the development of xfem fracture and mesh-free adaptivity

© 2006 Copyright by DYNAmore GmbH

The Development of XFEM Fracture and Mesh-free Adaptivity

C. T. Wu, Yong Guo and Hong Sheng Lu

LIVERMORE SOFTWARE TECHNOLOGYCORPORATION (LSTC), Livermore, USA

5. LS-DYNA Anwenderforum, Ulm 2006 Neue Methoden

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LIVERMORESOFTWARETECHNOLOGYCORPORATION

The Development of XFEM Fracture and The Development of XFEM Fracture and MeshMesh--freefree AdaptivityAdaptivity

C. T. Wu, Yong Guo and Hong Sheng Lu

LIVERMORE SOFTWARE TECHNOLOGYCORPORATION (LSTC)

[email protected]@lstc.com,, [email protected]@lstc.com,, [email protected]@lstc.com

LSLS--DYNA Forum 2006DYNA Forum 2006October, 2006October, 2006

UlmUlm

1

Copyright © 2006 by LIVERMORE SOFTWARE TECHNOLOGY CORPORATION

LS-DYNA

Out of Lagrangian description

Problem Looking for Solution

Multi-Physics : shear bandshear band + history dependent large deformation + failure

Numerical : multimulti--resolutionresolution + avoid mesh tangle + failure mechanicsSpectral element method

The variationl multiscale methodPartition of unity method

ALEEulerian

Mesh-freeAdaptivity

Damage mechanicsCohesive model

Discrete element methodInterface Element, XFEM …

2

Neue Methoden 5. LS-DYNA Anwenderforum, Ulm 2006

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LS-DYNAAdaptive Mesh-free Method

Adaptive Mesh-free Method = EFG Fast Transformation Method

+ Adaptive Lagrnagian Particles with Eulerian Kernel

+ Consistent Mesh-free Interpolation for State Variable Transfer

3

Motivation

Improve the large deformation analysis that is beyond the Lagrangian description.

Current Practice

Solving the forging and extrusion problems.

Formulation


LS-DYNA

• Momentum equation

• Continuity equation

vx∇−= ρρ�

bv x +⋅∇=�ρ ( ) sssII

II Vm

dddvd

xv

vbvvv

x ⋅∇−=

⋅+⋅+∇−=⋅

∑∫ ∫∫∫

�

� δδδρδ :

( )sII

Iss xv x∑ ∇⋅−= ρρ�

Particle

Stress point

I

( )t,xr I

1. EFG Fast Transformation Method

∑∑∑∈∈∈

∈∈

≡⋅=I

I[m]

IJ

JI[m]J

I

[n]I

h )()()()(u

xx

xxxxxx ˆ

4

Particles Stress points

AccelerationVelocityPosition

DensityStrain rateStressesInternal energy

Direct variables computation


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LS-DYNA

5

2. Adaptive Lagrangian Particles with Eulerian Kernel

fCf

f

vvC

)t

∇⋅+∂∂

=

−= +−

(χ�

)(),(

)(),(

1

1

frametimereferencet

t

frametimematerialt

t

n

n

xv

Xxv X

χ∂∂

=

∂∂

=

++

+−

( )n[m]

I t( )1n

[m]I tX, +

( )1n[m]

I t,x +−

Lagrangian Kernel

Eulerian Kernel Particle

Stress point

• Momentum equation

• Continuity equation

j

j

jj x

v

xC

t ∂∂

−=∂∂+

∂∂ ρρρ

χ

ij

ij

j

ij

i bxx

v�

t

v +∂∂

=∂∂+

∂∂

Convective velocity C due to Eulerian kernel


LS-DYNA

Operator Split

( ) ( )−+=

+ −Δ+=∂

∂⋅Δ+≈ ∑ vvxxxfx

Cff n

xx

- ttt n-

JJ

J

Jθθ ,;

