Approximation of Nonsmooth Optimization Problems and Elliptic Variational Inequalities with

181 Pages·2017·1.84 MB·English

by  Simon Rösel

Approximation of Nonsmooth Optimization Problems
and Elliptic Variational Inequalities with Applications to
Elasto-Plasticity
D I S S E R TAT I O N
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat.)
im Fach Mathematik:
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultät
der Humboldt-Universität zu Berlin
von
Dipl.-Math. Simon Rösel
Präsidentin der Humboldt-Universität zu Berlin:
Prof. Dr.-Ing. Dr. Sabine Kunst
Dekan der Mathematisch-Naturwissenschaftlichen Fakultät:
Prof. Dr. Elmar Kulke
Gutachter:
1. Prof. Dr. Michael Hintermüller
2. Prof. Dr. Roland Glowinski
3. Prof. Dr. Batmanathan Dayanand Reddy
Tag der mündlichen Prüfung: 7.2.2017

IwouldliketoexpressmysincereappreciationandgratitudetomyadviserProf.Dr.MichaelHintermüller
forhisongoingguidanceandsupportthroughoutmydoctoralstudies(andevenbefore). Iamespecially
indebtedtohimforprovidinganinspiringresearchenvironmenttostudythebeautifulfieldof
infinite-dimensionaloptimizationundervariousfascinatingaspects,rangingfromfunctionalanalysisover
finiteelementmethods
tomanifoldapplications.
IwouldalsoliketothankmycolleaguesintheresearchgroupofProf.Dr.MichaelHintermüller. In
particular,IhavetomentionDr.CarlosN.Rautenbergforthefruitfulcollaborationonthesignificanceof
densitypropertiesofconvexsets,andDr.DonatWegnerformanyinspiringdiscussionsandhelpful
comments,especiallyinthebeginningofmydoctoralstudies.
Finally,IwouldliketoexpressmydeepestgratitudetomyfiancéeFranziska,myparentsRoswithaand
Gert-SteffenandmysisterDeborah. YouhavesupportedmemorethanIcouldhaveeverwishedfor.

Contents
Introduction 1
I Preliminaries: FunctionalAnalysis, OptimizationandVariationalInequal-
ities 5
1 Functional Analysis 7
1.1 GeneralNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 SomeFrequentlyUsedFunctionSpaces . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Spacesofsmoothfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Smoothnessofdomains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 Spacesderivedfromdistributiontheory . . . . . . . . . . . . . . . . . . . . . 10
1.2.4 Sobolevspacesonmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Optimization in Banach Spaces 21
2.1 AbstractExistenceResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 ConvexOptimizationandFenchelDuality . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 TheSemismoothNewtonMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Basicpropertiesandcalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Meshindependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Γ-andMosco-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Variational Inequalities 33
3.1 SomeGeneralities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 ApproximationofVariationalInequalities . . . . . . . . . . . . . . . . . . . . . . . . 33
II Density of Convex Intersections and Applications 37
4 Motivation 39
4.1 OptimizationProblemswithConvexConstraints . . . . . . . . . . . . . . . . . . . . 39
4.2 EllipticVariationalInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Quasi-monotoneperturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 Galerkinapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.3 Singularperturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Density Results for Pointwise Constraint Sets in Sobolev Spaces 47
5.1 ConeConstraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 ContinuousObstacles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 DiscontinuousObstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3.1 AcounterexampleforobstaclesinSobolevspaces . . . . . . . . . . . . . . . 52
5.3.2 LowersemicontinuousobstaclesandLp-spaces . . . . . . . . . . . . . . . . 58
5.3.3 LowersemicontinuousobstaclesandSobolevspaces . . . . . . . . . . . . . 59
5.3.4 SupersolutionsofellipticPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 62
v
Contents
6 Applications 67
6.1 FiniteElements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.1.1 OnthesignificanceofMosco-convergence . . . . . . . . . . . . . . . . . . . 67
6.1.2 FiniteElementdiscretizedconvexsets . . . . . . . . . . . . . . . . . . . . . . 69
6.2 ImageRestoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
III An Infinite-Dimensional Semismooth Newton Solver for Elasto-Plastic
Contact Problems 79
7 Quasi-Static Hardening Plasticity and Contact Condition 81
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 Quasi-StaticPlasticitywithHardening . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.3 ContactCondition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.4 FunctionSpaceSettingandVariationalFormulation . . . . . . . . . . . . . . . . . . 85
7.4.1 Elasto-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.4.2 Elasto-plasticcontactproblem . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8 The Time-Discretized Elasto-Plastic Contact Problem 89
8.1 ProblemFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.1.1 Thegeneralcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.1.2 ThecaseofvonMisesplasticityandlinearhardening . . . . . . . . . . . . . 90
8.1.3 Areducedformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.2 AFenchelDualProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.2.1 Fencheldualityset-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.2.2 ComputationoftheFenchelconjugates . . . . . . . . . . . . . . . . . . . . . 94
8.2.3 Primal-dualoptimalityconditions . . . . . . . . . . . . . . . . . . . . . . . . 96
8.2.4 Interpretationofthedualvariables . . . . . . . . . . . . . . . . . . . . . . . . 97
9 A Duality-Based Path-Following Strategy 99
9.1 TheRegularizedProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.2 ASemismoothNewtonMethodfortheRegularizedProblem . . . . . . . . . . . . . 104
9.3 TheDiscreteSolver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9.3.1 Regularizationsetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9.3.2 ThediscretesemismoothNewtonalgorithm . . . . . . . . . . . . . . . . . . 109
9.3.3 Aninexactpath-followingalgorithm. . . . . . . . . . . . . . . . . . . . . . . 111
9.3.4 Numericaltests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
IV Duality Results and Regularization Schemes for Perfect Plasticity 121
10 Perfect Plasticity 123
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
10.2 ThePrandtl-ReussModelofPerfectPlasticity . . . . . . . . . . . . . . . . . . . . . . 124
10.3 FunctionSpaceSettingandVariationalFormulations . . . . . . . . . . . . . . . . . 126
10.3.1 Johnson’sweakformulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
10.3.2 Quasi-staticevolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
vi
Contents
11 The Time-Incremental Problem 133
11.1 ProblemStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
11.2 Inf-SupProblemFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
11.3 TheIncrementalStressProblemasaFenchelDualProblem . . . . . . . . . . . . . . 140
12 A Modified Visco-Plastic Regularization for the Time-Incremental Problems 143
12.1 Visco-PlasticRegularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
12.2 AModifiedVisco-PlasticRegularization . . . . . . . . . . . . . . . . . . . . . . . . . 145
12.3 AnInfinite-DimensionalDualSolverfortheRegularizedTime-IncrementalStress
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
12.3.1 Theshiftedproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
12.3.2 Tikhonovregularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
12.3.3 Aninfinite-dimensionalsemismoothNewtonmethod. . . . . . . . . . . . . 154
12.3.4 Outlookonadiscretesolver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Conclusion, Outlook and Some Related Open Problems 157
vii

