abqqus非线性分析原理及步骤.docx
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abqqus非线性分析原理及步骤.docx
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abqqus非线性分析原理及步骤
AbaqusAnalysisUser'sManual
7.1.1 Solvingnonlinearproblems
Products:
Abaqus/Standard Abaqus/CAE
References
∙“Convergenceandtimeintegrationcriteria:
overview,”Section7.2.1
∙“Commonlyusedcontrolparameters,”Section7.2.2
∙“Convergencecriteriafornonlinearproblems,”Section7.2.3
∙“Timeintegrationaccuracyintransientproblems,”Section7.2.4
∙“Configuringgeneralanalysisprocedures,”Section14.11.1oftheAbaqus/CAEUser'sManual
Overview
SolvingnonlinearproblemsinAbaqus/Standardinvolves:
∙acombinationofincrementalanditerativeprocedures;
∙usingtheNewtonmethodtosolvethenonlinearequations;
∙determiningconvergence;
∙definingloadsasafunctionoftime;and
∙choosingsuitabletimeincrementsautomatically.
Somestaticproblemsmaybecomeunstablebecauseofseverenonlinearity.Abaqus/Standardoffersasetofautomaticstabilizationmechanismstohandlesuchproblems.
Thesolutionofnonlinearproblems
Thenonlinearload-displacementcurveforastructureisshowninFigure7.1.1–1.
Figure7.1.1–1Nonlinearload-displacementcurve.
Theobjectiveoftheanalysisistodeterminethisresponse.Inanonlinearanalysisthesolutioncannotbecalculatedbysolvingasinglesystemoflinearequations,aswouldbedoneinalinearproblem.Instead,thesolutionisfoundbyspecifyingtheloadingasafunctionoftimeandincrementingtimetoobtainthenonlinearresponse.Therefore,Abaqus/Standardbreaksthesimulationintoanumberoftimeincrementsandfindstheapproximateequilibriumconfigurationattheendofeachtimeincrement.UsingtheNewtonmethod,itoftentakesAbaqus/Standardseveraliterationstodetermineanacceptablesolutiontoeachtimeincrement.
Steps,increments,anditerations
∙Thetimehistoryforasimulationconsistsofoneormoresteps.Youdefinethesteps,whichgenerallyconsistofananalysisprocedure,loading,andoutputrequests.Differentloads,boundaryconditions,analysisprocedures,andoutputrequestscanbeusedineachstep.Forexample:
Step1:
Holdaplatebetweenrigidjaws.
Step2:
Addloadstodeformtheplate.
Step3:
Findthenaturalfrequenciesofthedeformedplate.
∙Anincrementispartofastep.Innonlinearanalyseseachstepisbrokenintoincrementssothatthenonlinearsolutionpathcanbefollowed.Yousuggestthesizeofthefirstincrement,andAbaqus/Standardautomaticallychoosesthesizeofthesubsequentincrements.Attheendofeachincrementthestructureisin(approximate)equilibriumandresultsareavailableforwritingtotherestart,data,results,oroutputdatabasefiles.
∙Aniterationisanattemptatfindinganequilibriumsolutioninanincrement.Ifthemodelisnotinequilibriumattheendoftheiteration,Abaqus/Standardtriesanotheriteration.WitheveryiterationthesolutionthatAbaqus/Standardobtainsshouldbeclosertoequilibrium;however,sometimestheiterationprocessmaydiverge—subsequentiterationsmaymoveawayfromtheequilibriumstate.InthatcaseAbaqus/Standardmayterminatetheiterationprocessandattempttofindasolutionwithasmallerincrementsize.
Convergence
Considertheexternalforces,P,andtheinternal(nodal)forces,I,actingonabody(seeFigure7.1.1–2(a)andFigure7.1.1–2(b),respectively).Theinternalloadsactingonanodearecausedbythestressesintheelementsthatareattachedtothatnode.
Figure7.1.1–2Internalandexternalloadsonabody.
Forthebodytobeinequilibrium,thenetforceactingateverynodemustbezero.Therefore,thebasicstatementofequilibriumisthattheinternalforces,I,andtheexternalforces,P,mustbalanceeachother:
Thenonlinearresponseofastructuretoasmallloadincrement,
isshowninFigure7.1.1–3.Abaqus/Standardusesthestructure'stangentstiffness,
whichisbasedonitsconfigurationat
and
tocalculateadisplacementcorrection,
forthestructure.Using
thestructure'sconfigurationisupdatedto
.
Figure7.1.1–3Firstiterationinanincrement.
Abaqus/Standardthencalculatesthestructure'sinternalforces,
inthisupdatedconfiguration.Thedifferencebetweenthetotalappliedload,P,and
cannowbecalculatedas
where
istheforceresidualfortheiteration.
If
iszeroateverydegreeoffreedominthemodel,pointainFigure7.1.1–3wouldlieontheload-deflectioncurveandthestructurewouldbeinequilibrium.Inanonlinearproblem
willneverbeexactlyzero,soAbaqus/Standardcomparesittoatolerancevalue.If
islessthanthisforceresidualtoleranceatallnodes,Abaqus/Standardacceptsthesolutionasbeinginequilibrium.Bydefault,thistolerancevalueissetto0.5%ofanaverageforceinthestructure,averagedovertime.Abaqus/Standardautomaticallycalculatesthisspatiallyandtime-averagedforcethroughoutthesimulation.Youcanchangethis,andallothersuchtolerances,byspecifyingsolutioncontrols(see“Convergencecriteriafornonlinearproblems,”Section7.2.3).
