Introduction to stochastic finite element methods _SFEM

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1、LiteraturereviewJianwenFENGPh.DcandidateatSwanseaUniversity,UnitedKingdomAdvisedbyDr.ChenfengLiFundedbyZienkiewiczScholarshipProgramBackgroundRandomnessisubiquitousMacro-scaleMicro-scaleMechanics,heat,dielectric,magnetic…StochasticfiniteelementUncertaintyfromconstitutiveUncertaintyfromr

2、elationship&geometryexternalforcesWhatisthestochasticoutput?OutlinePreliminaryknowledgeonstatisticsRepresentationofrandomfieldsStochasticequationsolveProbabilitytheoryOnerandomvariableXCumulativedistributionfunctionProbabilitydensityfunction(PDF)MomentsTworandomvariablesJointCu

3、mulativedistributionfunctionJointProbabilitydensityfunctionMarginalprobabilitydensityfunctionMomentsNrandomvariable:straightforwardTwospecificrelationsbetweentworandomvariables:UncorrelationandindependenceUncorrelation(whitenoise)IndependenceIndependenceismuchmorestrongerthanunco

4、rrelationIndependentUncorrelationRandomfields(Randomprocesses)Randomfields:afamilyofrandomvariablesTherelationamongthoserandomvariablesareverycomplexJointprobabilitydistributionfunctionofALLtheserandomvariablesisrequiredtouniquelydeterminetherandomfieldThejointprobabilitydistributio

5、nisavailableNEITHERanalyticallyNORnumericallyWeaklystationaryprocessStronglystationaryprocess:•PDFinvariantwithtranslationWeaklystationaryprocess:correlationfunctioninvariantwithtranslationCovXt1,Xt2Rt1t2RR:correlationfunctionWeaklystationaryprocessAdvantagesOnlyafe

6、wsamplesareneededtoobtainstatisticalinformation(ergodicity)DOFreducedsignificantlyCovoperatoriseasiertouseDisadvantagesTruncateddescriptionofarandomfieldOnlyGaussianrandomfieldscanbecompleteddepictedviafirstandsecondordermomentsGaussianrandomfieldsJointdistributionofarbitraryvaria

7、tesareMulti-Gaussian(normal)distributionMarginaldistributionisGaussian(Normaldistribution)Probabilisticpropertiesarecompletelydeterminedbyitsfirst-andsecond-ordermomentsForGaussianfields,uncorrelation=independenceConvenienttouse,butonlyallowrelativelyconcentratedfluctuation