17 EXtended FEM

7/25/2019 17 EXtended FEM

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Code_Aster, Salome-Meca course materialGNU FDL licence (http://www.gnu.org/copyleft/fdl.html)

The eXtended Finite Element Method


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2 - Code_Aster and Salome-Meca course material GNU FDL Licence

Outline

1. Motivation

2. Theoretical aspects

Level Sets Method

X-FEM enrichments

3. Application to Code_Aster

Crack definition (Level Sets)

Enriched Finite Elements (X-FEM)

Post-processing in fracture mechanics

Visualisation

4. Advanced tools

Mesh refinement

Crack propagation


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Goals

Study of cracked structures without explicitly meshing crack

Automatic crack propagation on a single mesh

Finite Element Method (FEM)

Crack is explicitly meshed

A long time (human intervention) is needed to mesh complexstructures

Re-meshing is required if changing the crack geometry(parametric study) or position (propagation)

eXtended Finite Element Method (X-FEM)

Simple mesh (does not respect the crack geometry)

Unique mesh for entire propagation process

Simple extension of FEM (relatively suitable forimplementation in a EF software)

Motivation


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Outline

2. Theoretical aspects

Level Sets Method

X-FEM enrichments


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Representation of crack by Level sets

Main idea: interface = iso-zero of a distance function

In mechanics: used for modelling interfaces between materials,

edges, voids, cracks

LS > 0

LS < 0

LS = 0

Iso-values of LS for a sphericalinterface


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Crackplane(LSN = 0)

Crack(LSN = 0 LST < 0)

xLSN(x)

LST(x)

Iso-values of LST

Iso-zero of LSN

Representation of crack by Level sets

2 level setsare needed


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X-FEM enrichment

Finite elements enriched withHeavisidefunction

Finite elements enriched with

asymptotic functions (~r)

X-FEM = Two enrichmentsEnrichment with a Heaviside function to represent a discontinuous field (displacement

jump) across the interfaceNB : The actual formulation is more complex but not detailed here for the sake of simplicity

Enrichment with asymptotic functions near the crack tip

( )( ) ( ) ( )i ii i

N N H= + i iu x a x x xb ( ) ( )1,4 ,ii

N B r+ 1,4i xc

Classical part(continuous)

Heavisideenrichment

(discontinuous)

Asymptoticenrichment(singular)

=

sin

2cos,sin

2sin,

2cos,

2sin rrrrB

where

=+= lst

lsnlstlsnr 122 tan, (signed distance functions)


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Outline

3. Application to Code_Aster

Crack definition (Level Sets)

Enriched Finite Element (X-FEM)

Post-processing in fracture mechanicsVisualisation


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Crack Definition (Level sets)

Definition of a crack (or an interface) with thecommand DEFI_FISS_XFEM

Choice of the meshChoice of a crack or a interface

Definition of the crack or the interface with level sets

Definition of the enrichment zone (optional)

Definition of the type of enrichment (optional)

Mesh

Finite ElementEnrichment

ComputationResolution

Post-processing

SIF,propagation

Visualisation

Crack definition(level sets)

sane mesh


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Choice of a mesh

FISS=DEFI_FISS_XFEM(MAILLAGE=MAILLA,

)

The sane mesh is provided via the keywordMAILLAGE (mandatory)

Level sets will be computed at all the nodes of this mesh

Mesh



Post-processing

SIF,propagation

Visualisation


sane mesh


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A crack An interface

Choice of a crack (fissurein French) or an interface


TYPE_DISCONTINUITE = / 'FISSURE', [DEFAULT]

/ 'INTERFACE'

)

Mesh



Post-processing

SIF,propagation

Visualisation


sane mesh


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Choice of a crack (fissure in French) or an interface


TYPE_DISCONTINUITE = / 'FISSURE',

/ 'INTERFACE'

DEFI_FISS=_F()

)

Level sets defined under the keyword DEFI_FISS

4 available methods:

Analytical functions

Pre-defined geometries

Projection onto group of elements

Lecture of pre-computed LSN and LST fields

Mesh



Post-processing

SIF,propagation

Visualisation


sane mesh

(only this method isemployed in this workshop)


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x

zy

H

a


Method 1: analytical functions

LSN=FORMULE(NOM_PARA=('X','Y','Z'),VALE='Z-H')

LST=FORMULE(NOM_PARA=('X','Y','Z'),VALE=Y-a')


DEFI_FISS=_F( FONC_LN=LSN,

FONC_LT=LST,),

)

Formula of LSN and LST are available only for very simple crackgeometries

Rapidity

Formula of LSN and LST not available forcomplex crack geometries

Mesh



Post-processing

SIF,propagation

Visualisation


sane mesh


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Method 2: pre-defined geometries


