ACOMPUTATIONALFRAMEWORKBASEDONANEMBEDDEDBOUNDARYMETHODFORNONLINEARMULTI-PHASEFLUID-STRUCTUREINTERACTIONSADISSERTATIONSUBMITTEDTOTHEINSTITUTEFORCOMPUTATIONALANDMATHEMATICALENGINEERINGANDTHECOMMITTEEONGRADUATESTUDIESOFSTANFORDUNIVERSITYINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYKevinGuanyuanWangDecember2011AbstractNonlinear uid-structure interaction (FSI) is a dominating feature in many importantengineeringapplications. Examplesincludeunderwaterimplosions, pipelineexplo-sions, apping wings for micro aerial vehicles, and shock wave lithotripsy. Due to theinherent nonlinearity and system complexity, such problems have not been thoroughlyanalyzed,whichgreatlyhinderstheadvanceofrelatedengineeringelds.This thesis focuses onthedevelopment, verication, andvalidationof auid-structure coupled computational framework for the solution of nonlinear multi-phaseFSI problems involving high compressions and shock waves, large structural displace-mentsanddeformations,self-contact,andpossiblytheinitiationandpropagationofcracksinthestructure.First, anembeddedboundarymethodfor solving3Dmulti-phasecompressibleinviscidowsonarbitrary(i.e. structuredandunstructured)nonbody-conformingCFDgridsispresented. Keycomponentsinclude: (1)robustandecientcompu-tational algorithmsfortrackingopen, closed, andcrackinguid-structureinterfaceswithrespecttothexed,nonbody-conformingCFDgrid;(2)anumericalalgorithmbasedontheexactsolutionoflocal, one-dimensional uid-structureRiemannprob-lemstoenforcetheno-interpenetrationtransmissionconditionattheuid-structureivinterface;and(3)twoconsistentandconservativealgorithmsforenforcingtheequi-libriumtransmissionconditionatthesameinterface.Next, the multi-phase compressible ow solver equipped with the aforementionedembeddedboundarymethodis carefullycoupledwithanextendednite elementmethod(XFEM)basedstructuresolver,usingapartitionedprocedureandprovablysecond-order explicit-explicit and implicit-explicit time-integrators. In particular, theinterface tracking algorithms in the embedded boundary method are adapted to track-ing embedded discrete interfaces with phantom elements and carrying implicitly rep-resentedcracks.Finally,theresultinguid-structurecoupledcomputationalframeworkisappliedto the solution of several challenging FSI problems in the elds of aeronautics, under-waterimplosionsandexplosions, andpipelineexplosionstoassessitsperformance.Inparticular,twolaboratoryexperimentsareconsideredforvalidationpurpose: therst one concerns the implosive collapse of an air-lled aluminum cylinder; the secondonestudiesthedynamicfractureofpre-awedaluminumpipesdrivenbydetonationwaves. In both cases, the numerical simulation correctly reproduces in a quantitativesensetheimportantfeaturesintheexperiment.vAcknowledgementsFirstandforemost, I oermydeepestgratitudetomyadvisor, ProfessorCharbelFarhat,forhisconstantencouragement,support,andguidancethroughthelastveyears. Without him, this workwouldnot have beenpossible. I alsothankhimfor providingme withunique opportunities for interactionwithindustryandtheacademicworld.I would like to thank the other members of my thesis committee, Professor AdrianLew and Professor Gianluca Iaccarino, for reviewing this work and providing me withhelpfuladviceandcomments.I wishtoexpress mywarmandsincere thanks toProfessor RonFedkiwandProfessor Stelios Kyriakides for their detailed suggestions and constructive commentsonthedevelopmentandvalidationofthecomputationalframeworkproposedinthiswork. I am also grateful to Professor Jean-Frederic Gerbeau and Dr. Michel LesoinnefortheirprecioushelpduringtheirvariousvisitstoStanford.Iwouldliketoexpressthankstomyformerandcurrentresearchcollaborators:Dr. Arthur Rallu, Dr. Phil Avery, Jon Gretarsson, Alex Main, Dr. Jeong-Hoon Song,PatrickLea, andDr. Liang-Hai Lee. Byworkingwiththem, I havebeenabletosignicantlyextendmyknowledge of computational mathematics andmechanicalengineering. I am also very grateful to all my colleagues in the Farhat Research Group,viwith whom I have spent ve years of joyful time. Special thanks to Dr. Charbel Bou-Moslehforgeneratingnumeroushigh-qualitycomputational meshesandDr. JulienCortialforproofreadingthisdocument.I want to acknowledge the Oce of Naval Research (ONR), which sponsored thiswork.Finally,Iwishtothankmyfriendsandfamilyfortheirunconditionalsupport.viiContentsAbstract ivAcknowledgements vi1 Introduction 11.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Underwaterimplosions . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Pipelineexplosions . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Highlyexibleaeronauticalsystems . . . . . . . . . . . . . . . 51.1.4 Shockwavelithotripsy . . . . . . . . . . . . . . . . . . . . . . 81.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Thesisaccomplishmentsandoutline. . . . . . . . . . . . . . . . . . . 111.3.1 Thesisaccomplishments . . . . . . . . . . . . . . . . . . . . . 111.3.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 MathematicalModels 142.1 Fluidmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.1 Governingequations . . . . . . . . . . . . . . . . . . . . . . . 15viii2.1.2 Equationsofstate . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Structuremodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Governingequations . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Constitutivelaws . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Cohesivecrackmodelsandfracturecriterion . . . . . . . . . . 202.3 Interfaceconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 Impermeableuid-structuretransmissionconditions. . . . . . 222.3.2 Immiscibleuid-uidinterfaceconditions. . . . . . . . . . . . 232.4 One-dimensionalmodels: Riemannproblems . . . . . . . . . . . . . . 242.4.1 Single-phaseuidRiemannproblem. . . . . . . . . . . . . . . 242.4.2 Two-phaseuidRiemannproblem . . . . . . . . . . . . . . . 272.4.3 Fluid-structureRiemannproblem. . . . . . . . . . . . . . . . 283 ComputationalFramework 313.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Partitionedprocedureforuid-structureinteractionproblems . . . . 373.3 Finitevolumebasedsingleandmulti-phasecompressibleowsolver . 393.3.1 Finitevolumesemi-discretization . . . . . . . . . . . . . . . . 393.3.2 Numericaltreatmentofuid-uidinterface. . . . . . . . . . . 423.3.3 Timeintegration . . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Finiteelementbasedstructuralsolver. . . . . . . . . . . . . . . . . . 443.4.1 Finiteelementsemi-discretization . . . . . . . . . . . . . . . . 443.4.2 Timeintegration . . . . . . . . . . . . . . . . . . . . . . . . . 463.5 Numericalmethodsfordynamicfracture . . . . . . . . . . . . . . . . 473.6 Embedded/immersedboundarymethodforuid-structureinteractions 49ix4 TrackingtheEmbeddedFluid-StructureInterface 544.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Aprojection-basedapproach. . . . . . . . . . . . . . . . . . . . . . . 594.2.1 ClosestpointontheembeddedinterfacetoagivenCFDgridpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2.2 Signed distance between a CFD grid point and its closest pointontheembeddedinterface . . . . . . . . . . . . . . . . . . . . 614.2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Acollision-basedapproach . . . . . . . . . . . . . . . . . . . . . . . . 674.3.1 Collision-basedinterfacetrackingalgorithm . . . . . . . . . . 684.3.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.3 Trackingacrackinginterface. . . . . . . . . . . . . . . . . . . 704.4 Distributedboundingboxhierarchy(scoping) . . . . . . . . . . . . . 724.5 Numericalexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.5.1 Example1: anAGARDwing . . . . . . . . . . . . . . . . . . 754.5.2 Example2: acircularcylinder . . . . . . . . . . . . . . . . . . 784.5.3 Example3: apairofultra-thintriangularwings . . . . . . . . 785 EnforcingtheFluid-StructureTransmissionConditions 885.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2 Enforcementoftheno-interpenetrationTransmissionCondition . . . 905.2.1 Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.2.2 Someimplementationaldetails. . . . . . . . . . . . . . . . . . 955.2.3 Vericationsonone-dimensionaluid-structureproblems . . . 965.3 EnforcementoftheEquilibriumTransmissionCondition . . . . . . . 1075.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107x5.3.2 A numerical algorithm for load computation based on the localreconstructionofembeddedinterfaces. . . . . . . . . . . . . . 1095.3.3 Areconstruction-freealgorithmforloadcomputation . . . . . 1215.3.4 Numericalaccuracystudy . . . . . . . . . . . . . . . . . . . . 1326 Applications 1396.1 VericationforatransientowpastaheavingAGARDwing. . . . . 1406.2 VericationforasteadyowaroundanaircraftforHALEights . . 1456.3 Applicationtoultra-thinappingwings . . . . . . . . . . . . . . . . . 1526.4 Validationfortheimplosivecollapseofanair-backedaluminumcylin-dersubmergedinwater. . . . . . . . . . . . . . . . . . . . . . . . . . 1586.4.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586.4.2 Fluid-structurecoupledsimulation . . . . . . . . . . . . . . . 1606.5 Validationfortheexplosionofanaluminumpipe . . . . . . . . . . . 1786.5.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.5.2 Fluid-structurecoupledsimulations . . . . . . . . . . . . . . . 1816.5.3 Currentmodellimitationandpossiblefuturework . . . . . . . 1846.6 Applicationtounderwaterexplosions . . . . . . . . . . . . . . . . . . 1917 ConclusionsandPerspectivesforFutureWork 1967.1 Summaryandconclusions . . . . . . . . . . . . . . . . . . . . . . . . 1967.2 Perspectivesforfuturework . . . . . . . . . . . . . . . . . . . . . . . 198AExact solution of the one-dimensional uid-structure Riemann prob-lem 200Bibliography 209xiListofTables4.1 Tracking an embedded AGARD wing: CPU performance results on 32processors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.1 AGARD wing in heaving motion: CPU performance results on 32 pro-cessors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.2 Statisticsofvenonbody-conformingCFDgridscreatedforthesim-ulationofasteadyowaroundanaircraft. . . . . . . . . . . . . . . . 148xiiListofFigures1.1 Typicalimplodablevolumesattachedtoasubmarine. Left: universalmodularmast. Right: unmannedunderseavehicle,ordrone. . . . . . 41.2 RupturedpipesectionsfromtheHamaokaNuclearPowerStationac-cident(left)andtheSanBrunonatural gastransmissionpipelineac-cident(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 AappingwingMAVdevelopedbyDARPA. . . . . . . . . . . . . . . 61.4 NASAHeliosPrototypeAircraft(HP03)conguration. . . . . . . . . 81.5 NASAHeliosPrototypeAircraft(HP03)athighwingdihedral (left)andfallingtowards the pacic ocean(right). MishaphappenedonJune26,2003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Extracorporealshockwavelithotripsy(left)andakidneystone(right). 102.1 Typicalsolutionstructureofasingle-phaseuidRiemannproblem. . 262.2 Typical solution structure of a two-phase uid Riemann problem. EOS-LandEOS-RrefertotheEOSthatgovernstheuidmediumontheleftandrightofthecontactdiscontinuity,respectively. . . . . . . . . 282.3 Solution structure of a uid-structure Riemann problem in the presenceofararefaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30xiii3.1 Denition of a control volume Ci, its boundary surface Ci, and a facetCijofCi(viewofhalfentitiesforanhexahedraldiscretization). . . 393.2 Thephantomnodeformulation: eachcrackedelementisreplacedbytwophantomelementswithadditionalphantomnodes. . . . . . . . . 494.1 Domain setting of an Eulerian embedded method for uid-structure in-teraction: extended uid domain F, structural domain S, embeddedsurfaceE,andoutwardnormalnEtoE. . . . . . . . . . . . . . . . 584.2 Signs of the barycentric coordinates of the projection point in dierentregions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3 Determinationof thesigneddistance(V i , Vi) whenV i , theclosestpointtoVion TEh ,liesonanedgeofatriangle. . . . . . . . . . . . . 654.4 Determinationof thesigneddistance(V i , Vi) whenV i , theclosestpointtoVion TEh ,isthevertexofatriangle. . . . . . . . . . . . . . . 664.5 Illustration of Algorithm4.1 and Algorithm4.2 for three distinc-tivecases. Apointingreen(blue)colorrepresentsaCFDgridpointlyingtheuid(structure) regionof thecomputational domain. Anedgeinredrepresentsauid-structureintersectingedge. . . . . . . . 714.6 Validandinvalidintersectionsinaphantomelement. . . . . . . . . . 724.7 CFD subdomains and distributed bounding box hierarchy: scoping forsubdomain10(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.8 The AGARD445.6wing: embeddeddiscrete interface (top) andacutview at z= 0 of inviscid non body-conforming CFD grid Th (bottom). 804.9 TheAGARD445.6wing: uid-structureintersectingedgesfoundbyAlgorithm4.1(Left)andAlgorithm4.2(Right). . . . . . . . . 81xiv4.10 Tracking an embedded AGARD wing: scope decomposition of the em-bedded discrete interface for 32 uid computational subdomains (eachcolor designates arelevant component of theinterfacefor aspecicsubdomainfor whichaboundingboxhierarchyis computedontheassignedCPU). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.11 Twodierentviewsofthesurfaceofacircularcylinder. . . . . . . . . 824.12 A circular cylinder: TEh(magenta color, top) and a cut-view of the nonbody-conformingCFDgridatz= 0of Th(bottom). . . . . . . . . . 834.13 Acircularcylinder: intersectingedgesidentiedbyAlgorithm4.1(left)andAlgorithm4.2(right)foralocalregionneartheinterface. 