The singular dynamic method for constrained second order hyperbolic equations Application to dynamic contact

Yves Renard1

Description

Niveau: Supérieur, Doctorat, Bac+8
The singular dynamic method for constrained second order hyperbolic equations. Application to dynamic contact problems Yves Renard1 Abstract The purpose of this paper is to present a new family of numerical methods for the approximation of second order hyperbolic partial differential equations submitted to a convex constraint on the solution. The main application is dynamic contact problems. The principle consists in the use of a singular mass matrix obtained by the mean of different discretizations of the solution and of its time derivative. We prove that the semi-discretized problem is well-posed and energy conserving. Numerical experiments show that this is a crucial property to build stable numerical schemes. Keywords: hyperbolic partial differential equation, constrained equation, finite element methods, variational inequalities. 65M60, 35L87, 74M15, 35Q74. Introduction An interesting class of hyperbolic partial differential equations with constraints on the so- lution consists in elastodynamic contact problems for which the vast majority of traditional numerical schemes show spurious oscillations on the contact displacement and stress (see for instance [12, 9, 10]). Moreover, these oscillations do not disappear when the time step decreases. Typically, they have instead tended to increase. This is a characteristic of order two hyperbolic equations with unilateral constraints that makes it very difficult to build stable numerical schemes. These difficulties have already led to many research under which a variety of solutions were proposed.

• semi-discretized problem


• problem


• standard scheme


• rnw


• sup w?f


• posed space


• dynamic method


• problem ?


• method using

Sujets

• Renard
• Problem
• Dynamic method

Informations

• Publié par: mijec
• Nombre de lectures: 10
• Langue: English
• Poids de l'ouvrage: 15 Mo

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Thesingulardynamicmethodforconstrainedsecondorder hyperbolicequations.Applicationtodynamiccontact problems Yves Renard 1 Abstract Thepurposeofthispaperistopresentanewfamilyofnumericalmethodsforthe approximationofsecondorderhyperbolicpartialdifferentialequationssubmittedtoa convexconstraintonthesolution.Themainapplicationisdynamiccontactproblems. Theprincipleconsistsintheuseofasingularmassmatrixobtainedbythemeanof differentdiscretizationsofthesolutionandofitstimederivative.Weprovethatthe semi-discretizedproblemiswell-posedandenergyconserving.Numericalexperiments showthatthisisacrucialpropertytobuildstablenumericalschemes. Keywords:hyperbolicpartialdifferentialequation,constrainedequation,finiteelement methods,variationalinequalities. 65M60,35L87,74M15,35Q74. Introduction Aninterestingclassofhyperbolicpartialdifferentialequationswithconstraintsontheso- lutionconsistsinelastodynamiccontactproblemsforwhichthevastmajorityoftraditional numericalschemesshowspuriousoscillationsonthecontactdisplacementandstress(see forinstance[12,9,10]).Moreover,theseoscillationsdonotdisappearwhenthetimestep decreases.Typically,theyhaveinsteadtendedtoincrease.Thisisacharacteristicoforder twohyperbolicequationswithunilateralconstraintsthatmakesitverydifficulttobuild stablenumericalschemes.Thesedifficultieshavealreadyledtomanyresearchunderwhich avarietyofsolutionswereproposed.Someofthemconsistsinaddingdampingterms(see [24]forinstance),butwithalossofaccuracyonthesolution,ortoimplicitthecontact stress[7,6]butwithalossofkineticenergywhichcouldbeindependentofthediscretiza- tionparameters(seethenumericalexperiments).Someenergyconservingschemeshave alsobeenproposedin[11,25,17,16,9,2,10].Unfortunately,theseschemes,althoughmore satisfactorythanthemostotherschemes,leadtolargeoscillationsonthecontactstress. Besides,mostofthemdonotstrictlyrespecttheconstraint. Inthispaper,weproposeanewclassofmethodswhoseprincipleistomakedifferent approximationsofthesolutionandofitstimederivative.Comparedtotheclassicalspace semi-discretization,thiscorrespondstoasingularmodificationofthemassmatrix.Inthis sense,itisinthesameclassofmethodsthanthemassredistributionmethodproposedin [12,13]forelastodynamiccontactproblems.Indeed,inthislattermethod,themassmatrix haszerocomponentsforallthenodesonthecontactboundary(whichlimitsitsapplication toconstraintsonaboundary).Thesingularmodificationofthemassmatrixjustifythe proposedterminology“singulardynamicmethod”.Themainfeatureistoprovideawell- posedspacesemi-discretization.Thenumericaltestsshowthatithasacrucialinfluence 1 Universite´deLyon,CNRS, INSA-Lyon,ICJUMR5208,F-69621,Villeurbanne,[e-mail] 1