~

1. Lagrnagian phase :

2. Transport phase : 0( =∇⋅+∂∂

fCf

)t χ

: Ponthot and Belytschko 1998

A

B

: TB stress recovery scheme – conservative– consistent

is [m] order accurate; Currently m=1

Nodal valueRemapped value

-[m]Jn

[m]J )t,( =x

( )∑∑ =∂

∂=∇J

JnI[m]

JIJ

I

[m]J t fxffx

f~

,;~

( ) ( )

( ) JJ

[m]J

[m]J[m]

J[m]J

fxf

xxxxx

x--

~

)(

∑ ++

++

≈

+∂

∂⋅−−= − �

6


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LS-DYNACurrent EFG Methodology for Large Deformation Analysis

• Rubber materials : Lagrangian kernel

• Foam materials : Semi-Lagrangian and Eulerian kernel

•

• Fluid materials and E.Q.S. materials : Eulerian kernel (trial version)

7

(ls971)

( ls970, ls971)

( ls971)

Metal materials : Adaptivity + Eulerian kernel ( ls971)

EFG EFG + Semi-Lagrangian Kernel


LS-DYNA3. Mesh-free Interpolation for Data Transfer

Current variable update :

( )J[m]

IIJ

nns

ns

ns

xA

fAAfAf

=

=≈ −+++ −

βαβααα1111 ~

( )1n[m]

I t,x +−

Particle

Stress point

( )1n[m]

I t,x ++

Old Particle

Old Stress point

New Particle

New Stress point

( )1n[m]

S t,x ++

I I

S

8

Stress recovery scheme is conservative, consistent and monotonic !


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LS-DYNA

Analytical SolutionEFG SolutionInterpolation Solution

fudx

du

dx

d =+− )( )1,0(∈x

0)1()0( == uu4sinh

4sinh2 xxu −=

Interpolation of the State Variable in 1-D

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-3 -2.5 -2 -1.5 -1 -0.5 0

Log (Nodal space)

Lo

g (

Err

or

no

rm)

EFG solution

Interpolation solution

Slope 1.68

Convergence in derivative of solution

4sinh

4sinh1522 x

xf +−=

9


LS-DYNA

V

5827 nodes

EFG

EFGAdaptivity

ForceVolume change (1.7%)

13661 nodes

Wheel Forging Simulation

10


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LS-DYNAWheel Forging Simulation (Explicit)

Movie

11


LS-DYNAExtrusion Simulation

2769 nodes

10410 nodes

EFG Adaptive EFG

Internal energyVolume change (0.6%)

12


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LS-DYNA

Movie

Extrusion Simulation

13


LS-DYNAThe Upsetting Process with Thermal

Initial Temperature 20 0CLow carbon steel Ck105

No heat transfer to the environment

Internal Energy Force

14


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LS-DYNAThe Upsetting Process (Implicit with Thermal)

Plastic Strain

15


LS-DYNAXFEM Fracture

XFEM Fracture = Extended FEM (Partition of Unity) with Level Set Theory

+ Phantom Nodes Approach

+ Initially-Rigid Cohesive Law

16

Motivation

Modeling arbitrary dynamic cracks.

Current Practice

Solving 2D plain-strain problems.

Formulation

- Ted Belytschko et al. (Northwestern University), Pablo D. Zavattieri (GM R&D)


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LS-DYNAModeling Material Failure and Delamination/Debonding

1. Weak Discontinuities : discontinuous deformation gradients

2. Strong Discontinuities : discontinuous displacement

3. Weak + Strong Discontinuities

• Continuum damage constitutive equation + Nonlocal strain smoothing + Material erosion• Implicit Cracks: Crack is an assumed width• Polynomial basis is inadequate to represent the fine scale. • Time step tents to be very small in explicit analysis with fine mesh

• Cohesive model + (Interface element, or elemental enrichment EFEM, or nodal enrichment XFEM, or EFG)

• Explicit Cracks : remove the influence of mesh size and orientation• No direct correlation between the strain softening and critical energy release rate.• Time step:

h

kt

ρω

ω2

;2

maxmax

=≤Δ

• loss of uniqueness as a criterion for changing from continuum damage mechanics to cohesive law

