Table Of ContentInternational Series of Numerical Mathematics
169
Volker Schulz
Diaraf Seck
Editors
Shape Optimization,
Homogenization and
Optimal Control
DFG-AIMS workshop held at the AIMS
Center Senegal, March 13-16, 2017
ISNM
InternationalSeries ofNumerical Mathematics
Volume169
ManagingEditors
G.Leugering,Erlangen,Germany
M.Hintermueller,Berlin,Germany
AssociateEditors
V.N.Starovoitov,Novosibirsk,Russia
N.Kenmochi,Kita-ku,Kyoto,Japan
R.W.Hoppe,Houston,USA
Z.Chen,Beijing,China
HonoraryEditor
K.-H.Hoffmann,Garching,Germany
Moreinformationaboutthisseriesathttp://www.springer.com/series/4819
Volker Schulz • Diaraf Seck
Editors
Shape Optimization,
Homogenization and Optimal
Control
DFG-AIMS workshop held at the AIMS
Center Senegal, March 13-16, 2017
Editors
VolkerSchulz DiarafSeck
FBIVAbt.Mathematik DépartementdeMathématiquesdela
Universita¨tTrier UniversitéCheikhAntaDiopdeDaka
Trier,Germany Dakar-Fann,Senegal
ISSN0373-3149 ISSN2296-6072 (electronic)
InternationalSeriesofNumericalMathematics
ISBN978-3-319-90468-9 ISBN978-3-319-90469-6 (eBook)
https://doi.org/10.1007/978-3-319-90469-6
LibraryofCongressControlNumber:2018951124
MathematicsSubjectClassification(2010):49Q10,35B27,49J15,49J20,65K10
©SpringerInternationalPublishingAG,partofSpringerNature2018
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Preface
This volume contains the proceedings of the DFG-AIMS Workshop on shape
optimization,homogenizationandcontrol,heldatAIMSSénégalinMbour,Séné-
gal, on March 13–16, 2017. This workshop joined mathematicians from Sénégal,
Germany and various other countries to discuss the topics of shape optimization,
homogenizationandcontrolandtoexploremutualinterestsandoptionsoffurther
jointresearch.
The contributionsin this volumegive an insightinto currentresearchactivities
inAfrica,Germanyandinternationallyinthosefields.Seedsforcollaborationcan
be found in the first four papers in the field of homogenization. Modelling and
optimalcontrolin partialdifferentialequationsis the topic of the nextsix papers,
again mixed from Africa and Germany. Finally, new results in the field of shape
optimizationarediscussedinthefinalinternationalthreepapers.
This workshop has been supported by the Deutsche Forschungsgemeinschaft
(DFG),whichisthelargestself-governingfundingorganisationforbasic research
inGermanyandinEurope,andbytheAfricanInstituteforMathematicalSciences
(AIMS)inSenegal,whichisoneofsixcentresofapan-Africannetworkofcentres
of excellence for postgraduate education, research and outreach in mathematical
sciences.
We wish to thank all the contributors to this workshop for making it an
outstandingscientific and intellectual event. We are convincedthat this workshop
and the contributions collected in this volume show that future collaborations in
the field of mathematics between Africa and Germany are already visible on the
horizon.
Trier,Germany VolkerSchulz
Dakar-Fann,Senegal DiarafSeck
v
Contents
HomogenizationinMagnetic-Shape-MemoryPolymerComposites....... 1
SergioConti,MartinLenz,MatthäusPawelczyk,andMartinRumpf
HomogenizationofStochasticParabolicEquations
inVaryingDomains.............................................................. 19
MogtabaA.Y.MohammedandMamadouSango
Two-Scale Convergence: Obviousness of the Choice of Test
Functions?NotAlways.......................................................... 41
HubertNnang
CorrectorProblemandHomogenizationof NonlinearElliptic
MonotonePDE ................................................................... 57
JeanLouisWoukeng
Instantaneous Optimal Controlof Friction Dominated Flow
inaGas-Network................................................................. 75
GünterLeugeringandGisèleMophou
MultilevelIterationsfortheCombinedMovingHorizonEstimation
andNonlinearModelPredictiveControlforPDEModels.................. 89
EkaterinaKostinaandGregorKriwet
SomeRecentDevelopmentsinOptimalControlofMultiphaseFlows..... 113
MichaelHintermüllerandTobiasKeil
LocalizedModelReductioninPDEConstrainedOptimization............ 143
MarioOhlberger,MichaelSchaefer,andFelixSchindler
Numerical Simulation for a Dimensionless Coupled System
ofShallow WaterEquationwithLongTerm Dynamicof Sand
DunesEquation .................................................................. 165
MouhamadouA.M.T.BaldéandDiarafSeck
vii
viii Contents
CouplingtheNavier-StokesEquationswithaShortTermDynamic
ofSandDunes .................................................................... 191
IbrahimaFaye,MariamaNdiaye,andDiarafSeck
BranchingStructuresinElasticShapeOptimization........................ 213
NoraLüthen,MartinRumpf,SaschaTölkes,andOrestisVantzos
ShapeandTopologicalDerivativesviaOneSidedDifferentiation
oftheMinimaxofLagrangianFunctionals ................................... 227
MichelC.Delfour
OnOptimizationTransferOperatorsinShapeSpaces...................... 259
VolkerH.SchulzandKathrinWelker
Homogenization in
Magnetic-Shape-Memory
Polymer Composites
SergioConti,MartinLenz,MatthäusPawelczyk,andMartinRumpf
Abstract Magnetic-shape-memory materials (e.g. specific NiMnGa alloys) react
with a large change of shape to the presence of an external magnetic field. As
an alternative for the difficult to manufacture single crystal of these alloys we
study composite materials in which small magnetic-shape-memory particles are
embeddedinapolymermatrix.Themacroscopicpropertiesofthecompositedepend
strongly on the geometry of the microstructure and on the characteristics of the
particlesandthepolymer.