844.14 Apairofultra-thintriangularwings: discreteembeddedinterface. . . 854.15 A pair of ultra-thin triangular wings: computational uid domain andtheembeddeddiscreteinterface. . . . . . . . . . . . . . . . . . . . . . 864.16 Apairofultra-thintriangularwings: nonbody-conformingCFDgridcutviewaty= 5mm(left)andcutviewatz= 0mm(right). . . . 864.17 Apair ofultra-thintriangular wings: intersectingedges andnode sta-tusesidentiedbyAlgorithm4.2. . . . . . . . . . . . . . . . . . . 875.1 Twocontrolvolumesontheleftandrightsidesofaregionofanem-beddeddiscreteinterface(two-dimensionalcase,quadrilateralmesh). 945.2 Fluidmediaandmaterial interfacesnearacontrol volumeboundaryfacet(Cij): dierentcases. . . . . . . . . . . . . . . . . . . . . . . . 965.3 A3Dnonbody-conformingCFDgridforthesimulationof 1Duid-structureproblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98xv5.4 Aone-dimensional stienedgas thinshell perfect gas prob-lem: analytical andnumerical solutionsforthedensityeldattimet=0.007s. Top: theentirecomputational domain; bottom-left: thevicinityoftherarefaction;bottom-right: thevicinityoftheshock. . . 995.5 Aone-dimensional stienedgas thinshell perfect gas prob-lem: analytical andnumerical solutionsforthevelocityeldattimet=0.007s. Top: theentirecomputational domain; bottom-left: thevicinityoftherarefaction;bottom-right: thevicinityoftheshock. . . 1005.6 Aone-dimensional stienedgas thinshell perfect gas prob-lem: analytical andnumerical solutionsforthepressureeldattimet=0.007s. Top: theentirecomputational domain; bottom-left: thevicinityoftherarefaction;bottom-right: thevicinityoftheshock. . . 1015.7 Aone-dimensionalbarotropicliquidthinshellperfectgasprob-lem: analytical andnumerical solutionsforthedensityeldattimet=0.007s. Top: theentirecomputational domain; bottom-left: thevicinityoftherarefaction;bottom-right: thevicinityoftheshock. . . 1025.8 Aone-dimensionalbarotropicliquidthinshellperfectgasprob-lem: analytical andnumerical solutionsforthevelocityeldattimet=0.007s. Top: theentirecomputational domain; bottom-left: thevicinityoftherarefaction;bottom-right: thevicinityoftheshock. . . 1035.9 Aone-dimensionalbarotropicliquidthinshellperfectgasprob-lem: analytical andnumerical solutionsforthepressureeldattimet=0.007s. Top: theentirecomputational domain; bottom-left: thevicinityoftherarefaction;bottom-right: thevicinityoftheshock. . . 104xvi5.10 Aone-dimensionalperfectgassolidbodyproblem: twononbody-conformingCFDgridswith40and80gridpointsinlength-direction. 1055.11 A one-dimensional perfect gas solid body problem: pressure eld inattimet = 0.01s,computedonvenonbody-conformingCFDgrids. 1065.12 A one-dimensional perfect gas solid body problem: relative error inthepressureeldattimet = 0.01s. . . . . . . . . . . . . . . . . . . . 1075.13 Spatial discretization of a two-dimensional uid computational domainusingabody-conformingCFDgridwithtriangularelements. . . . . . 1085.14 Reconstructionof the embeddeddiscrete interface (two-dimensionalcase,quadrilateralmesh). . . . . . . . . . . . . . . . . . . . . . . . . 1115.15 Atwo-dimensionalacademicexample. . . . . . . . . . . . . . . . . . . 1165.16 Threetypicalsituationsarisingfromthechoiceofasurrogateembed-ded interface E: (a) an ideal situation,(b) a situation where is notaone-to-onemapping, and(c)asituationwherethevariationof thenormalto Eisnonsmoothandleadstolossofaccuracy. . . . . . . 1255.17 Illustrationof anembeddeddiscreteinterfaceconsistingof onlyonetriangleandthecorrespondingsurrogatesurface. . . . . . . . . . . . 1265.18 Surrogate embedded discrete interface Ein the context of a nite vol-ume method with dual control volumes (two-dimensional case, quadri-lateralmesh). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.19 Aprescribedpressureeldincomputationaldomain[0, 2] [0, 2]. . . 1335.20 Ideal case: coarsest meshinset S1(uniform, quadrilateral, satisfy-ingAssumption5.1)andreconstructed(left)andsurrogate(right)interfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134xvii5.21 Realistic case: coarsest mesh in set S2 (arbitrary, triangular, not satis-fying Assumption5.1) and reconstructed (left) and surrogate (right)interfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.22 Idealcase: performanceofAlgorithm5.3andAlgorithm5.4forloadcomputation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.23 Realisticcase: performanceofAlgorithm5.3andAlgorithm5.4forloadcomputation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.1 The AGARD445.6wing: embeddeddiscrete interface (left) andacutviewatz= 0ofthenonbody-conformingCFDgrid Th(right). . 1416.2 Thinwinginheavingmotion: comparisonof the lift time-historiespredictedbySimulation1.1,1.2,and1.4. . . . . . . . . . . . . . . . . 1436.3 Thinwinginheavingmotion: comparisonof the lift time-historiespredictedbySimulation1.2,1.3,and1.4. . . . . . . . . . . . . . . . . 1446.4 AsideviewoftheHeliosPrototypevehicleHP03. . . . . . . . . . . 1466.5 Twodierentviewsof asurfacegridwith853gridpointsand1, 687triangularelementsforanaircraftwithhighaspectratio. . . . . . . . 1476.6 Theuidcomputational domainandtheembeddeddiscretesurface(coloredinmagenta). . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.7 Acut-view(atx=0in)ofGrid4showingthreelevelsoflocalmeshrenement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.8 Acut-view(atx = 0ft)oftheuidpressurevariationwithrespecttothefree-streampressure,predictedbythesimulationonGrid1 . . . 1496.9 Streamlines near the aircraft, shownas ribbons. The color ontheribbonsrepresentthemagnitudeofvelocity. . . . . . . . . . . . . . . 1506.10 Totalliftpredictedbysimulationsusingdierentgrids. . . . . . . . . 151xviii6.11 Ultra-thinexibleappingwings: descriptionandstructuralmodeling. 1536.12 A pair of ultra-thin triangular wings: computational uid domain andtheembeddeddiscreteinterface(magenta). . . . . . . . . . . . . . . . 1546.13 Apairofultra-thintriangularwings: nonbody-conformingCFDgridcutviewaty= 5mm(left)andcutviewatz= 0mm(right). . . . 1546.14 Ultra-thinexibleappingwings: time-historyof atipdisplacementalongthez-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1566.15 Ultra-thin exible apping wings: snapshots of the uid pressure (cutviewat y= 20 mm) and structural deformation at six dierent time-instances.1576.16 Schematic drawing of a cylindrical implodable with end caps designatedbystripes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.17 Photographsofthecollapsedcylinder(courtesyofSteliosKyriakides). 1616.18 Pressuretime-historyrecordedbysensor1(courtesyof SteliosKyri-akides). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.19 CSDgridforModel1.1and1.2. Theelasto-plasticshellelementsforthe cylinder and the rigid shell elements for the plug are distinguishedbycolor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.20 Model1.1foranaluminumcylinderandasteelplug. . . . . . . . . . 1646.21 Model1.2foranaluminumcylinderandasteelplug. . . . . . . . . . 1656.22 ThestructuregridforModel2. . . . . . . . . . . . . . . . . . . . . . 1686.23 Model2foranaluminumcylinderandasteelplug. . . . . . . . . . . 1696.24 Underwaterimplosionproblem: theCFDdomainandtheembeddeddiscreteinterface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.25 Acut-viewat z =0of CFDgrid Thfor theunderwater implosionproblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171xix6.26 Pressure time-histories at a sensor location,predicted by Simulation 1andmeasuredintheexperiment. . . . . . . . . . . . . . . . . . . . . 1726.27 Pressure time-histories at a sensor location,predicted by Simulation 2andmeasuredintheexperiment. . . . . . . . . . . . . . . . . . . . . 1736.28 Pressure time-histories at a sensor location,predicted by Simulation 3andmeasuredintheexperiment. . . . . . . . . . . . . . . . . . . . . 1746.29 Pressure at a sensor location for 103sec t 1.2103sec, predictedbySimulation2and3,togetherwiththeexperimentaldata. . . . . . 1756.30 Transversal andaxial views of the deformedstructure predictedbySimulation1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1756.31 Transversal andaxial views of the deformedstructure predictedbySimulation2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.32 Transversal andaxial views of the deformedstructure predictedbySimulation3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.33 Snapshotsof theuidpressure(twocut-viewsatx=0inandz =0 in) and structural deformation at eight dierent time-instances. Thestructureisshowninwireframessuchthattheairinsidethestructureisvisible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.34 Schematicdrawingofthesetupofapipelineexplosionexperiment. . 1796.35 Setupofapipelineexplosionexperiment.[28] . . . . . . . . . . . . . 1806.36 Photographsshowingcrackpropagationandtheburstof detonationproduct throughthe crackopening. Time zerocorresponds totheignitionsparkinthedetonationtube. . . . . . . . . . . . . . . . . . . 1866.37 TheCSDgridforapre-awedaluminumpipe. . . . . . . . . . . . . . 187xx6.38 Theinitial crackprescribedontheCSDmodel forapre-awedalu-minumpipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1876.39 TheembeddingCFDdomainandtheembeddedCSDmodel. . . . . . 1886.40 Two cut-views of the non body-conforming CFD grid for a uid-structurecoupledpipeexplosionsimulation. Left: cut-viewaty=0m; right:cut-viewatx = 0.457m . . . . . . . . . . . . . . . . . . . . . . . . . 1886.41 Snapshotsoftheuidpressure(cut-viewaty=0m)andstructuraldeformationat sixdierent time-instances. Top-left: t =0s; top-right: t = 4.0 105s, middle-left: t = 8.0 105s;middle-right: t =1.6104s; bottom-left: t = 3.2104s; bottom-right: t = 4.0104s.1896.42 Time-historyof overpressureatasensorlocation(0.24mabovethenotch): experimentalsignal(left)andsimulationresult(right). . . . . 1906.43 Thepeakoverpressureatasensorlocation(0.24mabovethenotch)as a function of initial notch length: experimental and simulation results.1906.44 TheCFDdomainandtheembeddedCSDmodel foranunderwaterexplosionsimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1926.45 Acut-view(y=0m)of thenonbody-conformingCFDgridforanunderwaterexplosionsimulation. . . . . . . . . . . . . . . . . . . . . 1936.46 Thestructural deformationandeectiveplasticstrainatT=1.0 103sec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1946.47 The uidpressure eld (left) andthe domains of water andair (right)at1.78104sec(top),6.25104sec(middle),and1.0103sec(bottom). Acut-viewfory=0isshown. Thestructuregeometryisshowninwireframe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195xxiA.1 Solution structure of a uid-structure Riemann problem in the presenceofararefaction(left)orashock(right). . . . . . . . . . . . . . . . . . 205xxiiChapter1IntroductionFluid-structureinteractiondescribesalargeclassofphysicalproblemswhereauidowcausesthedeformationofastructure,andtheresultingchangeofcongurationin turn inuences the uid ow, ultimately leading to a two-way coupling. It is a fairlycommonphenomenonbothinnatureandaroundhumanworld. Examplesincludethe apping ight of insects, such as fruit y [2], and the generation of sound by reedinstrument, such as harmonica. The rst scientic study of uid-structure interactiondates back to the 1820s when Friedrich Bessel investigated experimentally the motionof a pendulum in a uid. He discovered that a pendulum moving in a uid had longerperiodthaninavacuumevenifthebuoyancyeectsweretakenintoaccount. Thisphenomenonwasthendescribedasaddedmassasitseemedlikethesurroundinguidincreasedthemassof thependulum. IthaddrawnimmediateattentionfrommanycontemporarymathematiciansincludingSimeonPoisson, GeorgeGreen, andGeorge Stokes. In 1853 George Stokes published a theoretical study [1] and concludedthat theeectivemass of thecylinder movingintheuidincreasedduetothedynamiceectof surroundinguidbytheamountof hydrodynamicmassequal to1CHAPTER1. INTRODUCTION 2themassofuiditdisplaced.