Numerics of Unilateral Contacts and Friction: Modeling and Numerical Time Integration in Non-Smooth Dynamics

185 Pages·2009·3.771 MB·English

by  Christian Studer (auth.)

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Numerics of Unilateral Contacts
and Friction

Modeling and Numerical Time Integration in
Non-Smooth Dynamics

Christian Studer

123
Dr. Christian Studer
IMES-Center of Mechanics,
ETH Zurich,
8092 Zurich,
Switzerland
E-mail: [email protected]

ISBN: 978-3-642-01099-6 e-ISBN: 978-3-642-01100-9

DOI 10.1007/978-3-642-01100-9

Lecture Notes in Applied and Computational Mechanics ISSN 1613-7736
e-ISSN 1860-0816

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Preface
Mechanicsprovidesthelinkbetweenmathematicsandpracticalengineeringappli-
cations. It is one of the oldest sciences, and many famous scientists have left and
will leave their mark in this fascinating field of research.Perhapsone of the most
prominentscientistsinmechanicswasSirIsaacNewton,whowithhis“lawsofmo-
tion”initiatedthedescriptionofmechanicalsystemsbydifferentialequations.And
still today, more than 300 years after Newton, this mathematical concept is more
actualthanever.
Therisingcomputerpowerandthedevelopmentofnumericalsolversfordiffer-
entialequationsallowedengineersallovertheworldtopredictthebehavioroftheir
physicalsystemsfastandeasyinannumericalway.Andthetrendtocomputational
simulationmethodsisstillfurtherincreasing,notonlyinmechanics,butpractically
inallbranchesofscience.Numericalsimulationwillprobablynotsolvetheworld’s
engineeringproblems,butitwillhelpforabetterunderstandingofthemechanisms
ofourmodels.
Withtheriseofcomputerpower,demandsinmechanicalmodelingobtainedan
additionalfocus.In“pre-computationalmechanics”scientistsaimedatshortandin-
telligentformulations,forexampleordinary(linear)differentialequationsexpressed
inminimalsetsofcoordinates.The“post-computationalmechanics”alsoconsiders
questionslikehowtoprovidesimpleandgeneralmodelingstructureswhicharealso
suitedforacomputer.Amachine,whichcanprocesslargeamountsofdata,butis
limited within its intelligence,i.e. rightthe opposite of a human being. Questions
arise how such different “mechanisms” like computersand human beings can in-
teract and complement each other. An example of this change in demands is the
conceptof differentialalgebraic equations,which are probablynot the suited tool
for “hand-evaluation”but are one of the favored ways how computers can set up
generalstructuresforconstrainedmechanicalproblems.
The rising computer power has also brought concepts into focus, for which
excellenttheoreticalframeworkexistedbutwhichhavebeenso fartoodatainten-
sive for practicalevaluation.One such field of increasingresearchactivityis non-
smooth mechanics,which deals with mechanicalsystems whose time evolutionis
notsmoothanymore.Suchmodelingapproacheshelpforthebetterunderstanding
VI Preface
of systems with friction and impacts, which are omnipresentin our world, but of
whichstilllittleisknownfromthemodelingandnumericaltreatmentpointofview.
Perhapsthemostimportantcontributioninnon-smoothmechanicsoriginatesfrom
JeanJacquesMoreauinthelate80’s.Hereplacedtheequationsofmotionandthe
constraintequationsknownfromclassicalmechanicsbymeasuredifferentialequa-
tionsincombinationwithinequalities.Doingso,Moreauwasablenotonlytodeal
with classical smooth mechanics but also to account for non-smooth events like
stick-sliptransitionsorimpacts.Inthissense,Moreaudidnotextendclassicalme-
chanics to some non-smoothspecial cases but set up a complete new formulation
whichincorporatesclassicalmechanics.Hisexcellenttheoreticalworkwasaccom-
paniedbyanumericalintegrationscheme,whichlinksthemathematicalframework
withpracticalapplications.
InmyyearsasresearchassistantattheCenterofMechanicsattheETHZurichI
obtainedthechancetostudythisrelativelynewfieldofmechanics.Itriedtobridge
thegapbetweenhighlytheoreticalmathematicaltheoriesinnon-smoothmechanics
andpracticalapplicationinsoftwarecodes,Itriedtoidentifythestructureofnon-
smooth mechanics which allowed for simple and elegant implementation codes.
Duringthistime,IwasaccompaniedbymanypeoplewhomI wouldliketothank
for their kind support of my work, which resulted finally in this book. Especially
I wantto thankmysupervisorProf.Dr.-Ing.Dr.-Ing.habil. ChristophGlockerfor
guidingmyresearchandforthemanyinterestingdiscussionsaboutmechanicsand
the non-smoothness of our world. Prof. Glocker managed to create a fruitful un-
complicatedatmospherewhichallowedmetofollowmyownideasandtodevelop
freelywithinhissupport.Hewasalsopresentindifficulttimesandencouragedme
withhisenthusiasmwhenspeakingaboutmechanics.Hisbroadknowledgeinnon-
