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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

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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

AmethodologyforadaptivefiniteelementanalysisTowardsanintegratedcomputationalenvironmentG.H.Paulino,I.F.M.Menezes,J.B.CavalcanteNeto,L.F.MarthaAbstractThisworkintroducesamethodologyforselfadaptivenumericalprocedures,whichreliesonthevariouscomponentsofanintegrated,objectoriented,computationalenvironmentinvolvingpre,analysis,andpostprocessingmodules.Abasicplatformfornumericalexperimentsandfurtherdevelopmentisprovided,whichallowsimplementationofnewelements/errorestimatorsandsensitivityanalysis.AgeneralimplementationoftheSuperconvergentPatchRecoverySPRandtherecentlyproposedRecoverybyEquilibriuminPatchesREPispresented.BothSPRandREParecomparedandusedforerrorestimationandforguidingtheadaptiveremeshingprocess.Moreover,theSPRisextendedforcalculatingsensitivityquantitiesoffirstandhigherorders.ThemeshregenerationprocessisaccomplishedbymeansofmodernmethodscombiningquadtreeandDelaunaytriangulationtechniques.Surfacemeshgenerationinarbitrarydomainsisperformedautomaticallyi.e.withnouserinterventionduringtheselfadaptiveanalysisusingeitherquadrilateralortriangularelements.TheseideasareimplementedintheFiniteElementSystemTechnologyinAdaptivityFESTAsoftware.TheeffectivenessandversatilityofFESTAaredemonstratedbyrepresentativenumericalexamplesillustratingtheinterconnectionsamongfiniteelementanalysis,recoveryprocedures,errorestimation/adaptivityandautomaticmeshgeneration.Keywordsfiniteelementanalysis,errorestimation,adaptivity,hrefinement,sensitivity,superconvergentpatchrecoverySPR,recoverybyequilibriuminpatchesREP,objectorientedprogrammingOOP,interactivecomputergraphics.1IntroductionThisworkpresentsanintegratedobjectorientedcomputationalenvironmentforselfadaptiveanalysesofgenerictwodimensional2Dproblems.Thisenvironmentincludesanalysisprocedurestoinsureagivenlevelofaccuracyaccordingtocertaincriteria,andalsotheprocedurestogenerateandmodifythefiniteelementdiscretization.Thiscomputationalsystem,calledFESTAFiniteElementSystemTechnologyinAdaptivity,involvesfivemaincomponentsseeshadedboxesinFigure1Agraphicalpreprocessor,fordefiningthegeometryoftheproblem,theinitialfiniteelementmeshtogetherwithboundaryconditions,andthemainparametersusedinaselfadaptiveanalysis.Herethegeometricalmodelisdissociatedfromthefiniteelementmodel.Afiniteelementmoduleforsolvingthecurrentboundaryvalueproblem.Thecodehasbeendevelopedsothatitishighlymodular,expandable,anduserfriendly.Thus,itcanbeproperlymaintainedandcontinued.Moreover,otherusers/developersshouldbeabletomodifythebasicprogrammingsystemtofittheirspecificapplications.Anerrorestimationandsensitivitymodule.Discretizationerrorsareestimatedaccordingtoavailablerecoveryprocedures,e.g.ZienkiewiczandZhuZZ,superconvergentpatchrecoverySPRandrecoverybyequilibriuminpatchesREP.Sensitivitiesofvariousorders1st.,2nd.orhigherarecalculatedbymeansofaprocedureanalogoustotheSPR.Theuserchoosesthedesirederrorestimatorandsensitivityorder.Ameshregenerationratherthanmeshenrichmentprocedure,basedonthecombinationofquadtreeandDelaunaytriangulationtechniques.Accordingtothemagnitudeoftheerror,calculatedinthepreviousmodule,anewfiniteelementmeshisautomaticallygeneratedi.e.withnouserintervention,usingeithertriangularorquadrilateralelementshrefinement.ComputationalMechanics231999361–388SpringerVerlag1999361G.H.PaulinoDepartmentofCivilandEnvironmentalEngineering,UniversityofIllinoisaturbanachampaign2209NewmarkLaboratory,205NorthMathewsAvenue,Urbana,IL618012352,U.S.A.I.F.M.Menezes,J.B.CavalcanteNeto,L.F.MarthaTeCGrafComputerGraphicsTechnologyGroup,PUCRio,RiodeJaneiro,R.J.,22453900,BrazilJ.B.CavalcanteNeto,L.F.MarthaDepartmentofCivilEngineering,PUCRio,RiodeJaneiro,R.J.,22453900,BrazilCorrespondencetoG.H.PaulinoG.H.PaulinoacknowledgesthesupportfromtheUnitedStatesNationalScienceFoundationNSFunderGrantNo.CMS9713798.I.F.M.MenezesacknowledgesthefinancialsupportprovidedbytheFAPERJ,whichisaBrazilianagencyforresearchanddevelopmentinthestateofRiodeJaneiro.G.H.PaulinoandI.F.M.MenezesalsoacknowledgetheDepartmentofCivilandEnvironmentalEngineeringatUCDavisforhospitalitywhilepartofthisworkwasperformed.J.B.CavalcanteNetoandL.F.MarthaacknowledgethefinancialsupportprovidedbytheBrazilianagencyCNPq.Theauthorsalsothankananonymousreviewerforprovidingrelevantsuggestionstothiswork.Finally,apostprocessormodule,wherealltheanalysisresultse.g.deformedshape,sensitivityandstresscontourscanbevisualized.Essentially,FESTAisacomputationallaboratorywhichoffersabasicplatformfornumericalanalysisandfurtherdevelopment,e.g.implementationofnewerrorestimators,elements,ormaterialmodelsCavalcanteNetoetal.1998.Objectorientedprogrammingandintegrationofpre,analysis,andpostprocessingmodulesmakeFESTAasoftwarewellsuitedforbothpracticalengineeringapplicationsandfurtherresearchdevelopment.Theremainderofthispaperisorganizedasfollows.AmotivationtotheworkandabriefliteraturereviewareprovidedinSect.2.Afterwards,Sect.3presentssometheoreticalbackgroundonselfadaptivesimulationsandanoverviewofthegraphicalinterfaceusedintheFESTAsoftware.Section4introducesthemathematicalformulationoftheSPRusingweightedleastsquaresystems,theREP,andthesensitivitymethod.AdiscussionabouttheautomaticmeshgenerationtechniquesusedinthisworkisgiveninSect.5.RelevantinformationregardingtheimplementationofFESTAispresentedinSect.6,especiallyaspectsrelatedtotheSPRandREPrecoveries.Inordertoassesstheeffectivenessoftheproposedcomputationalsystem,representativenumericalexamplesaregiveninSect.7.Finally,inSect.8,conclusionsareinferredanddirectionsforfutureresearcharediscussed.2MotivationandrelatedworkNormalpracticetosolveengineeringproblemsbymeansoftheFiniteElementMethodFEMortheBoundaryElementMethodBEMinvolvesincreasingthenumberofdiscretizationpointsinthecomputationaldomainandresolvingtheresultingsystemofequationstoexaminetherelativechangeinthenumericalsolution.Ingeneral,thisprocedureistimeconsuming,itdependsontheexperienceoftheanalyst,anditcanbemisleadingifthesolutionhasnotenteredanasymptoticrange.Ideally,witharobustandreliableselfadaptivescheme,onewouldbeabletospecifyaninitialdiscretemodelwhichissufficienttodescribethegeometry/topologyofthedomainandtheboundaryconditionsBCs,andtospecifyadesirederrortolerance,accordingtoanappropriatecriterion.Then,thesystemwouldautomaticallyrefinethemodeluntiltheerrormeasurefallsbelowtheprescribedtolerance.Theprocessshouldbefullyautomaticandwithoutanyuserintervention.ThisisthemaingoalwhichmotivatedthedevelopmentofFESTA.Thisapproachincreasestheoverallreliabilityoftheanalysisproceduresinceitdoesnotdependontheexperience,orinexperience,oftheanalyst.TheneedfordevelopingbetterpreprocessingtechniquesfortheFEM,forperformingautomatedanalysis,andforobtainingselfadaptivesolutionswhichisbecomingatrendforcommercialFEMsoftwarehavedriventhedevelopmentofautomaticmeshgenerationalgorithms,i.e.algorithmswhicharecapableofdiscretizinganarbitrarygeometryintoaconsistentfiniteelementmeshwithoutanyuserintervention.Severalalgorithmsfor2Dgeometrieshavebeendevelopede.g.Baehmannetal.1987BlackerandStephenson1991Zhuetal.1991Potyondyetal.1995bBorouchakiandFrey1998,andapproachesforthreedimensional3Dgeometrieshaveappearedmorerecentlye.g.Cassetal.1996EscobarandMontenegro1996Bealletal.1997Lo1998.Thepresentworkfocusonautomatic2Dmeshgenerationinconnectionwithadaptivesolutions.Efficienttechniquesforgeneratingallquadrilateralandalltriangularmeshesareconsideredindetail.Althoughthealgorithmspresentedhereincouldbeextendedtomixedmeshes,i.e.mesheswithbothtriangularandquadrilateralelementssee,forexample,BorouchakiandFrey1998,thistopicisnotwithinthescopeofthiswork.Thereexistavastliteratureonerrorestimationandadaptivity,andthereaderisdirectedtotheappropriatereferences1.ThevolumeseditedbyBrebbiaandAliabadi1993andBabusˇkaetal.1986reviewadaptivetechniquesfortheFEMandtheBEM.ThebookeditedbyLadevezeandOden1998presentsacompilationofpapersfromtheworkshopofNewAdvancesinAdaptiveComputationalMechanics,heldatCachan,France,17–19September1997,whichdealtwiththelatestadvancesinadaptivemethodsinmechanicsandtheirimpactonsolvingengineeringproblems.Severalissuesofjournalshavealsobeendedicatedtoadaptivity,e.g.volume121996,number2ofEngineeringwithComputers,volume151992,numbers3/4ofAdvancesinEngineeringSoftware,andvolume361991,number1oftheJournalofComputationalandAppliedMathematics.SurveysoftheliteratureinFEMincludearticlesbyNoorandBabusˇka1987,OdenandDemkovicz1989,StrouboulisandHaque1992a,b,BabusˇkaandSuri1994,andAinsworthandOden1997.Mackerle1993,1994hascompiledalonglistofreferencesonmeshgeneration,refinement,erroranalysisandadaptivetechniquesforFEMandBEMthatwerepublishedfrom1990to1993.TheZZ,SPR,REP,...VisualizationPostprocessorFinalDiscretizationMeshRegenerationGraphicalPreprocessorGeometry,Topology,BCsFiniteElementSolverConvergenceFESTAITERATIVEMESHDESIGNCYCLENYErrorEstimatorFig.1.SimplifieddiagramoftheFESTAinteractivemeshing1Thelistofpapersreferredhereisjustasmallsamplingoftheliterature,consideringarticlesofparticularinteresttothepresentwork,andisnotintendedtobearepresentativesurveyoftheliteratureinthefield.362volumeeditedbyBabusˇkaetal.1983presentsadaptivetechniquesfortheFEMandtheFiniteDifferenceMethodFDM.RelativelyrecenttextbooksintheFEMemphasizethefieldofadaptivesolutiontechniques.Forexample,thebookbyZienkiewiczandTaylor1989includesaChapteronErrorEstimationandAdaptivityChapter14,whichissupplementedbythepapersbyZienkiewiczandZhu1992a,b,1994.Moreover,thebookbySzabo´andBabusˇka1991isprimarilydedicatedtothissubject.Thefirstpapersonadaptivefiniteelementsappearedintheearlyseventies.Sincethen,anexplosivenumberofpapersonthesubjecthavebeenpublishedinthetechnicalliterature.BabusˇkaandRheinboldt1978presentedapioneeringpaperabouterrorestimatesbyevaluatingtheresidualsoftheapproximatesolutionandusingthemtoobtainlocal,moreaccurateanswers.Theydevelopedthemathematicalbasisofselfadaptivetechniques.Withtheconceptofaposteriorierrorestimates,onecandevelopaselfadaptivestrategyfortheFEMsuchthatonlycertainelementsshouldberefined.Zienkiewiczetal.1982presentedahierarchicalapproachforselfadaptivemethods.Intheearly1980s,computergraphicstechniquesstartedtobeusedasstandardtoolsbymeshgenerationprograms.Shephard1986publishedapaperwheregeometricmodelingandautomaticmeshgenerationtechniqueswereusedinconjunctionwithselfadaptivemethods.ZienkiewiczandZhu1987introducedanerrorestimatorbasedonobtainingimprovedvaluesofgradientsstressesusingsomeavailablerecoveryprocesses.Easytobeimplementedinanyfiniteelementcode,thistypeoftechnique,basedonaveragingandonthesocalledL2projection,hasbeenusedtorecoverthegradients,andreasonableestimatorswereachieved.In1992,thistechniquewascorrected/improvedbythesameauthors,leadingtothesocalledSuperconvergentPatchRecoverySPRZienkiewiczandZhu1992a,b,1994.Thismethodisastresssmoothingprocedureoverlocalpatchesofelementsandisbasedonadiscreteleastsquaresfitofahigherorderlocalpolynomialstressfieldtothestressesatthesuperconvergentsamplingpointsobtainedfromthefiniteelementcalculation.Attemptstoimprovefurthertherecoveryprocesscanbefoundinvariousreferences,e.g.Wibergetal.1994,WibergandAbdulwahab1993,BlackerandBelytschko1994,Tabbaraetal.1994,andLeeetal.1997.Essentiallytheseimprovedtechniquesincorporateequilibriumandboundaryconditionsontherecoveryprocess.AnexhaustivestudybyBabusˇkaetal.1994a,bshowed,throughnumerousexamples,theexcellentperformanceandsuperiorityoftheSPRoverresidualtypeapproaches.Recently,BoroomandandZienkiewicz1997havepresentedanewsuperconvergentmethodsatisfyingtheequilibriumconditioninaweakform,whichdoesnotrequireanyknowledgeofsuperconvergentpoints.ThenewrecoverytechniquehasbeencalledRecoverybyEquilibriuminPatchesREP.BothSPRandREPareofparticularinteresttothepresentwork.Asindicatedabove,thegeneralfieldofadaptivityisbroadandhasadvancedsignificantlyinrecentyears.Forinstance,Paulinoetal.1997haveproposedanewclassoferrorestimatesbasedontheconceptofnodalsensitivities,whichcanbeusedinconjunctionwithgeneralpurposecomputationalmethodssuchasFEM,BEMorFDM.RannacherandSuttmeier1997havesuggestedafeedbackapproachforerrorcontrolintheFEM.MahomedandKekana1998havepresentedanadaptiveprocedurebasedonstrainenergyequalisation.Moreover,asummaryofrecentadvancesinadaptivecomputationalmechanicscanbefoundinthebookeditedbyLadevezeandOden1998.Quantificationofthequalityofamodelwithrespecttoanotherone,takenasthereference,isofprimaryimportanceinnumericalanalysis.Thisisthecasewithwellestablishedmethods,suchastheFEM,oremergingmethods,suchastheelementfreeGalerkinEFGChungandBelytschko1998,thesymmetricGalerkinBEMPaulinoandGray1999,ortheboundarynodemethodMukherjeeandMukherjee1997.IntegrationofconceptsregardingerrorestimationandadaptivityintheFEM,withinamoderncomputationalenvironment,isthefocusofthepresentwork.3TheoreticalandcomputationalaspectsWheneveranumericalmethodisusedtosolvethegoverningdifferentialequationsofaboundaryvalueproblem,errorisintroducedbythediscretizationprocesswhichreducesthecontinuousmathematicalmodeltoonehavingafinitenumberofdegreesoffreedom.Thediscretizationerrorsaredefinedasthedifferencebetweentheactualsolutionanditsnumericalapproximation.Bydefinition,thelocalerrore//1isameasureofthedifferencebetweentheexact/andanapproximatesolution/.Here,/isanalogoustoaresponsequantitye.g.displacementsinatypicalnumericalsolutionprocedure.Selfadaptivemethodsarenumericalschemeswhichautomaticallyadjustthemselvesinordertoimprovethesolutionoftheproblemtoaspecifiedaccuracy.Thetwobasiccomponentsinadaptivemethodsareerrorestimationandadaptivestrategy.Thesecomponentsarediscussedbelow.Ingeneral,therearetwotypesofdiscretizationerrorestimatesaprioriandaposteriori.Althoughaprioriestimatesareaccuratefortheworstcaseinaparticularclassofsolutionsofaproblem,theyusuallydonotprovideinformationabouttheactualerrorforagivenmodel.Aposterioriestimatesuseinformationobtainedduringthesolutionprocess,inadditiontosomeaprioriassumptionsaboutthesolution.Aposterioriestimates,whichcanprovidequantitativelyaccuratemeasuresofthediscretizationerror,havebeenadoptedhere.Inthecontextofadaptivestrategies,extensionmethodshavebeenpreferredoverothersapproachese.g.dualorcomplementarymethodsandarethefocusofthiswork.Thesemethodsincludeh,p,andrextensions.Thecomputerimplementationisreferredtoastheh,p,andrversions,respectively.Inthehextension,themeshisautomaticallyrefinedwhenthelocalerrorindicatorex363ceedsapreassignedtolerance.Thepextensiongenerallyemploysafixedmesh.Iftheerrorinanelementexceedsapreassignedtolerance,thelocalorderoftheapproximationisincreasedtoreducetheerror.Therextensionnoderedistributionemploysafixednumberofnodesandattemptstodynamicallymovethegridpointstoareasofhigherrorinthemesh.Anyoftheseextensionscanalsobecombinedinaspecialstrategy,forexample,hp,rh,amongothers.3.1ErrorestimationandadaptiverefinementAspointedoutbyseveralauthorse.g.ZienkiewiczandTaylor1989,thespecificationoflocalerrorinthemannergiveninEq.1isgenerallynotconvenientandoccasionallymisleading.Thusmathematicalnormsareintroducedtomeasurethediscretizationerror.Theexactdiscretizationerrorinthefiniteelementsolutionisoftenquantifiedonthebasisoftheenergynormforthedisplacementerror,jjejj,whichcanbeexpressed,intermsofstresses,asjjejj2ZXrexrTD1rexrdX2whererexandraretheexactandthefiniteelementstressfields,Distheconstitutivematrix,andXistheproblemdefinitiondomain.Thebasicideaoferrorestimatorsistosubstitutethefieldrex,whichisgenerallyunknown,bythefieldr,obtainedbymeansofrecoveryprocedurese.g.ZZ,SPRorREP.Therefore,theexpressionforcomputingtheapproximateestimatedrelativeerrordistributionjjejjescanbeexpressedasjjejj2esZXrrTD1rrdX3Takingintoaccountthefiniteelementdiscretizationandconsideringaspecificfiniteelementi,Eq.3canberewrittenasjjejjies2ZXirrTD1rrjJjdXi4wherestandardisoparametricelementshavebeenassumedrdenotestherecoveredstressfield,jJjisthedeterminantoftheJacobiantransformationmatrix,andXiistheelementdomain.Theenergynormfortheerrorcanbeevaluatedoverthewholedomainorpartofit.Thecontributionofalltheelementsinthemeshisgivenbyjjejj2Xmi1jjejji25wheremisthetotalnumberofelements,ireferstotheelementwithdomainXi,andmi1XiX.TherelativepercentageerrorintheenergynormgexforthewholedomainorpartofitcanbeobtainedasgexjjejjexjjUjjex6wherejjUjjexisthesquarerootoftwicethestrainenergy,anditisgivenbyjjUjjexZXrTD1rdX127Theadaptiverefinementstrategyhextensionisdiscussednext.Theerrorestimatorwilldefinehowthediscretizationmodelwillberefinedorcoarsened.Asimplecriteriontoachieveasolutionerrorwithanacceptablelevelforthewholedomaincanbestatedasgesgmax8wheregmaxisthemaximumpermissibleerror,andgesisgivenbygesjjejjesjjUhjj2jjejj2es129wherejjUhjjistheenergynormobtainedfromthefiniteelementsolution.Asoundcriterionforanoptimalmeshconsistsofrequiringthattheenergynormerrorbeequidistributedamongelementsbecauseitleadstomesheswithhighconvergencerates.Thus,foreachelementi,jjejjies1012Amoreefficientprocedure,whichisadoptedhere,consistsofdesigningacompletelynewmeshregenerationwhichsatisfiestherequirementni1013inthelimitofmeshrefinement.ByassumingacertainrateofconvergenceZienkiewiczandTaylor1989,thevalueoftheerrorrationicanbeusedtodecidethenewsizeoftheelement.Thushhin1pi14wherehiistheinitialsizeoftheelement,histhefinalsize,andpisthepolynomialorderoftheapproximation.Thisproceduredoesnotaccountforsingularityeffects,suchasthosegeneratedbythesharpcornersandcracks.Atreatmentofsingularbehavior,withvaryingdegreeofsophistication,canbefoundinZienkiewiczandTaylor1989,Szabo´andBabusˇka1991,Babusˇkaetal.1994,andCoorevitsetal.1994.TheratioerrorEq.11isimplementedinthenumericalanalysismodule,anditisexportedtotheselfadaptivemoduleofFESTA,wherenewelementsizesarecalculatedtosatisfytheerrorcriterion.Thisisdone364

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