If
islessthanthecurrenttolerancevalue,Pand
areconsideredtobeinequilibriumand
isavalidequilibriumconfigurationforthestructureundertheappliedload.However,beforeAbaqus/Standardacceptsthesolution,italsochecksthatthelastdisplacementcorrection,
issmallrelativetothetotalincrementaldisplacement,
.If
isgreaterthanafraction(1%bydefault)oftheincrementaldisplacement,Abaqus/Standardperformsanotheriteration.Bothconvergencechecksmustbesatisfiedbeforeasolutionissaidtohaveconvergedforthattimeincrement.
Ifthesolutionfromaniterationisnotconverged,Abaqus/Standardperformsanotheriterationtotrytobringtheinternalandexternalforcesintobalance.First,Abaqus/Standardformsthenewstiffness,
forthestructurebasedontheupdatedconfiguration,
.Thisstiffness,togetherwiththeresidual
determinesanotherdisplacementcorrection,
thatbringsthesystemclosertoequilibrium(pointbinFigure7.1.1–4).
Figure7.1.1–4Seconditeration.
Abaqus/Standardcalculatesanewforceresidual,
usingtheinternalforcesfromthestructure'snewconfiguration,
.Again,thelargestforceresidualatanydegreeoffreedom,
iscomparedagainsttheforceresidualtolerance,andthedisplacementcorrectionfortheseconditeration,
iscomparedtotheincrementofdisplacement,
.Ifnecessary,Abaqus/Standardperformsfurtheriterations.FormoredetailsonconvergenceinAbaqus/Standard,see“Convergencecriteriafornonlinearproblems,”Section7.2.3.
ForeachiterationinanonlinearanalysisAbaqus/Standardformsthemodel'sstiffnessmatrixandsolvesasystemofequations.Therefore,thecomputationalcostofeachiterationisclosetothecostofconductingacompletelinearanalysis,makingthecomputationalexpenseofanonlinearanalysispotentiallymanytimesgreaterthanthecostofalinearanalysis.SinceitispossiblewithAbaqus/Standardtosaveresultsateachconvergedincrement,theamountofoutputdataavailablefromanonlinearsimulationcanalsobemuchgreaterthanthatavailablefromalinearanalysisofthesamegeometry.
Automaticincrementationcontrol
Bydefault,Abaqus/Standardautomaticallyadjuststhesizeofthetimeincrementstosolvenonlinearproblemsefficiently.Youneedtosuggestonlythesizeofthefirstincrementineachstepofthesimulation,afterwhichAbaqus/Standardautomaticallyadjuststhesizeoftheincrements.Ifyoudonotprovideasuggestedinitialincrementsize,Abaqus/Standardwillattempttoapplyalloftheloadsdefinedinthestepinasingleincrement.ForhighlynonlinearproblemsAbaqus/Standardwillhavetoreducetheincrementsizerepeatedlytoobtainasolution,resultinginwastedCPUtime.Itisadvantageoustoprovideareasonableinitialincrementsizebecauseonlyinmildlynonlinearproblemscanalloftheloadsinastepbeappliedinasingleincrement.
Thenumberofiterationsneededtofindaconvergedsolutionforatimeincrementwillvarydependingonthedegreeofnonlinearityinthesystem.Withthedefaultincrementationcontrol,theprocedureworksasfollows.Ifthesolutionhasnotconvergedwithin16iterationsorifthesolutionappearstodiverge,Abaqus/Standardabandonstheincrementandstartsagainwiththeincrementsizesetto25%ofitspreviousvalue.Itthenattemptstofindaconvergedsolutionwiththissmallertimeincrement.Ifthesolutionstillfailstoconverge,Abaqus/Standardreducestheincrementsizeagain.Thisprocessiscontinueduntilasolutionisfound.Ifthetimeincrementbecomessmallerthantheminimumyoudefinedormorethan5attemptsareneeded,Abaqus/Standardstopstheanalysis.
Iftheincrementconvergesinfewerthan5 iterations,thisindicatesthatthesolutionisbeingfoundfairlyeasily.Therefore,Abaqus/Standardautomaticallyincreasestheincrementsizeby50%if2 consecutiveincrementsrequirefewerthan5 iterationstoobtainaconvergedsolution.
Whilethedefaultautomaticincrementationcontrolissuitableformostanalyses,youcanchangeallthedefaultswhennecessarybyspecifyingsolutioncontrols;see“Commonlyusedcontrolparameters,”Section7.2.2,and“Timeintegrationaccuracyintransientproblems,”Section7.2.4.
Automaticstabilizationofunstableproblems
Nonlinearstaticproblemscanbeunstable.Suchinstabilitiesmaybeofageometricalnature,suchasbuckling,orofamaterialnature,suchasmaterialsoftening.Iftheinstabilitymanifestsitselfinaglobalload-displacementresponsewithanegativestiffness,theproblemcanbetreatedasabucklingorcollapseproblemasdescribedin“Unstablecollapseandpostbucklinganalysis,”Section6.2.4.However,iftheinstabil
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