Mass redistribution method for finite element contact problems in elastodynamics

Description

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


• let also


• ?v ?

Sujets

• Renard
• Jordan
• INSA de Lyon
• Sabatier
• Institut de Mathématiques de Toulouse

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 9
• Langue: English
• Poids de l'ouvrage: 6 Mo

Extrait

Massredistributionmethodforfiniteelementcontactproblemsin elastodynamics HouariBoumediène Khenous 1 ,Patrick Laborde 2 ,Yves Renard 3 . Abstract Thispaperdealswiththenumericalstabilizationofthesemi-discretizedfiniteelementuni- lateralcontactprobleminelastodynamics.Itiswellknownthatitisill-posedduetothe nonpenetrationconditiononthefiniteelementnodeslyingonthecontactboundary.Weintro- duceanewmethodbasedonaredistributionofthemassmatrixsuchthatthereisnoinertia onthecontactboundary.Thisleadstoamathematicallywell-posedandenergyconserving semi-discretizedproblem.Finally,somenumericaltestsarepresented. keywords:elasticity,unilateralcontact,timeintegrationschemes,energyconservation,stability, redistributedmassmatrix. Introduction Inthispaper,weareinterestedinthestudyofthenumericalinstabilitiescausedbythespacesemi- discretizationofcontactproblemsinelastodynamics.Forthesakeofsimplicity,welimitourselfto thesmalldeformationsframework. Theunderlyingcontinuouselastodynamiccontactproblem(purelyhyperbolicproblem)isvery difficultfromamathematicalviewpoint.Asfarasweknow,someexistenceresultshaveonlybeen establishedforaclosebutscalarandtwodimensionalproblemin[14,11],andinthevectorcase withamodifiedcontactlawin[23].However,nouniquenessresultisknowninthepurelyhyperbolic framework. Thesemi-discretizedproblembyfiniteelementsisitselfill-posed,whichleadstonumerical instabilitiesoftimeintegrationschemes.Thus,manyauthorsadapteddifferentapproachesto overcomethisdifficulty.Torecovertheuniquenessinthediscretizedcase,oneoftheapproacheswell adaptedtorigidbodiesistointroduceanimpactlawwitharestitutioncoefficient[20].However,this approachseemsnotsatisfactoryfordeformablebodies.Ontheotherhand,theunilateralcontact conditionleadstosomedifficultiesintheconstructionofenergyconservingschemes[20,13,12,5] becauseofthepresenceofoscillationsofthedisplacementandofthenormalstressonthecontact boundary.Awaytoavoidanoisybehaviorofthesolutionistoimplicitthecontactforce[24,3]. Asaresult,nodescomingtocontactarestuck.Thedrawbackofthismethodisthatthekinetic energyofthecontactingnodesiscancelledeachtimeanewcontactoccurs.Anotherwellknown approachisthepenaltymethodwhichintroducesimportantoscillationsthathavetobereduced withadampingtechnique[24].Eventhoughitispossibletobuildenergyconservingschemes withapenalizedcontactcondition[2,5],thisleadstoimportantoscillationsofthenormalstress. Oneofthekeypointstoavoidoscillationsistotrytoenforcethecomplementarityconditionwith 1 InstitutdeMathématiquesdeToulouse,UMRCNRS5219,INSAdeToulouse,ComplexescientifiquedeRangueil, 31077Toulouse,France,[e-mail],[e-mail] 2 InstitutdeMathématiquesdeToulouse,UMRCNRS5219,UniversitéPaulSabatier,118routedeNarbonne, 31062Toulouse,France,[e-mail] 3 PôledeMathématiques,INSAdeLyon,UniversitédeLyon,InstitutCamilleJordan,UMRCNRS5208,20rue AlbertEinstein,69621Villeurbanne,France,[e-mail] 1