NumericalsimulationofuidstructureinteractionusingtheniteelementmethodP.R.F.Teixeiraa,A.M.Awruchb,*aDepartmentofMaterialsandConstruction,FederalUniversityofRioGrande-FURG,96201-900RioGrande,RS,BrazilbGraduatePrograminCivilEngineering,AppliedandComputationalMechanicsCenter,FederalUniversityofRioGrandedoSul,90035-190PortoAlegre,RS,BrazilReceived20March2002;receivedinrevisedform8December2003;accepted31March2004Availableonline20July2004AbstractAnalgorithmtosimulate3Duidstructureinteractionproblemsusingtheniteelementtechniqueispresented in this work. A two-step TaylorGalerkin scheme and linear tetrahedra elements are employed toanalyzetheuidow, whichmaybehighor slightlycompressible. AnarbitraryLagrangeanEulerian(ALE)formulationisadopted,whichmustbecompatiblewiththemotionof theuidstructure interface.Afractional methodwithvelocitycorrectionisusedforincompressibleuids. Thestructureisanalyzedusingtriangularelementswiththreenodesandsixdegreesoffreedomineachnode(threedisplacementcomponents and three rotation components). Geometrically non-linear eects are included. The NewmarkmethodisemployedtointegrateintimethedynamicequilibriumequationsusinganupdatedLagrangeandescription. Thealgebraicsystemof equations is solvedusingtheconjugatedgradient methodandanincremental-iterativeschemeisusedtosolvethenon-linearsystemresultingfromnitedisplacementsandrotations. Thecodeisoptimizedtotakeadvantagesofvectorprocessors. Somecasesstudieshavebeenconsideredfor validationof the computational algorithm. Atwo-dimensional supersonic owover aclamped at plate is analyzed in order to study the aeroelastic behavior of this plate. Vibrations due to windactionof aninatedmembraneaswell asvortexinducingvibrationsinapanel immersedinaslightlycompressibleuidarealsostudied.2004ElsevierLtd.Allrightsreserved.*Correspondingauthor.Tel.:+55-51-3316-3587;fax:+55-51-3316-3999.E-mailaddress:amawruch@vortex.ufrgs.br(A.M.Awruch).0045-7930/$-seefrontmatter 2004ElsevierLtd.Allrightsreserved.doi:10.1016/j.compuid.2004.03.006Computers&Fluids34(2005)249273www.elsevier.com/locate/compuid1.IntroductionImportant progresshasbeenobtainedinthesolutionof complexuidstructureinteractionproblemsinrecent years, allowingthestudyof multidisciplinaryapplicationsindierent engi-neering areas. This progress is due mainly to the increase of the speed of modern computers, to theevolutionof multidisciplinarysolutionalgorithms andof pre- andpost-processingtools. Themethods of simulation of uidstructure interaction problems are divided basically in two groupscalledpartitionedandmonolithicschemes[1,2].Inthepartitionedschemesthegoverningequa-tionsoftheuidandthestructureareintegratedintimealternatelyinanisolatedway. Inthemonolithicschemesthetwoeldsareconsideredasasingleentity,allowingtointegrateintimethetwosubdomainssimultaneously.InpartitionedschemessuchasproposedbySoriaandCa-sadei [3], Rifai etal. [4], Farhatet al. [5], Cebral andL ohner[6] andmanyotherauthors, thekinematic and dynamic boundary conditions in the interface are the unique information changedbetweenthe subdomains. Inthese schemes eachsub-domaincanbe solvedbydiscretizationtechniquesandbyecientsolutionalgorithmsinanindividual way. Newmethodsandmodelscanbeintroducedinamodularwayinordertogetmoreexibility.Besides,themeshesofeachsubdomaincanbebuilt without thecoincidenceof uidandstructural nodesat theinterfaces[6,7].Thesemodularcharacteristicsandexibilitymaybeextremelyconvenient.Analgorithmtosimulateuidstructureinteractionproblemsusingapartitionedschemeispresentedinthiswork.Atwo-stepexplicitTaylorGalerkinscheme[8,9],withlineartetrahedralniteelementsisemployed.AnarbritaryLagrangeanEulerian(ALE)descriptionisadoptedfortheuiddomain, whileforthestructural domainanupdatedLagrangeanformulationiscon-sidered. The ALE description was rst presented by Hirt et al. [10] and after this method was usedby several authors such as Hughes et al. [11], Donea et al. [12], Benson [13], Liu et al. [14] amongothers. Thestructureisanalyzedusinggeneralizedconformingtriangularplatesandshell ele-mentswithdrillingdegreesof freedom[15]. Geometricallynon-lineareectsareincluded. TheNewmarkmethod[16] isemployedtointegrateintimethedynamicequilibriumequation. Thenon-linear and the algebraic systems are solved using an incremental-iterative scheme andthe conjugate gradient method, respectively. The code was vectorizedtotake advantages ofvectorial processors. Several problemsareusedtovalidatethemethodsdevelopedinthisworkandillustratethedierencebetweenlinearandnon-linearaeroelasticcomputations.2.Theuiddynamicsolver2.1.Theexplicittwo-stepTaylorGalerkinschemeforhighcompressibleowsIntheALEdescription, thecomputational frameisareferenceindependent of theparticlemovement and may be moving with an arbitrary velocity in the laboratory system (this motion iscalledthemesh motionintheniteelementformulation); thecontinuumviewfromthisref-erenceisdenotedasXvandthecoordinatesofanypointaredenotedasv[14]. Theequationsexpressingmass, momentumandenergyconservationinALEformulationmaybewritteninacompactformas250 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273oVotv oFioxi wioVoxii 1; 2; 3 1withFi FiA FiDandV qqv1qv2qv3qe______; FiA qviqv1vi pdi1qv2vi pdi2qv3vi pdi3viqe p______; FiD 0s1is2is3isijvj kijoT=oxj______i; j 1; 2; 3 2whereviandwiaretheuidandthemeshvelocitycomponentsinthedirectionof thespatialcoordinatexirespectively,qisthedensity,pisthethermodynamicpressure,sijarethecompo-nents ofthedeviatoric stress tensor, Tis thetemperature, eis thetotal specic energy and kijarethecomponentsoftheconductivitytensor;vectorVcontainstheconservationortheeldvari-ables and Fiare the components of the ux variables (FiAcontains the advective terms and FiDtheviscous terms). Finally, dij is the Kronecker delta. Eq. (1) is complemented by the equation of statefor an ideal gas and by the constitutive equations. Initial and boundary conditions must be addedtotheseequationsinordertodeneuniquelytheproblem.In the TaylorGalerkin scheme, conservation equations are expanded in time by Taylor series,andafter, space discretization isaccomplished by the classicalBubnovGalerkinscheme [1618].A two-step method is used, as indicated by Zienkiewicz et al. [8]; this approach can be interpretedas the nite element version of the LaxWendro scheme used in nite dierences [19]. In the rststep, corresponding to the time interval tn; tn1=2, the unknown vector V at t tn1=2is expandedinTaylorseries.UsingalinearshapefunctionNassociatedwitheachnodetointerpolateVn,aconstantshapefunctionPE 1associatedwithelementEtointerpolateVn1=2andapplyingtheclassical Galerkin weighted residual method to the expression resulting fromTaylor seriesexpansion,thefollowingequationisobtained:Xn1=2EVn1=2E_XnENdX_ _Vn Dt2_XnEoNoxidX_ _Fni__XnENwnioNoxidX_ _Vn_i 1; 2; 3 3where Xn1=2Eistheelementvolume,Vn1=2Eisaconstantvalueatelementlevelandtheupperbarindicatesnodalvariables.In the second step, the unknown vector at t tn1is expanded in Taylor series. Using again thesame shape functions, andapplyingthe BubnovGalerkinmethod, the followingequationisobtainedforthesecondstep:_Xn1ENTNdX_ _Vn1_XnENTNdX_ _VnDt_Xn1=2EoNoxidX_ _Fn1=2i_ wn1=2iVn1=2_ Dt_Cn1=2ENlidC_ _Fn1=2Bi_ wn1=2BiVn1=2B_i 1; 2; 3 4P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 251where Cn1=2Eis the boundary of the element domain Xn1=2Eand liis the cosine of the angle formedby the outward normal axis to Cn1=2Ewith the positive direction of the reference axis xi. Index B isreferred to values at the boundary of the element domain. In Eq. (4), the consistent mass matrix issubstituted by the lumped mass matrix, and then this equation is solved iteratively. The proposedschemeisconditionallystable,andthelocalstabilityconditionisapplied.Inordertostabilizenumericallythesolution, speciallyinthepresenceofstrongshocks, itisnecessary to add numerical damping to the ow solver. In this work the viscosity model, as givenbyPeraireetal. [20], isadopted. Anarticial viscosityisaddedexplicitlytothenon-smoothedsolution,asfollowsVn1s Vn1 Mn1L1D 5where Vn1sand Vn1are the smoothed and non-smoothed solution at t tn1respectively. Mn1Listheassembledlumpedmassmatrixat t tn1ThevectorDisgivenbyD

