Level Set Methods for Fluid-Structure Interaction

203 Pages·2022·7.319 MB·English

by  Georges-Henri Cottet, Emmanuel Maitre, Thomas Milcent

Applied Mathematical Sciences
Georges-Henri Cottet
Emmanuel Maitre
Thomas Milcent
Level Set Methods
for Fluid-Structure
Interaction
Applied Mathematical Sciences
FoundingEditors
F.John
J.P.LaSalle
L.Sirovich
Volume 210
SeriesEditors
AnthonyBloch, Departmentof Mathematics,Universityof Michigan,Ann Arbor,
MI,USA
C. L. Epstein, Department of Mathematics, University of Pennsylvania, Philadel-
phia,PA,USA
AlainGoriely,DepartmentofMathematics,UniversityofOxford,Oxford,UK
LeslieGreengard,NewYorkUniversity,NewYork,NY,USA
AdvisoryEditors
J. Bell, Center for Computational Sciences and Engineering, Lawrence Berkeley
NationalLaboratory,Berkeley,CA,USA
P. Constantin, Department of Mathematics, Princeton University, Princeton, NJ,
USA
R.Durrett,DepartmentofMathematics,DukeUniversity,Durham,CA,USA
R. Kohn, Courant Institute of Mathematical Sciences, New York University, New
York,NY,USA
R. Pego, Department of Mathematical Sciences, Carnegie Mellon University,
Pittsburgh,PA,USA
L.Ryzhik,DepartmentofMathematics,StanfordUniversity,Stanford,CA,USA
A.Singer,DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,USA
A.Stevens,DepartmentofAppliedMathematics,UniversityofMünster,Münster,
Germany
S.Wright,ComputerSciencesDepartment,UniversityofWisconsin,Madison,WI,
USA
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Georges-Henri Cottet (cid:129) Emmanuel Maitre (cid:129)
Thomas Milcent
Level Set Methods for
Fluid-Structure Interaction
Georges-HenriCottet EmmanuelMaitre
LaboratoireJeanKuntzmann LaboratoireJeanKuntzmann
UniversitéGrenobleAlpes,CNRS, UniversitéGrenobleAlpes,CNRS,
Grenoble-INP* Grenoble-INP*
Grenoble,France Grenoble,France
ThomasMilcent
InstitutdeMécaniquedeBordeaux
ArtsetMétiers,UniversitéBordeaux,
CNRS
Bordeaux,France
ThisworkwassupportedbyAgenceNationaledelaRecherche(ANR-19-CE46-0001,ANR-17-CE23-
0024)
ISSN0066-5452 ISSN2196-968X (electronic)
AppliedMathematicalSciences
ISBN978-3-031-08658-8 ISBN978-3-031-08659-5 (eBook)
https://doi.org/10.1007/978-3-031-08659-5
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Foreword
Fluid-structureinteractionproblemsoccurinmanyfieldsofengineering:mechanics
of immersed structures, aeroelasticity, biomechanics, lubrication effects in pipe
flows,particletransport,sedimenttransport,anderosionofshorelines.
From a mathematical and numerical point of view, these problems represent a
real challenge insofar as fluids and solids are naturally described by models of
a different nature: Eulerian models for fluids and Lagrangian models for solids.
These models require adapted discretization methods. The traditional approachto
deal with these problems, known as ALE (for Arbitrary Lagrangian Eulerian), is
basedonthesevariousmodelizations.ItusesaEuleriandiscretizationoffluidsand
a Lagrangian discretization of solids, with adapted meshes and suitable coupling
conditionstotranslatethecontinuityofvelocityandforcesattheinterfaces.
In the last 15 years, alternatives to these methods have appeared, based on
Eulerian models of the two types of media. Fluids and solids are considered as a
single system with constitutive laws that vary in space and time. The interest of
thesemethodsisthepossibilitythattheyoffertouseasinglenumericalmodeland
a single mesh for the whole system, with the drawback of giving a less detailed
descriptionoftheinterfaceandtheconditionsthatweimposeonit.
Theimmersedboundarymethods(IBMinshort)proposedbyPeskinin1972can
beseenasaclassofintermediatemethodsbetweenthetwoapproachesmentioned
above.Thesolidsareimmersedinthefluidwithouttheneedtofitafluidmeshon
them,butaretrackedinaLagrangianfashionusingmarkersadvectedbythefluid.
For a fairly complete review of these different methods, we refer to the recent
book by T. Richter [119] for the ALE and Eulerian methods, and to the review
article[92]fortheimmersedboundarymethods.
The purpose of this book is to describe Eulerian models of the fluid-structure
interactionbasedon LevelSetfunctions.LevelSetmethodsare wellknownsince
theworksofSethian[125]andOsher[112]inimageprocessingandthecomputation
of multi-phaseflows. More recently,they have also been used with success in the
context of shape optimization [4]. These methods allow implicit monitoring of
Lagrangianinterfaces(withphysicaladvectionfieldsformultiphaseflowsorvirtual
v
vi Foreword
flows in image processing or shape optimization) by solving partial differential
transportequations.
