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HyperWorksSolversNonlinearAnalysisOptiStructUser'sGuideStructuralAnalysis:NonlinearAnalysisNonlinearAnalysisNonlinearQuasi-StaticAnalysisLargeDisplacementNonlinearStaticAnalysisGeometricNonlinearAnalysisHyperWorksSolversNonlinearQuasi-StaticAnalysisOptiStructUser'sGuideStructuralAnalysisNonlinearAnalysis:NonlinearQuasi-StaticAnalysisNonlinearQuasi-StaticAnalysisThissolutionsequenceperformsquasi-staticnonlinearanalysis.Presently,thesourcesofnonlinearityincludeCONTACTinterfaces,GAPelements,andMATS1elastic-plasticmaterial.Smalldeformationtheoryisusedinthesolutionofnonlinearproblems,similartothewayitisusedwithLinearStaticAnalysis.Inertiareliefisalsopossible.Smalldeformationtheorymeansthatstrainsshouldbewithinlinearelasticityrange(some5percentstrain),androtationswithinsmallrotationrange(some5degreesrotation).Thisalsomeansthatthereisnoupdateofgap/contactelementlocationsororientationduetothedeformations–theyremainthesamethroughoutthenonlinearcomputations.Theorientationmaychange,however,duetogeometrychangesinoptimizationruns.批注[张剑龙1]:非线性准静态分析大位移非线性静力分析几何非线性分析批注[张剑龙2]:非线性准静态分析该解决方案序列执行准静态非线性分析。目前，非线性的来源包括CONTACT接触，GAP单元和MATS1弹性塑料材料。小变形理论被用于解决非线性问题，类似于线性静态分析的方法。惯性救济也是可能的。小变形理论意味着应变应在线弹性范围内（约5％应变），并在小旋转范围（约5度转角）内旋转。这也意味着由于变形而不存在间隙/接触单元位置或方向的更新-它们在整个非线性计算中保持不变。然而，由于优化运行中的几何变化，方向可能会改变。NonlinearSolutionMethodThebasicNewtonmethodisusedforthesolutionofnonlinearproblems.Theprincipleofthismethodisillustratedforaone-dimensionalprobleminthefigurebelowandcanbeformulatedasfollows:Consideranonlinearproblem:Where,uisthedisplacementvector,Pisthegloballoadvector,andL(u)isthenonlinearresponseofthesystem(nodalreactions).Notethatforalinearproblem,L(u)wouldsimplybeKu(asdescribedintheLinearStaticAnalysissection).ApplicationofNewton'smethodtothisequationleadstoaniterativesolutionprocedure:Where,Intheaboveformulas,Knrepresentsaslopematrix,definedasatangenttotheL(u)curveatapointun,andRnisthenonlinearresidual.Repeatingthisprocedureiteratively,undercertainconvergenceconditions,leadstosystematicreductionofresidualRnandhence,convergence.批注[张剑龙3]:非线性解法基本的牛顿法用于解决非线性问题。该方法的原理如下图所示，用于一维问题，可以表达如下：考虑一个非线性问题：其中，u是位移矢量，P是全局负载矢量，L（u）是系统的非线性响应（节点反应）。注意，对于线性问题，L（u）将简单地是Ku（如线性静态分析部分所述）。牛顿方法在这个方程中的应用导致迭代解法：其中，在上述公式中，Kn表示“斜率”矩阵，定义为在点un处的L（u）曲线的切线，Rn是非线性残差。在一定的收敛条件下迭代地重复这个过程，导致残差Rn的系统减少，从而导致收敛。注意，上述方案被稍微修改为等价格式，其中，代替计算，直接获得新解un+1：这种形式很容易通过在牛顿方程的两边添加Knun而产生，并且在实际实现中具有一定的优点。Notethattheaboveschemeissomewhatmodifiedtoanequivalentformatwherein,insteadofcalculating,thenewsolutionun+1isdirectlyobtained:ThisformisreadilyproducedbyaddingKnuntobothsidesofNewton'sequation,andhascertainadvantagesinpracticalimplementations.IncrementalLoadingForalargeclassofproblemssatisfyingcertainstabilityandsmoothnessconditions,theNewton'siterativemethodisproventoconverge,providedthattheinitialguessissufficientlyclosetothetrueforce-displacementpathL(u).Hence,toimproveconvergenceforstronglynonlinearproblems,thetotalloadingPisoftenappliedinsmallerincrements,asshowninthefigurebelow.Ateachoftheintermediateloads,P1,P2,etc.,thestandardNewtoniterationsareperformed.Thisprocedure,knownasincrementalloading,helpstokeeptheconsecutiveiterationsclosertothetrueloadpath,therebyimprovingthechancesofobtainingafinal,convergedsolution(thoughusuallyattheexpenseofanincreasedtotalnumberofiterations).NonlinearConvergenceCriteriaInordertoassesswhetherthenonlinearprocesshasconverged,anumberofconvergencecriteriaareavailable.Thesecriteriaandrespectivetolerancescanbe批注[张剑龙4]:增量加载对于满足一定稳定性和平滑度条件的大类问题，假设初始猜测足够接近真实的力-位移路径L（u），牛顿迭代法是收敛的。因此，为了提高强非线性问题的收敛性，总负载P通常以较小的增量应用，如下图所示。在每个中间负载P1，P2等，执行标准牛顿迭代。被称为增量加载的这个过程有助于使连续的迭代保持更接近于真实的负载路径，从而提高获得最终收敛解的机会（尽管通常以增加的迭代次数为代价）。批注[张剑龙5]:非线性收敛准则为了评估非线性过程是否收敛，可以使用多个收敛准则。可以在NLPARM批量数据卡上选择这些标准和相应的公差。评估非线性收敛的基本原理是将解决方案的误差测量与预定的公差水平进行比较。当误差低于规定的公差时，该问题被认为是收敛的。在多个同时收敛标准的情况下，需要满足要收敛的解的所有标准。selectedontheNLPARMbulkdatacard.Thebasicprincipleinassessingnonlinearconvergenceistocompareanerrormeasureofthesolutionwithapre-determinedtolerancelevel.Whentheerrorfallsbelowtheprescribedtolerance,theproblemisconsideredconverged.Inacaseofmultiple,simultaneousconvergencecriteria,allcriterianeedtobesatisfiedforthesolutiontobeconverged.Therelativeerrorindisplacements(printedintheconvergencesummaryasEUI)iscalculatedas:Here,AisanormalizingvectorconsistingofsquarerootsofdiagonalelementsofstiffnessmatrixandthevectornormII.IIiscalculatedas:Furthermore,qisacontractionfactorthatcorrectstheincrementofsolutiontobetterrepresenttheactualerrorinthenonlinearsolution.Itisexpressedas:Inordertostabilizethebehaviorofqinpracticalcomputations,itisupdatediterativelyaccordingtotheformula:startingfrominitialvalueq1=0.99.Notethatthecontractionfactorismeaningfulwhenthesolutionisclosetohavingconverged–itthenreasonablywellestimatestheactualerrorremaininginthenonlinearsolution.Therelativeerrorintermsofloads(printedinconvergencesummaryasEPI)measurestherelativestrengthoftheresidualR,andiscalculatedas:TheloadvectorPinthisformulaincludesnodalreactionsduetoprescribeddisplacements.Therelativeerrorintermsofwork(printedinconvergencesummar