ECFLECCSEMnEC MnELVnE6whereEisanindexreferredtoaspecicelement, CFLEisthelocal Courantnumber, CCisaglobalconstantspeciedbytheuser, SEisacoecientofpressuredistribution(seeRefs.[8,9]),MnECand MnELare the consistent and lumped mass matrices, respectively. The constant CC must bespeciedwithcareinordertoavoidinterferencesof thearticial andphysical viscosities. CCvariesbetween2.0and10.0.2.2.Thetwo-stepTaylorGalerkinschemeforslightlycompressibleowsMass conservationfor slightlycompressible uids, assumingconstant entropy, maybe ex-pressedbythefollowingequation:oqot 1c2opot oUioxii 1; 2; 3 7where cisthesoundspeedand Ui qvi i 1; 2; 3.Expanding the momentum conservation equations in Taylor series, the following expression isobtainedfortherststep:Un1=2i Uni Dt2oUniot Uni Dt2ofnijoxj_osnijoxj opnoxi 12oDpoxi wnjoUnioxi_i; j 1; 2; 38where fij vjUi i; j 1; 2; 3and pn1=2 pn 1=2Dp,with Dp pn1 pn.UsingUn1=2i Uni Dt2ofnijoxj_osnijoxj opnoxi wnjoUnioxi_i; j 1; 2; 3 9Eq.(8)isgivenbythefollowingexpression:Un1=2i Un1=2i Dt4oDpoxii 1; 2; 3 10252 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273DiscretizingEq.(7)intimeandusingEq.(10),itisobtained:Dq 1c2Dp DtoUn1=2ioxi DtoUn1=2ioxi_ Dt4ooxioDpoxi_i 1; 2; 3 11Thesecondtimestepisgivenbythefollowingexpression:Un1i Uni DtoUn1=2iot Uni Dtofn1=2ijoxj_ osn1=2ijoxj opn1=2oxi wn1=2joUn1=2ioxi_i; j 1; 2; 3 12Thentheowisanalyzed, afterspacediscretization, bythefollowingalgorithm: (1)determineUn1=2iwithEq. (9); (2)determineDpwithEq. (11)andcalculatepn1 pn Dp; (3)determineUn1=2iwithEq.(10);(4)determine Un1iwithEq.(12).ConsideringthesameshapefunctionsusedincompressibleowsandapplyingtheclassicalGalerkinmethodforspacediscretization,thefollowingmatrixexpressionsareobtainedforEqs.(9),(11),(10)and(12),respectively:Xn1=2EUn1=2iE_XnNdX_ _Uni Dt2_XnoNoxjdX_ _

