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