83_The hierarchical concept in finite element analysis

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1、Coraputt'rs&Structure~Voi16.NuI--4.pp.53--65,19830045-7949!83{010053--13503.00/0PrintedinGreatBritainPergamonPressLtdTHEHIERARCHICALCONCEPTINFINITEELEMENTANALYSISO.C.ZIENKIEWlCZ,J,P.DES.R.GAGOandD.W.KELLYDepartmentofCivilEngineering,UniversityCollegeofSwa

2、nsea,SingletonPark,SwanseaSA28PP,WalesAbstract--Thehierarchicalconceptforfiniteelementshapefunctionswasintroducedmanyyearsagoasaconvenientdeviceformixedorderinterpolation.Itsfulladvantageshavenotbeenrealizeduntilamuchlatertime--andtheseincludeinaddition(a

3、)improvedconditioning;(b)easeofintroducingerrorindicatorsifsuccessiverefinementissought.Further,itispossibletousetheideastoconstructarangeoferrorestimatorswhichcomparewellwithalternativesandareideallysuitedforadaptiverefinementsofanalysis.Asthehierarchica

4、lelementsareequallysimpletoimplementas"standard"fixedorderelementsitisfeltthatmoreprogramswill.inthefurore,turntoutilizetheiradvantages.Thisisespeciallytrueinthefieldofnonlinearanalysiswhereeventodaycomputationaleconomiesarenecessary.INTRODUCTIONFourierse

5、riesexpansion.HereasnewtermsarecalledDespitethewideusageoffiniteelementstodaylittletheprevioustermsremainunalteredandasingletermisattentionhasbeengiventotheproblemofestimatingintroducedatatime,therefinementceasingwhentheerrorsoccurringintheanalysisandtoco

6、rrectingtheseaccuracyofsolutionisdeemedsufficient.errorsbyautomaticrefinement.TheseaspectsoftheWhythedifferencebetweenthetwoapproacheswhenproblemsareofobviousimportanceifintelligentuseofweareallawarethatboththefiniteelementprocessprogramsistobemadeinengin

7、eeringpractice.ANDtheseriessolutionsaremathematicallypreciselyofIntypicalapplicationtodaytheexperiencedanalystdecidesonthemeshofelementstobeusedandgeneratestheinformationnecessaryautomatically[l-3].Indistributinghisnodesheisguidedby(1)Previoussolutionswhi

8、chgavesatisfactoryans-werswhentestedagainstbenchmarkproblems.(2)Guidelinesgeneratedbypreviousinvestigationsintomesh"optimality"towhichmuchattentionhasbeengiveninthepast[4--8].//IWhenpresentedwiththecomputeroutputaqu