fnij__XnoNoxjdX_ _snij_XnoNoxidX_ _pn_XnNwnjoNoxjdX_ _Uni_i; j 1; 2; 3 13_Xn1=2NT1c2 NdX_ Dt24_Xn1=2oNToxioNoxidX _Cn1=2NT oNoxinidX_Dp Dt_Xn1=2oNToxidX_ _Un1=2iE__Cn1=2NTnidC_ _Un1=2iE_i 1; 2; 3 14Xn1=2EUn1=2iE Xn1=2EUn1=2iE Dt4_Xn1=2oNoxidX_ _Dp i 1; 2; 3 15_Xn1NTNdX_ _Un1i_XnNTNdX_ _Uni Dt_Xn1=2oNToxjdX_ _fn1=2ij_ wn1=2jUn1=2i_E Dt_Xn1=2oNToxjNdX_ _snij Dt_Xn1=2oNToxiNdX_ _pn_ Dp=2_ Dt_Cn1=2NTnjdC_ _fn1=2ij wn1=2jUn1=2iE Dt_Cn1=2NTNnjdC_ _snij Dt_Cn1=2NTNnidC_ __pn Dp=2_i; j 1; 2; 3 16P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 253In Eqs. (13)(16) the index Eindicates that the corresponding variables are taken with a constantvalue over the element domain. Eq. (14) is solvedusingthe conjugate gradient methodwithdiagonal pre-conditioning. Inthisequationtheboundaryintegral involvingthepressureincre-ment in the left hand side could be omitted. One of the reason for this simplication is apparent ifEq. (10) is examined. It is observedthat onboundaries where the velocities are prescribedoDp=oxi 0.Theotherreasonisthatontractionboundaries,nochangeoftractionoccursandthen Dp 0[18].2.3.ThealgorithmformeshmovementThemeshvelocityeldwiscomputedlookingforsmall elementdistortions, conservingpre-scribedvelocitiesinmovingandstationaryboundarysurfaces. Themeshmovement algorithmadoptedinthis workuses asmoothingprocedure for thevelocities basedinthese boundarysurfaces.Theupdatingofthemeshvelocityatapoint ioftheniteelementdomainisbasedonthe meshvelocity of the points j belonging tothe boundary surfaces inthe following way(seeFig.1):wi

