A spectral stochastic element free Galerkin method

INTERNATIONALJOURNALFORNUMERICALMETHODSINENGINEERINGInt.J.Numer.Meth.Engng2004;61:1957–1975Publishedonline15October2004inWileyInterScience().DOI:10.1002/nme.1140AspectralstochasticelementfreeGalerkinmethodfortheproblemswithrandommaterialparameterH.M.KimandJ.Inoue∗,†DepartmentofCivilEngineering,theUniversityofTokyo,Tokyo113-8656,JapanSUMMARYThispaperpresentsaspectralstochasticelementfreeGalerkinmethod(SSEFGM)fortheproblemsinvolvingarandommaterialproperty.Therandommaterialpropertyandresultingsystemresponsequantityarerepresentedbyaprobabilisticspectralexpansiontechniques(Karhunen–LoeveexpansionandPolynomicalChaosseries,respectively)andimplementedintotheelementfreeGalerkin(EFG)analysis.Numericalsolutionsin1Dlinearelasticproblemwithrandomelasticmodulusareintroduced,andcomparedwiththoseofMonteCarlosimulation(MCS)soastoprovidethevalidationoftheproposedapproach.ThepresentSSEFGMapproachcanproduceaprobabilisticdensitydistributionaswellasafirst-andsecond-orderstatisticalmoments(meanandvariance)ofresponsequantitybyasinglecalculation,whichisdistinguishedfromaniterativeMCS.Moreover,themethodisbasedonanelementfreeanalysissothatthereisnoneedofnodalconnectivities,whichusuallyrequiremoretimeandlabourioustaskthanmaincalculations.ThustheproposedSSEFGMapproachcanprovideanalternativeanalysistoolfortheproblemscontainsastochasticmaterialproperty,anddemandscomplexmeshstructures.Copyright
2004JohnWiley&Sons,Ltd.KEYWORDS:spectralmethod;elementfreeGalerkinmethod(EFGM);stochastic;MonteCarlosimulation;probabilitydensitydistribution1.INTRODUCTIONNumericalmethodsfortheproblemswithuncertainstochasticvariablese.g.materialproperty,modelgeometryandexternalexcitations,havebeendevelopedduringpastdecades.Afiniteelementtechniqueisawell-knownmethodforgeneralengineeringproblemsandmoststochas-ticfiniteelementmethodsfallbasicallyintotwocategories;MonteCarlosimulation(MCS)andperturbation-basedtechnique.TheperturbationmethodcanbemoreeffectiveratherthanMCS,particularlywhenaneventofinterestrarelyoccursanditsprobabilityofoccurrenceisextremelysmallthatagreatnumberofiterationsarerequiredforaninvestigation.Meanwhile,∗Correspondenceto:J.Inoue,DepartmentofCivilEngineering,theUniversityofTokyo,7-3-1,Hongo,Bunkyo,Tokyo113-8656,Japan.†E-mail:[e-mail]/grantsponsor:UniversityofTokyoReceived2June2003Revised2April2004Copyright
2004JohnWiley&Sons,Ltd.Accepted21May20041958H.M.KIMANDJ.INOUEperturbationmethodisalsolessgeneralsinceitessentiallyassumessmallstochasticvariabilitysothatthevalidityisusuallyrestrictedtothecaseswhereastochasticprocessexhibitssmallfluctuationaroundameanvalue.Spectralstochasticfiniteelementmethod(SSFEM),whichhasbeenproposedbyGhanemandSpanos[1],hasprovidedanattractivealternativefortheproblemsinvolvingrandomfieldswithcomparativelylargevariation.InthespectraldiscretizationwithintheSSFEM,statisticalprop-ertiesofrandomvariablesareexpectedtolessdependonachosenfiniteelementmeshincom-parisonwiththoseinotherstochasticfiniteelementmethods[2].TworepresentativeexpansionmethodsemployedintheSSFEMareKarhunen–Loeve(KL)expansionandpolynomialchaos(PC),whichare,respectively,usedforknownrandommaterialparameterandresultingunknownresponses.TheapplicationoftheSSFEMhasbeenmadeinvariouskindsofproblems,forexample,elasticity,soilmechanics,transportinporousmedia,andmaterialnon-linearity[3–5].Meanwhile,finiteelement(FE)methodoftenhasadifficultyinsimulatingproblemsinwhichtheelementsinthemodelbecomeextremelydistortedorcompressedunderlargedeformation,andtheproblemsdemandingre-meshing,forexample,aroundthecracktipincrackpropagationoriterativefeedbackanalysisduetoup-datedinsitumeasurementingeomechanics.Inrecentyears,ithasbeenwidelyacceptedthatameshgenerationisafarmoretimeconsumingandex-pensivetaskthangeneralassemblyandsolutionoftheFEanalysisparticularlyinlinearanalysis.Fromtheselimitations,anelementfreeGalerkinmethod(EFGM),whichrequiresonlynodaldatabutnotheircomplexconnectivity,hasbeendeveloped[6].Althoughthereexistseveraldif-ferentapproachintheelementfreeanalysisaccordingtothewaysofdefiningshapefunctions,aGalerkinmethodisemployedinthisstudy.Theelementfreeanalysisinvolvingstochasticmaterialpropertyhasoncebeenreported[7]buthasbeenlimitedtoperturbationbasedone.Inthepresentwork,elementfreeanalysisbasedonGalerkinmethodisextendedandcoupledwithstochasticanalysisusingspectralrepresentationforarandommaterialproperty,whichistermedhereasspectralstochasticelementfreeGalerkinmethod(SSEFGM).ThecurrentSSEFGMisdistinguishedbyitsabilitytoprovidetheprobabilisticdensitydistributionaswellasthestatisticalmomentsofsystemresponsewithoutspecificinformationofnodalconnectivity(meshgeneration).BriefintroductionofEFGMandstochasticmethod(KLandPCexpansion)arepresented,andnumericalsolutionsof1DlinearelasticbarbytheproposedSSEFGMareintroduced,inthefollowing.2.ELEMENT-FREEGALERKINMETHOD2.1.Movingleast-squaresapproximationConsidertheunknownfunctionu(x)inthedomain,
,overwhichitisdefined.Movingleastsquare(MLS)interpolantuh(x)ofthefunctionu(x)isdefinedbyuh(x)=m
jpj(x)aj(X)≡pT(x