外文翻译---一种适应的有限元分析方法面向一个整体计算环境 英文.pdf

Amethodologyforadaptivefiniteelementanalysis:TowardsanintegratedcomputationalenvironmentG.H.Paulino,I.F.M.Menezes,J.B.CavalcanteNeto,L.F.MarthaAbstractThisworkintroducesamethodologyforself-adaptivenumericalprocedures,whichreliesonthevariouscomponentsofanintegrated,object-oriented,computa-tionalenvironmentinvolvingpre-,analysis,andpost-processingmodules.Abasicplatformfornumericalexperimentsandfurtherdevelopmentisprovided,whichallowsimplementationofnewelements/errorestimatorsandsensitivityanalysis.AgeneralimplementationoftheSuperconvergentPatchRecovery(SPR)andtherecentlyproposedRecoverybyEquilibriuminPatches(REP)ispresented.BothSPRandREParecomparedandusedforerrorestimationandforguidingtheadaptiveremeshingprocess.Moreover,theSPRisextendedforcalculatingsensitivityquantitiesoffirstandhigherorders.Themesh(re-)generationprocessisaccomplishedbymeansofmodernmethodscombiningquadtreeandDelaunaytri-angulationtechniques.Surfacemeshgenerationinarbi-trarydomainsisperformedautomatically(i.e.withnouserintervention)duringtheself-adaptiveanalysisusingeitherquadrilateralortriangularelements.TheseideasareimplementedintheFiniteElementSystemTechnologyinAdaptivity(FESTA)software.Theeffectivenessandver-satilityofFESTAaredemonstratedbyrepresentativenu-mericalexamplesillustratingtheinterconnectionsamongfiniteelementanalysis,recoveryprocedures,erroresti-mation/adaptivityandautomaticmeshgeneration.Keywordsfiniteelementanalysis,errorestimation,ad-aptivity,h-refinement,sensitivity,superconvergentpatchrecovery(SPR),recoverybyequilibriuminpatches(REP),objectorientedprogramming(OOP),interactivecomputergraphics.1IntroductionThisworkpresentsanintegrated(object-oriented)com-putationalenvironmentforself-adaptiveanalysesofge-nerictwo-dimensional(2D)problems.Thisenvironmentincludesanalysisprocedurestoinsureagivenlevelofac-curacyaccordingtocertaincriteria,andalsotheproce-durestogenerateandmodifythefiniteelementdiscretization.Thiscomputationalsystem,calledFESTA(FiniteElementSystemTechnologyinAdaptivity),involvesfivemaincomponents(seeshadedboxesinFigure1):Agraphicalpreprocessor,fordefiningthegeometryoftheproblem,theinitialfiniteelementmesh(togetherwithboundaryconditions),andthemainparametersusedinaself-adaptiveanalysis.Herethegeometricalmodelisdissociatedfromthefiniteelementmodel.Afiniteelementmoduleforsolvingthecurrentboun-daryvalueproblem.Thecodehasbeendevelopedsothatitishighlymodular,expandable,anduser-friendly.Thus,itcanbeproperlymaintainedandcontinued.Moreover,otherusers/developersshouldbeabletomodifythebasicprogrammingsystemtofittheirspe-cificapplications.Anerrorestimationandsensitivitymodule.Discreti-zationerrorsareestimatedaccordingtoavailablere-coveryprocedures,e.g.ZienkiewiczandZhu(ZZ),superconvergentpatchrecovery(SPR)andrecoverybyequilibriuminpatches(REP).Sensitivitiesofvariousorders(1st.,2nd.orhigher)arecalculatedbymeansofaprocedureanalogoustotheSPR.Theuserchoosesthedesirederrorestimatorandsensitivityorder.Amesh(re-)generation(ratherthanmeshenrichment)procedure,basedonthecombinationofquadtreeandDelaunaytriangulationtechniques.Accordingtothemagnitudeoftheerror,calculatedinthepreviousmodule,anewfiniteelementmeshisautomaticallygenerated(i.e.withnouserintervention),usingeithertriangularorquadrilateralelements(h-refinement).ComputationalMechanics23(1999)361388Springer-Verlag1999361G.H.PaulinoDepartmentofCivilandEnvironmentalEngineering,UniversityofIllinoisaturbana-champaign2209NewmarkLaboratory,205NorthMathewsAvenue,Urbana,IL61801-2352,U.S.A.I.F.M.Menezes,J.B.CavalcanteNeto,L.F.MarthaTeCGraf(ComputerGraphicsTechnologyGroup),PUC-Rio,RiodeJaneiro,R.J.,22453-900,BrazilJ.B.CavalcanteNeto,L.F.MarthaDepartmentofCivilEngineering,PUC-Rio,RiodeJaneiro,R.J.,22453-900,BrazilCorrespondenceto:G.H.PaulinoG.H.PaulinoacknowledgesthesupportfromtheUnitedStatesNationalScienceFoundation(NSF)underGrantNo.CMS-9713798.I.F.M.MenezesacknowledgesthefinancialsupportprovidedbytheFAPERJ,whichisaBrazilianagencyforresearchanddevelopmentinthestateofRiodeJaneiro.G.H.PaulinoandI.F.M.MenezesalsoacknowledgetheDepartmentofCivilandEnvironmentalEngineeringatUC-Davisforhospitalitywhilepartofthisworkwasperformed.J.B.CavalcanteNetoandL.F.MarthaacknowledgethefinancialsupportprovidedbytheBrazilianagencyCNPq.Theauthorsalsothankananonymousreviewerforprovidingrelevantsuggestionstothiswork.Finally,apostprocessormodule,wherealltheanalysisresults(e.g.deformedshape,sensitivityandstresscontours)canbevisualized.Essentially,FESTAisacomputationallaboratorywhichoffersabasicplatformfornumericalanalysisandfurtherdevelopment,e.g.implementationofnewerrorestimators,elements,ormaterialmodels(CavalcanteNetoetal.1998).Object-orientedprogrammingandintegrationofpre-,analysis,andpost-processingmodulesmakeFESTAasoftwarewell-suitedforbothpracticalengineeringappli-cationsandfurtherresearchdevelopment.Theremainderofthispaperisorganizedasfollows.AmotivationtotheworkandabriefliteraturereviewareprovidedinSect.2.Afterwards,Sect.3presentssometheoreticalbackgroundonself-adaptivesimulationsandanoverviewofthegraphicalinterfaceusedintheFESTAsoftware.Section4introducesthemathematicalformula-tionoftheSPR(usingweightedleastsquaresystems),theREP,andthesensitivitymethod.AdiscussionabouttheautomaticmeshgenerationtechniquesusedinthisworkisgiveninSect.5.Relevantinformationregardingtheim-plementationofFESTAispresentedinSect.6,especiallyaspectsrelatedtotheSPRandREPrecoveries.Inordertoassesstheeffectivenessoftheproposedcomputationalsystem,representativenumericalexamplesaregiveninSect.7.Finally,inSect.8,conclusionsareinferredanddirectionsforfutureresearcharediscussed.2MotivationandrelatedworkNormalpracticetosolveengineeringproblemsbymeansoftheFiniteElementMethod(FEM)ortheBoundaryElementMethod(BEM)involvesincreasingthenumberofdiscretizationpointsinthecomputationaldomainandresolvingtheresultingsystemofequationstoexaminetherelativechangeinthenumericalsolution.Ingeneral,thisprocedureistimeconsuming,itdependsontheexperienceoftheanalyst,anditcanbemisleadingifthesolutionhasnotenteredanasymptoticrange.Ideally,witharobustandreliableself-adaptivescheme,onewouldbeabletospecifyaninitialdiscretemodelwhichissufficienttodescribethegeometry/topologyofthedomainandtheboundaryconditions(BCs),andtospecifyadesirederrortolerance,accordingtoanappro-priatecriterion.Then,thesystemwouldautomaticallyrefinethemodeluntiltheerrormeasurefallsbelowtheprescribedtolerance.Theprocessshouldbefullyauto-maticandwithoutanyuserintervention.ThisisthemaingoalwhichmotivatedthedevelopmentofFESTA.Thisapproachincreasestheoverallreliabilityoftheanalysisproceduresinceitdoesnotdependontheexperience,orinexperience,oftheanalyst.Theneedfordevelopingbetterpre-processingtech-niquesfortheFEM,forperformingautomatedanalysis,andforobtainingself-adaptivesolutions(whichisbe-comingatrendforcommercialFEMsoftware)havedriventhedevelopmentofautomaticmeshgenerationalgo-rithms,i.e.algorithmswhicharecapableofdiscretizinganarbitrarygeometryintoaconsistentfiniteelementmeshwithoutanyuserintervention.Severalalgorithmsfor2Dgeometrieshavebeendeveloped(e.g.Baehmannetal.1987；BlackerandStephenson1991；Zhuetal.1991；Potyondyetal.1995b；BorouchakiandFrey1998),andapproachesforthree-dimensional(3D)geometrieshaveappearedmorerecently(e.g.Cassetal.1996；EscobarandMontenegro1996；Bealletal.1997；Lo1998).Thepresentworkfocusonautomatic2Dmeshgenerationinconnec-tionwithadaptivesolutions.Efficienttechniquesforgen-eratingall-quadrilateralandall-triangularmeshesareconsideredindetail.Althoughthealgorithmspresentedhereincouldbeextendedtomixedmeshes,i.e.mesheswithbothtriangularandquadrilateralelements(see,forexample,BorouchakiandFrey1998),thistopicisnotwithinthescopeofthiswork.Thereexistavastliteratureonerrorestimationandadaptivity,andthereaderisdirectedtotheappropriatereferences1.ThevolumeseditedbyBrebbiaandAlia-badi(1993)andBabuskaetal.(1986)reviewadaptivetechniquesfortheFEMandtheBEM.ThebookeditedbyLadevezeandOden(1998)presentsacompilationofpa-persfromtheworkshopofNewAdvancesinAdaptiveComputationalMechanics,heldatCachan,France,1719September1997,whichdealtwiththelatestadvancesinadaptivemethodsinmechanicsandtheirimpactonsolvingengineeringproblems.Severalissuesofjournalshavealsobeendedicatedtoadaptivity,e.g.volume12(1996),number2ofEngineeringwithComputers,vol-ume15(1992),numbers3/4ofAdvancesinEngineeringSoftware,andvolume36(1991),number1oftheJournalofComputationalandAppliedMathematics.SurveysoftheliteratureinFEMincludearticlesbyNoorandBa-buska(1987),OdenandDemkovicz(1989),StrouboulisandHaque(1992a,b),BabuskaandSuri(1994),andAinsworthandOden(1997).Mackerle(1993,1994)hascompiledalonglistofreferencesonmeshgeneration,refinement,erroranalysisandadaptivetechniquesforFEMandBEMthatwerepublishedfrom1990to1993.The(ZZ,SPR,REP,.)Visualization(Postprocessor)FinalDiscretizationMeshRegenerationGraphicalPreprocessor(Geometry,Topology,BCs)FiniteElementSolverConver-gence?FESTAITERATIVEMESHDESIGNCYCLENYErrorEstimatorFig.1.SimplifieddiagramoftheFESTAinteractivemeshing1Thelistofpapersreferredhereisjustasmallsamplingoftheliterature,consideringarticlesofparticularinteresttothepresentwork,andisnotintendedtobearepresentativesurveyoftheliteratureinthefield.362volumeeditedbyBabuskaetal.(1983)presentsadaptivetechniquesfortheFEMandtheFiniteDifferenceMethod(FDM).RelativelyrecenttextbooksintheFEMemphasizethefieldofadaptivesolutiontechniques.Forexample,thebookbyZienkiewiczandTaylor(1989)includesaChapteronErrorEstimationandAdaptivity(Chapter14),whichissupplementedbythepapersbyZienkiewiczandZhu(1992a,b,1994).Moreover,thebookbySzabo´andBabuska(1991)isprimarilydedicatedtothissubject.Thefirstpapersonadaptivefiniteelementsappearedintheearlyseventies.Sincethen,anexplosivenumberofpapersonthesubjecthavebeenpublishedinthetechnicalliterature.BabuskaandRheinboldt(1978)presentedapioneeringpaperabouterrorestimatesbyevaluatingtheresidualsoftheapproximatesolutionandusingthemtoobtainlocal,moreaccurateanswers.Theydevelopedthemathematicalbasisofself-adaptivetech-niques.Withtheconceptofaposteriorierrorestimates,onecandevelopaself-adaptivestrategyfortheFEMsuchthatonlycertainelementsshouldberefined.Zienkiewiczetal.(1982)presentedahierarchicalap-proachforself-adaptivemethods.Intheearly1980s,computergraphicstechniquesstartedtobeusedasstandardtoolsbymeshgenerationprograms.Shep-hard(1986)publishedapaperwheregeometricmodel-ingandautomaticmeshgenerationtechniqueswereusedinconjunctionwithself-adaptivemethods.Zien-kiewiczandZhu(1987)introducedanerrorestimatorbasedonobtainingimprovedvaluesofgradients(stresses)usingsomeavailablerecoveryprocesses.Easytobeimplementedinanyfiniteelementcode,thistypeoftechnique,basedonaveragingandonthesocalledL2projection,hasbeenusedtorecoverthegradients,andreasonableestimatorswereachieved.In1992,thistechniquewascorrected/improvedbythesameauthors,leadingtothesocalledSuperconvergentPatchRecovery:SPR(ZienkiewiczandZhu1992a,b,1994).Thismethodisastress-smoothingprocedureoverlocalpatchesofelementsandisbasedonadiscreteleast-squaresfitofahigher-orderlocalpolynomialstressfieldtothestressesatthesuperconvergentsamplingpointsobtainedfromthefiniteelementcalculation.Attemptstoimprovefur-thertherecoveryprocesscanbefoundinvariousref-erences,e.g.Wibergetal.