smooth mechanics and his strong quest for simple, consistent and elegant math-
ematical formulations have been a valuable inspiration for me. I also would like
to thank Prof.Dr.habil.BernardBrogliato for reviewingmy scientific work. Prof.
Brogliatoiswellknowninthenon-smoothdynamicscommunityandheadsthere-
searchteamBIPOPattheINRIAinFrance.Furtherthanksgotoallmycolleagues
atthecenterofmechanics,whomadetheinstituteaninteresting,pleasantandopen
minded workplace. Special thanks go to Dr. ir. habil. Remco Leine who assisted
mewithmuchknowledgeandwhoalsoreviewedmywork,toUeliAeberhardwho
gavememathematicsupport,andtoMichaelMo¨llerwithwhomIhadmanydiscus-
sionsaboutnumericsandprogramming.FinallyIwanttothankmyfamilyandmy
friendsandespeciallyNadjaGerberforprovidingthemotivatingsocialbackground
andmentalsupport.
Zurich,2008 ChristianStuder
Contents
1 Introduction.................................................. 1
1.1 AimandScope ........................................... 1
1.2 LiteratureSurvey.......................................... 4
1.2.1 ModelingofUnilateralContactsandFriction............ 4
1.2.2 TimeDiscretization ................................. 5
1.2.3 SolversforInequalityProblems ....................... 6
1.2.4 Application ........................................ 7
1.3 Outline .................................................. 7
2 MathematicalPreliminaries .................................... 9
2.1 UsedNorms.............................................. 9
2.2 Derivatives............................................... 10
2.3 ConvexAnalysis .......................................... 10
2.3.1 ConvexSets,Indicator/SupportFunctions,Normal
Cones............................................. 10
2.3.2 ProximalPointandVectorDistanceFunctions........... 11
2.3.3 ProximalPointFunctionsforVariousConvexSets ....... 13
2.4 GlobalandLocalRepresentations............................ 14
2.5 DifferentialAlgebraicEquations............................. 15
3 Non-SmoothMechanics........................................ 17
3.1 EquationsofMotionintheSmoothCase ...................... 17
3.2 Non-ImpulsiveMotion ..................................... 19
3.2.1 Set-ValuedForceLaws .............................. 19
3.2.2 ExamplesofSet-ValuedForceLaws ................... 21
3.2.3 EquationsofMotionintheNon-SmoothCase ........... 24
3.3 Impacts.................................................. 26
3.4 EqualityofMeasures ...................................... 27
4 InclusionProblems............................................ 31
4.1 LinearComplementarityProblems ........................... 32
4.2 AugmentedLagrangianApproach............................ 32
VIII Contents
4.2.1 GeneralProblem.................................... 33
4.2.2 ApplicationtoNon-SmoothDynamics ................. 35
4.2.3 IterativeSolutionProcess ............................ 36
4.2.4 Choiceofr andConvergence......................... 39
i
4.2.5 UseofNon-DiagonalR’s ............................ 44
4.2.6 SuccessiveSolutionoftheSet-ValuedForceLaws ....... 48
4.2.7 Examples.......................................... 49
4.3 AlternativeApproaches .................................... 53
4.3.1 UncouplingtheSet-ValuedForceLaws................. 54
4.3.2 ExactRegularization ................................ 55
4.4 Summary ................................................ 57
5 Time-Stepping................................................ 59
5.1 DiscretizationofDifferentialAlgebraicEquations .............. 59
5.1.1 Index-1,AccelerationLevel .......................... 61
5.1.2 Index-2,VelocityLevel .............................. 62
5.1.3 Index-3,DisplacementLevel.......................... 64
5.1.4 Linearization of the Gap Function and Drift
Stabilization ....................................... 68
5.1.5 Conclusions........................................ 69
5.2 TimeEvolutionofNon-smoothSystems ...................... 71
5.2.1 Event-DrivenSchemes............................... 73
5.3 Time-SteppingMethods.................................... 74
5.3.1 GeneralDiscretizationTechnique...................... 75
5.3.2 Moreau’sMidpointRule ............................. 76
5.3.3 TheModifiedΘ-Method............................. 79
5.3.4 Time-SteppingbyPaoliandSchatzman................. 80
5.3.5 Time-SteppingbyStiegelmeyr,Funk,Foerg,Pfeifferetal.. 81
5.3.6 Time-SteppingbyAnitescu,Potra,Stewart,Trinkleetal... 82
5.3.7 Time-SteppingbyGGL.............................. 88
5.3.8 Time-SteppingbyPreconditioning..................... 90
5.3.9 DiscussionandConclusions .......................... 92
6 AugmentedTime-Stepping ..................................... 99
