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• FluidStructureInteractionUsingtheParticleFiniteElementMethod

S.R.Idelsohn(1,2),E.Oate (2),F.DelPin(1)and NestorCalvo(1)

email:sergio@ceride.gov.ar(2) InternationalCenterforNumericalMethodsinEngineering(CIMNE)

AbstractIn the present workanewapproach to solve fluidstructure interaction problems isdescribed. Both, the equations of motion for fluids and for solids have beenapproximatedusing amaterial (lagrangian) formulation. To approximate the partialdifferentialequationsrepresentingthefluidmotion,theshapefunctionsintroducedbytheMeshlessFiniteElementMethod(MFEM)havebeenused.Thus,thecontinuumisdiscretizedintoparticlesthatmoveunderbodyforces(gravity)andsurfaceforces(duetotheinteractionwithneighboringparticles).Allthephysicalpropertiessuchasdensity,viscosity,conductivity,etc.,aswellasthevariablesthatdefinethetemporalstatesuchasvelocity and position and also other variables like temperature are assigned to theparticlesandare transportedwith theparticlemotion. ThesocalledParticle FiniteElementMethod(PFEM)providesaveryadvantageousandefficientwayforsolvingcontact and freesurfaceproblems, highly simplifying the treatment of fluidstructureinteractions.

Key words: FluidStructure interaction, Particle methods, Lagrange formulations,IncompressibleFluidFlows,MeshlessMethods,FiniteElementMethod.

1IntroductionManyclassifications have been proposed to enclose the numerical formulations thatapproximate the continuum equations that govern incompressible fluid flows. Inparticulartheonedescribingthewaythatconvectionis treateddividesthenumericalformulationsintotwoclasses,namelymaterial(orlagrangian)formulationsandspatial(oreulerian)formulations.Thefirstonedescribesconvectionbyplacingasetofaxesover the material particles that moveaccordingly to the equations of motion. In the

• euleriancasetheaxesaresetfixedinspaceandconvectiontermsareincludedintheequationsdescribingthetransport ofthefluidflow.Thepresentworkwill describeamethodthatusesamaterialformulation.Theequationsofmotionforboth,thesolidandfluid donot present convection terms, implying that the convectioneffect is directlyobtainedbymovingthediscretedomain.

Manyauthorshavetakenadvantageoflagrangianformulationstodescribedifferenttypesof problems. The Smooth Particle Hydrodynamics (SPH) method developed byMonaghan(1977)[mon81,mon97]shouldbementionedasapioneermethodofthiskind.

ManyothermethodshavebeenderivedfromSPH.Onethathasshownremarkableresultsisthe MovingParticleSemiImplicitmethod(MPS)introducedbyKoshizukaandOka(1996)[kos96].Thesemethodsuseakernelfunctiontointerpolatetheunknowns.SPHusesaweakformulationwhileMPSusesastrongformofthegoverningequations.

Ramaswamy(1986)[ram87]proposedalagrangianfiniteelementformulationfora2Dincompressible fluid flow. In that paper the mesh was convected according to theequationsofmotionbutwithoutchangeoftopology,makingitratherlimitingwhentheelementsgothighlydistorted.Theequationsofmotionwerediscretizedinspacebyusingthefiniteelementmethodwithlinearshapefunctions.

Anotherpossibleclassificationfornumericalformulationsmaybetheonethatseparatesthe methods that make use of a standard finite element mesh (like those made oftetrahedraorhexahedra),andthemethodsthatdonotneedastandardmesh,namelythemeshlessmethods.Theformulationdescribedinthispapercanbeconsideredaparticularclassofmeshlessmethod.Again,SPHmightbecitedasonethefirstmeshlessmethods.

Indeed,afterMonaghansworkandinparticularinthepast20years,manyhavebeentheattemptstodeveloparobustmeshlessmethodthatcouldapproximatePDEsin2Dand3Dwithacceptableaccuracy,convergenceandspeed.Amongothers,themethodsbasedon MovingLeastSquare interpolations[nay92, bel94], PartitionofUnity [dua95],andtheonesbasedonthe naturalneighbor interpolationfunctions[bra95,suk98]maybelisted.

In this work the interpolationfunction usedby the Meshless Finite Element Method(MFEM)[ide03a]willbeimplemented.ThisfunctionusestheVoronodiagramofthecloudofpointstoconstructtheinterpolant.The extendedDelaunaytessellation (EDT)[ide03b]isappliedtoconnecttheneighboringparticles.TheEDTprovidespolyhedralelementsthataresliverfreein3D,avoidinginstabilitiesoftheDelaunaytessellationduetodistortedtetrahedra.TheMFEMshapefunctionsadaptautomaticallytothepolyhedraandinthecasethatthepolyhedronisasimplex,theshapefunctionbehavesexactlyasthelinearfiniteelementshapefunction.

• Fluidstructureinteraction(FSI)problemshavebeenofspecialinterestfordesignersandengineersinthepast20years.Thisexplainswhymorerobustandstableformulationshavebeendevelopedtoassisttheapproximationofcontactproblems.EmbeddedmethodshavebeendevelopedbyLhneretal.[loh03]whereasinglemeshisusedtopartitionthefluidaswell asthestructure.AlsoArbitraryLagrangianEulerian(ALE)formulations[Sou00] have given acceptable results when the displacements or the geometrydeformationsarenotexcessivelylarge.