Introduction
Thesubjectofthisthesisistheefficientsolutionofellipticvariationalinequalityproblemsand
optimization problems with convex constraints defined over some (infinite-dimensional) real
BanachspaceX. Givensomeoperator A: X → X∗,theabstractvariationalinequalityproblemis
definedasfollows:
Findu ∈ X : ⟨A(u),v−u⟩+j(v)−j(u) ≥ ⟨l,v−u⟩, ∀v ∈ X, (0.0.1)
where j: X → R∪{+∞}isanonsmoothconvexfunctionandl ∈ X∗ denotesaboundedlinear
functionalonX. Alargeclassofvariationalinequalitiesisdeterminedastheoptimalitycondition
tooptimizationproblemsofthetype
inf F(v)−⟨l,v⟩+j(v) overv ∈ X, (0.0.2)
whereF : X →RisaGâteauxdifferentiablebutnotnecessarilyconvexfunctionwithF′ = A. In
particular,thecaseofaconvexconstraintsetK ⊂ Xisformallycontainedintheaboveformulation
bysetting j(v) = +∞forallv ∈/ K. Theabstractproblems(0.0.1)and(0.0.2)ariseinamyriadof
applicationsinvolvingphysics,engineering,finance,lifesciencesandmanymore. Inthisthesis,
thefocusismainlyonapplicationsinquasi-staticassociativeelasto-plasticity,wherethevariable
vdescribesthematerialbehaviorsubjecttoagivenexternalloadingl. Materialpropertiesand
modelparametersdeterminetheoperator Aandthefunctionalj.
Ingeneral, theseproblemscannotbesolveddirectlyapartfromspecialcases. Ononehand,
thefactthattheaboveproblemsareposedinsomeinfinite-dimensionalBanachspacealready
underlinesthenecessityofanapproximationintermsofasuitablefinite-dimensionalproblem,
forexample, usingfiniteelementmethods. Ontheotherhand, itisoftenpreferabletoreplace
theoriginalproblembysomeperturbedversionevenifasuitablediscretizationisestablished.
Thismaybemotivatedbythefactthatthediscreteproblemisnotsolvablewithoutunreasonable
effortorthattheassociatedsolverdisplaysundesirablepropertieslikemesh-dependenceorlack
ofrobustness. Thus,theanalysisof(0.0.1)(or(0.0.2))isintimatelylinkedtotheinvestigationof
thebehavioroftheproblemwithrespecttocertainclassesofperturbationsincludingdifferent
discretizationandregularizationapproaches. Inthiscontext,themainparadigmofthisworkis
thattheunderstandingoftheunderlyinginfinite-dimensionalproblemstructureiscrucialforboth,
theanalysisoftheconsistencyofperturbationmethods,aswellasthepropertiesofcorresponding
solutionalgorithms. Here, wegenerallyemploythetermconsistencywheneversolutionsofa
perturbedversionconverge(inasensetospecify)toasolutionoftheoriginalproblem.
FollowingapreliminarydiscussionofbasictoolsfromFunctionalAnalysisandOptimization
in Part I, Part II begins with a unified consistency analysis of standard perturbation methods
comprising,amongothers,Galerkinapproximation,Moreau-Yosida-regularization,singularper-
turbationoracombinationofthelatter. Thiswillbeachievedbyintroducingthegeneralclassof
quasi-monotoneperturbations,both,onthelevelofthevariationalinequalityandtheconstrained
optimizationproblem. UsingtheconceptsofΓ-convergence[37]andMosco-convergence[95]in
thisgeneralframework,itturnsoutthatcertaindensitypropertiesofintersectionsoftheconvex
constraintsetKwithdensesubspacesYoftheBanachspaceXnaturallyariseasanecessaryand
sufficientconsistencycriterion. ThecorrespondingΓ-andMoscolimits,whichcanbeunderstood
asappropriatelimitproblemstoasequenceofperturbations,aredeterminedbytheclosureof
1
Introduction
theseintersections.
In the literature on variational inequality problems, density properties classically appear as
intermediate steps towards the convergence results of finite element methods under minimal
regularity[89,53,61]. Inthecontextofplasticityproblems,densityresultsrepresentanimportant