For 10 years or so, a certain number of works, in particular by the authors of
this book, have made it possible to use them to model the forces resulting from
deformations of elastic solids or to treat contacts between objects and leading to
numerical tools in fluid-structure interaction in a Eulerian framework. This book
reviews these different works. Its objective is to describe the models. It does not
focusonthenumericaldiscretizationmethods,exceptforwhatconcernstheexplicit
orimplicittimediscretizationswhichimpactthestabilityofthemodels.Infact,one
of the advantages of Eulerian methods is that they allow the use of conventional
methodsinfluiddynamics(finitedifferences,finitevolumes,etc.)withoutinvolving
questions specific to the coupling with solid structures. In this book, the spatial
discretization techniques used are only briefly mentioned to describe numerical
illustrations.
The plan of the book is as follows. In Chap.1, we review the techniques for
interfacecapturingorinterfacetracking.WeshowinparticularhowtouseLevelSet
functionsto translate surface forcesinto bulk forces, which is obviouslya central
pointinthesemethods,andwespecifyhowthesemethodsallowindimensiontwo
andthreetoaccountforcurvatures.WedeveloptheexamplesofLevelSetmethods
mentionedpreviouslyinimageprocessingandformultiphaseflows,andweaddress
the questionsof stability in Level Set methods in the example of the treatmentof
surface tension terms, questionswhich will be taken up in the Chap.3. Chapter 2
complements the reminders of Chap.1 by notions of differential calculus on the
trajectoriesin the Lagrangianand Euleriandescriptions. The conservationlaws in
thesedescriptionsarealsorecalled.
Chapter 3 deals with the first example of fluid-structure interaction, that of an
elasticmembraneinteractingwithafluid,firstinthecaseofamembranereactingto
thevariationofareatheninthegeneralcaseofamembranealsoreactingtoshear.
It also evokes the case of elastic curves immersed in a space of dimension three.
Finally,thischaptercontainsanexampleofcodewritteninFreeFEM++toallow
thereadertoexperiencethesemethodsforhimself.
Chapter 4 generalizesthe LevelSet approachto arbitraryelastic bodies,with a
distinctionbetween the cases of completelycompressibleor incompressiblefluid-
structuremedia.
Chapter 5 deals with the case of rigid solids, or solids deformable under the
actionofprescribedexternalforces.Inthesecases,thefluid-structureinteractionis
treated by the penalization method. Chapter 6 is concerned with the treatment by
LevelSetmethodsofcontactsbetweenobjects,whethertheseobjectsareelasticor
rigid. In particular, we describe a fast algorithm for processing multiple contacts.
Finally, an appendixdetails certain technical elementsof differentialcalculus and
givesthedemonstrationofcertainresultsusedinthebook.Italsogivessomebasic
toolsconcerningclassicalfinite-differencefortheresolutionoftransportequations.
Chapters 3–6 are largely independent of each other. Chapters 2 and 4 contain
the elements of continuum mechanics (covering solids and fluids) necessary for
the understandingof the bookso as to makeit accessible to studentsof Masterof
NumericalAnalysis.
Contents
1 LevelSetMethodsandLagrangianInterfaces ........................... 1
1.1 InterfaceTrackingorInterfaceCapturing ............................. 1
1.2 LevelSetMethodsandGeometryofSurfaces......................... 4
1.3 LevelSetMethodsandGeometryofCurvesinR3.................... 8
1.4 Expression of Surface Forces Using the Level Set
Function ................................................................. 10
1.4.1 Example1:ImageProcessing.................................. 15
1.4.2 Exemple2:SurfaceTension.................................... 17
1.5 NumericalAspectsI:ConsistencyandAccuracy ..................... 19
1.5.1 Redistancingofϕ............................................... 21
1.5.2 Renormalizationofϕ........................................... 23
1.5.3 ComparisonoftheTwoApproaches........................... 24
1.5.4 TowersMethodtoApproximateSurfaceIntegrals............ 27
1.6 NumericalAspectsII:Stability......................................... 28
1.6.1 ExplicitScheme................................................. 31
1.6.2 ImplicitScheme................................................. 34
1.6.3 Semi-ImplicitScheme.......................................... 35
2 MathematicalToolsforContinuumMechanics .......................... 39
2.1 CharacteristicsandFlowsAssociatedwithaVelocityField.......... 39
2.2 ChangeofVariables..................................................... 43
2.3 ReynoldsFormulas...................................................... 46
2.4 ConservationofMass................................................... 50
2.4.1 MassConservationinEulerianFormulation .................. 50
2.4.2 MassConservationinLagrangianFormulation............... 50
2.5 ConservationofMomentum............................................ 51
2.5.1 MomentumConservationinEulerianFormulation........... 51
2.5.2 MomentumConservationinLagrangianFormulation........ 51
vii
viii Contents
3 InteractionofanIncompressibleFluidwithanElasticMembrane.... 53
3.1 FromtheImmersedBoundaryMethodtoLevelSetMethods........ 54
3.2 ImmersedMembrane:CaseWithoutShear............................ 57
3.2.1 LevelSetFormulationoftheElasticDeformationof
aHypersurfaceImmersedinaIncompressibleFluid ......... 57
3.2.2 Level Set Formulationof Elastic Energyand
