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Publié par | universitat_stuttgart |
Publié le | 01 janvier 2010 |
Nombre de lectures | 61 |
Langue | English |
Poids de l'ouvrage | 16 Mo |
Extrait
Robustnumericalalgorithmsfor
dynamicfrictionalcontactproblems
withdierenttimeandspacescales
VonderFakulta¨tMathematikundPhysikderUniversita¨tStuttgart
zurErlangungderWu¨rdeeinesDoktorsder
Naturwissenschaften(Dr.rer.nat.)genehmigteAbhandlung
Hauptberichter:
Mitberichter:
Vorgelegtvon
CorinnaHager
ausBerlin
Hauptberichter:Prof.Dr.B.Wohlmuth
Mitberichter:Prof.Dr.T.Laursen
Prof.Dr.A.Klawonn
Tagdermu¨ndlichenPru¨fung:17.November2010
Institutfu¨rAngewandteAnalysisundNumerischeSimulation
Universita¨tStuttgart
0102
39D
(Diss.
Universit
ta¨
Stuttgart)
Acknowledgments
Thisthesissummarizesthemainresultsofmyresearchactivitiesduringthelastfour
yearsatthechair“NumerischeMathematikfu¨rHo¨chstleistungsrechner”oftheInstitut
fu¨rAngewandteAnalysisundNumerischeSimulationattheUniversita¨tStuttgart.
Firstofall,IwouldliketoexpressmysinceregratitudetomysupervisorProf.Dr.Bar-
baraWohlmuthforhersurpassingguidanceandmentoringduringthistime.Besidesher
numerousactivitiesandengagements,shehasalwaysbeenwillingtoprovideadviceand
assistanceforallkindsofquestions.Heroutstandingdedicationtotheresearchactivities
oftheworkgrouphasbeentrulyinspiringtome.Inparticular,Iwouldliketothank
herforgivingmethepossibilitytopresentmyworkonsomeinternationalconferences
andforputtingmeintotouchwithseveraldesignatedexperts,whichhasleadtofruitful
andinterestingdiscussions.
SpecialthanksgotoProf.Dr.TodLaursenandProf.Dr.AxelKlawonnfortheir
willingnessandeorttowritetherefereereportsforthisthesis.Further,Iwouldliketo
thankProf.Dr.HelmutHarbrechtforhisparticipationintheoraldefense.Iammuch
obligedtoProf.Dr.PatrickLeTallecforinvitingmetoEcolePolytechniqueandtohim
andDr.PatriceHauretformanystimulatingandinspiringdiscussions.Manythanks
alsotoProf.Dr.LucaPavarinoandDr.MarilenaMunteanufortheirkindwelcomeat
Universit`adiMilano,aswellastoProf.Dr.RainerHelmigandAndreasLauserforthe
goodcooperation.
DuringmyemploymentattheIANS,Igreatlyenjoyedthecooperativeandfriendlyat-
mosphereinourworkgroup.IwouldliketothankDr.StephanBrunßen,Dr.YufeiCao,
Dr.BerndFlemisch,Dr.ArpirukHokpunna,Dr.AndreasKlimke,Dr.BishnuLamich-
hane,JuliaNiemeyer,Dr.IrynaRybak,Dr.EvgenySavenkov,MarcSchlienger,Dr.Igor
Shevchenko,BritSteiner,AlexanderWeißandespeciallyDr.StefanHu¨eberfortheir
help,patienceandadviceinvarioustopics.
Iamdeeplyindeptedtomyparentsfortheircontinuoussupportandunderstanding,
aswellasforprovidingmewiththepossibilitytopursuitanacademiccareer.Thanks
alsotomyfriendsfortheirmoralbackupandtheirwelcomedistractions.
Finally,IwouldliketoexpressmydeepestappreciationtoJohannesforhisconstant
encouragementandsupport.Withouthisloveandpatience,itwouldnothavebeen
possibletocompletethiswork.