[3]Since the dawn of the 20th century,uid-structure interaction has been an activeresearchandengineeringtopic,primarilyintermsofaeroelasticity,whichdealswiththe behavior of an elastic body or vehicle in an airstream wherein there is a signicantreciprocalinteractionbetweendeformationandow[4],andthecloselyrelatedeldof hydroelasticity, whichfocuses onoatingandsubmergedoceanstructures andvessels. AfterWorldWarII,theadventandgrowthofdigitalcomputerhavegreatlytransformed these elds [5]. For example,after the emergence of computational uiddynamics(CFD) (1970s) andniteelement methodsfor structures(1960s), CFD-basedaerodynamicsimulationscanbecoupledtofull-orderstructuralniteelementmodelsforanaccuratepredictionofairplaneutter[6].Up to these days, linear uid-structure interaction problems have been thoroughlystudiedformanyengineeringapplications[5, 6]. Examplesincludeairplaneutter,propellantsloshing,andthevibrationofbridgesandtallbuildingsinducedbywind.These problems are characterized by smooth ows and linear vibrations of the struc-ture.Thepresentthesisfocusesonhighlynonlinearuid-structureinteractionswhichinvolve large, possiblyplastic structural deformations, strong acoustic andshockwaves, multi-phase ows, and the initiation and propagation of cracks in the structuremedium. Duetononlinearityandsystemcomplexity,theseproblemshavenotbeenthoroughlyanalyzed,whichgreatlyhinderstheadvanceofrelatedengineeringelds.Several specicengineeringapplicationsthatmotivatedthisworkaredescribedinSection1.1. TheobjectivesandaccomplishmentsofthisthesisarethenprovidedinSection1.2and1.3,respectively.CHAPTER1. INTRODUCTION 31.1 Motivations1.1.1 UnderwaterimplosionsThe implosive collapse of an air-lled underwater structure can lead to high compres-sionsandstrongshockwaveswhichformapotential threattoanearbystructure.Inparticular, thecollapseof anair-backedimplodablevolume1external but closetoasubmarine, suchas auniversal modular mast (seeFigure1.1left) or anun-manned undersea vehicle (see Figure 1.1right), may lead to the damage or failure ofthesubmarine. Withthenumberofthesevolumesexpectedtoincrease,underwaterimplosionhasbecomeaconcerntotheNavy,whichnowrequiresanimprovedcapa-bility to design and qualify submarine external payloads for implosion avoidance andplatformsurvivability.From a physical point of view, the implosion of an air-lled underwater structure isa transient, high-speed, nonlinear, multi-phase uid-structure interaction problem. Itis characterized by ultrahigh compression and shock waves, large structural displace-ments anddeformations, self-contact, andpossiblytheinitiationandpropagationof cracksinthestructure. Thedevelopmentof acomputational frameworkforthisproblem is a formidable challenge. It requires not only incorporating in the computa-tionsmaterialfailuremodels,butalsoaccountingforallpossibleinteractionsoftheexternalliquidnamely,watertheinternalgasnamely,airandthegivennonlinearelasto-plasticstructure.1Implodable volume is dened by US Navy as any non-compensated pressure housing containingacompressibleuidatapressurebelowtheexternal ambientseapressure(atanydepthdowntomaximumoperatingdepth)whichhasthepotentialtocollapse[100].CHAPTER1. INTRODUCTION 4Figure1.1: Typical implodablevolumesattachedtoasubmarine. Left: universalmodularmast. Right: unmannedunderseavehicle,ordrone.1.1.2 PipelineexplosionsAs a common type of accidents in gas transmission systems and nuclear power plants,pipelineexplosions can, andusuallydo, leadtodisastrous consequences includingbothcasualtyandpropertylosses. Forexample,onNovember7,2001,asegmentofthe steam condensing pipeline at the Hamaoka Nuclear Power Station (Japan) Unit-1ruptured (see Figure 1.2left), most likely due to the detonation of hydrogen accumu-latedinthepipe[7]. Fortunatelytherewerenoinjuriesorlossoflife. However,thesurroundingareawasseverelydamaged, andthereactorhadtobeshutdowntem-porarily. More recently, on September 9, 2010, a segment of a natural gas transmissionpipeline ruptured in a residential area in San Bruno, California (see Figure 1.2right).CHAPTER1. INTRODUCTION 5Thereleasednaturalgasexploded,killingeightandinjuringmore[8].Figure 1.2: Ruptured pipe sections from the Hamaoka Nuclear Power Station accident(left)andtheSanBrunonaturalgastransmissionpipelineaccident(right).Intheseaccidents,alotofdamagesandharmstostructures(e.g. buildings)andpersonnel are caused by the high blast overpressure in explosion waves. To predict theblastoverpressureinsuchevents,acomputationalapproachmusttakeintoaccounttheinteractionbetweenanonlinearuidowcarryingstrongshocksandastructureundergoinglarge,plasticdeformationsandcracking.1.1.3 HighlyexibleaeronauticalsystemsDuringthelastdecade,therehavebeengrowinginterestsinthedesignandanalysisofaeronauticalsystemswithappingorexiblewingsforbothmilitaryandcivilianapplications.Demonstratedbyyingbirds andinsects, appingwings canbeadvantageousover xedor rotarywings inmaneuverabilityandlift generation, particularlyforsmallvehiclesandatlowspeed. Therefore,theyhavegreatpotentialinmicroaerialCHAPTER1. INTRODUCTION 6vehicles (MAV) designedfor surveillanceandreconnaissancepurposes. As anex-ample, DARPAs2hummingbird(Figure 1.3) was recentlyrecognizedbyTIMEMagazineasoneofThe50BestInventionsof2011. Becausethewingstructuresareoftenexibleandtendtodeformduringight, theuidandstructuredynamicsofsuchsystemsarecloselylinkedtoeachother[9]. Asaresult,uid-structurecoupledanalysis is necessary for studying the aerodynamics, lift/drag generation, and controlofappingwingMAVs.Figure1.3: AappingwingMAVdevelopedbyDARPA.Inaconceptuallydierent yet physicallyrelatedarea, aircrafts equippedwithhighly exible wings are currently under investigation for high altitude long endurance(HALE) ights. Flying at approximately 70, 000 ft with an expected endurance of onemonthorlonger, theseaircraftsarebeingconsideredformanymilitaryandcivilianapplicationsincludingsurveillance,reconnaissanceandtelecommunication. Inorderto achieve high eciency at such high altitude, their wings are characterized by large2DARPAstandsforTheDefenseAdvancedResearchProjectsAgency.CHAPTER1. INTRODUCTION 7aspectratioandlowweight. Asaresult, theyexhibithighexibilityinight. Oneexample of such aircrafts is the Helios prototype vehicle, a propeller-driven, remotelypiloted aircraft developed by NASA (see Figure 1.4). Its wing has a span of 247 ft andachordlengthof8ft,correspondinglytoaveryhighaspectratio(30.9)3. OnJune26, 2003, during a test ight, this aircraft encountered turbulence and morphed intoan unexpected, persistent, high dihedral conguration [29] (depicted in Figure 1.5left). Consequently, it becameunstableandnallycrashed(Figure1.5right). Apost-morteminvestigationof thismishap[29], conductedlaterbyNASA, concludethatoneoftherootcausesisLackof adequateanalysismethodsledtoaninaccurateriskassessment of theeectsof congurationchangesleadingtoaninappropriatedecisiontoyanaircraftcongurationhighlysensitivetodisturbances.Indeed, the physics behind this mishap is characterized by a challenging uid-structureinteractionprobleminvolvinglargestructural deformationsinducedbyaturbulentow. Toanalyzethisproblem,theinvestigationcommitteerecommend:Developmoreadvanced, multidisciplinary(structures, aeroelastic, aerodynam-ics, atmospheric, materials, propulsion, controls, etc)time-domainanalysismethodsappropriatetohighlyexible,morphingvehicles.Develop multidisciplinary (structures, aerodynamic, controls, etc) models, whichcan describe the nonlinear dynamic behavior of aircraft modications or performincremental ight-testing.3Forcomparison,theaspectratioofthewingofBoeing777-200is8.68.CHAPTER1. INTRODUCTION 8Figure1.4: NASAHeliosPrototypeAircraft(HP03)conguration.1.1.4 ShockwavelithotripsyShockwavelithotripsy(SWL)isanon-invasivetreatmentofkidneystones. Extra-corporeallygeneratedshockwavesarefocusedonthestoneinordertopulverizeitintograins, whichcanthentravel throughtheurinarytractandpassfromhumanbody(seeFigure1.6). Currently, asignicantpercentageof kidneystones(69%inUSin2000[10])aretreatedwiththisprocedure. Giventhatasignicantfractionofhuman body is water, SWL highlights a complex uid-structure interaction problem,whichinvolvesshockwavesandcavitationintheuidmedium, aswell ascrackingand comminution of the structure (kidney stone). Due to the lack of knowledge in thephysics behind it, particularly the mechanism of uid-induced stone comminution, nofundamental improvementsinSWLtechnologyhavebeenaccomplishedinthepasttwodecadestowardsbetter treatmenteciencyandreducedtissueinjury[11]. Astatistical studyevenrevealsanunfortunatetrendofmovingawayfromSWLtoCHAPTER1. INTRODUCTION 9Figure 1.5: NASA Helios Prototype Aircraft (HP03) at high wing dihedral (left) andfallingtowardsthepacicocean(right). MishaphappenedonJune26,2003.moreinvasivetherapies[10].Currently, thereexists nouid-structurecoupledcomputational methodwhichcan simultaneously capture all the aforementioned features involved in SWL. Indeed,recentcomputationalstudiesonSWLhavebeenfocusingoneithertheuidpartoftheproblem, particularlytheowdynamicsof cavitationbubblesinteractingwithshockwavesand/orrigidstructures[1114]; orthestructurepartof theproblem,particularlythestressgrowthanddistributioninakidneystonehitbyacousticandshockwaves[15, 16].1.1.5 SummaryAt the time of this writing, all the problems presented above are active research topics.Moregenerally, theyrepresentalargeclassof challenginguid-structureproblemswhichinvolveoneormoreofthefollowingfeatures:large,possiblyplastic,structuraldeformations;initiationandpropagationofcracksinthestructuremedium;CHAPTER1. INTRODUCTION 10Figure1.6: Extracorporealshockwavelithotripsy(left)andakidneystone(right).strongshockwavesintheuidmedium;multipleuidmediadirectlyincontact.To the authors best knowledge,there currently exists no computational approach ineitheracademiaorindustrythatsimultaneouslyaddressesallthesechallenges.1.2 ObjectivesThecurrentworkaimsatthedevelopmentofahigh-delityuid-structurecoupledcomputational frameworkwhichaccounts for thenonlinear features highlightedintheapplicationspresentedinSection1.1. Tothisend, thefollowingtasksmustbeCHAPTER1. INTRODUCTION 11performed:developanecientandaccurateembeddedboundarymethodforsolvingonxed, nonbody-conformingCFDgridsmulti-phasecompressibleuidsinter-actingwithstructuresundergoinglargedeformationsandcracking;developa uid-structure coupledcomputational frameworkbycoupling themulti-phase compressible ow solver equipped with the aforementioned embed-dedboundarymethod,withastate-of-the-art,extendedniteelementmethod(XFEM)basedstructuralsolver4.demonstratetheeectivenessandrobustnessof theaforementionedcomputa-tionalframeworkforchallengingapplications;validate the proposed computational framework for challenging applications us-ingrelevantlaboratoryexperimentaldata.1.3 Thesisaccomplishmentsandoutline1.3.1 ThesisaccomplishmentsThemajoraccomplishmentsofthisthesisissummarizedasfollows.Developmentofanembeddedboundarymethodforsolvingonxed,arbitrary(i.e. structured and unstructured), non body-conforming CFD grids multi-phasecompressibleows interactingwithstructures undergoinglargedeformationsandcracking. Keycomponentsofthismethodinclude:4this solver is developed and provided by Prof. Ted Belytschkos research group at NorthwesternUniversity.CHAPTER1. INTRODUCTION 12(a) robust andecient computational algorithms for trackingopen, closed,and cracking embedded interfaces with respect to a xed, three-dimensional,structuredorunstructurednonbody-conformingCFDgrid;(b) a numerical algorithm based on the solution of local, one-dimensional uid-structureRiemannproblemsforenforcingtheno-interpenetrationtrans-missionconditionattheuid-structureinterface;(c) twoconsistentandconservativealgorithmsforenforcingtheequilibriumtransmissionconditionattheuid-structureinterface.Assessment of the order of spatial accuracy of the developed embedded bound-arymethodforacademicexamples.Development of a uid-structure coupledcomputational framework using apartitionedprocedureandprovablysecond-orderexplicit-explicitandimplicit-explicit time-integrators. This frameworklinks twomajor components: (1)the multi-phase compressible ow solver equipped with the aforementioned em-beddedboundarymethod; and(2)astate-of-the-art, extendedniteelementmethod (XFEM) based structural solver. In particular, the interface tracking al-gorithms in the embedded boundary method are adapted to tracking embeddeddiscreteinterfaceswithphantomelementsandcarryingimplicitlyrepresentedcracks.Validation of the developed computational framework for two laboratory exper-iments:(a) the implosive collapse of an air-lled aluminum cylinder submerged in wa-ter;CHAPTER1. INTRODUCTION 13(b) thedynamicfractureof apre-awedaluminumpipedrivenbyinternaldetonation.Performance assessment of the developedcomputational frameworkfor fourengineeringapplicationsintheeldsofaeronauticsandunderwaterexplosion.1.3.2 OutlineThedocumentisorganizedasfollows. Chapter2describesthemathematicalformu-lationsconsideredinthisworkformodelingthenonlinearuid-structureinteractionproblems of interest. Chapter 3 presents the design of the uid-structure coupled com-putationalframeworkaswellasthenumericalmethodsunderlyingeachcomponent.Chapter 4 details two robust, ecient, and accurate numerical algorithms for trackinganembeddeduid-structureinterfacewithrespecttoaxed, nonbody-conformingCFD grid. In addition, the performance of these algorithms are demonstrated and as-sessed. Chapter 5 presents the numerical algorithms used by the embedded boundarymethodfortheenforcementoftransmissionconditionswhichholdattheembeddeduid-structureinterface. Numerical accuracyanalysis of thesealgorithms arealsoincluded. InChapter6, thecomputational frameworkisassessedinthecontextofsix realistic uid and uid-structure problems in the elds of aeronautics, underwaterimplosion, underwater explosion, and pipeline explosion. Finally, Chapter 7 providesconclusionsaswellasperspectivesforfuturework.Chapter2MathematicalModelsThepresentchapterdescribesthemathematical modelsemployedinthisthesisforanalyzing highly nonlinear uid-structure problems in general, and the physical prob-lems describedinSection1.1 inparticular. The organizationof this chapter isstraightforward. Auid-structureinteractionproblemcanbeconceptuallydividedinto two sub-problems: a uid sub-problem and a structure sub-problem. The math-ematical models for solvingtheuidandstructuresub-problems arepresentedinSections2.1and2.2,respectively. Section2.3dealswithmodelingtheinteractionofdierent materials across an interface. Finally, three one-dimensional models for uidanduid-structureproblems aredescribedinSection2.4. Thesesimpleandwell-understoodmodelsplayanimportantroleinthedesignofseveralkeyalgorithmsinthecomputationalframework,tobediscussedinChapter3,4and 5.14CHAPTER2. MATHEMATICALMODELS 152.1 Fluidmodel2.1.1 GoverningequationsLetF(t) 13bethetime-dependentowdomainofinterest. Withinthisthesis,theuidsdynamicsareassumedtobecompressibleandinviscid,hencegovernedbythe Euler equations, which account for conservations of mass, momentum, and energy.Theirstrongconservativeformcanbewrittenas:Wt+

