Robust numerical algorithms for dynamic frictional contact problems with different time and space scales [Elektronische Ressource] / vorgelegt von Corinna Hager
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Robust numerical algorithms for dynamic frictional contact problems with different time and space scales [Elektronische Ressource] / vorgelegt von Corinna Hager

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Robust numerical algorithms fordynamic frictional contact problemswith different time and space scalesVon der Fakult¨at Mathematik und Physik der Universit¨at Stuttgartzur Erlangung der Wu¨rde eines Doktors derNaturwissenschaften (Dr. rer. nat.) genehmigte AbhandlungVorgelegt vonCorinna Hageraus BerlinHauptberichter: Prof. Dr. B. WohlmuthMitberichter: Prof. Dr. T. LaursenProf. Dr. A. KlawonnTag der mu¨ndlichen Pru¨fung: 17. November 2010Institut fur Angewandte Analysis und Numerische Simulation¨Universitat Stuttgart¨2010D93 (Diss. Universit¨at Stuttgart)AcknowledgmentsThis thesis summarizes the main results of my research activities during the last fouryears at the chair“Numerische Mathematik fur Hochstleistungsrechner” of the Institut¨ ¨fu¨r Angewandte Analysis und Numerische Simulation at the Universita¨t Stuttgart.Firstofall,Iwouldliketoexpress mysincere gratitudetomysupervisorProf.Dr.Bar-bara Wohlmuth for her surpassing guidance and mentoring during this time. Besides hernumerous activities and engagements, she has always been willing to provide advice andassistance forall kinds ofquestions. Her outstandingdedication totheresearch activitiesof the work group has been truly inspiring to me. In particular, I would like to thankher for giving me the possibility to present my work on some international conferencesand for putting me into touch with several designated experts, which has lead to fruitfuland interesting discussions.

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Published 01 January 2010
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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

v

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

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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
thelocalinequalityconditionsforplasticityandcontactintermsofnondierentiable
equalities.Thesenonlinearcomplementarity(NCP)functionscanbecombinedwiththe
equationsforthebulkmaterialtoformasetofnonlinearsemismoothequationswhich
arethensolvedbymeansofageneralizedformoftheNewtonmethodforsemismooth
systems.Duetothelocalstructureoftheinequalityconstraints,thisiterativescheme
canbeimplementedasanactivesetstrategywheretheactivesetsareupdatedineach
Newtoniteration.Further,theadditionaldualdegreesoffreedomcaneasilybeeliminated
usinglocalstaticcondensation,suchthatonlyasystemofthesizeofthedisplacement
needstobesolvedineachstep.Weremarkthatthewell-knownradialreturnmethodis
aspecialcaseofthisgeneralframeworkiftheplastichardeninglawsarelinear.
However,theconvergencepropertiesoftheNewtoniterationstronglydependonthe
choiceoftheNCPfunction.Inthiscontext,weshowthatthefunctioncorresponding

xi

Abstract

totheradialreturnmethodisnotoptimal,andwepresentafamilyofmodiedNCP
functionswhichallowforbetterconvergenceresults.
Anotherimportantissuefortherobustsimulationofdynamiccontactproblemsis
relatedtotheinertiaterms.Ifstandardtimediscretizationschemeslikethetrapezoidal
ruleareused,thecontactstressoftenshowsspuriousoscillationsintimethatbecome
worsewhenthetimestepisrened.Inordertoavoidthiseect,weemployamodied
massmatrixwherenomassisassociatedwiththecontactnodes.Bythis,theoriginal
semi-discretesystemdecouplesintoanalgebraicequationintimeforthecontactnodes
andanordinarydierentialequation(ODE)intimefortheothernodes.Thisinturn
leadstomuchsmootherresultsforthecontactstress.Wepresentanecientwayof
obtainingthemodi