A DIRECTIONAL ERROR ESTIMATOR FOR ADAPTIVE FINITE

COMPUTATIONALMECHANICSNewTrendsandApplicationsS.Idelsohn,E.O~nateandE.Dvorkin(Eds.)cCIMNE,Barcelona,Spain1998ADIRECTIONALERRORESTIMATORFORADAPTIVEFINITEELEMENTANALYSISLaviniaBorges,RaulFeijooy,ClaudioPadrayyandNestorZouainCOPPE/EE-MechanicalEngineeringDepartmentFederalUniversityofRiodeJaneiroCx.Postal68503RiodeJaneiro,RJ,BrasilCep:21945.970e-mail:[e-mail]@serv.com.ufrj.bryLaboratrioNacionaldeComputac~aocientca(LNCC/CNPq)LauroMuller455-Botafogo,RiodeJaneiro-RJ-BrasilCep:22290.160E-mail:[e-mail]:~tacsomyyInstitutoBalseiro(CAB)CentroAtomicodeBariloche,8400,Bariloche,Argentina.E-mail:[e-mail]:FiniteElements,MeshGeneration,Errorestimator,AdaptiveAnalysis,LimitAnalysisAbstract.Wepresentanerrorestimatorbasedonrst-andsecond-orderderivativesrecoveryforniteelementadaptiveanalysis.Atrst,webrieflydiscusstheabstractframeworkoftheadoptederrorestimationtechniques.Somepossibilitiesofderivativesrecoveryareconsidered,includingtheproposalofadirectionalerrorestimator.Usingthedirectionalerrorestimatorproposed,anadaptiveniteelementanalysisisperformedwhichgivesanadaptedmeshwheretheestimatederrorisuniformlydistributedoverthedomain.Theadvantagesofadaptingmeshesarewellknown,butweplaceparticularemphasisontheanisotropicmeshadaptationprocessgeneratedbythedirectionalerrorestimator.Thismeshadaptationprocessgivesimprovedresultsinlocalizingregionsofrapidorabruptvariationsofthevariables,whoselocationisnotknownapriori.Weapplytheaboveabstractformulationtoanalyzethebehaviouroftherecoverytechniqueandtheproposedadaptiveprocessforsomeparticularfunctions.Finally,weapplytheproceduretosomeniteelementmodelsforlimitanalysis.1LaviniaBorges,RaulFeijoo,ClaudioPadraandNestorZouain1INTRODUCTIONThemainobjectiveofthispaperittopresentanerrorestimatorbasedonrst-andsecond-orderderivativesrecoveryforniteelementadaptiveanalysis,includingtheproposalofadirectionalerrorestimator.Usingtheproposeddirectionalerrorestimator,weperformanadaptiveniteelementanalysis,whichgivesanadaptedmeshwheretheestimatederrorisuniformlydistributedoverthedomain.Theadvantagesofadaptingmeshesarewellknown,butweplaceparticularemphasisontheanisotropicmeshadaptationprocess,generatedbyadirectionalerrorestimatorbasedontherecoveringofsecondderivativesoftheniteelementsolution.Thegoalofthisapproachistoachieveamesh-adaptivestrategyaccountingformeshsizerenement,aswellasredenitionoftheorientedelementstretching.Thisway,alongtheadaptationprocess,themeshturnsalignedwiththedirectionofmaximumcurvatureofthefunction.Thismeshadaptationprocessgivesimprovedresultsinlocalizingregionsofrapidorabruptvariationsofthevariables,whoselocationisnotknownapriori1;2;3;4.So,wecanobtainanaccuraterepresentationofshocks,boundarylayers,wakesandotherdiscontinuities.Inthelightoftheabstracttheory,itispresentedanadaptivemesh-renementpro-cessforsomeclassicalexamplesinlimitanalysis.AsimilarapproachisadoptedinapplicationsofComputacionalFluidDynamicsproblemsbyReginaC.deAlmeidaetal.inthepaper\AdaptiveFiniteelementComputacionalFluidDynamicsusinganAnisotropicErrorEstimator,alsopresentedinthisevent.Limitanalysisdealswiththedirectcomputationoftheloadproducingplasticcollapseofabody-aphenomenonwhere,underconstantstresses,kinematicallyadmissibleplasticstrainratestakeplace.Localizedplasticdeformationsorslipbandsarepresentinmostcollapsesituations(seeforexampleBorgesetal.18andpaperstherein).Accuracyinthenumericalsolutionoflimitanalysisisseriouslyaectedbylocalsingularitiesarisingfromtheselocalizedplasticdeformations.Onepossibleapproachinordertoovercomethisproblemistoaddmoregrid-pointswherethesolutionpresentsthosesingularities.So,itbecomesnecessarynotonlytoidentifytheseregions,butalsotoobtainagoodequilibriumbetweentherenedandunrenedregionsforanoptimaloverallaccuracy5.Inlimitanalysis,anapriorierrorestimate,asprovidedbythestandarderroranalysisintheniteelementmethod,isofteninsucienttoassurereliableestimatesofthecomputedsolutionaccuracy.Thisisduetothefactthatitonlyyieldsinformationontheasymptoticerrorbehaviourandrequiresregularityconditionsofthesolution,whicharenotsatisedinthepresenceofsingularitiessuchastheabovementionedones.Thosefactsdisclosetheneedofanestimatorwhichcanaposterioribeextractedfromthecomputednumericalsolution.Wediscussinthispaper,rstly,theabstractframeworkoftheadoptederrorestima-tiontechniques,byconsideringthederivativesrecoveryandtheproposalofadirectionalerrorestimator.Togetherwithlimitanalysisapplications,wealsoselectsomeadaptivemesh-renementsolutionsforinterpolationproblemsinordertoshowthatthepro-2LaviniaBorges,RaulFeijoo,ClaudioPadraandNestorZouainposedadaptivestrategyusingouranisotropicerrorestimatorrecoversoptimaland/orsuperconvergencerates.2ESTIMATORSBASEDONDERIVATIVESRECOVERYInrecoverybasederrorestimationmethodsthegradientsand/orHessiansofsolu-tions,obtainedonagivenmesh,aresmoothedandafterthatthesmoothedsolutionisusedinerrorestimation.Itiswellknownthatthederivativesoftheuhfunctionissuperconvergentinsomeinteriorpointsofthemes