T(W) = 0, inF(t) (2.1)whereW = (, vx, vy, vz, E)T, (2.2)

=_x,y,z_T, (2.3)

T(W) = (Tx(W), Ty(W), Tz(W))T, (2.4)Tx=____________vxp + v2xvxvyvxvzvx(E + p)____________, Ty=____________vyvxvyp + v2yvyvzvy(E + p)____________, Tz=____________vzvxvzvyvzp + v2zvz(E + p)____________, (2.5)denotes theuiddensity, Eis its total energyper unit volumeandis givenbyE= e +12(v2x + v2y + v2z),whereedenotestheinternalenergyperunitmass,pdenotestheuidpressure,and v= (vx, vy, vz)isthevelocityvector.CHAPTER2. MATHEMATICALMODELS 16Theclosureof theEulerequationsisobtainedwithanequationof state(EOS)whichrelatesthethermodynamicvariables,pande. ThreeEOSsareusedinthisthesisformodelingdierentuidmedia. TheywillbepresentedinSection2.1.2.Remark.1. TheEuler equations describedaboveareanappropriatemodel for uids in-volved in underwater implosions and pipeline explosions described in Section 1.1.1and 1.1.2. Indeed, theviolenceof thesephenomenaleadstolargemotions,shock waves, and strong acoustic waves in the uid media (liquid or gas). Hencethenonlinearityandcompressibilityof theuids(evenliquidwater)mustbeproperly modeled. On the other hand, the time scale in these problems (103secto101sec)issucientlysmallsuchthatthediusionofheatandmomentumcanbeneglected.2. Forowsinvolvedintheightsofhighlyexibleaeronauticalsystems,suchasthosedescribedinSection1.1.3, theEulerequationsareonlyacoarsemodel,astheviscouseectsintheseproblemscanbestrongandevendominatetheowdynamics. However, theycanstill reproducesomekeyfeaturesintheseproblems, such as uid-induced, large structural deformations, which bring sig-nicantchallengestonumerical simulations. Inthepresentwork, thisaspectofthemodel isemphasized, astwoapplicationsinvolvinghighlyexibleaero-nautical systems are considered in Chapter 6 to demonstrate the capability of acomputational frameworkforhandlingcomplexgeometryandlargestructuraldeformations.CHAPTER2. MATHEMATICALMODELS 172.1.2 EquationsofstateThreeEOSarepresentedhereformodelingdierenttypesofuidsinvolvedinthisthesis,namelyperfectgas,stienedgas,andTaitbarotropicliquid.PerfectGas: TheEOSforaperfectgascanbewrittenasp = ( 1)e (2.6)whereistheheatcapacityratiodenedastheratioof thespecicheatcapacityatconstantpressure(cp)tothespecicheatcapacityatconstantvolume(cv). Foraperfectgas,cpandcvareassumedtobeconstant.Theunderlyingphysicalmodelassumesthatthegasmoleculeshaveanegligiblevolume and the potential energy associated with intermolecular forces is also negligi-ble. ThisEOSiswidelyusedformodelingrealgasesatlowtonormalpressuresandnormaltohightemperatures. Forthemodelingofair,isusuallysetto1.4.StienedGas: TheEOSforastienedgascanbewrittenasp = ( 1)e pc(2.7)whereisanempirical constantwhichisusuallychoseninthewaythatadesiredshock speed is reproduced correctly by the EOS [31]. pcis another empirical constantrepresenting the intermolecular attractions. This EOS is a generalization of the EOSfor perfect gasasthelatter canbeobtainedbysettingpc=0. It hasbeenusedformodelingliquidsandevensolidmaterials, andtopropagatewavesinsuchstimedia[32]. Inthis thesis, this EOSis usedfor modelingthehighpressurewaterinunderwater implosionproblems. andpcare set to7.15and2.89 108Pa,respectively.CHAPTER2. MATHEMATICALMODELS 18Taitbarotropicliquid: TheEOSforaTaitbarotropicliquidcanbewrittenasp = p0 + __ 0_1_(2.8)where = p0 +k1k2, = k2(2.9)p0 and 0 are the pressure and density of a reference state, usually chosen to be thefar-eld state. k1and k2are two constants chosen to t the relation of bulk modulusandpressureusingananefunction: k1 + k2p = dpd. Forwater,k1andk2aremostcommonlysetto2.07 109kg.m3/s2and7.15respectively.TheTaitEOShasbeenusedformodelingliquids,especiallywater,underawiderange of temperatures and pressures [33]. Despite the fact that the ow is assumed tobeisentropic, ithasbeenwidelyusedineldssuchasunderwaterexplosionswhereshockwavesarepresent[34]. Forthisreason, itisconsideredinthisthesisasanoptionalEOSforthemodelingofwaterinunderwaterimplosions.2.2 StructuremodelThepresentthesisdoesnotinvolvethedesignofmathematicalmodelsforthestruc-ture sub-problem, nor the development of computational methods for analyzing them.However, since they represent one of the two sub-problems in any uid-structure inter-action problem, a brief summary of the structure model used in this thesis is providedhere.CHAPTER2. MATHEMATICALMODELS 192.2.1 GoverningequationsLet S(t) 13be the structural domain of interest. The governing equations of thedynamic equilibrium of the structural system are written in a Lagrangian formulationas[35]s usx s(us, us) = fextsin S(0) (2.10)snt=t on tu = u on uwhere usdenotes the displacement of the structure with respect to the reference con-guration (S(0)), s and s denote its density and Cauchy stress tensor, respectively,andfextsistheexternalforceactingonit. Thedotdenotesthetimederivative.tisthe applied traction on the Neumann boundary tand u is the applied displacementontheDirichletboundaryu. Aconstitutivelawthatexpressesthestresstensorintermsofdisplacementandvelocityisrequiredtoclosethissystem.2.2.2 ConstitutivelawsAstructureconstitutivelawconsistsoftwoaspects: therelationbetweenstrainanddisplacement (kinematics), and the relation between stress and strain in the structurematerial(kinetics).Geometric nonlinearityis always considered, as this thesis focuses onphysicalproblems involving large structural deformations (see Section 1.1 for examples). Con-sequently,thestrain-displacementrelationofastructuremediumisformulatedas

