Ψ)(Ψ)(Φ)(u

xx

xxxxxx ˆ

( ) sssII

II

Ω ΓΩΩ

VΦm

ΓddΩdΩvdΩ

σxv

τvbv σvv

x ⋅∇−=

⋅+⋅+∇−=⋅

∑

∫ ∫∫∫&

& δδδρδ :

bσv x +⋅∇=& ρ

( )sII

Iss Φ xv x∑ ∇⋅−= ρρ&


Local MAXENT (Ortiz and Arroyo, 2006)

( ) ( ) ( )

∑

∑

∑∑

=

=

==

=−

=

=≥

+−=

N

iii

N

ii

i

i

N

ii

N

iii

p

p

Nip

pppHMAXENT

1

1

1

2

1

)(

1

,...,1 ,0 subject to

log maximize

0xx

xxxp β

— for , H(p) is continuous and strictly convex in solution (well-behaved mass matrix, monotonicity, variation diminishing …) — less dependent— difficult to decide β

[ )+∞∈ ,0β

EFG Modified Maximum Entropy Method






( ) ( ) ( )

( )

( ) ( )( )

( )

( ) ( ) ]/[

)(

1

,...,1 ,0

be proven to is MAXENT ofsolution unique The

nodeat kernel of sizesupport theis nodeat function kernel theis

:function Zpartition theDefine

1

1

/

iii

N

iii

N

ii

i

fi

i

i

i

N

1ii

fwhere

p

psatisfying

NipZ

ep

iiwhere

eZ

i

rxxλλx,

0xx

λx,xλx,

rx

xλx,

λx,

rx-xλ ii

−⋅=

=−

=

=≥∀=

≡

∑

∑

∑

=

=

=

⋅

φ

φ

φ

• Non-negative approximation• Smoothness in irregular nodes• Less dependence• Kronecker-Delta at boundary

— implicit solve; 3~5 iterations

EFG Modified Maximum Entropy Method


Mesh-free Interpolation for Data Transfer in Adaptivity

Current variable update :

( )J[m]IIJ

nns

ns

ns

xΦA

fAAfAf

=

=≈ −+++−

βαβααα

1111 ~

( )1n[m]I t,xΨ +−

Particle

Stress point

( )1n[m]I t,xΨ ++

Old Particle

Old Stress point

New Particle

New Stress point

( )1n[m]S t,xΦ ++I IS






Essential Boundary Conditions

IEBT EQ. 1: Full transformation (default)EQ.–1: (w/o transformation)EQ. 2: Mixed transformationEQ. 3: Coupled FEM/EFGEQ. 4: Fast transformationEQ.–4: (w/o transformation)EQ. 5: Fluid particle (trial version)EQ. 7: Modified Maximum Entropy approximation

0.0111001.011.011.01Default

FIIIIFFFType

TOLDEFIDIMIEBTIDILAISPLINEDZDYDXVariable

Input Format*SECTION_SOLID_EFG

Card 2


VEFG

EFG Adaptivity

Force

Forging Simulation

5827 nodes13661 nodes

Volume change (1.7%)






Global Refinement Local Refinement


Extrusion Simulation

Force

Volume change (4.0%)

Comparisons of Implicit and Explicit Analysis






Global Refinement Local Refinement


Extrusion with Thermal Coupling

13969 nodes 15091 nodes 15003 nodes 15086 nodes

15997 nodes






2. The Stabilized EFG Method with Kernel Switch

The Stabilized EFG Method with kernel switchIs a one-point integration scheme + gradient type hourglass control.

Assumed strain method for nearly incompressible materials.

Designed especially for foam and rubber materials.

The speed is between FEM reduced integration element (#1) and full integration element (#2)

A switch to full integration (rubber) or Semi-Lagrangian kernel (foam) is allowed in large deformation range.

Available in SMP explicit and MPP explicit.


Gradient Type Stabilized EFG Method

[m]IΨI

Λ

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( )

( )

( ){ }

( ) ( )( ) ( )( ) ( )( )0,,,0,,,0,,,,,

][,

][,

][,

22

2

2

22

2

2

22

2

2

000

000

2][

,0][

,0][

,0][

0][

000000000000

,2,11

,

0

1

0

;

000000000

22

22

22

~

~

zzSFyySFxxSFSF

LagrangianTotal

NPIm

yx

xx

x

zxyxx

zxyxx

zxyxx

zzyyxx

zzyyxx

Ozzyyxx

zyxkjzikzyxkjyikzyxkjxikzyxkjikij

nullity of paritionn

II

mxI

n

II

mxI

n

I

mxI

T

T

T

mzI

myI

mxI

mI

mI

−+−+−+≈

=•⇒=Λ

≥

=⋅Ψ

=⋅Ψ

=Ψ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΨΨ

Ψ

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂∂

∂∂∂

−∂∂

−

∂∂∂

−∂∂∂

∂∂

−

∂∂∂

−∂∂

∂−

∂∂

=

−+−+−+=

=

−+−+−+=

+Ψ−+Ψ−+Ψ−+Ψ=Ψ

∑

∑

∑

Λ∈

Λ∈

Λ∈

σ

01B

BBBBBUBε

Method Strain Assumed

BBBBB

x

hourglass-anti

zyx0

[m]zI,

[m]yI,

[m]xI,

[m]I0

[m]I

43421L

444444 3444444 21

&

xB






Convective velocity C due to Semi-Lagrangian or Eulerian kernel

fCf

f

vv C

)t

∇⋅+∂

∂=

−= +−

(χ&)(),(

)(),(

1

1

frametimereferencett

frametimematerialt

t

n

n

χx

v

Xx

v X

χ∂

∂=

∂

∂=

++

+−

( )∑∑ =∂∂

=∇J

JnI[m]