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LS-DYNALiteratures in Strong Discontinuities

• Tie-break interface (force/stress-based failure + spring element, rigid rods, or other constraints)

• Cohesive Interface Element(Cohesive Zone model + Interface element, or contact forces)Tvergaard, V. and Hutchinson, J. W. (1993)Needlemam, A. (1997)Ortoz, M. and Pandolfi, A. (1999)

Borg, R., Nilsson, L. and Simonsson, K. (2002)Espinos, H. D. and Zavattieri, P. D. (2003)

• XFEM (Cohesive Zone model + level sets + extended finite element)Belystchko, T., Moes, S., Usui, S. and Parimi, C. (2001)Sukumar, Huang, Z., Prevost, J. H. and Suo, Z. (2004)Hettich, T. and Ramm, E. (2006)

• EFG (Cohesive Zone model + Moving least-square + EFG visibility)Rabczul, T. and Belytschko, T. (2004)Simonsen, B. C. and Li S. (2004) Brighenti, R (2005)

• Others (Virtual Crack Closure technique …)

18


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LS-DYNA1. Extended Finite Element with Level Set

19

fα x( ) = minx∈Γα

x − x sign n ⋅ x − x( )( )

fα x( ) = fαII

∑ NI x( )

Signed distance function

Approximation of crack in element

( )[ ]∑ += )()()()(),( XquXXu αfHttNt IIIh

Discontinuity

0),(0)( >=Γ∈ tgandfif XXX 0αα

In one-dimension

)1(22112211 −+++= HNqHNqNuNuu


LS-DYNA2. Phantom Nodes Approach

20

- Ted Belytschko et al. (2006)

Standard XFEM Phantom Nodes

)()())(1())(1(

2;1

)1()()1()(

2221

212

121

11

222

1121

2212

111

2222211111

aXHNuaXHNuaXHNuaXHNuu

uu

quuelement

quu

uuelement

HNuHNquHNuHNquu

−+−+−−+−−=

⎩⎨⎧

=+=

⎩⎨⎧

−==

+−−+−++=


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LS-DYNA

In Two-dimension

∫

∫∫∫∫

Γ

ΓΩ

Ω

Ω

Γ−=

Γ−+Ω−=

Ω−=

Ω−=

+−=

et

et

e

e

e

et

cecohe

et

eeeexte

eeinte

eekine

cohextkin

d

dfHdfH

dfH

dfH

,0

,00

0

0

,00

,000

0

00

int

)1(

))()1(())()1((

))()1((

))()1((

nNf

XtNXbNf

XBf

uXNNf

ffff

T

TT

T

T

τ

ρ

σ

ρ ��


LS-DYNA3. Initially-Rigid Cohesive Zone Model

• The potential crack propagation plane is idealized as a cohesive zone and is assumed to support a traction field T.

• The mechanical response of the cohesive interface is described through a constitutive law relating the traction field T with a separation parameter.

( )qT

∂∂= ,, tn δδφ

( )δTT =

maxT

∫= max

0)(

δδδ dTGIC maxδ

jumpntdisplaceme:δ

Initially-elasticInitially-rigid

22


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LS-DYNAMode-I Failure

Size: 5.0 X 10.0Displacement-control (1.2 in 0.3 sec)Tvergaard-Law I cohesive law

3.0;0.167;60.1

.3/.1

0.1

0.1

0.1

0.5max

==−==

==

==

νρλδδα

Ee

T

cr

t

n

Initially-rigidXFEM

Initially-elasticCohesive element

23


LS-DYNA

24

Material failure

Imperfection

Material A

Material B

Interface C

Failure in Particle Reinforced Rubber

24

Delamination/Debonding.


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LS-DYNAOngoing Works

Include local refinement and error estimator/indicator for solid

adaptivity.

Study the stability for XFEM

Consider crack closure (contact) and mass scaling in XFEM.

Extend XFEM into three dimensional case.

Implement MPP version.

25


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the development of xfem fracture and mesh-free adaptivity

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