s=12(u +uT+u uT). (2.11)CHAPTER2. MATHEMATICALMODELS 20wheresdenotesthesecond-orderGreenssymmetricstraintensorinthestructure,and denotesthetensorproduct.Asforkinetics,bothlinearelasticandnonlinearelasto-plasticmaterialsarecon-sideredinthisthesis. Indeed, thestructuresincertainhighlyexibleaeronauticalsystems, suchas theones describedinSection1.1.3, canbemodeledusinglinearelasticity, aspermanentdeformationsareusuallynegligible. However, inunderwa-terimplosionsandpipelineexplosions,whereirreversibledeformationsandcrackingappearasdominantfeaturesinthestructuremedium,anonlinearelasto-plasticma-teriallawismoreappropriate. Inparticular,theJ2-owtheoryplasticityisusedinthisthesisasitwasspecicallydevelopedformetals[35].2.2.3 CohesivecrackmodelsandfracturecriterionAs mentioned in Section 1.1, the present work is motivated to a large extent by uid-structureinteractionproblemsinwhichdynamiccrackingappearsinthestructuremedium. Themathematical toolsformodelingcrackpropagationsaresummarizedhere. First,acohesivecrackmodelisemployedtoensureanaccuratedissipationofenergyduetocracking. Itisassumedthatacrossacrack,thenormalcomponentofthestresstensorsatisesthefollowingcohesivelaw:+s nc= s nc= c(u), (2.12)inwhichncistheunitnormal tothecracksurface, andcisthecohesivetractionacrossit,expressedasafunctionofthejumpofdisplacementacrossthecrackinthenormaldirection(nc). Superscriptplusandminussignsrefertothetwosidesofthediscontinuity(crack). Inthisthesis, apiecewiselinearcohesivemodelisprescribed,CHAPTER2. MATHEMATICALMODELS 21i.e.c() =___k, 0 max0, > max(2.13)wheremaxisthemaximumcrackopeningdisplacement,andkisaconstantchosensuchthatthedissipatedenergyduetothecrackpropagationisequaltothefractureenergy,i.e.Gf=_max0c()d, (2.14)whereGfdenotesthefractureenergy.Next,afracturecriteriaisemployedtodeterminethepropagationdirectionatacracktip. Inthiswork,amaximumtensileprincipalstraincriteriaisemployed[36].Morespecically, whenthestrainat acracktipreaches afracturethreshold, thecrackisextendedatthistipalongthedirectionwheretheprincipaltensilestrainofanaveragestrainavgismaximized. Theaveragestrainavgisdenedby

avg=4_ 22_c0w(r)drdwhererandarethedistancefromthecracktipandtheanglewiththetangenttothecrack,respectively. w(r)isaweightfunction.2.3 InterfaceconditionsWithinthescopeof thisthesis, interactionsof dierentuidand/orstructurema-terialsareassumedtobelocalizedattheircommoninterfaces. Governingequationsrepresenting these localized interactions, or interface conditions, are described in thissectiontogetherwiththeirphysicalinterpretations. TwotypesofmaterialinterfacesCHAPTER2. MATHEMATICALMODELS 22are considered in this thesis, namely the impermeable interface between a uid and astructure, and the immiscible interface between two uids such as water and air. TheirunderlyinginterfaceconditionsarepresentedinSection2.3.1and 2.3.2respectively.2.3.1 Impermeableuid-structuretransmissionconditionsWithin this thesis,the uid andstructure media are assumedto be impermeable. Inother words, interpenetration of uid and structure particles is not allowed. To enforcethisassumptioninthemathematical model, thefollowingDirichlet andNeumanntransmission conditions are imposed at F(t)

S(t), the interface between the uidandstructuremedia: un = vn (2.15)s n = pn (2.16)wherendenotestheunitnormaltotheinterface.Eq. 2.15implies thecontinuityof thenormal component (withrespect totheinterface)of theuidandstructurevelocityeldsacrosstheinterface, henceinter-penetrationofuidandstructuredomainsareprohibited. Intheremainderofthisthesis, itisreferredtoastheno-interpenetrationcondition. Itisnotablethatforastaticstructure,itreducesto un = 0,which is well-known as the slip-wall boundary condition for an inviscid ow. Eq. 2.16impliestheequilibriumoftheinteractionforcebetweentheuidandthestructure,andisreferredtoastheequilibriumconditionintheremainderofthisthesis.CHAPTER2. MATHEMATICALMODELS 232.3.2 Immiscibleuid-uidinterfaceconditionsWithinthescopeofthisthesis,anyuid-uidinterfaceisassumedtobeimmiscible,meaningtheydonotmixwhenputincontact. Thisassumptionholdsperfectlyforthe underwater implosion problems described in Section 1.1.1, as the interface occursbetweenwaterandair. Inaddition, eectsof evaporationandsurfacetensionareignored. Again,thisisvalidforunderwaterimplosions. Ononehand,thetimescaleissucientlysmall(103to101sec)suchthatmassdiusion(evaporation)atthewater-airinterfaceisnegligible. Inaddition, theeectsof surfacetensionarealsonegligible, asthesurfacetensionisverysmall comparedtothehighwaterpressureintheseproblems.Consequently, in this thesis a uid-uid interface is modeled by a contact disconti-nuity,orfreesurface. Morespecically,let(1)F Fand(2)F Fbethedomainsof two uid media. If they are in contact, i.e. (1)F

(2)F,= , the following conditionsholdat(1)F

(2)F:v(1) n = v(2) n, (2.17)p(1)= p(2). (2.18)Inotherwords, thepressureeldsandthenormal component(withrespecttotheinterface)ofthevelocityeldsofthetwouidsarecontinuousacrosstheinterface.CHAPTER2. MATHEMATICALMODELS 242.4 One-dimensionalmodels: RiemannproblemsThree one-dimensional models for single-phase uid, two-phase uid, and uid-structureproblems are discussed here. These initial value problems are characterized by piece-wise constant initial state with a single discontinuity. Simple yet revealing, they havebeenthroughlystudied[37, 38, 40] andconsideredasfundamental toolsfor under-standingthenonlinearbehaviorof compressibleinviscidows, aswell asdesigninganddevelopingnumerical methodsforanalyzingthem. Inparticular, theyplayanimportant role in the design of several key algorithms in the computational frameworkpresentedinthisthesis,whichisdiscussedinChapter3,4,and5.2.4.1 Single-phaseuidRiemannproblemThestrongconservativeformofthethree-dimensionalEulerequationsarestatedinEq.2.1. Foratime-independentone-dimensionaluiddomainF= 1,theyreducetowt+xF(w) = 0, inF(2.19)wherew =_____vE_____, F(w) =_____vv2+ pv(E + p)_____. (2.20)Assuming a single EOS holds in the entire domain, the single-phase uid Riemannproblem (P1) is dened as nding the solution w(x, t) to Eq. 2.19, given the followingCHAPTER2. MATHEMATICALMODELS 25piecewiseconstantinitialcondition:w(x, 0) =___wL, x 0wR, x > 0, (2.21)wherewLandwRaretwodierentstates.This problem (P1) can be solved analytically using the method of characteristics.Details of the solution procedure can be found in [38] for the perfect gas EOS and [97]for more general EOS. In general,the solution of P1 is composed of three character-isticwaves: aleft-facingrarefactionorshock(referredtoas1-wave), aright-facingrarefactionorshock(referredtoas3-wave),andacontactdiscontinuity(referredtoas2-wave). Eachcharacteristicwaveinitiatesattheoriginandtravelsataconstantspeed. Therefore, the solution ofP1 is self-similar in the sense that it can be writtenas a function of only x/t. The solution at any time instance is given by wLto the leftof the 1-wave, an intermediate state (L, v, p) between the 1-wave and the 2-wave,anotherintermediatestate(R, v, p)betweenthe2-waveandthe3-wave,andwRto the right of the 3-wave. It is notable that the velocity and pressure are continuousacrossthecontactdiscontinuity, i.e. the2-wave. Asanexample, thestructureofasolutioninvolvingararefactionwaveontheleft(i.e. 1-wave)andashockwaveontheright(i.e. 3-wave)isshowninFigure2.1.Thesolutionof P1canbeobtainedinsixsteps:1. Determinewhethereachofthe1and3-wavesisararefactionorashockusinganentropycondition;2. Relates theintermediatestate(L, v, p) towLusingeither theRiemannCHAPTER2. MATHEMATICALMODELS 26xt3-wave(shock)1-wave(rarefaction)2-wave(contact)LLLpvRRRpvpvLpvR(0, 0)Figure2.1: Typicalsolutionstructureofasingle-phaseuidRiemannproblem.invariants (if the 1-wave is a rarefaction) or the Rankine-Hugoniot jump condi-tions(ifthe1-waveisashock);3. Relates theintermediatestate(R, v, p) towRusingeither theRiemanninvariants (if the 3-wave is a rarefaction) or the Rankine-Hugoniot jump condi-tions(ifthe3-waveisashock);4. SolvethesystemofequationsobtainedfromStep2and3fortheintermediatestates,morespecically: L,R,vandp.5. Determine the shockspeedfor anyshockwave usingthe Rankine-Hugoniotjumpconditions;6. Determine the structure of the solution through any rarefaction waves using theRiemanninvariants.Remarks:CHAPTER2. MATHEMATICALMODELS 27Foraperfectgasorastienedgas,theexistenceanduniquenessofsolutiontoP1isguaranteedbytheBethe-Weyl theorem, providedthattheformationofvoidsisallowed.FormostEOS(includingtheonesstatedinSection2.1.2), theexactsolutionofintermediatestatescannotbeobtainedinclosed-formexpression. However,approximate solutions canbe computedusingnumerical nonlinear equationsolvers such as Newtons method (also known as the Newton-Raphson method).2.4.2 Two-phaseuidRiemannproblemThetwo-phaseuidRiemannproblem(P2)isdenedasndingthesolutionoftheone-dimensionalEulerequations(Eq.2.19),togetherwithinitialconditionw(x, 0) =___wL, x 0wR, x > 0, (2.22)withtwodierentuidmediaontheleftandrightof theorigin. Thesetwouidscan be governed by either the same EOS with dierent parameters, or dierent EOS.Inbothcases,thesolutionstillhasthesamestructureasdiscussedinSection2.4.1,except that at any given time, the fraction of the computational domain on the left orrightofthecontactdiscontinuityisgovernedbyitsrespectiveEOS.Asanexample,the structure of solution involving a rarefaction traveling in the left uid and a shocktravelingintherightuidisshowninFigure2.2ProblemP2canstillbesolvedbytheprocedureoutlinedinSection2.4.1. How-ever,inStep2and3therelationbetweentwostatesacrossthe1or3-wavemustbeformulatedusingtheEOS(andparameterstherein)fortheuidmediuminwhichCHAPTER2. MATHEMATICALMODELS 28xt3-wave(shock)1-wave(rarefaction)2-wave(contact)LLLpvRRRpvpvLpvR(0, 0)EOS-LEOS-LEOS-REOS-RFigure 2.2: Typical solution structure of a two-phase uid Riemann problem. EOS-LandEOS-RrefertotheEOSthatgovernstheuidmediumontheleftandrightofthecontactdiscontinuity,respectively.thiswavetravels. Detailsofthesolutionprocedureareprovidedin [26, 39]. AlltheEOSpresentedinSection2.1.2arecovered.2.4.3 Fluid-structureRiemannproblemTheuid-structureRiemannproblem(P3) considers aone-dimensional compress-ibleandinviscidowwithamovingwall boundary. Aconstantinitial conditionisprescribedfortheow, whileaconstantvelocityisprescribedforthewall bound-ary. Alsoknownas the pistonproblem[40], this problemis simple yet powerful.First, it canbe solvedanalyticallyandfor some EOSthe solutionevenexists inclosed-formexpression. Second, itcanreveal thefundamental featuresintheinter-action of a three-dimensional compressible ow and a dynamic structure, such as thenonlinearityof theowintheformsof shocksandrarefactions, andamovingowCHAPTER2. MATHEMATICALMODELS 29boundary. Within this thesis,P3 is used in the design and development of a compu-tational technique for enforcing the no-interpenetration condition and simultaneouslyrecoveringtheuidpressureattheinterfacefortheenforcementoftheequilibriumcondition (Section5.2).Givenaconstant uidinitial state wLandaconstant wall velocityvwall, P3canbeformulatedasfollows. Consideringaone-dimensional time-dependentuiddomain F(t) = (, vwallt ], nd the solution w(x, t) to the one-dimensional EulerequationsstatedasEq.2.19,togetherwithinitialconditionw(x, 0) = wLforx F(0) = (, 0]andboundaryconditionv(vwallt, t) = vwallatthemovingwallboundaryfort 0.The solution of P3 consists of either a shock wave or a rarefaction wave (referredto as 1-wave) initiating from the origin and propagating to the left. At any time t > 0,theowtotheleftofthe1-waveisunperturbedthushasinitialstatewL,whiletheowbetweenthe1-waveandthewall boundaryis determinedbyanintermediatestate (, vwall, p). In the case of a rarefaction wave, the solution structure is showninFigure2.3.Detailedsolutionprocedures for perfect gas, stienedgas, andTait barotropicliquidareprovidedinAppendixA.Anoutlinecanbestatedas1. Determineifthewaveisashockorararefactionusinganentropycondition;CHAPTER2. MATHEMATICALMODELS 30xtrarefaction|||.|