nsj1aijwj

nsj1aij17where ns is the total number of points belonging to the boundary surfaces and aij are the inuencecoecients between the point i inside the domain and the point j of the boundary surface given bythefollowingexpression:aij 1d4ij18with dij being the distance between the points i and j. In other words, aij represents the weight thateachpoint joftheboundarysurfacehasonthevalueofthemeshvelocityatpoints iinsidethedomain. When dijis small, aijhas an high value, favouring the inuence of points i, located closertotheboundarysurfacecontainingpoint j. Fig. 1.Distancesfromtheboundarysurfacestoapointiintheuiddomain.254 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273In the present work a variation to simplify this method and safe computer time was adopted forthecaseinwhichthedisplacement of thebodysurfaceisnot signicantlyhigh. It consistsofchoosingpreviouslyfor eachpoint insidetheuiddomainat least twosurfacepoints havingimportantinuenceinthecalculationofthemeshvelocityaccordingtoEq. (17). Oneofthesepoints belongs to the moving body surface and the other one to the external xed surface. Each ofthese two points have the largest values of coecient aij among all the other points in its respectivesurfacebecauseeachoneistheclosestpointoftherespectivesurfacewithrespecttothenodewhere the mesh velocity will be determined. Therefore, this procedure uses the same Eq. (17), butconsideringonlyonepointineachboundarysurface,correspondingtothenodewhichhavethesmallest distance with respect of the point inside the uid domain where the mesh velocity will becomputed. The method can be generalized when there are two or more body surfaces. In this case,itmustbeusedapointofeachbodysurfacetocalculatethemeshvelocitieseldinsidetheuiddomain.3.ThestructuraldynamicsolverwithatriangularthinplateelementAgeneralizedconformingtriangularthinplateelement withsixdegreesof freedomineachnode (three displacements and three rotations components), in which the compatibility conditionsateachnodeandalongeachsideareapplied,isusedinthepresentwork(seeFig.2)[15].Thetotalstinessmatrixoftheelementisobtainedbytheoverlapofthemembranestinessmatrix with the bending stiness matrix. A typical membrane triangular element is adopted whereeach point has two degrees of freedom of translation uxiand uyi i 1; 2; 3 and one of rotation hzii 1; 2; 3 in the plane of the element middle surface. The membrane displacements are expressedinthefollowingway:um uxuy T Nmuem19whereuemisthenodalmembranegeneralizeddisplacementsvectorgivenby Fig. 2.Thethinplatetriangularelement.P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 255uemi uxiuyihzi Ti 1; 2; 3 20andNmisthemembraneinterpolationfunctiondenedasfollows:Nmi Li0 Nuhi0 LiNvhi_ _i 1; 2; 3 21being LitheareacoordinatesandNuhi 12LibmLj bjLm; Nvhi 12LicmLj cjLm;bi yj ym; ci xm xji; j; m 1; 2; 3 22Thetransversedisplacementeldisdiscretizedbyuz Nbueb23whereuebisthenodalbendinggeneralizeddisplacementsvector,whichiswritteninthefollowingway:uebi uzihxihyi Ti 1; 2; 3 24being uzithe nodal transverse displacements, and hxiand hyithe nodal rotations around the axes xand y,respectively.NbisthebendinginterpolationfunctiongivenbyNbi NiNxiNyi Ti 1; 2; 3 25withNi Li 2Fi 1 rjFj 1 rmFmNxi 12bmLiLj bjLmLi bj bmFi rjbj bmFj rmbm bjFm

