Pais(2010)IntroductiontotheExtendedFiniteElement(XFEM).ppt

IntroductiontotheExtendedFiniteElementMethod(XFEM),MatthewPaisIdahoNationalLaboratoryAugust25,2010,2,Overview,BasicConceptsLevelSetMethodExtendedFiniteElementMethod(XFEM)BasicformulationLevelsetrepresentationsofcracks,inclusions,andvoidsEnrichmentfunctionsforcracks,inclusions,andvoidsIntegrationofenrichedelementsApplicationstofluidsReanalysisofXFEMMotivationAlgorithmBasicanalysisofmethodInitialresultsConclusions,EnergyReleaseRate/SIF,G,energyreleaserateamountofenergyreleasedbyunitadvanceofcrackIngeneralacrackwillgrowinthedirectionwhichmaximizesGKi,stressintensityfactorcharacterizesthemagnitudeofamplificationofappliedstressatcracktipEspeciallyusefulforlinearelasticcasewhichpredictsr-1/2singularityatcracktipSinglevaluegivesstateofstressaroundthecracktip2D:KIandKII,3D:KI,KII,andKIII,3,ModeI:Opening,ModeII:In-PlaneShear,ModeIII:Out-of-PlaneShear,LevelSetMethod,Thelevelsetmethod1,2isafronttrackingmethodgivenintermsofthefrontvelocitywithrespecttotimeThelevelsetfunctioniscommonlydiscretizedandinterpolationbetweenpointsisperformedAmmenabletouseinFEenvironmentHasbeenusedinshapeoptimization3withintheFEframework,4,1Osheretal,1988,Frontspropagatingwithcurvaturedependentspeed,J.Comp.Phys.,79,12-49.2Stolarksaetal,2001,ModelingcrackgrowthbylevelsetsintheXFEM,Int.J.Num.Meth.Eng.,51,943-960.3Wangetal,2004,Structuralshapeandtopologyoptimizationinalevelsetbasedframework,Struct.Multi.Opt.,27,1-19.,5,ModelingCrackGrowthinFE,FiniteelementmeshcorrespondstothedomainofthegivenmaterialUsesingularelementsatcracktiptorepresenttheasymptoticcracktipdisplacementfield1Ascrackgrows,mustrecreatemesharoundthecracktip,whichcanbeexpensive2CreateschallengesintrackingtimehistoryofpointsnearcrackwhicharebeingremeshedDisplacement,stress,orstrain,1Barsoum,1976,Ontheuseofisoparametricfiniteelementsforlinearfracturemechanics,Int.J.Num.Meth.Eng.,10,25-37.2Malignoetal,2010,Athree-dimensionalnumericalstudyoffatiguecrackgrowthusingremeshing,Eng.Frac.Mech.,77,94-111.,6,eXtendedFiniteElementMethod(XFEM),Belytschko1StrongDiscontinuitiesCracksDaux2andSukumar3WeakDiscontinuitiesInclusionsVoidsDiscontinuousbehaviorembeddedintoelementsusinglocalenrichmentfunctions,additionalnodalDOFsDoesnotrequiremeshtoconformtodomainNoremeshingneededforevolvingdiscontinuitiesLevelsets4usedtotrackdiscontinuities,Crack,Inclusion,Void,1Belytschkoetal,1999,Elasticcrackgrowthinfiniteelementswithminimalremeshing,Int.J.Num.Meth.Eng.,45,601-620.2Dauxetal,2000,ArbitrarybranchedandintersectingcrackswithXFEM,Int.J.Num.Meth.Eng.,48,1741-1760.3Sukumaretal,2001,ModelingholesandinclusionbylevelsetsintheXFEM,Comp.Meth.App.Mech.Eng,190,6183-6200.4Osheretal,1988,Frontspropagatingwithcurvaturedependentspeed,J.Comp.Phys.,79,12-49.,7,GeneralXFEMApproximation,Uselowercasetorepresentenrichment,uppercaseforshiftedShift1enrichmenttorecoverFEMapproximationatnodeswhereenrichmentfunctionisnonzeroInterestedinenrichmentfunctionsrepresentingdiscontinuitiesSuperimposingcontinuousanddiscontinuousapproximations,EnrichmentFunction,AdditionalSpatialDOFatNodeI,TraditionalFEMApproximation,1Belytschkoetal,2001,Arbitrarydiscontinuitiesinfiniteelements,Int.J.Num.Meth.Eng.,50,993-1013.,LevelSetRepresentationofCracks,IntroducedbyStolarska1asanextensionoftheworkbySethian2IntersectionoftwolevelsetsdefinestheopensectionCanbeupdatedifdesiredusingwellknownmethodsNarrowband3,fastmarchingmethod4decreasecomputationaltime,1Stolarksaetal,2001,ModelingcrackgrowthbylevelsetsintheXFEM,Int.J.Num.Meth.Eng.,51,943-960.2Osheretal,1988,Frontspropagatingwithcurvaturedependentspeed,J.Comp.Phys.,79,12-49.3Adalsteinssonetat,1995,AFastLevelSetMethodforPropagatingInterfaces,J.Comp.Phys.,118,269-277.4Sethian,1996,AFastMarchingLevelSetMethodforMonotonicallyAdvancingFronts,Proc.Nat.Acad.Sci.,93,1591-1595.,9,CrackTipEnrichmentFunction,Cracktipenrichmentfunctions1embedthecracktipsingularityintotheenrichedelementAdditionalenrichmentfunctionsavailableforothercracktipconditionsBi-material,branching,cohesive,functionallygradedandorthotropicmaterials2,Crack,1Flemingetal,1997,Enrichedelement-freeGalerkinmethodsforcracktipfields,Int.J.Num.Meth.Eng.,40,1483-1504.2Belytschkoetal,2009,Areviewofextended/generalizedFEMformaterialmodeling,Int.J.Num.Meth.Eng.,17,043001.,10,HeavisideEnrichmentFunction,Crack,Heaviside1functionisusedinelementswhichhavetheirsupportcompletelycutbythecrackPlacesdiscontinuitydirectlyatcracklocationwithinelement,1Moesetal,1999,Afiniteelementmethodforcrackgrowthwithoutremeshing,Int.J.Num.Meth.Eng.,46,131-150.,11,CrackEnrichedElements,NodeswithHeavisideEnrichment(2additionalDOF),NodeswithCrackTipEnrichment(8additionalDOF),N,n,Nn,裂尖单元,贯穿单元,裂尖单元,常规单元,12,DiscreteEquations,c,u,t,b,n,to,WeightedResidual,Strain-DisplacementRelationship,EquilibriumEquation,ConstitutiveEquation,BoundaryConditions,13,WeightedResidualMethod,14,XFEMStiffnessMatrix,uaretraditionalDOFa,bareenrichedDOFKuuisindependentofcracklocation,traditionalFEstiffnessmatrixKua,Kaa,KabarecomponentswithHeavisideenrichmentKub,Kab,KbbarecomponentswithcracktipenrichmentKua,Kub,Kabareaddcouplingbetweentraditional,enrichedDOF,Example:Mixed-ModeEdgeCrack,H,1Sutradhar,Paulino,Gray.SymmetricGalerkinBoundaryElementMethod.Springer-Verlag,2008.,CrackOpeningwithGrowth,16,Displacementismagnifiedtoshoweffectofenrichment.,RecentWorkonCracks,Areaofcracktipenrichmenttoimproveconvergencerate1CorrectedXFEM2toremoveproblemswithblendingelementsIntroductionofharmonicenrichmentfunctions3tounifyenrichmentforbranching,homogenous,andintersectingcracksIntroductionofoptimizedenrichmentfunctions4forhomogeneousandcohesivecracksUseofXFEMforinterpretationofstructuralhealthmonitoring(SHM)datathroughoptimization5UseofXFEMforoptimizationofstructurewithrespecttofatiguelife6,1Labordeetal,2005,High-OrderXFEMforCrackedDomains,Int.J.Num.Meth.Eng.,64,354-381.2Fries,2007,ACorrectedXFEMApproximationWithoutProblemsinBlendingElements,Int.J.Num.Meth.Eng.,75,503-532.3Mousavietat,2010,AUnifiedTreatmentofMultiple,IntersectingandBranchedCracksinXFEM,Int.J.Num.Meth.Eng.,InPress.4Abbasetal,2010,AUnifiedEnrichmentSchemeforFractureProblems,WCCM/APCOM2010,Sydney,Australia.5