onthestabilityofstandardschemeandonthequalityoftheapproximation,especiallyfor thecomputationofLagrangemultiplierscorrespondingtotheconstraints. Theclassicalsemi-discretizations,forexamplewithfiniteelementmethods,giveaprob- lemintimewhichisameasuredifferentialinclusion(see[19,20,21,22]).Suchadifferential inclusionissystematicallyill-posed,unlessanadditionalimpactlawisconsidered.How- ever,theschemeobtainedwiththeadditionofanimpactlawin[21]leadsalsotospurious oscillations. Thesemi-discretizationweproposehereleadstoaproblemwhichisequivalenttoa regularLipschitzordinarydifferentialequation.Thus,timeintegrationschemesatleast convergeforafixedspacediscretizationwhenthetimesteptendstozero.Thiswork generalizesinasensethemethodspresentedin[13,8]. Theoutlineofthepaperisthefollowing.Section1isdevotedtothedescriptionof theabstracthyperbolicequationwithconstraintsandtheequivalentvariationalinequality. Section2presentsthenewapproximationmethodsandthemainresultsofwell-posedness andenergyconservation.Then,insection3,anon-trivialmodelproblemwhichcorresponds forinstancetothedynamicsofathinmembraneunderanobstacleconditionisdeveloped. Anexampleofwell-poseddiscretizationisalsobuiltinthissection.Section4briefly describesthefullydiscreteproblemobtainedwiththefinitedifferencemidpointscheme andpresentssomenumericalexperimentsonthismodelwhichshowsinparticularthat themidpointschemeisstablewithwell-posedsemi-discretizationsandunstableotherwise. Finally,inSection5,theproposedmethodisappliedtoanelastodynamiccontactproblem. 1Theabstracthyperbolicequation LetΩ ⊂ R d beaLipschitzdomainandH=L 2 (Ω)thestandardHilbertspaceofsquare integrablefunctionsonΩ.LetWbeaHilbertspacesuchthat W ⊂ H ⊂ W 0 , withdensecompactandcontinuousinclusionsandlet A:W → W 0 bealinearself-adjointellipticcontinuousoperator,i.e.whichsatisfies h Aw,v i W 0 ,W = h Av,w i W 0 ,W , ∀ v,w ∈ W, ∃ α>0, ∀ w ∈ W, h Aw,w i W 0 ,W ≥ α k w k 2 W , ∃ c>0, ∀ w ∈ W, k Aw k W 0 ≤ c k w k W . Weconsiderthefollowingproblem  Findu:[0,T] → Ksuchthat 2 u∂   2 (t)+Au(t) ∈ f − N K (u(t))fora.e.t ∈ (0,T], (1) t∂ u∂   u(0)=u 0 ,(0)=v 0 , t∂ whereKisaclosedconvexnonemptysubsetofW,f ∈ W 0 ,u 0 ∈ K,v 0 ∈ H,T>0and N K (u)isthenormalconetoKdefinedby(seeforinstance[5]foradetailedpresentation ofdifferentialinclusions)  ∅ ifu/ ∈ K, N K (u)= { f ∈ W 0 : h f,w − u i W 0 ,W ≤ 0, ∀ w ∈ K } ifu ∈ K. 2