Mass redistribution method for finite element contact problems in elastodynamics

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profil-urra-2012

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English

Poids de l'ouvrage

6 Mo

Mass redistribution method for finite element contact problems in elastodynamics Houari Boumediène Khenous1, Patrick Laborde2, Yves Renard3. Abstract This paper deals with the numerical stabilization of the semi-discretized finite element uni- lateral contact problem in elastodynamics. It is well known that it is ill-posed due to the nonpenetration condition on the finite element nodes lying on the contact boundary. We intro- duce a new method based on a redistribution of the mass matrix such that there is no inertia on the contact boundary. This leads to a mathematically well-posed and energy conserving semi-discretized problem. Finally, some numerical tests are presented. keywords: elasticity, unilateral contact, time integration schemes, energy conservation, stability, redistributed mass matrix. Introduction In this paper, we are interested in the study of the numerical instabilities caused by the space semi- discretization of contact problems in elastodynamics. For the sake of simplicity, we limit ourself to the small deformations framework. The underlying continuous elastodynamic contact problem (purely hyperbolic problem) is very difficult from a mathematical viewpoint. As far as we know, some existence results have only been established for a close but scalar and two dimensional problem in [14, 11], and in the vector case with a modified contact law in [23]. However, no uniqueness result is known in the purely hyperbolic framework.

lagrange finite

numerical instabilities

equivalent contact

contact condition

finite el- ement

continuous elastodynamic

lateral contact

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Publié par

profil-urra-2012

Nombre de lectures

Langue

English

Poids de l'ouvrage

6 Mo

MassredistributionmethodforﬁniteelementcontactproblemsinelastodynamicsHouariBoumedièneKhenous1,PatrickLaborde2,YvesRenard3.AbstractThispaperdealswiththenumericalstabilizationofthesemi-discretizedﬁniteelementuni-lateralcontactprobleminelastodynamics.Itiswellknownthatitisill-posedduetothenonpenetrationconditionontheﬁniteelementnodeslyingonthecontactboundary.Weintro-duceanewmethodbasedonaredistributionofthemassmatrixsuchthatthereisnoinertiaonthecontactboundary.Thisleadstoamathematicallywell-posedandenergyconservingsemi-discretizedproblem.Finally,somenumericaltestsarepresented.keywords:elasticity,unilateralcontact,timeintegrationschemes,energyconservation,stability,redistributedmassmatrix.IntroductionInthispaper,weareinterestedinthestudyofthenumericalinstabilitiescausedbythespacesemi-discretizationofcontactproblemsinelastodynamics.Forthesakeofsimplicity,welimitourselftothesmalldeformationsframework.Theunderlyingcontinuouselastodynamiccontactproblem(purelyhyperbolicproblem)isverydiﬃcultfromamathematicalviewpoint.Asfarasweknow,someexistenceresultshaveonlybeenestablishedforaclosebutscalarandtwodimensionalproblemin[14,11],andinthevectorcasewithamodiﬁedcontactlawin[23].However,nouniquenessresultisknowninthepurelyhyperbolicframework.Thesemi-discretizedproblembyﬁniteelementsisitselfill-posed,whichleadstonumericalinstabilitiesoftimeintegrationschemes.Thus,manyauthorsadapteddiﬀerentapproachestoovercomethisdiﬃculty.Torecovertheuniquenessinthediscretizedcase,oneoftheapproacheswelladaptedtorigidbodiesistointroduceanimpactlawwitharestitutioncoeﬃcient[20].However,thisapproachseemsnotsatisfactoryfordeformablebodies.Ontheotherhand,theunilateralcontactconditionleadstosomediﬃcultiesintheconstructionofenergyconservingschemes[20,13,12,5]becauseofthepresenceofoscillationsofthedisplacementandofthenormalstressonthecontactboundary.Awaytoavoidanoisybehaviorofthesolutionistoimplicitthecontactforce[24,3].Asaresult,nodescomingtocontactarestuck.Thedrawbackofthismethodisthatthekineticenergyofthecontactingnodesiscancelledeachtimeanewcontactoccurs.Anotherwellknownapproachisthepenaltymethodwhichintroducesimportantoscillationsthathavetobereducedwithadampingtechnique[24].Eventhoughitispossibletobuildenergyconservingschemeswithapenalizedcontactcondition[2,5],thisleadstoimportantoscillationsofthenormalstress.Oneofthekeypointstoavoidoscillationsistotrytoenforcethecomplementarityconditionwith1InstitutdeMathématiquesdeToulouse,UMRCNRS5219,INSAdeToulouse,ComplexescientiﬁquedeRangueil,31077Toulouse,France,khenous@insa-toulouse.fr,renard@insa-toulouse.fr2InstitutdeMathématiquesdeToulouse,UMRCNRS5219,UniversitéPaulSabatier,118routedeNarbonne,31062Toulouse,France,laborde@math.ups-tlse.fr3PôledeMathématiques,INSAdeLyon,UniversitédeLyon,InstitutCamilleJordan,UMRCNRS5208,20rueAlbertEinstein,69621Villeurbanne,France,yves.renard@insa-lyon.fr1

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