ingredientforrelaxationapproaches[7,38,15]. However,theconsiderationofdensityproperties
insuchageneralframeworkappearstobenovel.
InChapter5,wemainlyfocusonthecasewhereX = X(Ω)isausual(vector-valued)Lebesgue
orSobolevspaceoveraboundeddomainΩ ⊂ RN,Y = Y(Ω)isthesubspaceofcontinuousor
smoothvectorfieldsandKprescribesapointwiseboundonthenormofthefunctionvalueviaan
obstaclefunctionαdefinedonΩ,i.e.,
K = K(X(Ω)) = {w ∈ X(Ω) : |w(x)| ≤ α(x)a.e. inΩ}. (0.0.3)
WefurtherintroducetheproblemformulationandgivesomenewdensityresultsinLebesgue
andSobolevspacesforcontinuousobstaclesrelyingonthetheoryofmollification. Thesestate-
ments extend recent results from [69] which seems to be the only available reference which is
specificallydedicatedtoabstractdensitypropertiesofsetsofthetype(0.0.3). Withthehelpof
Mosco-convergenceofconvexsets,theseresultscarryovertolowersemicontinuousobstacles. A
differentapproachisproposedforupperboundsαwhicharesupersolutionsofanellipticpartial
differentialequation(PDE).Inthelattercase,thesmoothingisachievedbysolvingasequence
ofellipticPDEs. Itisfurthershownthatresultsofthistypecannotbeexpectedingeneralifthe
obstacleisjustaSobolevfunction,andforthispurposeaconcretecounterexampleisgiven. In
Chapter6,wemakeuseoftheprecedingdensityresultsforthecontinuoussettingtoprovethe
Mosco-convergenceofvarioustypesoffiniteelement-discretizationsofKinthespiritof[53]. Asa
consequence,thediscretizedproblemsareconsistentwiththecorrespondinginfinite-dimensional
limit problem; see, for instance, [95, 53] or Theorem 3.1 and Theorem 6.3. In the last section,
we return to the infinite-dimensional setting by studying density properties in the context of
dualizationintotalvariationbasedimagerestoration. ThemajorityoftheresultsinPartIIhave
beenobtainedwithinajointprojectwithM.HintermüllerandC.N.Rautenberg,andasofMay
2016theresultsarenotyetpublished.
InPartIII,weconsideranelasto-plasticcontactproblemunderthesmallstrainassumption.
Theproblemischaracterizedbythepresenceofarigidobstaclewhichrestrictsthedeformation
of the material under a loading process. In contrast to elasticity, plastic material behaviour is
irreversibleandthesetofadmissiblestressesisconstraintbydefinition. Whiletheliteratureon
discreteplasticityisextensive(seeChapter7formanyreferences),infinite-dimensionalalgorithms
areratherscarceandessentiallyonlycontainlinearlyconvergentsubspacecorrectionmethods[24].
InChapter9,aninfinite-dimensionalsemismoothNewtonmethodisproposedfortheone-step
time-discretizedcontactproblemofquasi-staticelasto-plasticitywithcombinedkinematic-isotropic
hardening. Neglectingfriction,thecombinationofbotheffects,contactandplasticity,leadstothe
problemofavariationalinequalityofthemixedkind,i.e.,thefunctionaljin(0.0.1)isnonsmooth
onitseffectivedomainandthelatterisaproperconvexsubsetofX.
We note that the semismooth Newton method based on the notion of Newton- or slant dif-
ferentiability [31, 75] has received considerable attention throughout the last decade as it has
provedtobearemarkablyefficientmethod,notablyforthesolutionofvariousproblemsinPDE-
constrainedoptimization[75,66,67]andofvariationalinequalities[41,68,87],tomentiononlya
few. Thefactthattheoriginalelasto-plasticcontactproblemdoesnotallowforaNewtondiffer-
entiablereformulationmotivatestheconsiderationofaspecialFencheldualproblem(Chapter8)
oftheso-calledprimalproblemofquasi-staticplasticity[61]. Thisdualproblemturnsouttobe
amenabletoaregularizationschemewheretheassociatedsolutionpathisinducedbyacoupled
2