\|LLLpv|||.|

\|--pvwall(0, 0)moving wallt v xwall =not involvedFigure2.3: Solutionstructureofauid-structureRiemannprobleminthepresenceofararefaction.2. Relatetheintermediatestate(, vwall, p)towLusingeithertheRankine-Hugoniotjumpconditionsifthewaveisashock, ortheRiemanninvariantsifitisararefaction;3. Solvethesystemof equations obtainedinStep2for theintermediatestate,morespecicallyandp;4. Determine the shock speed if the wave is a shock; or determine the structure oftherarefactionwaveifitisararefaction.Forperfectgasandstienedgas,theintermediatestatecanbewritteninclosed-formexpression. For Tait barotropic liquid, this is nolonger possible. However,approximationscanbecomputedusingnumericalnonlinearequationsolvers.Chapter3ComputationalFramework3.1 IntroductionThe present thesis focuses on the design, development, verication, and validation of acomputational framework for highly nonlinear multi-phase uid-structure interactionproblems. Toalargeextent, thisworkismotivatedbythesimulationof physicalproblems describedinSection1.1, whichinclude underwater implosions, pipelineexplosions, andtheights of certaintypes of highlyexibleaeronautical systems.Theyall share acommonfeature: the deformationof the structure is large, andinducedeitherpartiallyorfullybythedynamicuidow. Otherfeaturesappearinginatleastoneoftheseproblemsinclude:1. strong acoustic and shock waves in the uid medium (in underwater implosionsandpipelineexplosions);2. multipleuidmediaincontact(inunderwaterimplosionsandpipelineexplo-sions);31CHAPTER3. COMPUTATIONALFRAMEWORK 323. strongviscouseects,laminarorturbulent,intheuidmedium(intheightsofaeronauticalsystemswithappingwingsorhighlyexiblewings);4. plastic deformations in the structure medium (in underwater implosions, pipelineexplosions,andaircraftsforHALE);5. initiationandpropagationof cracksinthestructuremedium(inunderwaterimplosionsandpipelineexplosions);The present computational frameworkaccounts for all the features mentionedabove except the viscous eects that occur inthe ights of aeronautical systemsinvolving apping or exible wings. Adding high-delity models for capturing viscouseectsisfeasibleandinprogressbutconsideredoutsidethescopeofthisthesis(seeChapter7). Mathematical modelsusedinthepresentframeworkarepresentedinChapter2.Thedesignofthepresentcomputationalframeworkissummarizedhere. Firstofall, itisnecessarytosolvethedynamicuidandstructuresystemssimultaneouslyasacoupledsystem. Indeed, theaforementionedfeaturesintheuidandstructuremediaareresponsestotheinteractionof theuidandstructuresub-systems. Forexample, inanunderwaterimplosionproblem, thestrongacousticandshockwavespropagating in water are generated by the self-contact of the collapsed structure; whilethecollapseandself-contactofthestructurearecausedbythehighwaterpressure.Similarly, inapipelineexplosionproblem, theinitiationandpropagationof cracksinthestructuremediumleadtostrongacousticandshockwavesenteringtheuidoutside the pipe, whereas the cracking of the pipe is caused by the high dynamic uidpressureinsideit.CHAPTER3. COMPUTATIONALFRAMEWORK 33Dependingonwhetherthesemi-discretizeduidandstructuresystemsaretime-integratedjointlyasonesystemorseparatelyastwosystems,computationalformu-lations for transient uid-structure coupled systems can be categorized in two classes,namelymonolithicapproaches[4143]andpartitionedapproaches[4446, 48, 49].Inamonolithicapproach, theuidandstructuregoverningequationsarecom-binedandsolvedusingasinglepreferredtime-integrator. Thiscanbeappealingasitisrelativelyeasytoachievethedesirednumerical properties, includingaccuracy,stability, andconservation, fortheentirecoupledsystem. However, itnecessitateswritingafully-integrateduid-structuresolver, thusprecludingtheuseof existinguidandstructuresoftwares. Additionally,inordertointegratetheuidandstruc-turesystemstogether, modelingassumptionsarecommonlyneededinatleastonesystem,whichinevitablyreducesmodeldelity.Bycontrast,inapartitionedapproach,theuidandstructuresystemsaresemi-discretizedandtime-integratedbydierentschemescarefullytailoredtotheircorre-sponding mathematical models, and coupled through explicit inter-system communi-cations. Thisstrategyenablestheexploitationofo-the-shelfsoftwarecomponentsspecicallydevelopedfor eachsystem, andmakes replacements relativelypainlesswhenbetter mathematical models andnumerical methods emerge. However, it isrelativelydicult toachievesomenumerical properties, particularlyconservation,fortheentirecoupledsystem. Thelossofconservationcanbetroublesomeforsomeproblems, as discussed in [51, 52]. A more detailed comparison, or debate, over mono-lithicandpartitionedapproachescanbefoundin[48] and[43], withadvocatesonbothsides.CHAPTER3. COMPUTATIONALFRAMEWORK 34Inthepresent computational framework, apartitionedprocedureis employed.Makingthisdecisionissimpleandstraightforward. Asmentionedearlierinthissec-tion, this computational framework targets uid-structure interactions with a numberof highly nonlinear features which are very challenging for numerical simulations. De-veloping a single software that includes numerical techniques accounting for each andevery one of these features requires a tremendous amount of work. Moreover, if bettermathematical models and numerical methods emerge during the time of software de-velopment, they can not be easily incorporated into the software. On the other hand,usingapartitionedprocedure, availablesoftwaremodules, suchas anite-volumecompressibleowsolver, canbeeasilyincorporatedintothecomputational frame-workand,ifdesired,replacedinfuturewhenbettersoftwaresbecomeavailable. ThepartitionedprocedureusedhereispresentedinSection3.2.The uid sub-system, modeled with the Euler equations, is solved by the softwarepackage AERO-F, astate-of-the-art nite-volume compressible owsolver, whichoperates on unstructured tetrahedron grids. Developed by Prof. Charbel Farhat andcollaborators, it has been validated for many applications in the past two decades [63,64]. Oneofitsdistinguishingfeaturesisthetreatmentofmulti-phaseows,whichishandled via a ghost uid method based on a two-phase Riemann solver. This approachhasbeenvalidatedinthecontextofunderwaterimplosionofairbubbles[27]. ThisowsolverisbrieydescribedinSection3.3.Thestructuresub-systemisdiscretizedbytheniteelementmethod. Twoal-ternativesolversaresupportedinthepresentcomputational framework. TherstoneisthesoftwareAERO-S, developedbyProf. Charbel Farhatandcollaborators.It hasbeenvalidatedinvariousscienticandengineeringapplicationsinthepastdecade[101103]. ThesecondoneisDYNA3D, aniteelementprogramdevelopedCHAPTER3. COMPUTATIONALFRAMEWORK 35atLawrenceLivermoreNational Laboratory(LLNL)forstructural/continuumme-chanicsproblems[65]. Bothsolversarecapableofmodelinggeometricandmaterialnonlinearities,aswellascontact. AERO-Ssupportsbothexplicitandimplicittime-integratorswhereasDYNA3Dislimitedtoexplicitones. Themainalgorithmsusedby these two solvers are summarized in Section 3.4. Fracture dynamics is modeled viatheextendedniteelementmethod, orXFEM, followingthephantomnodeformu-lation[36]. UnderlyingalgorithmsaresummarizedinSection3.5. Thiscapabilityisincorporatedintothein-houseversionofDYNA3DbyProf. TedBelytschkosteam.Popularnumericaltechniquesdesignedforhandlingtheinteractionofauidandastructureingeneral,andenforcingtheuid-structuretransmissionconditions(seeSection2.3.1)inparticular, canbedividedintotwoclassesdependingonwhethertheCFDgridmoves/deformsaccordingtothemotion/deformationofthestructure.Therstclassofmethods, includingthewell-knownArbitraryLagrangian-Eulerian(ALE) method[64, 67, 68, 95], theclosely-relateddynamicmeshmethod[69], andthe co-rotational approach [79, 80], operate on dynamic, body-conforming CFD grids.Equippedwithameshmotioncomputationalgorithm(for examples, see[7072]),thistypeof methodsallowtheCFDgridtomoveordeforminthewaythatitisalways conformal to the dynamic wet surface of the structure. These methods can beappealingastheynaturallyaccommodatespatiallyhigh-orderschemesattheuid-structureinterface, andallowahighmeshresolutionforboundarylayersinviscousows. However, achievingeciency, robustness, andhighmeshqualityrequiresex-tremecarewhenthemotion/deformationofthestructureislargeandnotknownapriori [26]. Moreover, assoonasthestructureundergoessomekindof topologicalchange, suchas cracking, thechallengebecomes practicallyinsurmountable. TheCHAPTER3. COMPUTATIONALFRAMEWORK 36secondclassof methods, collectivelyreferredtointhisthesisasembeddedbound-arymethods, operateonxed, nonbody-conformingCFDgrids. Duringthelastfour decades, embeddedboundarymethods havegainedtremendous popularityinCFD anduid-structure interaction,under dierent names. These include immersedboundary [73, 74], embedded boundary [49, 75], immersed interface [85, 86], ctitiousdomain[19],andCartesian[76]methods. Theycanbeattractivebecausetheysim-plifyanumberofissuesrangingfrommeshingtheuiddomain,toformulatingandimplementing Eulerian-based algorithms for dicult uid-structure applications suchas those involving very large structural motions and deformations [49], or topologicalchanges [78]. However, because anonbody-conformingCFDgriddoes not con-tainanativerepresentationofthewetsurfaceofthestructure,embeddedboundarymethods also tend to complicate other issues such as the treatment of uid-structuretransmissionconditions.Inthepresentcomputational framework, anembeddedboundarymethodisde-veloped and used, since large deformation and cracking appear as main features of in-terest. This method is equipped with novel algorithms for tracking the uid-structureinterface with respect to the CFD grid,as well as enforcing the uid-structure trans-missionconditions. Thefundamental concepts of this methodaresummarizedinSection3.6, whilethedesignanddevelopmentofspecicalgorithmsaredetailedinChapter4and5.CHAPTER3. COMPUTATIONALFRAMEWORK 373.2 Partitioned procedure for uid-structure in-teractionproblemsInapartitionedprocedure, theuidandstructuresystemsaresemi-discretizedandtime-integratedbydierent numerical schemes that are tailoredtotheir dierentmathematical models, andcoupledthroughexplicit inter-systemcommunications.Thesenumerical schemescanbeadvancedintimeeithersimultaneously, oroneatatimeinastaggeredmanner. Intheformercase,theresultingpartitionedanalysisprocedure is usually referred to as a strongly coupled solution algorithm. In the lattercase, each uid or structure time-step can be completed either in one step or throughaloopofsub-iterations. Astaggeredsolutionprocedurewithcarefullydesignedsub-iterationsissometimesalsoconsideredasastronglycoupledalgorithm, whereasastaggered,sub-iterationfreeprocedureisreferredtoasalooselycoupledalgorithm.Withinthedomainofpartitionedprocedures, looselycoupledalgorithmscanbeadvantageouscomparedtotheirstrongcounterpartsbecauseofreducedcomputa-tional cost andsimpliedsoftwaredevelopment. Inthepast, theyhavebeensus-pectedforinherentissuesinaccuracyandstability. Indeed, forsomeschemes[27],thetime-accuracyof thecoupledschemeis foundtobeoneorder lower thantheones of the underlying uid and structure time-integrators; whereas its stability limittends tobemorerestrictivethantheones of theuid/structuretime-integrators.However, recentlyit hasbeenshownthat thesedecienciescanbeovercomewithcarefullydesignedprediction/correctionapproaches, withoutsubstantial increaseofcomputational cost. First, provablysecond-order explicit-explict, implicit-explict,and implicit-implict schemes have been developed using second-order time-integratorsforboththeuidandstructuresystems[27, 48]. Second, itisshownin [27] thatCHAPTER3. COMPUTATIONALFRAMEWORK 38atleastforaspecicnumerical example, thetime-stepstabilitylimitofacarefullydesigned explicit-explicit scheme is equal to the lowest of the time-step stability limitsoftheindividualuidandstructureschemes.Looselycoupledpartitionedprocedures developedin[48] and[27] areadoptedinthepresentthesis. Itshouldbenotedthattheappealingaccuracyandstabilitypropertiesofthesealgorithmsreportedtherearemathematicallyprovedandexperi-mentallyveriedforanALEframework. Whethertheyholdforthepresentcompu-tational framework, in which an embedded boundary method is used instead, has notbeen rigorously investigated yet. This work is considered as a possible future researchdirectionanddiscussedinChapter7.Finally, the basic cycle of a loosely coupled partitioned procedure within one time-stepcanbesummarizedasfollows:1. transferthemotionofthewetsurfaceofthestructuretotheuidsystem;2. advanceintimetheuidsystemwithaspecictime-integrator;3. compute the uid-induced force load acting on the wet surface of the structure,andsendittothestructuresystem;4. advanceintimethestructuresystemwithatime-integratorthatmaybedif-ferentfromtheoneusedfortheuid.CHAPTER3. COMPUTATIONALFRAMEWORK 393.3 Finite volume based single and multi-phase com-pressibleowsolver3.3.1 Finitevolumesemi-discretizationAsdiscussedpreviouslyinSection2.1.1, thestrongconservativeformof thethree-dimensionalEulerequationsWt+