We present a variational model based on micromagnetism and elasticity, and
derive via homogenization an effective macroscopic model under the assumption
that the microstructure is periodic. We then study numerically the resulting cell
problem,and discuss the effectof the microstructureon the macroscopicmaterial
behavior. Our results may be used to optimize the shape of the particles and the
microstructure.
Keywords Homogenization · Magneticshapememory · Micromagnetism ·
Calculusofvariations · Shapeoptimization
S.Conti((cid:2))
InstitutfürAngewandteMathematik,UniversitätBonn,Bonn,Germany
e-mail:[email protected]
M.Lenz
InstitutfürNumerischeSimulation,UniversitätBonn,Bonn,Germany
e-mail:[email protected]
M.Pawelczyk
InstitutfürGeometrie,TechnischeUniversitätDresden,Dresden,Germany
e-mail:[email protected]
M.Rumpf
InstitutfürNumerischeSimulation,UniversitätBonn,Bonn,Germany
e-mail:[email protected]
©SpringerInternationalPublishingAG,partofSpringerNature2018 1
V.Schulz,D.Seck(eds.),ShapeOptimization,Homogenization
andOptimalControl,InternationalSeriesofNumericalMathematics169,
https://doi.org/10.1007/978-3-319-90469-6_1
2 S.Contietal.
MathematicsSubjectClassification(2010). Primary74Q05;Secondary74F15
1 Introduction
Thepeculiarmechanicalbehaviorofmagneticshapememory(MSM)alloysrenders
them interesting for many applications. At the same time, they give rise to a
new class of models, whose mathematical study is challenging. MSM alloys are
multiferroic materials, which combine a magnetic phase transition with a shape-
memory one. The deformation driven by a magnetic field deformation is due to
a coupling between the two order parameters,which arises from the differencein
magneticanisotropybetweenthe elastic variants. A typicalexampleis a NiMnGa
alloy[15,21,26,29,30].
Shape-memorymetalsarecharacterizedbyaphasetransformationfromahigh-
temperature, high-symmetry phase, called austenite, to a low-temperature, low-
symmetryphase, called martensite. Since the pointgroupof the martensite lattice
issmallerthanintheaustenitephase,thereareanumberofdifferentvariantsofthe
martensiticphase,whichareobtainedfromtheausteniteviadifferentbutsymmetry-
related eigenstrains. Whereas in ceramic materials elastic phase transformations
typically generate spontaneousstrains of less than 1%, in alloys the deformations
canbesignificant.ForexampleinNiMnGasinglecrystalsthedeformationbetween
two martensitic variants is about 11%. The practical exploitation of the shape-
memoryeffectforactuationhasbeenhinderedbythefactthatthetransitionbetween
austenite and martensite can often be induced only by slow heating and cooling
processes.Furthermore,itisdifficulttoselectonevariantovertheother.Thisleads
in typical situations to low-temperature states characterized by a fine mixture of
thedifferentvariants,inwhichmartensiteappearswithoutsignificantmacroscopic
deformation.
The couplingof the structuralphase transformationto a magneticfield renders
the materials much easier to control. The difference in anisotropy between the
different variants of the martensitic phase leads to a coupling between the mag-
netizationandtheeigenstrainandprovidesasimplemechanismforselectingoneof
thevariantsovertheothers.Atleastincleansinglecrystals,largestrainsupto11%
havebeeninducedbytheapplicationofexternalmagneticfields[21,26,29,30].
Itishoweverpracticallyverydifficulttoproducesinglecrystalsofgoodquality.
Additionally, single crystals tend to be brittle. At the same time, in polycrystals
the deformation is blocked by incompatibilities at the grain boundaries. It has
thereforebeen proposedto use small single-crystalMSM particlesembeddedin a
softermatrix[12,13,16,25,27],althoughmagneticallyinducedstrainsaretypically
smallerthaninsinglecrystals[17,18,28].Thecompositegeometryopenstheway
foroptimizationoftheparticleshapeandingeneralofthemicrostructure.
Inthispaperwereviewrecentmathematicalprogressandextendtheanalysisof
MSM-polymercomposites.A modelbasedonmicromagnetismandelasticity was
developedin [7], and will be presented in Sect.2 below.Under the assumptionof