Stuttgart,July2010
CorinnaHager
iii
vi
Contents
Abstract
Zusammenfassung
xi
ix
IIntroductionandproblemformulation15
1Continuummechanics17
1.1Dynamicelasticity..............................17
1.2Plasticity...................................20
1.3Frictionalcontact...............................22
2Discretizationtechniques25
2.1Weakformulation...............................25
2.2Spatialdiscretization.............................27
2.3Timestepping.................................30
2.4Reformulationoftheinequalityconstraints.................32
IINonlinearsolversforfrictionalcontactandplasticity37
3SemismoothNewtonmethods39
3.1Semismoothfunctions............................39
3.2Newtonmethodforsemismoothfunctions.................42
3.3Abstractframework..............................43
4Applicationtoplasticityandfrictionalcontact47
4.1Applicationofabstractframeworktoplasticity...............47
4.1.1SemismoothNewtonscheme.....................48
4.1.2Numericalresultsforplasticity...................53
4.2Applicationofabstractframeworktofrictionalcontact..........57
4.2.1SemismoothNewtonscheme.....................58
4.2.2Numericalresultsforcontact....................63
4.3Combinationoftheschemes.........................65
4.3.1Combinedalgorithm.........................66
4.3.2Numericalresultsforplasticcontactproblem............67
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Contents
IIIDAEsolversfordynamicnormalcontact71
5Massmodicationtechniques73
5.1Whymassmodication?...........................74
5.1.1Indexreduction............................74
5.1.2Two-massoscillatingsystem.....................75
h¯5.2ConstructionofM..............................79
05.2.1QuadratureruleQ..........................81
15.2.2QuadratureruleQ..........................81
5.2.3Propertiesofthequadraturerules..................83
h5.3Dierentinterpretationof¯m(,)......................84
ih5.3.1InterpolationoperatorI.......................85
0h5.3.2InterpolationoperatorI.......................86
15.4Numericalresults...............................87
5.4.1Nonlinearbeamin2D........................87
5.4.2Frictionlesstwo-bodycontactin2D.................89
5.4.3Frictionaltwo-bodycontactin2D..................91
5.4.4Comparisonwithstabilizedpredictor-correctorscheme......94
5.4.5Frictionaltwo-bodycontactin3D..................95
6Apriorierrorestimates99
6.1Semi-discretesystem.............................100
6.2Fullydiscretesystem.............................106
6.3Numericalresults...............................110
IVIterativesolversforproblemswithdierentscales113
7Overlappingdomaindecomposition115
7.1Settingandproblemformulation.......................117
7.1.1Problemstatement..........................117
7.1.2Spatialdiscretization.........................118
7.1.3Timediscretization..........................122
7.1.4Schurcomplementformulation....................122
7.2Iterativecouplingalgorithm.........................124
7.2.1Derivation...............................124
7.2.2Errorpropagation...........................125
7.2.3Conditionnumberanalysis......................128
7.2.4Stoppingcriteria...........................133
7.3Numericalresults...............................134
7.3.1Geometryandparameters......................134
7.3.2Algebraicerrorforstaticcase....................134
7.3.3Algebraicerrorfordynamiccase...................137
7.3.4ComparisonwithDirichlet–Neumannalgorithm..........138
iv
Contents
7.3.5Algebraicerrorfornonnestedtracespaces.............139
7.3.6Alternativecouplingalgorithm....................142
7.3.7Nearlyincompressiblematerial...................143
8ODDMfornonlinearproblems145
8.1Nonlinearsetting...............................145
8.2Approximatesolutionschemes........................146
8.2.1Nestediterations...........................146
8.2.2Coarsegridapproximations.....................149
8.3Numericaltests................................150
8.3.1Geometricallyconformingsettingin2D...............150
8.3.2Geometricallyconformingsettingin3D...............152
8.3.3Tireapplicationin2D........................154
8.3.4Tireapplicationin3D........................156
9Localtimesubcycling161
9.1Continuousoutput..............................161
9.2Timesubstepping...............................165
9.3Approximatesolutionscheme........................176
9.4Numericalresults...............................176
9.4.1Discretizationerroroftimesubcycledsystem............176
9.4.2AlgebraicerrorofAlgorithm4....................180
9.4.3Tireapplicationin2D........................182
10Concludingremarks
Bibliography
581
781
ivi
Contents
iiiv
Abstract
Inmanytechnicalandengineeringapplications,numericalsimulationisbecomingmore
andmoreimportantforthedesignofproductsortheoptimizationofindustrialpro-
ductionlines.However,thesimulationofcomplexprocessesliketheformingofsheet
metalortherollingofacartireisstillaverychallengingtask,asnonlinearelasticor
elastoplasticmaterialbehaviourneedstobecombinedwithfrictionalcontactanddy-
namiceects.Inaddition,theseprocessesoftenfeatureasmallmobilecontactzone
whichneedstoberesolvedveryaccuratelytogetagoodpictureoftheevolutionofthe
contactstress.Inordertobeabletoperformanaccuratesimulationofsuchintricate
systems,thereisahugedemandforarobustnumericalschemethatcombinesasuitable
multiscalediscretizationofthegeometrywithanecientsolutionalgorithmcapableof
dealingwiththematerialandcontactnonlinearities.Theaimofthisthesisistodesign
suchanalgorithmbycombiningseveraldierentmethodswhicharedescribedinthe
following.
Ourmaineldofapplicationisstructuralmechanics.Here,webasetheimplemen-
tationonniteelementmethodsinspaceandimplicitnitedierenceschemesintime.
Theconditionsforbothplasticityandfrictionalcontactaregivenintermsofasetof
localinequalityconstraintswhichareformulatedbyintroducingadditionalinnerordual
degreesoffreedom.Fortheplasticcontributions,thedualvariablesaredenedwith
respecttotheelements,whereasthecontactmultipliersareassociatedwiththepotential
contactnodes.Asthemeshesaregenerallynon-matchingatthecontactinterface,we
employmortartechniquestoincorporatethecontactconstraintsinavariationallycon-
sistentway.Byusingbiorthogonalbasisfunctionsforthediscretemultiplierspace,the
contactconditionscanbeenforcednode-wise,andatwo-bodycontactproblemcanbe
solvedinthesamewayasaone-bodyproblem.
Thenextstepintheconstructionofanecientsolutionalgorithmistoreformulate