Fluid-StructureCouplingintheIncompressibleCase ........ 61
3.2.3 GeneralizationtoCompressibleFlows ........................ 64
3.2.4 TakingintoAccountCurvatureForces ........................ 66
3.2.5 KortewegModelsandExistenceofSolutions................. 68
3.3 ImmersedMembrane:TheCasewithSurfaceShear.................. 70
3.3.1 LevelSetApproachforSurfaces............................... 70
3.3.2 AnEulerianTensortoMeasureSurfaceDeformation........ 72
3.3.3 InvariantsandAssociatedElasticForce ....................... 73
3.3.4 EnergyandCouplingModel ................................... 77
3.4 CurvesImmersedinR3................................................. 79
3.4.1 AnEulerianTensortoMeasureStrainsAlongCurves........ 79
3.4.2 InvariantsandAssociatedElasticForce ....................... 80
3.5 ExplicitandSemi-implicitTimeDiscretizations...................... 82
3.5.1 ExplicitSchemes................................................ 82
3.5.2 Semi-implicitScheme.......................................... 83
3.5.3 NumericalValidation........................................... 84
3.6 NumericalIllustrationsandSampleCode ............................. 87
3.6.1 Shear-FreeMembrane.......................................... 87
3.6.2 MembranewithShear .......................................... 92
4 ImmersedBodiesinaFluid:TheCaseofElasticBodies................ 99
4.1 HyperelasticMaterialsinLagrangianFormulation ................... 100
4.1.1 PrincipleofMaterialIndifference.............................. 101
4.1.2 IsotropicMaterials.............................................. 102
4.1.3 ComputationoftheStressTensorinaLagrangian
Framework...................................................... 103
4.2 HyperelasticMaterialsinEulerianFormulation....................... 104
4.2.1 ComputationoftheStressTensorinanEulerian
Framework...................................................... 104
4.2.2 ElasticConstitutiveLawsforElasticMedia................... 106
4.2.3 EulerianElasticityintheIncompressibleCase................ 106
4.2.4 EulerianElasticityintheCompressibleCase ................. 107
4.3 Fluid-StructureCouplingModelintheIncompressibleCase......... 108
4.3.1 ModelandConstitutiveLawintheIncompressibleCase..... 109
4.3.2 NumericalIllustrations......................................... 110
4.4 Fluid-StructureCouplingintheCompressibleCase .................. 116
4.4.1 ModelandConstitutiveLawintheCompressibleCase ...... 119
4.4.2 NumericalScheme.............................................. 120
4.4.3 NumericalIllustration .......................................... 122
Contents ix
5 ImmersedBodiesinIncompressibleFluids:TheCaseofRigid
Bodies ......................................................................... 125
5.1 The Penalization Method for Flow Around Bodies
withGivenVelocity..................................................... 126
5.2 TheCaseoftheTwo-WaysFluid-SolidInteraction................... 127
5.3 RemarksontheNumericalImplementation........................... 130
5.4 ExtensionsofthePenalizationMethod................................ 132
5.5 NumericalIllustrations.................................................. 133
5.5.1 KissingandTumblingofTwoSpheres........................ 134
5.5.2 FlowsAroundOscillatingObstacles........................... 135
5.5.3 AnguilliformSwimmers........................................ 138
6 ComputingInteractionsBetweenSolidsbyLevelSetMethods ........ 143
6.1 LevelSetMethodtoModelInteractionForces........................ 144
6.1.1 PointRepulsionModel......................................... 144
6.1.2 SurfaceRepulsionModelbyLevelSetMethod............... 145
6.1.3 TakingintoAccountCohesionandDampingForces ......... 146
6.1.4 NumericalIllustrations......................................... 147
6.2 AnEfficientMethodforDealingwithContactsBetween
MultipleObjects ........................................................ 149
6.2.1 Motivation....................................................... 149
6.2.2 TheAlgorithm .................................................. 150
6.2.3 ComputationalEfficiencyoftheMethod...................... 155
6.2.4 NumericalIllustrations......................................... 156
7 Annex.......................................................................... 161
7.1 ExamplesofCurvatureCalculationsUsingaLevelSetFunction.... 161
7.1.1 TheCaseoftheEllipsoid....................................... 161
7.1.2 TheCaseoftheTorus .......................................... 162
7.2 JustificationoftheResultsUsedforMembraneswithShear......... 163
7.2.1 ProofoftheResultsConcerningtheZ Invariant............. 163
1
7.2.2 AnalyticalIllustrationsforZ .................................. 168
2
7.3 Justification of the Results Used for the Curves
ParameterizedinR3..................................................... 173
7.3.1 ProofoftheResultsConcerningtheInvariantZ ............. 174
3
7.3.2 AreaandCo-areaFormulas.................................... 177
7.3.3 Volume Approximationof Line Integralsand
CalculationoftheElasticForce................................ 178
7.4 WENOSchemesfortheTransportEquation .......................... 182
7.5 SomeIdeastoGoFurther............................................... 186
CreditsofFiguresReproducedwithPermission ............................. 189
References......................................................................... 191