T(W) = 0, inF(t). (3.1)ischosenasafundamental mathematical model. Theexactdenitionsof thecon-servativestatevectorWandtheuxfunction

TareprovidedinSection2.1.1andomitted here. In the present computational framework, Eq. 3.1 is semi-discretized bya classical nite volume method. For the sake of completeness, the basic steps of thismethodareoutlinedbelow.Ci Vj Cij@ Ci@ Vi nij Figure3.1: DenitionofacontrolvolumeCi,itsboundarysurfaceCi,andafacetCijofCi(viewofhalfentitiesforanhexahedraldiscretization).CHAPTER3. COMPUTATIONALFRAMEWORK 40Let ThdenoteastandarddiscretizationoftheowdomainofinterestF,wherehdesignatesthemaximallengthoftheedgesofthisdiscretization. ForeveryvertexVi Th,i = 1, , NV,acell(orcontrolvolume)Ciisconstructed. Forexample,ifThconsistsofhexahedra, Ciisdenedastheunionofthesub-hexahedraresultingfrom the subdivision by means of the median planes of each hexahedron of ThhavingViasavertex(seeFigure3.1). TheboundarysurfaceofCiisdenotedbyCi, andtheunitoutwardnormaltoCiisdenotedby ni= (nix, niy, niz). Theunionofallofthecontrolvolumesdenesadualdiscretizationof Ththatveriestheproperty:NV_i=1Ci= Th. (3.2)Inotherwords, theunionof thecellsexactlycoversthecomputational domain.Usingthestandardcharacteristicfunctionassociatedwithacontrol volumeCi, astandard variational approach, and integration by parts, Eq. (2.1) can be transformedintoitsweakerform1_CiWhtd +

jK(i)_Cij

T(Wh)nij d = 0, (3.3)where Whdenotes a discrete approximation of the uid state vector Win a semi-discrete space, K(i) denotes the set of adjacent vertices of Vi, Cijdenotes a segmentof Cithat is denedas Cij=Ci

Cj, andnijis the unit outwardnormaltoCij. Theweaker formabovesuggests that inpractice, thecomputations canbe performedinaone-dimensional manner, essentiallybyevaluatinguxes alongnormal directions to boundaries of the control volumes. For this purpose, Ciis split1Eq. 3.3 can also be obtained by directly applying conservations of mass, momentum, and energywithincontrolvolumeCi[40].CHAPTER3. COMPUTATIONALFRAMEWORK 41incontrol volumeboundaryfacetsCijconnectingthecentroidsof thehexahedrasharingverticesViandVj(Figure3.1). Thetotal uxacrosstheboundariesof Ci,expressedas

jK(i)_Cij

T(Wh)nij din(3.3),isapproximatedasfollows

jK(i)_Cij

T(Wh)nij d =

jK(i)Fij(Wi, Wj, EOS, nij). (3.4)whereWiandWjarecell-averagedstatevectorsdenedasWi=1[Ci[_CiWh d, Wj=1[Cj[_CjWh d. (3.5)Fijis the numerical ux function evaluated using Roes approximate Riemannsolver [81] andMUSCL(MonotonicUpwindSchemeConservationLaw) [84] withlinearreconstruction. CombiningEqs.3.4and3.5withEq.3.3yieldsdWidt+[Ci[

jK(i)Fij(Wi, Wj, EOS, nij) = 0. (3.6)Therefore, the semi-discretized Euler equations for a discretization Thof the uiddomaincanbeexpressedinacompactformasdWdt+F(W) = 0, (3.7)whereWandFdenotethecell-averagedstatevectorandthenumericaluxvectorfortheentiregrid.CHAPTER3. COMPUTATIONALFRAMEWORK 423.3.2 Numericaltreatmentofuid-uidinterfaceAs mentioned in Section 2.3.2, dierent uids are assumed immiscible and every uid-uidinterfaceismodeledbyacontactdiscontinuity, orfreesurface. Therefore, thevelocityandpressureof theowarecontinuous across theinterface, whereas thedensityistypicallydiscontinuous. Suchmulti-phaseowswithimmiscibleuid-uidinterfacesrequireextracomputationaleortsinaowsolverinorderto:locatethedynamicallyevolvinguid-uidinterface;applytheinterfaceconditions(Eqs.2.17and2.18).Numerical approaches for this type of problems abound in the literature. A reviewof thepopularoptionsisprovidedin[26]. Inthepresentframework, thelevel setmethod[82, 83] is employedtolocatethedynamicuid-uidinterface, whereas atwo-phase Riemann solver based ghost uid method [25] is used to apply the interfaceconditions.Usingthelevel set method, thedynamicinterfaceis implicitlyrepresentedbythezeroiso-valueofthelevel setfunction, whichsatisesthefollowingadvectionequation:t+v = 0, inF(3.8)wherevdenotesthevelocityeldoftheow, isinitializedateachpointinbythesigneddistancefromthispointtotheinitiallocationoftheinterface.The two-phase Riemannsolver basedghost uidmethodapplies the interfaceconditionbymodifyingtheusual numerical ux(Eq. 3.4)locallybetweentwogridpointsresidingindierentuidmedia. Morespecically, if apairof adjacentgridpoints, namely Viand Vj, reside in dierent uid media, the numerical ux across CijCHAPTER3. COMPUTATIONALFRAMEWORK 43isdenedasij= ij(Wi , Wj , EOSi, EOSj, nij), (3.9)where Wiand Wjare the two constant states in the exact solution (see Section 2.4.2for details) of the one-dimensional uid-uid Riemann problem, initialized by WiandWj.3.3.3 TimeintegrationTwosecond-orderaccuratetime-integratorsfortheuidsystemareconsidered.Second-order explicit Runge-Kutta: to advance the semi-discretized uid system(Eq.3.7)fromtimetntotn+1,theexplicitsecond-orderRunge-KuttaschemecanbewrittenasK1= tnF(Wn), (3.10)W= WnK1, (3.11)K2= tnF(W), (3.12)Wn+1= Wn12(K1 +K2), (3.13)wheretn= tn+1tndenotesthetime-stepsize.Second-orderimplicitthreepointbackwarddierence: toadvancethesemi-discretized uid system (Eq. 3.7) from time tnto tn+1, the implicit second-order threepointbackwarddierenceschemecanbewrittenas3Wn+14Wn+Wn1= tnF(Wn+1). (3.14)CHAPTER3. COMPUTATIONALFRAMEWORK 44Eq. 3.14isanimplicitsystemofnonlinearequationsinWn+1. Inthepresentcom-putationalframework,itissolvednumericallyusingNewtonsmethod.3.4 FiniteelementbasedstructuralsolverWithinthis section, the displacement eldof the structure systemis assumedtobecontinuousinbothspaceandtime. Inparticular, thismeansfracture, whichischaracterized by a strong spatial discontinuity in displacement, is not considered here.The special treatment for modeling dynamic fracture will be presented in Section 3.5.Under the above assumption, the equations governing the dynamic equilibrium of thestructuresubsystem(Eq. 2.11)aresemi-discretizedusingastandardniteelementmethod. Thetime-integrationiscompletedusingschemesof theNewmarkfamily.Theycanbeeitherexplicitorimplicitdependingonthechoiceof twoparameters.Basicideasofthisprocessaresummarizedinthissection.3.4.1 Finiteelementsemi-discretizationThestandardniteelement methodis appliedtothespatial discretizationof thestructuregoverningequations, whosestrongformulationhasbeenpresentedinSec-tion2.2.1andisremindedhere u x (u, u) = fextin S(3.15)First,aweakformulationisobtainedbymultiplyingEq.3.15byatestfunction,fol-lowedbyanintegrationbyparts. Itcanbestatedasfollows.CHAPTER3. COMPUTATIONALFRAMEWORK 45Findsolutionu(x, t) |= u(x, t)[u(x, t) 11(S), u= uonu, suchthatu(x) |0= v(x, t)[v(x, t) 11(S), v = 0onu,itsatises_Su ud +_Su : (u, u)d =_Sufextd +_tu td. (3.16)Followingthestandardniteelementapproach[88], thecomputational domainSissubdividedintonon-overlappingelementsdenotedbyei, i=1, 2, ..., Ne. TheunionofalltheelementsdenesadiscretizationofS,whichsatisesNe_i=1ei= S.Basingonthisdiscretization, approximatesolutionofEq. 3.16issoughtinanite-dimensionalsubspaceof |,denedas|= u(x, t) =Nn

I=1NI(x)uI(t), u(x, t) = uonu,whereNnisthenumberofnodesand NI(x) 11(S) [I= 1, 2, ..., Nnisasetofprescribedshapefunctions. Correspondingly,thetestfunctionsarealsorestrictedtoanite-dimensionalsubspaceof |0,denedas|0= v(x) =Nn