TheapproximationfortheFSIproblemdependsbasicallyonthecouplingofthefluidandstructureequations.BasedonthiscouplingFSIproblemsmaybedividedintoproblemswithweakinteractionandproblemswithstronginteraction.Thelaterarefoundwhenelasticdeformationofthesolidtakesplace.Theweakinterpolationcasehappenswhenlargerigiddisplacementsarepresent.Thissituationistypical inshiphydrodynamics,whenarigidbodymovesaccordingtotheforcesgivenbythepressurefieldobtainedfromthefluiddynamicproblem.Theseforcesappliedtotherigidbodywillaccelerateit,changingitsvelocityandtherefore,itsposition.

FSIproblemshavebeenclassicallysolvedinapartitionedmannersolvingiterativelythediscretizedequationsfortheflowandthesoliddomainseparately.Thesolutionofboth,fluid flowandsolid, with thesamematerial formulation, open thedoor to solve theglobalcoupledprobleminamonolithicfashion.Nevertheless,inthispapertherigidsolidwillstillbesolvedseparatelyfromthefluid.Apartitionedmethod[pip95,mok01]oriterativemethod[rug00,rug01,zha01ischosentosolvethecouplingbetweenthefluidandsolid.Theadvantagetouseamaterialformulationforboth,solidandfluidpartswillbeusedhereonlytobetterreproducebreakingwavesorseparateddropsinthefluid,whicharephenomenaimpossibletoreproduceusingaspatialformulation.

Thelayoutofthepaperisthefollowing:inthenextsectionthebasiclagrangianequationsof motion for the fluidandsolid domains aregiven. Next the discretization methodchosentosolvetheincompressiblefluidflowequationsandthesoliddynamicsintimeequationsaredetailed.Thealgorithmfortherecognitionoftheboundarynodesandthetreatmentofthefreesurfaceinthefluidisexplained.FinallytheefficiencyoftheParticleFinite Element Method for solving a variety of fluidstructure interaction problemsinvolvinglargemotionofthefreesurfaceinthefluidisshown.

2Equationsofmotion

2.1Fluiddynamicproblem:updatingthefluidparticlepositionsThefluidparticlepositionswillbeupdatedviasolvingthelagrangianformoftheNavierStokesequations.

• Let X i theinitialpositionofaparticleatimet=t0andlet x i thefinalposition.Beenui x j , t =ui thevelocityoftheparticleinthefinalpositionthefollowingapproximaterelationcanbewritten:x i=X i f ui , t , Dui /Dt . (1)

Conservation of momentum and mass for incompressible Newtonian fluids in thelagrangianframeofreferencearerepresentedbytheNavierStokesequationsandthecontinuityequationinthefinal xi position,asfollows:

Massconservation:

0=

+

ixiu

DtD . (2)

Momentumconservation:

ifijjx

pixDt

iDu +

+= , (3)

where isthedensity, p thepressure, ij thedeviatoricstresstensor, f i thesourceterm(usuallythegravity)and

DDt

representsthetotalormaterialtimederivative.

ForNewtonianfluidsthestresstensor ij maybeexpressedasafunctionofthevelocityfieldthroughtheviscosity byij= ui x ju j x i 23 ul x l ij . (4)

Fornearincompressibleflowsui xi

• x j

ij= x j ui x j u j xi = x j ui x j x j u j xi = x j ui x j x i u j x j x j ui x j .

(7)

Usingeq.(7),themomentumequationscanbefinallywrittenas:

DuiDt

= xi

p

x jij f i xi

p x j ui x j f i (8)

Note: eq.(3) or the equivalent for incompressible fluid floweq.(8) are nonlinear. Ineulerianformulationsthenonlinearityisexplicitlypresentintheconvectiveterms.Inthislagrangianformulation,thenonlinearityisduetothefactthateqs.(3)and(8)arewritteninthefinalpositionsoftheparticles,whichareunknown.Thereareotherswaytowritelagrangianformulations,forinstancestayingintheinitialposition[aub04].Inallcases,theequationsarenonlinear.

BoundaryconditionsOntheboundaries,thestandardboundaryconditionsfortheNavierStokesequationsare:

ij jpi= ni on nii uu = on ntii uu = on t

where i and i are the components of the normal and tangent vectors to theboundary.

2.2Soliddynamicsproblem:updatingtherigidbodypositionInthispaper,thestructurewillbeconsideredasarigidsolid.Then,theequationsofmotionforarigidbodyare:

mDU iDt

=F i (9)

whereFiaretheresultantoftheexternalforces(surfaceforces,gravityforce,etc.),whoselineofactionpassesthroughthemasscenterofthebody,Uiisthevelocityofthemasscenterandmthetotalmassofthesolid.

The actual motion of the rigid body consists in the superposition of the translationproducedbytheresultantforceFiandtherotationproducedbythecoupleTisatisfying:

• DM iDt

=T i , (10)