显示算法与隐式算法

以下内容转自abaqus版面的总结:显式一般用于动态问题的分析，对于大型问题，或复杂的接触情况可能需要几百万的增量步的计算，所用时间可能是几天或更长.而隐式的增量步长要长得多，一般用于静态问题的求解.显式算法另0explicitmethodusedirectiterativemethod,whichhassmallcostineachtimeincrementbutrequirerelativelysmallincrement.Abaquspre-determinethetimeincrementbasedonwavepropagationspeedandminimummeshsize.Thismethodcouldbeefficientforhighlynonlinearandcontactproblem.Forquasi-staticproblem,properlyadjustmodelparameterasdensityandtotaltimeisimportanttoachievegoodcomputationtime.standard-隐式算法Implicitmethodusenewtonmethodforiteration,whichmeanshighcostforeachtimeincrementbutcouldmeanlargetimeincrement.Convergencecouldbeaprobleminthiscase.Itcouldbeefficientforlinearandsomenonlinearproblem.Morematerials,elementsandproceduresareavailableinstandard.所谓显式和隐式，是指求解方法的不同，即数学上的出发点不一样。并不是说显式只能求动力学问题，隐式只能求静力学问题，只是求解策略不通。显式求解是对时间进行差分，不存在迭代和收敛问题，最小时间步取决于最小单元的尺寸。过多和过小的时间步往往导致求解时间非常漫长，但总能给出一个计算结果。解题费用非常昂贵。因此在建模划分网格时要非常注意。隐式求解和时间无关，采用的是牛顿迭代法（线性问题就直接求解线性代数方程组），因此存在一个迭代收敛问题，不收敛就的不到结果。两者求解问题所耗时间的长短理论上无法比较。实际应用中一般感觉来说显式耗时多些。由于两者解题的出发点，所以一般来说显式用于求解和时间相关的动力学问题。隐式用来求解和时间无关的静力学问题。但也不是绝对的。比如，用隐式求解时，为了克服迭代不收敛，改用显式算，但是要多给点时间，这样虽然克服了不收敛的问题，但是求解的时间费用也是相当客观的。另外，隐式也可以求解动力学问题。abaqus就有一个用隐式积分求解动力学的选项。n+1个时间步的量可以由第n个时间步的量直接求得，称为显式例如：an+1=bn+cnbn+1=an+cncn+1=an+bn优点是计算量比较小缺点是有累积误差n+1个时间步的量不可以由第n个时间步的量直接求得，称为隐式例如：an+1+bn+1=cnbn+1+cn+1=anan+1+cn+1=bn优点是计算量比较大，需要通过方程组求解缺点是没有累计误差显示算法不进行刚度矩阵的重新计算，只在开始形成以后不变，是时间的显示积分。而隐式算法没进行一次计算都要重新计算刚度矩阵，然后进行迭代，是无条件收敛的。Q:Whatisthedifferencebetweenimplicitandexplicitdynamics?(DifferencebetweenregularANSYSandANSYS/LS-DYNA?)A:Forcomputers,matrixmultiplicationisn'tdifficult.Matrixinversionisthemorecomputationallyexpensiveoperation.Theequationswesolveinnonlinear,dynamicanalysesinANSYSandinLS-DYNAare:[M]{a}+[C]{v}+[K]{x}={F)Hence,inANSYS,weneedtoinvertthe[K]matrixwhenusingdirectsolvers(frontal,sparse).IterativesolversuseadifferenttechniquefromdirectsolverswhichIwon'tgetintohere,but,basically,theinversionof[K]istheCPU-intensiveoperationforany'regular'ANSYSsolver,directoriterative.Wethencansolvefordisplacements{x).Ofcourse,withnonlinearities,[K(x)]isalsoafunctionof{x),soweneedtouseNewton-Raphsonmethodtosolvefor[K]aswell.(materialnonlinearitiesandcontactgetthrowninto[K(x)])InLS-DYNA,ontheotherhand,wesolveforaccelerations{a}first.Now,inLS-DYNA,weassumethatthemassmatrixislumped.Thisbasicallyforcesustouselower-orderelements--thatiswhy,forallexplicitdynamicscodes(ANSYS/LS-DYNA,MSC.Dytran,ABAQUS/Explicit),wecanonlyuselower-orderelements.Also,thebenefitofdoinglumpedmassisthat,ifwesolvefor{a),then[M],iflumped,isadiagonalmassmatrix.Thismeansthatinversionof[M]istrivial(diagonaltermsonly)--anotherwaytoviewitisthatwenowhaveNsetof*uncoupled*equations.Hence,wejusthavetodomatrixmultiplication,whichisn'tnearlyasCPU-intensive.It'salsoworthwhiletonotethat[K]doesnotneedtobeinverted,andaccountingformaterialnonlineartiesandcontactiseasier.Now,asfortimeintegration,theterms'implicit'and'explicit'refertotimeintegration--forexample,ifyoumightrecallsomethinglikebackwardEulermethod,thatisanexampleofanimplicittimeintegrationscheme,whereascentraldifferenceorforwardEulerareexamplesofexplicittimeintegrationschemes.Itrelatestowhenyoucalculatethequantities--eitherbasedoncurrentorprevioustimestep.Inanycase,thisisaverysimplifiedexplanation,andthemainpointisthatimplicittimeintegrationisunconditionallystable,whereasexplicittimeintegrationisnot(thereisacriticaltimestepyourdelta(t)needstobesmallerthan).Asaresult,'regular'ANSYSallowsformuchlargertimesteps,butLS-DYNArequiresmuchtiniertimesteps.Also,LS-DYNArequiresverytinysteps,sothatiswhyitisusuallygoodforimpact/short-durationevents,notusuallythingslikemaybecreepwherethemodel'stimescalemaybeontheorderofhoursormore.Insummary:'Regular'ANSYSusesimplicittimeintegration.Thismeansthat{x}issolvedfor,butweneedtoinvert[K],whichmeansthateachiterationiscomputationallyexpensive.However,becausewesolvefor{x),itisimplicit,andwedon'tneedverytinytimesteps(i.e.,eachiterationisexpensive,butweusuallydon'tneedtoomanyiterationstotal).Theoveralltimescaledoesn'taffectusmuch(althoughthereareconsiderationsofsmallenoughtimestepsforpropermomentumtransfer,capturingdynamicresponse,etc.,butI'mgettingaheadofmyself).ANSYS/LS-DYNAusesexplicittimeintegration.Thismeansthat{a}issolvedfor,andinverting[M]istrivial--eachiterationisveryefficient.However,becausewesolvefor{a),thendetermine{x),itisexplicit,andweneedverysmalltimesteps(many,manyiterations)toensurestabilityofsolutionsinceweget{x}bycalculating{a}first.(i.e.,eachiterationischeap,butweusuallyneedmany,manyiterationstotal)Anyways,thisisaverysimplified(maybeover-simplified)explanation,butIhopeitmayhelpclearthedistinctionbetweenthesetwomethods.Forveryhigh-impact,nonlineareventsof*short*duration,ANSYS/LS-DYNAisusuallythebetterchoice.Foreventswhichareoflongduration,'regular'ANSYSisusuallythepreferredmethod.Forquasi-staticevents,therearewaysinwhichyoucanuseeithersolutionmethod(e.g.,mass-scalingforexplicit).However,youcansolvemostproblemswitheithermethod--I'mjustreferringtowhichonesaremore*efficient*.[NotethatIhaven'ttalkedaboutmode-superpositiontransientanalys