Optimal convergence analysis for the eXtended Finite Element Method

Serge Nicaise

Description

Optimal convergence analysis for the eXtended Finite Element Method Serge Nicaise 1, Yves Renard 2, Elie Chahine 3 Abstract We establish some optimal a priori error estimate on some variants of the eXtended Finite Element Method (Xfem), namely the Xfem with a cut-off function and the stan- dard Xfem with a fixed enrichment area. The results are established for the Lame system (homogeneous isotropic elasticity) and the Laplace problem. The convergence of the numerical stress intensity factors is also investigated. We show some numerical experiments which corroborate the theoretical results. Keywords: extended finite element method, error estimates, stress intensity factors. 1 Introduction Inspired by the Pufem [26], the Xfem (extended finite element method) was introduced by Moes et al. in 1999 [28, 27] for plane linear isotropic elasticity problems (Lame system) in cracked domains. The main advantage of this method is the ability to take into account the discontinuity across the crack and the asymptotic displacement at the crack tip by addition of special functions into the finite element space. It allows the use of a mesh which is independent of the geometry of the crack. This avoids the remeshing operations when the crack propagates and the corresponding re-interpolation operations which can cause numerical instabilities. In the original method, the asymptotic displacement is incorporated into the finite element space multiplied by the shape function of a background Lagrange finite element method.

• crack tip


• off function


• off function verifying


• called stress


• linear elasticity


• universite de valenciennes et du hainaut cambresis


• heaviside enrichment


• stress intensity


• functions

Sujets

• Renard
• Saint Nicaise
• Scherrer
• Chahine
• Laplace
• Linear elasticity

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 30
• Langue: English

Extrait

OptimalconvergenceanalysisfortheeXtendedFiniteElement Method SergeNicaise 1 ,YvesRenard 2 ,ElieChahine 3 Abstract WeestablishsomeoptimalapriorierrorestimateonsomevariantsoftheeXtended FiniteElementMethod(Xfem),namelytheXfemwithacut-offfunctionandthestan- dardXfemwithafixedenrichmentarea.TheresultsareestablishedfortheLame´ system(homogeneousisotropicelasticity)andtheLaplaceproblem.Theconvergence ofthenumericalstressintensityfactorsisalsoinvestigated.Weshowsomenumerical experimentswhichcorroboratethetheoreticalresults. Keywords :extendedfiniteelementmethod,errorestimates,stressintensityfactors. 1Introduction InspiredbythePufem[26],theXfem(extendedfiniteelementmethod)wasintroducedby Moe¨setal.in1999[28,27]forplanelinearisotropicelasticityproblems(Lame´system)in crackeddomains.Themainadvantageofthismethodistheabilitytotakeintoaccount thediscontinuityacrossthecrackandtheasymptoticdisplacementatthecracktipby additionofspecialfunctionsintothefiniteelementspace.Itallowstheuseofameshwhich isindependentofthegeometryofthecrack.Thisavoidstheremeshingoperationswhen thecrackpropagatesandthecorrespondingre-interpolationoperationswhichcancause numericalinstabilities.Intheoriginalmethod,theasymptoticdisplacementisincorporated intothefiniteelementspacemultipliedbytheshapefunctionofabackgroundLagrange finiteelementmethod.However,wedealalsowithavariant,introducedin[12],where theasymptoticdisplacementismultipliedbyacut-offfunction.Thisvariantissimilarto theclassicalsingularenrichmentmethodintroducedin1973byStrangandFix[32]butit additionallypreservestheindependenceofthemeshtothegeometryofthecrackwhichis indeedtheessentialcontributionofXfem. Anotherclassicalmethodtotakeintoaccountasingularbehaviorofthesolutionisthe dualsingularfunctionmethodintroducedbyM.Dobrowolskietal.in[5](seealso[19,10]) oramorerecentvariantthesingularcomplementmethodintroducedbyP.CiarletJr.et al.in[17](foraL-shapedomain,see[29]).Thesemethodsrequiretheuseofdualsingular functionswhichcanbedifficulttoobtaininsomesituations(evenfortheLame´system) orquiteimpossibletoobtainwhenjusttheasymptoticbehaviorisknown(fornon-linear elasticity[2]orMindlinplatemodelforinstance). TheXfemstrategycanbeadaptedtovarioussituations.Seeamongmanyotherrefer- ences[3,6,7,8,23,25,36,37,35,38].Inparticular,afictitiousdomainmethodcanbe 1 Universite´deValenciennesetduHainautCambre´sis,LAMAV,FRCNRS2956,InstitutdesSci- encesetTechniquesdeValenciennes,F-59313-ValenciennesCedex9France,email:Serge.Nicaise@univ- valenciennes.fr 2 Correspondingauthor.Universite´deLyon,CNRS INSA-Lyon,ICJUMR5208,LaMCoSUMR5259,F-69621,Villeurbanne,France,[e-mail] 3 LaboratoryforNuclearMaterials,NuclearEnergyandSafetyResearchDepartment,PaulScherrer InstituteOVGA/14,CH-5232VilligenPSI,Switzerland,[e-mail]. 1

derivedfromtheprincipleofXfem(see[24,4])anditispossibletoadaptsomestrategies whentheasymptoticbehaviorisunknnownoronlypartiallyknown(see[11,13,14]). Inthepresentpaper,weimprovetheresultsgivenin[12]concerningthevariantwhich usesacut-offfunction.Wealsogivesomeadditionalerrorestimatesconcerningthestress intensityfactorsandthestandardXfem.Thetheoreticalresultsareestablishedforboththe Lame´systemandtheLaplaceproblem.Somenumericalteststhatillustrateandconfirm thetheoreticalresultsarepresented. 2Themodelproblems TheanalysiswillbeperformedonacrackeddomainΩ⊂ R 2 fortwomodelproblems:The LaplaceequationandtheLame´system.ThecrackΓ C isassumedtobestraight.Inboth cases,theboundary∂ΩofΩispartitionedintoΓ D ,Γ N andΓ C (seeFig.1).ADirichlet conditionisprescribedonΓ D ,aNeumannoneonΓ N whileonthecrackΓ C weconsider anhomogeneousNeumanncondition. Figure1:ThecrackeddomainΩ. Theweakformulationofthe(scalar)Laplaceequationonthisdomainreadsasfollows:  Findu∈ Z Vsuchthata(u,v)=l(v)∀v∈V,  a(u,v)=∇u
∇vdx, Z Ω Z (1) l(v)=fvdx+gvdΓ, ΓΩ N V={v∈H 1 (Ω);v=0onΓ D }.   WhiletheoneoftheLame´(vectorial)system(linearelasticityproblemonthisdomainfor anisotropicmaterial)is:  Findu∈ Z Vsuchthata(u,v)=l(v)∀v∈V, a(u,v)=σ(u):ε(v)dx, Ω  ZZ  l(v)=f
vdx+g
vdΓ, ΓΩ N σ(u)=λtr(ε(u))I+2ε(u),   V={v∈H 1 (Ω; R 2 );v=0onΓ D }, T whereσ(u)denotesthestresstensor,ε(u)= 2 1 (∇u+∇u)isthelinearizedstraintensor, 2 )2(