6.1 IntegrationOrderofMoreau’sMidpointRule .................. 99
6.1.1 DefinitionoftheIntegrationOrder..................... 99
6.1.2 ExpansionoftheExactSolutionq(t)andu(t) ........... 102
6.1.3 ExpansionofMoreau’sTime-SteppingScheme.......... 105
6.1.4 LocalIntegrationOrder .............................. 107
6.1.5 GlobalIntegrationOrder ............................. 108
6.2 StepSizeAdjustmentforSwitchingPoints .................... 109
6.2.1 DeterminingtheDiscreteStateσˆ ...................... 110
6.2.2 LocalizingSwitchingPoints .......................... 112
6.3 HigherOrderIntegrationforSmoothTimeSteps ............... 114
6.3.1 Extrapolation....................................... 115
6.3.2 ExtrapolationAppliedtoMoreau’sMidpointRule........ 116
Contents IX
6.4 OverallAlgorithm......................................... 117
6.4.1 IntegrationOrder ................................... 117
6.4.2 Implementation..................................... 119
6.5 Examples ................................................ 120
6.5.1 PointMassFallingonaTable......................... 120
6.5.2 PointMassSlidingonaTable......................... 120
6.5.3 SingleDOFImpactOscillator......................... 123
6.5.4 TheWoodpeckerToy................................ 125
7 ThedynamYSoftware......................................... 129
7.1 Overview ................................................ 129
7.2 ModelingaPartialMechanicalSystem........................ 135
7.3 ModelingaNon-SmoothElement............................ 136
7.4 ModelingExternalForces .................................. 138
7.5 OneStepandSimulationObjects ............................ 139
7.6 Examples ................................................ 139
7.6.1 Bodiesin3DSpace ................................. 140
7.6.2 Non-CommonSet-ValuedLaws....................... 147
7.6.3 GranularMedia..................................... 154
7.6.4 Motorbike ......................................... 155
7.6.5 Elevator ........................................... 156
8 Summary .................................................... 161
Glossary ......................................................... 165
ProjectiontoanEllipseortoanEllipsoid ............................ 169
B.1 ProjectiontoEllipseContour................................ 169
B.2 ProjectiontoEllipsoidContour .............................. 170
References........................................................ 171
Index ............................................................ 179
Chapter 1
Introduction
1.1 Aim andScope
Reality is neither smooth nor non-smooth. Models of reality are smooth or non-
smoothdependingonthequestionsweask.Ifaphysicalsystemhasrapidlychang-
ingphases,thenitcanbeadvantageoustomodelthesysteminanon-smoothway.
Formechanicalsystemstheimpacttimesareusuallymuchsmallerthantheglobal
motionwhichisofinterest.Thismotivatesthestudyofrigidmultibodydynamics.
Theset-valuedforcelawswhichmodeltheconstitutivebehaviourofunilateralcon-
tactsandoffrictionleadtonon-smoothmodels.Usually,thepositionsareassumed
tobeabsolutelycontinuous,whilethevelocitiesareallowedtoundergojumpsand
are taken to be of bounded variation. Jumps in the velocities can not be accom-
plishedbyfinitebutonlybyimpulsiveforces.
Apointmassfallingdowntothegroundisasimpleexampleofamechanicalsys-
temwithoneunilateralcontact.Inaplanarnon-smoothmodeling,thepointmass’s
twodegreesoffreedomarereducedtoonewhenthepointmasstouchestheground.
If friction is considered additionally, then the degrees of freedom are reduced to
zerointhecaseofsticking.Thus,therearedifferentequationsofmotioninminimal
coordinates for these different configurations. In case of an impact, an additional
impactlawmustbeapplied.
Thisbookfocussesonnumericaltimeintegratorsforthedynamicsofnon-smooth
mechanicalmodels.Suchmodelsconsistofaset ofrigidbodieswhichmayinter-
actwith eachother.Similar to classical smoothmodels,the state of a non-smooth
mechanical model is described by a set of (minimal or non-minimal) generalized
positionsq=q(t)andgeneralizedvelocitiesu=u(t).Unlikesmoothmodels,the
time evolution of these states q and u is not smooth. While the generalized po-
sitions q are at least continuous, the generalized velocities u=q˙ a.e.1 may even
jump at certain time instances. As a consequence, the accelerations might not be
defined. Such non-smoothbehaviourresults from set-valuedforce laws which act
1The abbreviation a.e. stands for almost everywhere. It indicates that q˙ =u does not hold at
singletimeinstancesatwhichimpactsoccur.
C.Studer:NumericsofUnilateralContactsandFriction,LNACM47,pp.1–8.
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