24 Pages
English
Gain access to the library to view online
Learn more

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

-

Gain access to the library to view online
Learn more
24 Pages
English

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


Subjects

Informations

Published by
Reads 10
Language English
Document size 15 MB

Exrait

Thesingulardynamicmethodforconstrainedsecondorderhyperbolicequations.ApplicationtodynamiccontactproblemsYvesRenard1AbstractThepurposeofthispaperistopresentanewfamilyofnumericalmethodsfortheapproximationofsecondorderhyperbolicpartialdifferentialequationssubmittedtoaconvexconstraintonthesolution.Themainapplicationisdynamiccontactproblems.Theprincipleconsistsintheuseofasingularmassmatrixobtainedbythemeanofdifferentdiscretizationsofthesolutionandofitstimederivative.Weprovethatthesemi-discretizedproblemiswell-posedandenergyconserving.Numericalexperimentsshowthatthisisacrucialpropertytobuildstablenumericalschemes.Keywords:hyperbolicpartialdifferentialequation,constrainedequation,finiteelementmethods,variationalinequalities.65M60,35L87,74M15,35Q74.IntroductionAninterestingclassofhyperbolicpartialdifferentialequationswithconstraintsontheso-lutionconsistsinelastodynamiccontactproblemsforwhichthevastmajorityoftraditionalnumericalschemesshowspuriousoscillationsonthecontactdisplacementandstress(seeforinstance[12,9,10]).Moreover,theseoscillationsdonotdisappearwhenthetimestepdecreases.Typically,theyhaveinsteadtendedtoincrease.Thisisacharacteristicofordertwohyperbolicequationswithunilateralconstraintsthatmakesitverydifficulttobuildstablenumericalschemes.Thesedifficultieshavealreadyledtomanyresearchunderwhichavarietyofsolutionswereproposed.Someofthemconsistsinaddingdampingterms(see[24]forinstance),butwithalossofaccuracyonthesolution,ortoimplicitthecontactstress[7,6]butwithalossofkineticenergywhichcouldbeindependentofthediscretiza-tionparameters(seethenumericalexperiments).Someenergyconservingschemeshavealsobeenproposedin[11,25,17,16,9,2,10].Unfortunately,theseschemes,althoughmoresatisfactorythanthemostotherschemes,leadtolargeoscillationsonthecontactstress.Besides,mostofthemdonotstrictlyrespecttheconstraint.Inthispaper,weproposeanewclassofmethodswhoseprincipleistomakedifferentapproximationsofthesolutionandofitstimederivative.Comparedtotheclassicalspacesemi-discretization,thiscorrespondstoasingularmodificationofthemassmatrix.Inthissense,itisinthesameclassofmethodsthanthemassredistributionmethodproposedin[12,13]forelastodynamiccontactproblems.Indeed,inthislattermethod,themassmatrixhaszerocomponentsforallthenodesonthecontactboundary(whichlimitsitsapplicationtoconstraintsonaboundary).Thesingularmodificationofthemassmatrixjustifytheproposedterminology“singulardynamicmethod”.Themainfeatureistoprovideawell-posedspacesemi-discretization.Thenumericaltestsshowthatithasacrucialinfluence1Universite´deLyon,CNRS,INSA-Lyon,ICJUMR5208,F-69621,Villeurbanne,France.Yves.Renard@insa-lyon.fr1
onthestabilityofstandardschemeandonthequalityoftheapproximation,especiallyforthecomputationofLagrangemultiplierscorrespondingtotheconstraints.Theclassicalsemi-discretizations,forexamplewithfiniteelementmethods,giveaprob-lemintimewhichisameasuredifferentialinclusion(see[19,20,21,22]).Suchadifferentialinclusionissystematicallyill-posed,unlessanadditionalimpactlawisconsidered.How-ever,theschemeobtainedwiththeadditionofanimpactlawin[21]leadsalsotospuriousoscillations.Thesemi-discretizationweproposehereleadstoaproblemwhichisequivalenttoaregularLipschitzordinarydifferentialequation.Thus,timeintegrationschemesatleastconvergeforafixedspacediscretizationwhenthetimesteptendstozero.Thisworkgeneralizesinasensethemethodspresentedin[13,8].Theoutlineofthepaperisthefollowing.Section1isdevotedtothedescriptionoftheabstracthyperbolicequationwithconstraintsandtheequivalentvariationalinequality.Section2presentsthenewapproximationmethodsandthemainresultsofwell-posednessandenergyconservation.Then,insection3,anon-trivialmodelproblemwhichcorrespondsforinstancetothedynamicsofathinmembraneunderanobstacleconditionisdeveloped.Anexampleofwell-poseddiscretizationisalsobuiltinthissection.Section4brieydescribesthefullydiscreteproblemobtainedwiththefinitedifferencemidpointschemeandpresentssomenumericalexperimentsonthismodelwhichshowsinparticularthatthemidpointschemeisstablewithwell-posedsemi-discretizationsandunstableotherwise.Finally,inSection5,theproposedmethodisappliedtoanelastodynamiccontactproblem.1TheabstracthyperbolicequationLetΩRdbeaLipschitzdomainandH=L2(Ω)thestandardHilbertspaceofsquareintegrablefunctionsonΩ.LetWbeaHilbertspacesuchthatWHW0,withdensecompactandcontinuousinclusionsandletA:WW0bealinearself-adjointellipticcontinuousoperator,i.e.whichsatisfieshAw,viW0,W=hAv,wiW0,W,v,wW,α>0,wW,hAw,wiW0,Wαkwk2W,c>0,wW,kAwkW0ckwkW.WeconsiderthefollowingproblemFindu:[0,T]Ksuchthat2u2(t)+Au(t)fNK(u(t))fora.e.t(0,T],(1)tuu(0)=u0,(0)=v0,twhereKisaclosedconvexnonemptysubsetofW,fW0,u0K,v0H,T>0andNK(u)isthenormalconetoKdefinedby(seeforinstance[5]foradetailedpresentationofdifferentialinclusions)ifu/K,NK(u)={fW0:hf,wuiW0,W0,wK}ifuK.2