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1、COMMUNICATIONSINAPPLIEDNUMERICALMETHODS,Vol.1,3-9(1985)ITERATIVESOLUTIONOFMIXEDPROBLEMSANDTHESTRESSRECOVERYPROCEDURESO.C.ZIENKIEWICZUniversityofWales,Swansea,UKLIXI-KUIDalianInstituteofTechnologyS.NAKAZAWAMarcAnalysisResearchCorporationSUMMARYThisnoteintro
2、ducescertainstresssmoothingproceduresandthesocalled‘Loubignac-Cantin’iterationforrestorationofmomentumbalanceinthesmoothedstressfields.Itisshownthatthisiterationcorrespondspreciselytothesolutionofmixedformulationsinwhichthestresses(orstrains)andthedisplace
3、mentsareusedasprimaryvariables.Theprocedurehasaverywidefieldofapplicationandpromisestoaddconsiderableaccuracytof.e.m.resultsbyasmalladditionalexpenditure.INTRODUCTIONDisplacementformulationsofstressanalysisproblems(orequivalentflowproblems)result,asiswellk
4、nown,indiscontinuousandinaccuratestressfields.Varioustechniquesofstress‘a(chǎn)veraging’and‘smoothing’havebeenusedsincetheadventoffiniteelementmethods,themosteffectiveofthesebeingbasedona‘projection’techniq~el-~.Intheseweproceedasfollows:Firstsolve(usingthenotat
5、onofReference4)theequilibriumstatementk=Ku=fSecond,computestressesu=DBu(2)Third,assumeacontinuous(oratleastamorereasonable)stressfieldU*definedbyshapefunctionN*andnodalparametersU*Fourth,computeparametersO*sothatequalitybetweenUandU*issatisfiedinthemeansen
6、seiN*T(U-U*)dR=0(4)nThisyieldsimmediatelyonsubstitutionofequation(3)U*=M-’IN*TUdR=M-’(h*TDBdR)UR12074&8025/85/010003-O7$01.00ReceivedAugust198401985byJohnWiley&Sons,Ltd.40.c.ZIENKIEWICZ,LIXI-KUIands.NAKAZAWAwhereThecomputationof(5)isdoneseparatelyforeachco
7、mponentofstressandifM(whichhasthesamestructureasaclassical'mass'matrix)islumped4thecostofthisoperationistrivial.GoodresultsareobtainablewithbothlumpedandconsistentformsofM',buttheconsistentform,ingeneral,givesbetterresults.Figure1showssuchsmoothingappliedt
8、otwoelementsfordifferentdistributionsofU.Itisimmediatelyevidentthattheconsistentformgivesmorerealistic(statisticallyequivalent)valueswhereasthelumpedformappliedtoasmoothsolutionsimplyreducesitsvalues.'Nodala
