A study of incremental-iterative strategies for no

INTERNATIONALJOURNALFORNUMERICALMETHODSINENGINEERING,VOL.29,1365-1391(1990)ASTUDYOFINCREMENTAL-ITERATIVESTRATEGIESFORNON-LINEARANALYSESMURRAYJ.CLARKE*ANDGREGORYJ.HANCOCK’SchoolofCivilandMiningEngineering,UnioersityofSydney,N.S.W.,Australia,2006SUMMARYThepaperdescribesastudyofincremental-iterativesolutiontechniquesforgeometricallynon-linearanalyses.ThesolutionmethodsdocumentedarebasedonamodifiedNewton-Raphsonapproach,meaningthatthetangentstiffnessmatrixiscomputedatthecommencementofeachloadstepbutisthenheldconstantthroughouttheequilibriumiterations.Aconsistentmathematicalnotationisemployedinthedescriptionoftheiterativeandloadincrementationstrategies,enablingthesimpleinclusionofseveralsolutionoptionsinacomputerprogram.Theiterativestrategiesinvestigatedareiterationatconstantload,iterationatconstantdisplacement,iterationatconstant‘arc-length’,iterationatconstantexternalwork,iterationatminimumunbalanceddisplacementnorm,iterationatminimumunbalancedforcenormanditerationatconstant‘weightedresponse’.Theloadincrementationschemesinvestigatedincludestrategiesbasedonthenumberofiterationsrequiredtoachieveconvergenceinthepreviousloadstep,strategiesbasedonthe‘currentstiffnessparameter’andastrategybasedonaparabolicapproximationtotheload4eflectionresponse.Criteriafordetectingwhentheappliedexternalloadincrementshouldreversesignaredescribed.Achallengingexampleofacirculararchexhibitingsnap-through(loadlimitpoint)behaviourandsnap-back(displacementlimitpoint)behaviourissolvedusingseveraldifferentiterativeandloadincrementationstrategies.Theperformanceofthesolutionschemesisevaluatedandconclusionsaredrawn.1.INTRODUCTIONInrecentyears,considerableefforthasbeendevotedtothedevelopmentofnumericaltechniquesforthesolutionofnon-linearstructuralproblems.Ideally,asolutionmethodshouldbeabletotracetheentirepre-andpost-criticalstaticloadpathofastructure,whichmayincludebothsofteningandstiffeningbehaviour,thepresenceofloadanddisplacementlimitpointsandthepossiblebifurcationofthepath.Predictionofthepost-criticalorpost-ultimateresponseisusefulforstudiesofimperfectionsensitivityandmayleadtoanimprovedappreciationoftheoverallstructuralresponse.Thepurposeofthispaperistoexaminevariousmethodsofsolvingnon-linearfiniteelementequationswhichallowlimitpointstobetraversed.Comparisonsoftheaccuracyandreliabilityofthevariousmethodsarealsogiven.Solutionmethodsfornon-linearproblemswhichhavebeenformulatedusingvariationalprinciplesinconjunctionwithadiscretizationtechniquesuchasthefiniteelementmethodmaybeclassifiedintothree‘levels’accordingtothemethodofmathematicalformulation.’.Thefirstlevelofformulationrepresentsaproblembyapotentialwhichdependsonthegeneralizedloadsanddisplacements.Asolutionisobtainedwhenthepotentialhasastationaryvalue.Themost*PostgraduateStudent‘AssociateProfessor0029-5981/90/071365-27S13.5001990byJohnWiley&Sons,Ltd.Received10March1989Revised30September19891366M.J.CLARKEANDG.J.HANCOCKwidelyused'firstlevel'formulationistheprincipleofstationarypotentialenergy.Thesecondlevelofformulationisobtainedbyexpressingdirectlytheconditionoftotalequilibrium,whichstatesthattheexternalandinternalforcesmustbalance.Itcanalsobeobtainedusingtheprincipleofvirtualworkorbysettingthefirstvariationoftheenergypotentialtozero.Thethirdlevelofformulationexpressestheconditionofincrementalequilibrium.Itcanbeobtainedbyemployingtheprincipleofvirtualworkinitsincrementalformorbysettingthesecondvariationoftheenergypotentialtozero.Theincrementalformulationiswidelyusedandhasbeenadoptedinthepresentwork.EarlymethodsofsolvingtheincrementalequilibriumequationsusedsimpleEulerintegra-ti~n.~However,thistechniquerequiresthatloadincrementsbekeptsmallsothatthesolutiondoesnotdeviatetoofarfromthesatisfactionofequilibrium.Largerloadstepscanbeaccom-modatedifthepurelyincrementalmethodiscombinedwithconventionalormodifiedNewton-Raphsoniterations3toenablethetotalequilibriumequationstobesatisfiedtowithinaspecifiedtolerance.ConventionalNewton-typeiterativestrategiesholdtheloadparameterconstantwhilstiteratingtoconvergence.Passingloadlimitpointsisthereforeextremelydifficultowingtothenearsingularnatureofthetangentstiffnessmatrixintheneighbourhoodofaloadlimitpoint.Severaldifferenttechniqueshavebeenusedtotracetheresponsebeyondthelimitpoint.BerganetaL2andBergaq4utilizingthe'currentstiffnessparameter',suppressequilibriumiterationsinthecriticalzoneuntilthelimitpointhasbeentraversed.Alternatively,thetechniqueofaddingfictitiousspringstotheincrementalstiffnessmatrixtoenableittoremainpositivedefiniteinasnap-throughwasdevelopedforframesbyWrightandGaylord'andhasbeenappliedtoarchesbySharifiandPOPOV.~Themaindisadvantagesofthemethodarethattrialanderrorisoftenrequiredintheselectionoftheappropriatespringsandthedifficultyinmathematicallyjustifyingthemethodwhenseveralspringsareaddedtoacomplexmulti-degree-of-freedomsystem.2Anothercommonlyusedstrategyhasbeentoincrementtheloadparameteruntilthelimitpointisreachedandthentotraversethelimitpointbyincrementingacharacteristicdisplacementbyaprescribedamountandevaluatingthecorrespondingloadl