JIJ

I

[m]J tΨΨ fxffx

f ~, ;~

Lagrnagian phase :

Transport phase :

-[m]Jn

[m]J Ψ)t,(Ψ =x

0( =∇⋅+∂

∂fC

f)

t χ

( ) ( )

( ) JJ

[m]J

[m]J[m]

J[m]

J

Ψ

ΨΨΨ

fxf

xxxxx x

--

~

)(

∑ ++

++

≈

+∂

∂⋅−−= − L

Stress recovery scheme is conservative, consistent and monotonic !

Semi-Lagrange or Eulerian Kernel

( )n[m]I tΨ( )1n[m]I tX,Ψ +

( )1n[m]I t,xΨ +−

Lagrangian Kernel

Particle

Stress point

Semi-Lagrangian Kernel in Foam Material


Domain Integration Schemes

IEBT EQ. 1: Full transformation (default)EQ.–1: (w/o transformation)EQ. 2: Mixed transformationEQ. 3: Coupled FEM/EFGEQ. 4: Fast transformationEQ.–4: (w/o transformation)EQ. 5: Fluid particle (trial version)EQ. 7: Modified Maximum Entropy approximation

0.0111001.011.011.01Default

FIIIIFFFType

TOLDEFIDIMIEBTIDILAISPLINEDZDYDXVariable

IDIM EQ. 1: Local boundary condition method (default)EQ. 2: Two-points Guass integrationEQ.-1: Stabilized EFG method

Input Format*SECTION_SOLID_EFG

Card 2






Deformation tolerance for the activation of Semi-Lagrangian kernel

TOLDEF= 0.0 : Lagrangian kernel> 0.0 : Semi-Lagrangian kernel< 0.0 : Eulerian kernel

0.1




Original EFG EFG + Semi-Lagrangian Kernel

Foam Compression using Stabilized EFG Method and Semi-Lagrangian Kernel

Comparison of final deformation

Low Density FoamStabilized EFG explicit analysisSwitched to Semi-Lagrangian (TOLDEF=0.01)


3. EFG Failure Analysis

Meshfree Failure AnalysisIs a discrete approach.

Crack initiation and propagation are governed by cohesive law.

Crack currently is cell-by-cell propagation and is defined by visibility.

Minimized mesh sensitivity and orientation effects.

Applied to quasibrittle materials.






Discrete Cracks

Crack in Meshfree: Visibility Criterion (Belytschko et al.1996)

( ) ( ) ( )( ) ( )( )

( ) ( ) ( ) ( ) ( )ηη

ηη

ηη

ηηηη

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂Ψ∂

⊗+∂Ψ∂

⊗+∂

Φ∂⊗=

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ+Ψ+Φ=

∑ ∑∑

∑ ∑∑

+ −

+ −

Ω∈ Ω∈=

Ω∈ Ω∈=

XX

XuX

XuXx

uXuXXx

JJ0 0

0 0

21

21

2

1

2

1

J J

JJFEMI

II

J JJJJJI

I

FEMI+Ω0

−Ω0( )ηX

Intrinsic (Implicit crack) : no additional unknowns

Initially-rigid Cohesive Law: Redefined Displacement Jump (Sam, Papoulia and Vavasis 2005)

2

0

22

0 )()( ⎟⎠⎞⎜

⎝⎛

++⎟⎠⎞⎜

⎝⎛

+= ttt

nn

n uuδδβδδλ

crt

tt

crn

nn

tnefs

TuTTuT

TTTT

λα

δλλ

λδλλ

αβ

−−

=−

−=

=⎟⎠⎞

⎜⎝⎛+≡

11

11 maxmax

max2

22

and

01.0== crλλ005.0== crλλ

T


Minimization of Mesh Size Effect in Mode-I Failure Test

Coarse elements Fine elements

Failure is limited in this area

01.0== crD λ005.0== crD λ

Tn

D=0.01 D=0.005






EFG 3D Edge-cracked Plate under Loading

101 x 31 x 6 nodes

ElasticEFG FractureLinear Cohesive LawExplicit analysis

front

back

Resultant Displacement Contour

Failure Contour

d


EFG Glass under Impact

101 x 101 x 4 nodesElastic + RubberEFG FractureLinear Cohesive LawExplicit analysis

front

back

Failure Contour






Rigid ball Metal ball


Conclusions

• Adaptive method is attractive in metal forging/extrusion simulation.

• Stabilized method designed for foam and rubber materials can be used to improve the efficiency in explicit analysis.

• Failure analysis using cohesive model and visibility can be applied to brittle and semi-brittle materials.

• Strong discontinuities formulation including XFEM will be our next focus for a more general failure analysis.



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