DEFI_FISS=_F( FORM_FISS='ELLIPSE',

DEMI-GRAND_AXE = 0.4,

DEMI-PETIT_AXE = 0.2,

CENTRE = (0.5, 0.5, 0.5),

VECT_X = (0. , 1. , 0.2 ),

VECT_Y = (-1., 0. , 0.4 ),

))

This method is highly recommended

If a crack geometry is not available in the catalogue, propose a newdevelopment to the Code_Asterteam

Low risk of user error

Rapidity

Automatic crack frontorientation

Mesh



Post-processingSIF,

propagation

Visualisation


sane mesh

Half major axis

Half minor axis


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Method 2: pre-defined geometries

FORM_FISS= SIMP (

# type of cracks :

3d : "ELLIPSE","CYLINDRE","DEMI_PLAN" (half-plane)

Mesh



Post-processingSIF,

propagation

Visualisation


sane mesh

2d : "DEMI_DROITE ", "SEGMENT"


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x

zy

H

a

Groups of 2D elements :MA_FISS

Groups of 1D elements :MA_FOND

Capability of building complex cracks

Time consuming

Method 3: projection onto groups of elements


DEFI_FISS=_F( GROUP_MA_FISS=MA_FISS,

GROUP_MA_FOND=MA_FOND),

)

Mesh



Post-processingSIF,

propagation

Visualisation


sane mesh


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Method 4: lecture of a pre-computed LSN and LSTfields

TBLSN=LIRE_TABLE(UNITE=38,

FORMAT='ASTER',

NUME_TABLE=1,

SEPARATEUR=' ',TITRE='level set normale',)

CHLSN=CREA_CHAMP(OPERATION='EXTR',TABLE=TBLSN,

TYPE_CHAM='NOEU_NEUT_R',MAILLAGE=MAILLAG1)


DEFI_FISS=_F( CHAM_NO_LSN=CHLSN,

CHAM_NO_LST=CHLST),

)

Mesh



Post-processingSIF,

propagation

Visualisation


sane mesh


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optional keyword

Definition of enrichment zoneRestriction of the crack definition zone with GROUP_MA_ENRI


DEFI_FISS=_F(GROUP_MA_FISS=MA_FISS,


GROUP_MA_ENRI = M_LEFT,)

advanced

Mesh



Post-processingSIF,

propagation

Visualisation


sane mesh

crackCrack extension

(un-intentional)

Extension of the crack

front

M_LEFT, M_RIGHT,


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RAYON_ENRI / NB_COUCHES

TOPOLOGIQUE GEOMETRIQUE

Definition of the type of enrichment


DEFI_FISS=_F(GROUP_MA_FISS=MA_FISS,


GROUP_MA_ENRI = M_VOL,

TYPE_ENRI_FOND = / 'TOPOLOGIQUE' [DEFAULT]

/ 'GEOMETRIQUE'

optional keyword

advanced

Mesh



Post-processingSIF,

propagation

Visualisation


sane mesh


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Maillage

Enrichissementdes lments

finis

Calcul -Rsolution

Post-traitementFICs, critrepropagation

Visualisation

Dfinition de lafissure (level sets)

# Crack definitionFISS = DEFI_FISS_XFEM(MAILLAGE =MAILLA,

...

DEFI_FISS=_F(...),)

# Sane model definitionMOD_SAIN =AFFE_MODELE(MAILLAGE =MAILLA,

...

AFFE = _F(...),)


Mesh



Post-processingSIF,

propagation

Visualisation


Available modelisations: 3D, C_PLAN, D_PLAN,AXIS

Linear or Quadratic elements

(but X-FEM quadratic elements are still unstable in Version 12)

The crack and the sane model must be defined on the same mesh


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Maillage


finis

Calcul -Rsolution


Visualisation


# Crack definitionFISS = DEFI_FISS_XFEM(MAILLAGE =MAILLA,

...

DEFI_FISS=_F(...),)

# Sane model definitionMOD_SAIN =AFFE_MODELE(MAILLAGE =MAILLA,

...

AFFE = _F(...),)

# Creation of a model with enriched elements (X-FEM)MOD_FISS=MODI_MODELE_XFEM(MODELE_IN =MOD_SAIN,

FISSURE = FISS)


MOD_FISS must be used from now on (loadings, material)

MOD_SAIN is the sane modelFISS is the crack

FISSURE accepts also a list of

cracks / interfaces

Mesh



Post-processingSIF,

propagation

Visualisation



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Specific additionalloading for X-FEM

only necessary withcontact

No new keyword:

The crack is knownviathe model

(enrich elements)

if contact on crack faces: MUMPSsolver

Resolution

Computation for non-linear studies in quasi-static : STAT_NON_LINE, MECA_STATIQUE,