(1994),WibergandAbdulwahab(1993),BlackerandBelytschko(1994),Tabbaraetal.(1994),andLeeetal.(1997).Essentiallytheseimprovedtechniquesincorporateequilibriumandboundaryconditionsontherecoveryprocess.AnexhaustivestudybyBabuskaetal.(1994a,b)showed,throughnumerousexamples,theexcellentperformanceandsuperiorityoftheSPRoverresidual-typeapproaches.Recently,BoroomandandZienkiewicz(1997)havepre-sentedanewsuper-convergentmethodsatisfyingtheequilibriumconditioninaweakform,whichdoesnotrequireanyknowledgeofsuperconvergentpoints.ThenewrecoverytechniquehasbeencalledRecoverybyEquilibriuminPatches:REP.BothSPRandREPareofparticularinteresttothepresentwork.Asindicatedabove,thegeneralfieldofadaptivityisbroadandhasadvancedsignificantlyinrecentyears.Forinstance,Paulinoetal.(1997)haveproposedanewclassoferrorestimatesbasedontheconceptofnodalsensi-tivities,whichcanbeusedinconjunctionwithgeneralpurposecomputationalmethodssuchasFEM,BEMorFDM.RannacherandSuttmeier(1997)havesuggestedafeedbackapproachforerrorcontrolintheFEM.MahomedandKekana(1998)havepresentedanadaptiveprocedurebasedonstrainenergyequalisation.Moreover,asummaryofrecentadvancesinadaptivecomputationalmechanicscanbefoundinthebookeditedbyLadevezeandOden(1998).Quantificationofthequalityofamodelwithrespecttoanotherone,takenasthereference,isofprimaryimpor-tanceinnumericalanalysis.Thisisthecasewithwelles-tablishedmethods,suchastheFEM,oremergingmethods,suchastheelementfreeGalerkin:EFG(ChungandBel-ytschko1998),thesymmetricGalerkinBEM(PaulinoandGray1999),ortheboundarynodemethod(MukherjeeandMukherjee1997).IntegrationofconceptsregardingerrorestimationandadaptivityintheFEM,withinamoderncomputationalenvironment,isthefocusofthepresentwork.3TheoreticalandcomputationalaspectsWheneveranumericalmethodisusedtosolvethegov-erningdifferentialequationsofaboundaryvalueproblem,errorisintroducedbythediscretizationprocesswhichreducesthecontinuousmathematicalmodeltoonehavingafinitenumberofdegreesoffreedom.Thediscretizationerrorsaredefinedasthedifferencebetweentheactualsolutionanditsnumericalapproximation.Bydefinition,thelocalerrore/1isameasureofthedifferencebetweentheexact(/)andanapproximatesolution(/).Here,/isanalogoustoare-sponsequantity(e.g.displacements)inatypicalnumericalsolutionprocedure.Self-adaptivemethodsarenumericalschemeswhichautomaticallyadjustthemselvesinordertoimprovethesolutionoftheproblemtoaspecifiedaccuracy.Thetwobasiccomponentsinadaptivemethodsareerrorestima-tionandadaptivestrategy.Thesecomponentsarediscus-sedbelow.Ingeneral,therearetwotypesofdiscretizationerrorestimates:aprioriandaposteriori.Althoughapriories-timatesareaccuratefortheworstcaseinaparticularclassofsolutionsofaproblem,theyusuallydonotprovideinformationabouttheactualerrorforagivenmodel.Aposterioriestimatesuseinformationobtainedduringthesolutionprocess,inadditiontosomeaprioriassumptionsaboutthesolution.Aposterioriestimates,whichcanprovidequantitativelyaccuratemeasuresofthediscreti-zationerror,havebeenadoptedhere.Inthecontextofadaptivestrategies,extensionmethodshavebeenpreferredoverothersapproaches(e.g.dualorcomplementarymethods)andarethefocusofthiswork.Thesemethodsincludeh-,p-,andr-extensions.Thecomputerimplementationisreferredtoastheh-,p-,andr-versions,respectively.Intheh-extension,themeshisautomaticallyrefinedwhenthelocalerrorindicatorex-363ceedsapreassignedtolerance.Thep-extensiongenerallyemploysafixedmesh.Iftheerrorinanelementexceedsapreassignedtolerance,thelocalorderoftheapproxima-tionisincreasedtoreducetheerror.Ther-extension(node-redistribution)employsafixednumberofnodesandattemptstodynamicallymovethegridpointstoareasofhigherrorinthemesh.Anyoftheseextensionscanalsobecombinedinaspecialstrategy,forexample,h-p-,r-h-,amongothers.3.1ErrorestimationandadaptiverefinementAspointedoutbyseveralauthors(e.g.ZienkiewiczandTaylor1989),thespecificationoflocalerrorinthemannergiveninEq.(1)isgenerallynotconvenientandocca-sionallymisleading.Thusmathematicalnormsareintro-ducedtomeasurethediscretizationerror.Theexactdiscretizationerrorinthefiniteelementsolutionisoftenquantifiedonthebasisoftheenergynormforthedis-placementerror,jjejj,whichcanbeexpressed,intermsofstresses,asjjejj2ZXrexrTD1rexrdX2whererexandraretheexactandthefiniteelementstressfields,Distheconstitutivematrix,andXistheproblemdefinitiondomain.Thebasicideaoferrorestimatorsistosubstitutethefieldrex,whichisgenerallyunknown,bythefieldr,ob-tainedbymeansofrecoveryprocedures(e.g.ZZ,SPRorREP).Therefore,theexpressionforcomputingtheap-proximate(estimated)relativeerrordistributionjjejjescanbeexpressedasjjejj2esZXrrTD1rrdX3Takingintoaccountthefiniteelementdiscretizationandconsideringaspecificfiniteelementi,Eq.(3)canbere-writtenasjjejjies2ZXirrTD1rrjJjdXi4wherestandardisoparametricelementshavebeenas-sumed；rdenotestherecoveredstressfield,jJjisthede-terminantoftheJacobiantransformationmatrix,andXiistheelementdomain.Theenergynormfortheerrorcanbeevaluatedoverthewholedomainorpartofit.Thecontributionofalltheelementsinthemeshisgivenbyjjejj2Xmi1jjejji25wheremisthetotalnumberofelements,ireferstotheelementwithdomainXi,andmi1XiX.Therelativepercentageerrorintheenergynorm(gex)forthewholedomainorpartofitcanbeobtainedasgexjjejjexjjUjjex6wherejjUjjexisthesquarerootoftwicethestrainenergy,anditisgivenbyjjUjjexZXrTD1rdX1=27Theadaptiverefinementstrategy(h-extension)isdis-cussednext.Theerrorestimatorwilldefinehowthedis-cretizationmodelwillberefinedorcoarsened.Asimplecriteriontoachieveasolutionerrorwithanacceptablelevelforthewholedomaincanbestatedasgesgmax8wheregmaxisthemaximumpermissibleerror,andgesisgivenbygesjjejjesjjUhjj2jjejj2es1=29wherejjUhjjistheenergynormobtainedfromthefiniteelementsolution.Asoundcriterionforanoptimalmeshconsistsofrequiringthattheenergynormerrorbeequidistributedamongelementsbecauseitleadstomesheswithhighconvergencerates.Thus,foreachelementi,jjejjies<gmaxjjUhjj2jjejj2esm!1=2em10Bydefiningtherationijjejjiesem11itisobviousthatrefinementisneededifni>1:012Amoreefficientprocedure,whichisadoptedhere,con-sistsofdesigningacompletelynewmesh(re-generation)whichsatisfiestherequirementni1:013inthelimitofmeshrefinement.Byassumingacertainrateofconvergence(ZienkiewiczandTaylor1989),thevalueoftheerrorrationicanbeusedtodecidethenewsizeoftheelement.Thushhin1=pi14wherehiistheinitialsizeoftheelement,histhefinalsize,andpisthepolynomialorderoftheapproximation.Thisproceduredoesnotaccountforsingularityeffects,suchasthosegeneratedbythesharpcornersandcracks.Atreatmentofsingularbehavior,withvaryingdegreeofsophistication,canbefoundinZienkiewiczandTay-lor(1989),Szabo´andBabuska(1991),Babuskaetal.(1994),andCoorevitsetal.(1994).Theratioerror(Eq.(11)isimplementedinthenu-mericalanalysismodule,anditisexportedtotheself-adaptivemoduleofFESTA,wherenewelementsizesarecalculatedtosatisfytheerrorcriterion.Thisisdone364