I=1NI(x)vI, v(x, t) = 0onu.Plugging u(x, t) =Nn

I=1NI(x)uI(t) andu(x) =NI(x), I =1, 2, ..., NnintoEq. 3.16, it can be shown that the discrete nodal displacement vector u = [u1(t), ..., uNn(t)]TCHAPTER3. COMPUTATIONALFRAMEWORK 46satisesanonlinearODEsystemwhichcanbecompactlywrittenasM u(t) +fint(u, u) = fext(t), (3.17)where M denotes the mass matrix,fintdenotes the vector of internal forces,and fextdenotesthevectorofexternalforces.3.4.2 TimeintegrationSchemes from the Newmark family [87] are employed to integrate the semi-discretizedstructure governing equation (Eq. 3.17). Eq. 3.17 is linearized at every time step. ThegeneralformofthelinearizedsystemcanbewrittenasM u(t) +C u(t) +Ku = f , (3.18)wherematricesM, C, K, andvectorf areconstantwithineachtime-step. Toin-tegratethissystemfromtntotn+1, thefamilyofNewmarkschemescanbewrittenas(M+ tnC+ (tn)2K) un+1= fn+1C( un+ (1 )tn un) K_un+ tn un+ (12 )(tn)2 un_(3.19) un+1= un+ (1 )tn un+ tn un+1(3.20)un+1= un+ tn un+ (tn)2(12 ) un+ (tn)2 un+1(3.21)CHAPTER3. COMPUTATIONALFRAMEWORK 47whereandaretwoconstantparameters. Forexample, with=0.5and=0,the explicit central dierence scheme is recovered, whereas the implicit midpoint ruleisobtainedbysetting= 0.5and= 0.25.3.5 NumericalmethodsfordynamicfractureDynamic fracture in the structure medium is modeled by the extended nite elementmethod (XFEM) [89,90], which has been implemented in the local version of DYNA3DbyProf. TedBelytschkosteam, followingthephantomnodeformulation[36, 91].This thesis does not involve anyworkonthe designanddevelopment of specicalgorithmsinthisarea. However, giventhesignicanceoffracturemodelinginthepresent computational framework, a brief summary of involved numerical methods isprovidedhere.Acrackinastructure mediumis representedas astrongdiscontinuityinitsdisplacement eld. It cannot becapturedautomaticallyinclassical niteelementanalysiswhichreliesoncontinuousshapefunctions. Thekeyideaof XFEMistoenrichtheapproximationbasiswithshapefunctionsthat arediscontinuousacrossthecrack. Morespecically, theXFEMapproximationof thedisplacementeldisgivenby[36]u(x, t) =Nn

I=1NI(x)_uI(t) + H(f(x))qI_=Nn

I=1NI(x)uI(t) +Nn

I=1NI(x)H(f(x))qI, (3.22)CHAPTER3. COMPUTATIONALFRAMEWORK 48whereH()istheHeavisidestepfunctiondenedbyH(r) =___1, x > 00, x 0. (3.23)f(x)isalevelsetfunctionusedtodenethelocationofthecrack[92]. Itispositiveononesideof thecrackandnegativeontheotherside. Therefore, theenrichmentshape functions NI(x)H(f(x)) are discontinuous across the crack. uIandqIaretheregularandenrichmentnodalvariables,respectively. Inpractice,theenrichmentshape functions and associated nodal variables need to be introduced only in elementstraversedbythecrack,orso-calledcrackedelements.Thephantomnodeformulationemployedinthepresent computational frame-work[36] usesatransformationof thenodal variableswhichleadstoasuperposedelementformalism. Morespecically,withinacrackedelement,letu1I=___uIiff(xI) < 0uI qIiff(xI) > 0(3.24)andu2I=___uI+ qIiff(xI) < 0uIiff(xI) > 0(3.25)bethenewnodal variables, thedisplacementeldexpressedinEq. 3.22canbere-writtenasu(x, t) =

IS1u1I(t)NI(x)H(f(x)) +

IS2u2I(t)NI(x)H(f(x)), (3.26)whereS1andS2aretheindexsets of thenodes of superposedelement 1and2,CHAPTER3. COMPUTATIONALFRAMEWORK 49respectively. Intermsof implementation, eachcrackedelementisreplacedbytwophantomelementswithadditional phantomnodes(Figure3.2). Thecrackingpathwithineachphantomelement is trackedimplicitlyusingalocal distancefunctiondenedatthenodes.V2V1V3V4eV1V2V5V6V3V4V8V7real nodephantom nodef(x) > 0f(x) < 0f(x) > 0f(x) < 0e(1)e(2)fflocal distance functioncrackFigure3.2: Thephantomnodeformulation: eachcrackedelementisreplacedbytwophantomelementswithadditionalphantomnodes.3.6 Embedded/immersed boundary method for uid-structureinteractionsDebuted in 1972 for the coupled uid-structure simulation of blood ows through elas-tic heart valves [73], immersed/embedded boundary methods have gained tremendouspopularityduringthelastfourdecadesinComputational FluidDynamics(CFD),CHAPTER3. COMPUTATIONALFRAMEWORK 50underdierentnames. Theseincludeimmersedboundary[73, 74],embeddedbound-ary [49,75], immersed interface [85,86], ctitious domain [19], and Cartesian [76] meth-ods. Alloftheseandotherrelatedmethodswhicharepopularnowadaysforalargevarietyofowsimulationsaroundxed[17, 19, 49, 7476], moving[18, 20, 21, 49, 96],anddeformable[22, 49, 53, 54, 73, 78]bodies,arecollectivelyreferredtointhisthesisasembeddedboundarymethods. Theyareparticularlyattractivefordynamicuid-structureinteraction(FSI)problemscharacterizedbylargestructural motionsanddeformations[27] ortopological changes[78], forwhichmostarbitraryLagrangian-Eulerian(ALE)methods[63, 64, 68]areoftenunfeasible.Embeddedboundarymethodssimplifythegriddingtaskastheyoperateonnonbody-conforminggrids. Mostof themaredesignedforcomputationsonCartesiangrids[18, 2022, 53, 54, 73, 74, 78],butsomehavealsobeentailoredforcomputationsonunstructuredmeshes[49, 93]. However,embeddedboundarymethodscomplicatethetreatmentof sliporno-slipwall boundaryconditionsingeneral [1721, 75, 76],anduid-structuretransmissionconditions(seeSection2.3.1)inparticular[22, 49,53, 54, 73, 78]. Thisisessentiallybecauseanonbody-conformingCFDgriddoesnotcontainanativerepresentationof thewetsurfaceof thebodyof interest. Indeed,recentdevelopmentsinembeddedboundarymethodshavefocusedmostlyonthesetwo issues,albeitprimarily onthetreatmentof thevelocitywallboundary conditionfor incompressible viscous ows past rigid and motionless obstacles (for example,seethereviewpaper[74]). Inthiscontext, recentlyproposedalgorithmsforinterfacetreatment havefocusedeither onsomeformof interpolation [55] withparticularattentiontonumerical stability[56] orhigher-orderaccuracy[20, 55, 57], orontheconcept of aghost cell [58, 59], somevariant of thepenaltymethod[60], andthemirroringtechnique[61].CHAPTER3. COMPUTATIONALFRAMEWORK 51For dynamic uid-structure applications, twotransmissionconditions must bedealt withat the intersectionof the embeddedstructural surface andembeddinguidmesh(seeSection2.3.1). Therst oneis theno-interpenetrationcondition.Thediscretizationof thisconditionissimilartothatof thevelocitywall boundaryconditionforowspastrigidandmotionlessobstacles. Forthisreason,virtuallyallmethodsmentionedaboveforthetreatmentofwall boundaryconditionsinembed-dedboundarymethodscanbeappliedforthatpurpose. Thesecondtransmissionconditionexpressesequilibriumatauid-structureinterfacebetweentheuidandstructural surface tractions. In practice, it leads to the computation of the generalizedand/ortotal ow-inducedloadonthewetsurfaceof thestructure. Forembeddedboundary methods, this computation shares with standard lift and drag computationsthesamedicultyofintegratingthepressureandviscoustractionsoftheowonasurfacenotexplicitlyrepresentedinthecomputational uidmodel. Typically, thisdicultyhasbeenaddressedintheliteratureseparatelyfromthatassociatedwiththe semi-discretization of the no-interpenetration transmission condition, albeit usinginmanycasessimilartechniquesbasedoninterpolationand/orextrapolation, withorwithoutresortingtotheexplicitcomputationoftheintersectionoftheembeddedinterfacesandembeddingmesh[60, 62].Theembeddedboundarymethodusedbythepresentcomputational frameworkisequippedwithanewapproach[49] forthetreatmentof uid-wall interfacesforboth purely uid and uid-structure applications. This approach is a departure fromthe methods outlined above and related published works in that it treats the velocityandpressureconditionsontheembeddedinterfacessimultaneously,ratherthandis-jointly. Furthermore,insteadofrelyingforthispurposeexclusivelyoninterpolationor extrapolation, theproposedmethodenforces theappropriatevalueof theuidCHAPTER3. COMPUTATIONALFRAMEWORK 52velocityat awall andrecovers thevalueof theuidpressureat this wall viatheexactsolutionoflocal,one-dimensional,uid-structureRiemannproblems. Thisap-proachisdetailedinSection5.2. Inaddition,twonovel,consistentandconservativemethodologiesforevaluatingow-inducedforcesandmomentsonrigidandexibleembeddedinterfacesarealsoincorporatedinthiscomputationalframework. Oneofthemisbasedonthelocal reconstructionof theembeddeddiscreteinterfaces. Theotheroneisbasedonthelevel setconcept. Itisparticularlyattractivebecauseitrigorously allows the substitution of an embedded discrete interface by a simpler sur-rogate, which simplies implementation and at the same time reduces computationalcost. ThesealgorithmsarepresentedinSection5.3.Typically, all interfacetreatmenttechniquesrequiretrackingthepositionof theembeddedinterface(orthecollectionof interfacesrepresentingtheentirewetsur-faceor thebodyof interest) withrespect tothenonbody-conformingCFDgrid.Inthisaspect,mostcomputationalmethodsdescribedintheliteraturehavefocusedprimarilyonclosedembeddedinterfaces(e.g. surfacesof solidbodies)andCarte-siangrids [17, 18, 2022]. However, the closedinterface assumptionis limitingasmanyFSI problems, suchas appingwings andcrackedpipes, involve openthinshell surfaces. Furthermore, theCartesiangridassumptionforbidstheuseof moststate-of-the-artowsolvers,suchasAERO-F,whichoperateonunstructuredgrids.The present computational framework is equipped with two robust and ecient inter-facetrackingalgorithmscapableofoperatingonstructuredaswell asunstructuredthree-dimensional CFDgrids. Therstoneisbasedonaprojectionapproach. Itisfasterbutstill restrictedtoclosedinterfacesandresolvedenclosedvolumes. Thesecond algorithm is based on a collision approach with motivation from the computerCHAPTER3. COMPUTATIONALFRAMEWORK 53graphicscommunity[53, 94]. Whilereasonablyslower, itcanhandleopenshell sur-facesandunderresolvedenclosedvolumes. Bothcomputational algorithmsexploittheboundingboxhierarchytechniqueandits parallel distributedimplementationtoecientlystoreandretrievetheelementsof thediscretizedembeddedinterface.ThesealgorithmsarepresentedinChapter4indetails.Chapter4TrackingtheEmbeddedFluid-StructureInterfaceTheembeddedboundarymethodintroducedinSection3.6operatesonxed, nonbody-conformingCFDgridswhichdonothaveanativerepresentationofthestaticor dynamic wall boundary in general, and the uid-structure interface in particular1.Instead, this wall boundary or uid-structure interface is represented as a Lagrangiansurfacegrid,commonlyreferredtoasthediscreteembeddedsurface,whichisgener-atedindependentlyfromtheCFDgrid. Thissurfacegridcanbestaticordynamic.Inthelattercase, itsmotion/deformationcanbeeitherprescribed, asinaforcedmotion/deformationsimulation, orcomputedbyinterpolatingorextrapolatingthedisplacement eldof adynamicstructure, as inatwo-waycoupleduid-structuresimulation. Inbothcases,inordertodiscretizeandenforcethewallboundarycon-ditionsortheuid-structuretransmissionconditions(seeSection2.3.1fordetails),special computational techniques must be employed to nd the location of the discrete1Thischapterisbasedonapublishedjournalarticle[50].54CHAPTER 4. TRACKINGTHE EMBEDDEDFLUID-STRUCTURE INTERFACE55embeddedsurfacewithrespecttothenonbody-conformingCFDgrid.Focusingontwo-waycoupleduid-structureproblems,thischapterpresentstworobust, ecient, andaccuratecomputational algorithmsfortrackingadiscreteem-beddedinterfacewithrespecttoanarbitrary, i.e. structuredorunstructured, CFDgrid. Thisinter