THER_LINEAIRE

CHXFEM=AFFE_CHAR_MECA(MODELE=MOD_FISS,

LIAISON_XFEM='OUI')

RESU=STAT_NON_LINE(CHAM_MATER=CHAMPMAT,

EXCIT=(_F(CHARGE=CH1,),

_F(CHARGE=CHXFEM)),

MODELE=MOD_FISS,

SOLVEUR=_F(METHODE='MUMPS'))

Maillage


finis

Calcul -Rsolution


Visualisation


Mesh



Post-processingSIF,

propagation

Visualisation



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Resolution

Focus on the sub-element cutting processContext: numerical integration of rigidity terms and right-hand side force terms forelements cut by the crack

On an element cut through by the crack, the integrand might be discontinuous

Numerical integration with Gauss method on the whole element is not possible

Solution: sub-element cutting (only for integration purpose)

[ ] [ ] [ ][ ]

=

elementssub sese

t

e dBDBK

with[ ] [ ] [ ][ ] = e et

e dBDBK [ ] ( ) ( ) ( ) ( ) 4,1, = += FFHB kkji


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Post-processing in fracture mechanics

PK=POST_K1_K2_K3( MODELISATION='3D',

MATER=ACIER,

FISSURE = FISS,

RESULTAT = UTOT1,

NB_POINT_FOND= 15,

ABSC_CURV_MAXI = 0.3)

TABFIC=CALC_G(RESULTAT=RESU,

OPTION='CALC_K_G',

THETA=_F(FISSURE=FISS,

R_INF=0.1,

R_SUP=0.3,

NB_POINT_FOND= 15))

Limit the post-processing on fewpoints along the crackfront reduction of theoscillations

Idem for meshed cracks (FEM)POST_K1_K2_K3

CALC_G

Maillage


finis

Calcul -Rsolution


Visualisation


Mesh



Post-processingSIF,

propagation

Visualisation



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Post- processing for visualisation

Impossible to visualise the crack opening on the computationalmesh (sane mesh)

Need for a visualisation tool


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MA_VISU = POST_MAIL_XFEM(MODELE =MOD_FISS)

MO_VISU =AFFE_MODELE(MAILLAGE =MA_VISU, AFFE=_F())

RES_VISU = POST_CHAM_XFEM(MODELE_VISU =MO_VISU,

RESULTAT = RESU)

IMPR_RESU(RESU=_F(RESULTAT = RES_XFEM))

Post- processing for visualisation

Creation of a mesh for visualisation purpose

Creation of result fields on this mesh

Maillage


finis

Calcul -Rsolution

Post-traitementFICs, critre

propagation

Visualisation


Mesh

Finite ElementsEnrichment


Post-processingSIF,

propagation

Visualisation



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Outline

4. Advanced tools

Mesh refinement

Crack propagation


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Mesh refinement

von Mises stresses near crack tip

r

x

y

Stress is proportional to 1/r, with r the distance to thecrack front

For both crack representations using FEM or X-FEM, mesh

needs to be fine enough for accurate SIF computation

Importance of the mesh refinement near the crack front


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Mesh refinement

A sane mesh needs to be refined before launching the FEanalysis (adaptation a priori)

Refinement near the crack tip (2D) or near the crack front (3D)

Automatic operation using the distance to the crack front (computed by level sets)

Only elements nearest to the crack from are refined

X-FE analysisInitialsanemesh

Level sets

Distance tothe crack-front

Refinementsoftware

(HOMARD)

Refined sane mesh

Automatic mesh refinement loop

SIFs


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Mesh refinement in Code_Aster

mesh = LIRE_MAILLAGE()

crack = DEFI_FISS_XFEM()

error = RAFF_XFEM()

refmesh =

MACR_ADAP_MAIL()

Level sets computation on the initial mesh(DEFI_FISS_XFEM)

Computation of the refinement criteria(RAFF_XFEM)

goal: to localize the area near the crack front

r = distance to the crack front

Near crack front, error 0Far crack front, error -

Mesh refinement (MACR_ADAP_MAIL)Homard will refine the elements with themathematically largest error

For ex., 10% of the elements with the largest error willbe refined

lstlsnr=

+=error

22


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Mesh refinement in Code_AsterMA=LIRE_MAILLAGE()

MO=AFFE_MODELE(MAILLAGE=MA,

AFFE=_F(TOUT=OUI,PHENOMENE='MECANIQUE',

MODELISATION='3D'))

# crack definition

FISS = DEFI_FISS_XFEM(MODELE=MO,DEFI_FISS=_F(FORM_FISS = 'ELLIPSE',

DEMI_GRAND_AXE = 2.,

DEMI_PETIT_AXE = 2.,

CENTRE = (0.,0.,0.),

VECT_X = (1.,0.,0.),

VECT_Y = (0.,1.,0.),),

GROUP_MA_ENRI='CUBE')

# error fieldCH_ERR=RAFF_XFEM(FISSURE=FISS)

# Mesh refinement (call of HOMARD software)

MACR_ADAP_MAIL(ADAPTATION='RAFFINEMENT',CHAM_GD =CH_ERR,

CRIT_RAFF_PE = 0.1,

TYPE_VALEUR_INDICA='V_RELATIVE',

NOM_CMP_INDICA = 'X1',

MAILLAGE_N = MA,

MAILLAGE_NP1 = CO(MA2),)

Refined mesh

Possibility touse python fora loop


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Examples of mesh refinement


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Mesh refinement

1) File for refinement: ref.comm

IN : coarse mesh + crack geometry

OUT : refined mesh

2) File for computation: comput.comm

IN : refined mesh + crack geometry

OUT : results

1) File for refinement + computation:ref_comput.comm

IN : coarse mesh + crack geometry

OUT : refined mesh + results

Inconvenient : re-launch the refinement

if changing the crack

Inconvenient : time of refinement for

each computation

Practical advisesIt is better to refine a few number of elements at several times than a lot of elementsonce

All the calls toMACR_ADAP_MAIL must be done in one .comm file

Warning: the refined mesh depends on the crack geometry


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Fatigue Propagation (constant amplitude)

common to meshedand non-meshed

cracks

Specific to X-FEM +level sets framework

Propagation of a 2d crack in aplate under tension with a hole

Computation of fatigue propagation: 3 ingredients

A propagation law (gives the increment of crack advance)

A bifurcation criterion (gives the bifurcation angle)

An algorithm for the crack update


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Ingredient n1: propagation law

Paris law

C and m : material parameters

Equivalent stress intensity factor

Propagation driven by a maximal advance increment : amax

Ingredient n2: bifurcation criterion

Maximum Hoop Stress (Erdogan, Sih)

( )

+

= 8

4

1arctan2

2

II

III

II

I

K

KKsign

K

K

2

2

222

1

1IIIIIIeq KKKK

+

++=

( )m

eqKCdN

da

= .


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Ingredient n3: crack update

Crack update = level sets update (X-FEM + level set framework)

3 methods available in Code_Asterfor the level sets update(operator PROPA_FISS)

nave approach: MAILLAGE method

SIMPLEX method

UPWIND method

GEOMETRIQUE method


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Initial crack

bifurcation

angle:

Advance

increment: da

Ingredient n3: crack update nave approach: MAILLAGE method

level sets update = adding a segment/anelementary surface (due to crack advancement)to the crack discretization, then computation oflevel sets by orthogonal projection of the nodes onthis new crack location

Already in DEFI_FISS_XFEM(projection onto

groups of elements)


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Ingredient n3: crack updateSIMPLEX and UPWIND methods

Numerical methods for level set update

SIMPLEX methodEvaluation of the contribution of each element to the level set update on a node, then selection of the

positives contributions to get a monotone schemeMain limitation: developed only for simplex elements (triangles in 2D and tetrahedrons in 3D)

UPWIND method

Selective finite differences scheme

Auxiliary regular grid is required for complex geometry

GEOMETIQUE method

This technique uses a vector propagation of the crack front and its local basis and an estimationof the distances in this local basis.

Faster than the other techniques and capable of modeling 3D non-planar crack growth.

( ) ( )xfH =+


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Propagation of a quarter-disk crack withautomatic mesh adaptation


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Conclusion on X-FEM in Code_Aster

Available in Code_Aster(in version 11)Behaviour law: linear elasticity, non-linear, elasto-plasticity

Modelisations: plane strain, plane stress, axi-symmetrical, 3D

Loadings: pressure on crack faces; displacement boundary conditions near the crack(symmetry conditions); rotation, gravity

Contact on crack faces

Finite elements: linear or quadratic

One or multiple crack, crack net, crack branching

Post-processing in fracture mechanics: CALC_G and POST_K1_K2_K3

Visualisation: displacements and stresses on a cracked mesh

Crack propagation (2d, 3d in-plane and out-of-plane)

Modal analysis

Under development/validationFriction on crack faces (or interface)

Cohesive Zone Modelling


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References

Propagation and crackbranching

General user documentation[U2.05.02] how to use X-FEM

Operator documentation[U4.82.08] Operator DEFI_FISS_XFEM

[U4.41.11] OperatorMODI_MODELE_XFEM

[U4.82.11] Operator PROPA_FISS

[U4.82.21] Operator POST_MAIL_XFEM[U4.82.22] Operator POST_CHAM_XFEM

General user documentation[R7.02.12] eXtended Finite Element Method


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17 EXtended FEM