Generalized finite element method using

Generalizedfiniteelementmethodusingmesh-basedhandbooks:applicationtoproblemsindomainswithmanyvoidsTheofanisStrouboulisa,*,LinZhanga,IvoBabu
sskabaDepartmentofAerospaceEngineering,TexasA&MUniversity,CollegeStation,TX77843-3141,USAbTexasInstituteofComputationalandAppliedMathematics,UniversityofTexasatAustin,Austin,TX78712,USAReceived12May2003;accepted12May2003AbstractThispaperdescribesanewversionofthegeneralizedfiniteelementmethod,originallydeveloped[Int.J.Numer.MethodsEngrg.47(2000)1401;Comput.MethodsAppl.Mech.Engrg.181(2000)43;Thedesignandimplementationofthegeneralizedfiniteelementmethod,Ph.D.thesis,TexasA&MUniversity,CollegeStation,Texas,August2000;Comput.MethodsAppl.Mech.Engrg.190(2001)4081],whichiswellsuitedforproblemssetindomainswithalargenumberofinternalfeatures(e.g.voids,inclusions,cracks,etc.).Themainideaistoemployhandbookfunctionsconstructedonsubdomainsresultingfromthemesh-discretizationoftheproblemdomain.TheproposednewversionoftheGFEMisshowntoberobustwithrespecttothespacingofthefeaturesandiscapableofachievinghighaccuracyonmesheswhicharerathercoarserelativetothedistributionofthefeatures.
2003ElsevierB.V.Allrightsreserved.1.IntroductionThegeneralizedfiniteelementmethod(GFEM)wasintroducedin[1–4]asacombinationofthepartitionofunitymethod(PUM)proposedin[5–8],withtheclassicalfiniteelementmethod(FEM)(see[9,10]).ThetwomainattributesoftheGFEM,asdevelopedin[1–4],arethecapabilitiesofusing:1.Domainindependentmeshes:TheGFEMapproximationcanbeconstructedonmesheswhicharenon-overlappingpartitionsofanydomainX0whichcoverstheproblemdomainX,namelyX
X0,andwhenX0ischosentohavesimplegeometry,e.g.X0isarectangle,theycanbemuchsimplertoconstruct.Incontrast,themeshesusedintheclassicalFEMarenon-overlappingpartitionsoftheproblemdomainXintosimplesubdomains,andifXhascomplexgeometry,maybedifficulttogenerate.ThecompleteinformationabouttheproblemdomainXentersintotheGFEMthroughspecialintegrationmeshesconstructedtoreflectthelocalgeometryoftheproblemdomainineachelementoftheGFEMmeshbyanautomatedadaptive*Correspondingauthor.E-mailaddress:[e-mail](T.Strouboulis).0045-7825/03/$-seefrontmatter
2003ElsevierB.V.Allrightsreserved.doi:10.1016/S0045-7825(03)00347-5Comput.MethodsAppl.Mech.Engrg.192(2003)3109–3161(see[1]).2.Enrichmentbyhandbookfunctions:TheGFEMapproximationcanbeenrichedbyhandbookfunctionswhicharesolutionsoflocalboundary-valueproblemsreflectingthelocalgeometryoftheproblemdomainXandtheboundaryconditionsoftheproblemofinterest,e.g.corners,voids,inclusions,cracks,curvedNeumannorDirichletboundaries,etc.Thehandbookfunctionsmaybeobtainedpriortothesolutionoftheproblemofinterest,eitherthroughananalytical[1,2]oranumerical[3,4]construction.In[3,4]weshowedthattheGFEMwithjudiciouslyselectedhandbookfunctionsiscapableinachievinghighaccuracyforproblemswithrathercomplexgeometry,whileemployingrathercoarsemeshes.Letusillustratetheabovepointsthroughsomesampleresults.Foranextensivesetofsimilarresultssee[3,4].LetXbethedomainshowninFig.1(a),letCdenoteitsouterboundary,andletusconsidertheNeumannboundaryvalueproblem:Du¼0inX;ouon¼g¼defrð2xyÞnontheouterboundaryC;ouon¼0ontheboundaryofthevoids:8:ð1:1ÞFig.1.Illustrationofthe‘‘meshless’’characteroftheGFEMormorepreciselyitsabilitytoconstructtheapproximationonameshwithgeometryindependentoftheproblemdomain.(a)TheproblemdomainXwhichhasseveralinternalvoidsandcracks;(b)theemployedGFEMmeshobtainedbysubdividinguniformlyasquaredomainX0,whichincludestheproblemdomainXinitsinterior,X
X0,andbyemployingfournestedrefinementsofthesquareswhichoverlapareentrantcorner;(c)theintegrationmeshemployedinthecomputation;and(d)detailoftheintegrationmesh.3110T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.192(2003)3109–3161Aswehaveseenin[3,4],wemaycomputeGFEMsolutionsoftheaboveproblemusingameshofsquareelementsconstructedfromanestedsubdivisionofasquaredomainX0whichoverlapsthedomainX,i.e.wehaveX
X0,asshowninFig.1(b).TheinformationabouttheproblemdomainXandtheappliedboundaryconditionsisincludedintotheGFEMthroughlocalintegrationmeshes,theadaptivecon-structionofwhichcanbeeasilyautomatedineachelement.Fig.1(c)showstheintegrationmeshusedinthecomputationgivenbelow.Letusunderlinethattheintegrationmeshisconstructedelementbyelementbyanautomaticrefinementalgorithmanditdoesnotneedtobeconformingattheelementinterfaces.Inordertoshowtheeffectofthehandbookfunctionsontheaccuracy,wecomputedtheGFEMsolutionofthemodelproblem(1.1)onthemeshofFig.1(b)usingthe1.biquadratic(p¼2)FEMbasis,2.biquadratic(p¼2)FEMbasiswithhandbookfunctionsofdegree1,phandbook¼1(see[3,4]),forthevoids,cornersandcracksaddedatnlayers¼0(see[3,4])aroundeachfeature.WeanalyzedtheresultsbyemployingasexactsolutionanoverkillGFEMsolutionobtainedonthesamemesh,usingbiquintic(p