A TRANSIENT DOMAIN DECOMPOSITION METHOD FOR THE AN

COMPUTATIONALMECHANICSNewTrendsandApplicationsS.Idelsohn,E.O~nateandE.Dvorkin(Eds.)cCIMNE,Barcelona,Spain1998ATRANSIENTDOMAINDECOMPOSITIONMETHODFORTHEANALYSISOFSTRUCTURALDYNAMICPROBLEMSAlexandreS.Hansen,FernandoA.RochinhaMechanicalEngineeringDepartmentEE-COPPE-UFRJCx.Postal68503,CEP21945-970,RIODEJANEIRO-BRAZILe-mail:[e-mail],[e-mail]:DomainDecomposition,Dynamics,InexactUzawa,ParallelProcessingAbstract.Weproposeandstudy,forstructuraldynamicproblems,adomaindecom-positionmethodbasedonanaugmentedLagrangianformulation.ThismethodusesaninexactUzawaalgorithmtosolvethelinearsystemrequiredatanytimeincrement.Theproposedalgorithmisinteresting,forparallelandvectorapplications,becauseitdoesn'tneedscalarproductsandhaslessoperationsbyiterationthantheconjugategradientmethodappliedtothedualproblem.ThreedimensionalelasticitydinamicproblemsaresolvedbytheproposedalgorithmandsomecomparisonswiththeconjugategradientmethodandUzawaalgorithmareaddressed.1AlexandreS.Hansen,FernandoA.Rochinha1INTRODUCTIONInthisworkweareinterestedinthetransientresponseofanelastic,isotropic,threedimensionalsolidsubjecttosmalldeformationsundertheactionofatime-dependentforcesystem,whichisgovernedbythevariationalformbelowGiveninitialconditions,u0;_u0,ndthedisplacementeldu2Usuchthat8v2VZΩv:¨u+ZΩrs(v):(21rs(u)+2(trrsu)I)=ZΩb:v+Zγq:v(1)where,1and2arepropertiesofthematerial,bandqare,respectively,thebobyandsurfaceforces,rs(:)denotesthesymmetricpartofthegradientof(:),_(),¨()denotestherstandsecondderivativeswithrespecttothetimeandU,Vare,respectively,thesetofadmissibledisplacementsandthespaceofadmissiblevariations.Thenumericalsolutionoftheaboveequationcanbeobtainedbythecombinationofimplicittime-integrationmethodswiththeFiniteElementspatialdiscretizationwhichreducesthisproblemtothesolutionoflinearsystemswhosesizedependsonthespa-tialdiscretization.Inordertosolveecientlythisproblemusingparallelcomputingwemustconsiderdomaindecompositiontechiniques1.Inparticular,werefertothetransientFETImethodology2inwhichtheoriginalproblemisreplacedbysmalleronesrestrictedeachonetoasub-domainwiththeinterfacedisplacementcompatibilitybetweenneighborssub-domainsenforcedbyLagrangemultipliers.Applyingthismethodology,afterdiscretizationintimeandspaceandgiventhedisplacementsandlinearmomentumvectorsforeachsub-domain(s)atthetimestepn(usnandvsnrespectively),thealgorithmprogresseswiththeequationsbelow(Ms+
t24Ks)usn+1=2=
t24Fsn+1=2+Msusn+
t2vsn−
t24BsTn+1=2s=NsXs=1Bsusn+1=2=0(2)and_vsn+1=2=Fsn+1=2−Kusn+1=2usn+1=2usn+1=2−usnvsn+1=vsn+
t_vsn+1=2(3)whereKsandMsarethestinessandmassmatricesforthesub-domains,Fsn+1=2istheaveragebetweenthenodalforcesvectorforthesub-domainsattimestepsnandn+1andBsarematriceswhichentailthedisplacementcompatibility.2AlexandreS.Hansen,FernandoA.RochinhaAsinthediscreteevolutionequations(3)allthecomputationsarerestricttoeachsub-domain,anecientparallelimplementationwillbemainlydependentonthesolutionofthesystemofequations(2),whichcorrespondstoasaddlepointproblem.Inthepresentworkweproposetomodifythesystem(2),addingapenaltytermleadingtoanaugmentedLagrangianformulation,andsolvethenewsystemwithaninexactUzawaalgorithm3;4Insection2,thepenaltytermisintroducedandsomepropertiesoftheUzawaal-gorithmareaddressed.TheinexactUzawaalgorithmproposedforthisproblemispresentedonthesection3,alongwiththeconvergenceproof.Insection4,wepresentsomenumericalexamplessolvedbytheproposedalgorithmandsomecomparisonswiththeconjugategradientmethodappliedtothedualproblemareadressed.2AUGMENTEDLAGRANGIANFORMULATIONTheequations(2)canberecastonthematricialformbelow5Findthepair(x;y)2RnRmsuchthatK−BTB0xy=F0(4)whereKnnistheresultofassemblingtheindividualsub-domainmatricesanthussymmetric,strictlypositiveandwithablock-diagonalstructure.Throughoutthispaperxandystandsforvectorscontaing,respectively,theunionofdisplacementsandlagrangemultipliersdenedsoneachsub-domain.Theabovesystemcorrespondstotheoptimalconditionsofaminimizationproblemthatcanberewrittenonanaugmentedlagrangianformulationfromwhichweobtainthepenalizedlinearsystem5A−BTB0xy=F0(5)withA=K+1BTBand0isthepenalizationparameter.Althoughthesolutionofthesystems(4)and(5)arethesame,wehaveamotivationtoworkwiththepenalizedsystemduetotheconvergencefactorfortheUzawaalgorithmgivenby5kexi+1ksc1+c01keyik(6)keyi+1kc21+1c3keyik(7)3AlexandreS.Hansen,FernandoA.Rochinhawhereexi+1=xi+1−x,eyi+1=yi+1−y,xi;yiareapproximationsforthesolutionx;yatiterationiandc1,c01,c2,c3arepositiveconstantsdependingonlyonKandB.Fromthisestimatewecansaythat,onexactcomputation,theconvergencecanbeasfastaswewantonlybyreducing.Unfortunately,forthedomaindecompositionproblem,thepenaltyterm(1BTB)doesn'thaveablock-diagonalstructureassociatedwiththesub-domains,likethematrixK,whichwillcausedicultiesonapplyingtheUzawaalgorithmonparallelmachinesaswewillseeonthenextsection.3THEINEXACTUZAWAALGORITHMAPPLIEDONTHEPENAL-IZEDPROBLEM3.1UzawaAlgorithmTheUzawaalgorithmappliedtothesystem(5)ispresentedbelowGivenyievaluatexi+1andyi+1solvingthelinearsystemAxi+1=F−BTyi(8