THE FINITE ELEMENT IMMERSED BOUNDARY METHOD Adviso

UNIVERSIT`ADEGLISTUDIDIPAVIADIPARTIMENTODIMATEMATICATHEFINITEELEMENTIMMERSEDBOUNDARYMETHODAdvisor:Ph.D.ThesisProf.DanieleBOFFILucaHELTAIOctober200623AFederico,BeppeeRoy.4Contents1ContinuumMechanics191.1BasicPrinciples...........................201.1.1TheReferenceandDeformedConfigurations.......201.1.2TransportTheorem.....................241.1.3ConservationofVolume..................241.1.4ConservationofMass...................251.1.5ConservationofMomenta.................261.2Navier-StokesEquations......................301.2.1StrongFormulation.....................301.2.2VariationalFormulation..................331.2.3EnergyEstimates......................341.3Elasticity..............................351.3.1Hyper-Elasticity......................361.3.2Incompressibility......................401.3.3Viscoelasticity.......................421.3.4StrongFormulation.....................441.3.5VariationalFormulation..................451.3.6EnergyEstimates......................472Fluid-StructureInteraction492.1ClassicalFormulation........................512.2ImmersedBoundaryMethod....................542.3OriginalFormulation........................562.3.1Hyper-ElasticFormulation.................582.3.2Co-dimensionOne.....................642.3.3VariationalFormulation..................662.3.4FullyVariationalFormulation...............682.3.5EnergyEstimates......................712.4TheoreticalResults.........................722.4.1APrioriEstimates.....................722.4.2FixedPointApproach...................7756CONTENTS2.4.3CharacteroftheSolutionforaCo-DimensionOneModelProblem813NumericalApproximation873.1FiniteElementImmersedBoundaryMethod............883.2TimeApproximation........................913.2.1BackwardEuler/BackwardEuler..............933.2.2ForwardEuler/BackwardEuler..............943.3StabilityAnalysis..........................953.3.1StabilityoftheSpaceDiscretizedProblem........963.3.2StabilityoftheSpaceandTimeDiscretizedProblem...963.4AnAlternativeFiniteElementApproach..............1054NumericalExperiments1114.1ConvergenceAnalysis.......................1134.1.1StaticTwo-DimensionalBalloon..............1154.1.2DynamicTwo-DimensionalBalloon............1174.1.3Areaconservation.....................1204.1.4StaticTwo-DimensionalVisco-ElasticShell........1244.2NumericalStability.........................1314.2.1Co-dimensionone,two-dimensionaldomain.......1334.2.2Co-dimensionzero,two-dimensionaldomain.......1374.2.3Co-dimensionone,three-dimensionaldomain.......1404.2.4Co-dimensionzero,three-dimensionaldomain......1414.3Classicaltwo-dimensionalmodelproblems............1434.3.1FluidDrivenTwo-DimensionalBalloon..........1464.3.2BallooninLidDrivenCavity................1474.3.3ParametrizationDependencies...............1494.3.4NonClosedImmersedBoundary..............1494.3.5InteractionBetweenImmersedBoundaryandDomainBoundary1514.3.6InteractionBetweenTwoImmersedBoundaries......1524.3.7Nozzle...........................1534.3.8DynamicTwo-DimensionalVisco-ElasticShell......1544.3.9Two-DimensionalVisco-Elastic“Radial”Shell......1564.4MovingMembranesinThree-DimensionalFluid.........1594.4.1DynamicThree-DimensionalBalloon...........1614.4.2ThreeDimensionalCube..................1614.4.3InteractionBetweenImmersedBoundaryandDomainBoundary1615Conclusions165CONTENTS76Appendix1696.1ListofSymbols...........................1696.2ElementaryVectorandTensorRelations..............1726.3JumpConvention..........................1728CONTENTSAcknowledgmentsFirstly,IwouldliketothankmySupervisor,ProfessorDanieleBoffi.IcouldnothaveimaginedhavingabetteradvisorandmentorformyPhD,andwithouthiscommon-sense,knowledgeandperceptivenessIwouldneverhavemadeit.Thank-youtoProfessorLuciaGastaldiforhercracking-of-the-whipandforhav-ingputupwithmy“approximate”mathematicalknowledgeintheselastyearsofworkingtogether.Thank-youtoProfessorCharlesS.Peskinforthecountlessdiscussionsandenlighteningcommentsthatdeepenedmypassionforthisdiverseanddifficultsubject,throughoutmyentirestayinNewYork.Thank-youtomyreviewer,ProfessorRicardoH.Nochetto,formanagingtoreadthewholemanuscriptsothoroughlyinsuchashorttime,andforhiseye-openingandseverecomments.ThisthesishasbeenpartiallysupportedbyaFulbrightgrantinstitutedbytheItalianFulbrightCommission.Iamgratefultothisinstitutionforthewonderfulopportunitytheygaveme.IwouldalsoliketothankalltherestoftheacademicstaffoftheDepartmentofMathematicsattheUniversityofPavia.Youhavebeenamodeltofollownotonlyfromtheacademicpointofview.AspecialonegoestoFrancoBrezzi(doIneedtosaywhy?)andtoAnnalisaBuffa,forherattemptto“naturalize”mymathematicalknowledge.MuchrespecttomyPhDfriendsinItaly,withwhomIhadcountlesslunchesandconstructivediscussions.ThankstoLuis,BoyceandBarneyforputtingupwithmy“Italianity”,andtoLaura,EvaandCiaforputtingupwiththefourofusforsomanyendlessevenings.Onadifferentnote,IwouldliketothankthecoffeeproducersofItalyfor910CONTENTSkeepingmethinking,theLivignocoffeetradersformakingmefeelokayaboutdrinkingsomuchcoffeeandthemusicofPinkFloyd,FrancoBattiato,FrancescoDeGre