Waismanetal,2009,DetectionandQuantificationofFlawsinStructuresbyXFEM,10thUSNat.Cong.Comp.Mech.,Columbus,OH.6Edke,ShapeOptimizationfor2DMixed-ModeFractureUsingXFEMandLSM,Struct.Mutli.Opt.,InPress.,18,LevelSetRepresentationofInclusion/Void,IntroducedbySukumar1asanextensionoftheworkdonebyDaux2DefinenegativevaluestobeinteriorofinclusionorvoidCanbeupdatedifdesiredusingwellknownmethodsNarrowband3,fastmarchingmethod4decreasecomputationaltime,1Sukumaretal,2001,ModelingHolesandInclusionbyLevelSetsintheXFEM,Comp.Meth.App.Mech.Eng,190,6183-6200.2Dauxetal,2000,ArbitraryBranchedandIntersectingCrackswithXFEM,Int.J.Num.Meth.Eng.,48,1741-1760.3Adalsteinssonetat,1995,AFastLevelSetMethodforPropagatingInterfaces,J.Comp.Phys.,118,269-277.4Sethian,1996,AFastMarchingLevelSetMethodforMonotonicallyAdvancingFronts,Proc.Nat.Acad.Sci.,93,1591-1595.,19,InclusionEnrichmentFunction,Interface,1Moesetal,2003,AComp.ApproachtoHandleComplexMicrostructureGeometries,Comp.Meth.App.Mech.Eng,192,3163-3177.,=LevelSetatNodeI,Inclusionfunction1isusedinelementswithmultiplematerialsNoneedtoshiftenrichmentasitiszeroatallnodes,20,VoidEnrichmentFunction,Material,1Dauxetal,2000,ArbitrarybranchedandintersectingcracksintheXFEM,Int.J.Num.Meth.Eng.,48,1741-1760.,Voidfunction1isusedinelementswhichcontainvoidboundaryNoadditionaldegreesoffreedom,modifiesdisplacementdirectlyOnlyperformnumericalintegrationinregionswhichcontainmaterial,Void,21,IntegrationofEnrichedElement,Forenrichedelement,subdivide1quadrilateralintotrianglesandintegrateovereachtriangletoavoiddifficultieswithintegratingdiscontinuousdomain.,1Sukumaretal,2003,ModelingQuasi-StaticCrackGrowthwithXFEM,PartI:ComputerImplementation,Int.J.Sol.Str.,40,7513-7537.,RecentWorkonIntegration,Mousavi1presentedgeneralizedgaussquadraturerulesoverarbitrarypolygonswhereoptimizationisusedtoidentifythelocationandweightsofthegausspointsMousavi2presentedtheDuffytransformationfromatriangle(tetrahedron)toasquare(cube)forintegrationNatarajan3presentedatransformationtoaunitdiskbasedonSchwarz-ChristoffelmappingPark4presentedatransformationfromatriangle(tetrahedron)toasquare(cube)wherethesingularityisplacedattheoriginofthesquare(cube)inthetransformedspace,22,1Mousavietal,2009,GeneralizedGaussianQuadratureRulesonArbitraryPolygons,Int.J.Num.Meth.Eng.,82,99-113.2Mousavietal,2010,GeneralizedDuffyTransformationforIntegratingVertexSingularities,Comp.Meth.,45,127-140.3Natarajanetat,2009,NumericalIntegrationOverArbitraryPolygonalDomainsBasedonSchwarz,Int.J.Num.Meth.Eng.,80,103-134.4Parketal,2009,IntegrationofSingularEnrichmentFunctionsintheGFEM/XFEM,Int.J.Num.Meth.Eng.,78,1220-1257.,XFEMforFluidApplications,1Fries.2009.TheintrinsicXFEMforTwo-FliudFlows,Int.J.Num.Meth.Eng.,60,437-471.2Friesetal.2009.OnTimeIntegrationintheXFEM.Int.J.Num.Meth.Eng.,79,69-93.3Fries.http:/www.xfem.rwth-aachen.de.,TwoincompressiblefluidsDensityandviscositydiscontinuousacrossinterfaceVelocity(strong)andpressure(weak)discontinuitiesStationarymesh,RisinggasbubbleinfluidTopbubblemoredensethanbottom,XFEMinABAQUS,XFEMimplementationofcohesivecrackmodelusingphantomnodemodel1Limitations2OnlySTATICanalysisOnlylinearcontinuumelementsNoparallelprocessingNocontourintegrals(availablein6.9-EF3andnewer)NofatiguecrackgrowthOnlyonecrackwithinanelementAcrackmaynotturnmorethan90degreeswithinanelementAcrackmaynotbranchNoimplementationinABAQUS/Explicit2,1Songetal,2006,DynamicCrackandShearBandPropagationwithPhantomNodes,Comp.Meth.App,Mech.Eng,193,3524-3540.22009,ExtendedFiniteElementMethod(XFEM),ABAQUS6.9UpdateSeminar,DassaultSystems.32009,ABAQUS6.9ExtendedFunctionality(EF)Overview,ABAQUS6.9-EFUpdateWebinar,DassaultSystems.,25,XFEMinMATLAB,Created2Dplanestress/strainXFEMcodeAllowsrectangulardomainwitharbitraryboundaryandloadingconditionsHomogeneousandbi-materialcracks,voidandinclusionenrichmentsNarrowbandlevelsetforcracks,fulllevelsetforinclusionsandvoidsIntegrationofenrichedelementsbytriangularsubdivisionContourintegralsforcalculatingSIFsFinitecrackgrowthincrementorParisLawtogrowcrackPlottingoflevelsets,mesh,displacementandstressesBenchmarksforenrichments,fatiguecrackgrowth,andoptimizationimplementation1Dand2DMATLABGoogle:AbaqusXFEMorMATLABXFEM,ReanalysisofXFEM,FatigueCrackGrowth,Fatiguecracksformastheresultofrepeatedloadingbelowtheyieldstress,eventuallyleadingtofailureHighcyclefatigue,Nfail106+cyclesModelspredictgrowthwithanordinarydifferentialequation,da/dN=f(K)Manymodelsprovidef(K)invaryinglevelsofcomplexity,N,K,IncidentsfromFatigueFailure,1954SouthAfricanAirwaysFlight201,21deaths1954BOACFlight781,35deaths1957CebeDouglascrash,25deaths1968HelicoptercrashinCompton,21deaths1980AlexanderL.Keillandoilplatform,123deaths1985JapanFlight123,521deaths1988AlohaAirlinesFlight243,1death1989UnitedFlight232,112deaths1992ElAlFlight1862,43deaths1998ICEtraincrash,101deaths2002ChinaAirlinesFlight611,225deaths2005ChalksOceanAirwaysFlight101,20deaths2007MissouriAirNationalGuardcrashed,pilotejected2009SouthwestAirlinesFlight2294,footballsizedholeMetalfatiguehasbeenandstillisachallengeforengineeringapplicationsandcanleadtothelossoflife.,28,29,Motivation,ForwardEulermethodisconditionallystable,becomesunstableforsomeaorNSeriesoflinearapproximationstoexponentialfunctionCrackpathbecomesfunctionofh,aorN,30,ObservationofKforQuasi-StaticGrowth,Forsameboundaryconditions:KuuareequalforAandBKaa,Kuagrow,oldportionconstantKbb,Kub,Kabchangebasedontiplocation,A,B,31,ReanalysisTechniques,ExistingreanalysistechniquesmaybeappliedtoXFEMformodelingquasi-staticcrackgrowthDevelopedindesignandoptimizationfieldsConsidersmallchangestosystemofequationsbymodifyingKSavingsinassemblyandfactorizationofthesystemofequationsApproximatemethodsIterativesolver1foraddingDOFstosystemofequationsExactmethodsIncrementalCholesky2factorizationSherman-Morrison-Woodbury3formula,1Wuetal,2006,Staticreanalysisofstructureswithaddeddegreesoffreedom,Comm.Num.Meth.Eng.,22,269-281.2Chapraetal,2002,Numericalmethodsforengineers,4thedition,McGrawHill,NewYork,NY.3Woodbury,1950,Invertingmodifiedmatrices,Mem.Rpt.42,Stat.Res.Gr.,PrincetonUniversity,4ppMR38136.,32,UpdatingStiffnessMatrix,Constant,33,UpdatingFactorizedStiffnessMatrix,Constant,Factor,Appe