Nyi 12cmLiLj cjLmLi cj cmFi rjcj cmFj rmcm cjFm

26Fi LiLi 0:5Li 1; ri 1l2jml2im_ l2ij_; lij x2ij y2ij_;xij xi xj; yij yi yji; j; m 1; 2; 3The linear stiness matrix at element level due tomembrane andbendingeects are given,respectively,bythefollowingexpressions:KLmi _XeBTmiDmBmidX; KLbi _XeBTbiDbBbidX i 1; 2; 3 27where Xeis the element domain. The constitutive matrices Dmand Dbare given, respectively, byDm Eh1 m2 D; Db Eh3121 m2D; D 1 m 0m 1 00 0 1 m=2____28where histheelementsthickness, mthePoissonsratioand EtheYoungsmodulus.Finally,thestraindisplacementrelationsare256 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273Bmi 14Xe2bi0 bibmLj bjLm0 2cicicmLj cjLm2ci2bicibm bicmLj cibj bicjLm____i; j; m 1; 2; 3 29andBbi Ni;xxNxi;xxNyi;xxNi;yyNxi;yyNyi;yy2Ni;xy2Nxi;xy2Nyi;xy____i 1; 2; 3 30Thenon-lineargeometricstinessmatrixisgivenbyKNLi _XeGTGiTiGGidX i 1; 2; 3 31whereTicontainsthemembraneinternalforcesandisgivenbyTi TxxiTxyiTyxiTyyi_ _i 1; 2; 3 32withTxxi

3k1Dm1; k Bmik; 1_ _ umi1Tyyi

3k1Dm2; k Bmik; 2_ _ umi2 i 1; 2; 3Txyi

3k1Dm3; k Bmik; 3_ _ umi1 Tyxi33whileGGiisgivenbyGGi Ni;xNxi;xNyi;xNi;yNxi;yNyi;y_ _i 1; 2; 3 34Thedynamicequilibriumequationsofthestructural systemareobtainedbythespacediscreti-zationofthevirtualworkequations,whichmustbesolvedforeachtimestepusinganumericalintegrationscheme. Inthepresent workthewell-knownNewmarkmethodis used[16], withparameters d 1=2 anda 1=4. Ingeometrically non-linear problems the solutionof thesebalanceequations is accomplishedfor eachtimestepwithanincremental iterativeprocedureusing an updated Lagrangean formulation. The resulting system of algebraic equations are solvedusingthegradientconjugatemethodwithincompleteCholeskyfactorization[21,22].4.TheuidstructureinteractionalgorithmCommonly, uidandstructure elds have dierent scales of time. The global time stepisusuallycommandedbytheuid. AlthoughtheuseofthesametimestepfortheuidandtheP.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 257structuremayprovidesomeimplementationadvantages,theprocedurewithsubcyclesoffactornSF DtS=DtF(where DtSand DtFarethetimeintervalsadoptedforthestructureandtheuid,respectively) may oer substantial computational advantages, including an economy of CPU timeduetothesmallernumberoftimestepsinthestructuralanalysisandaneconomyintheinfor-mation transfer. The partitioned algorithm with subcycles adopted in the present work consists inthefollowingsteps(seeFig.3):(a) Settheinitialconditionsforthestructureandtheuid.(b) Updatethestructuredisplacements,velocities,accelerationsandstresses.(c) Updatetheuidvelocities,specicmass,pressureandtotalenergyusingsubcycles.(c.1) Computethenewmeshtakenintoaccountthestructuralmotion.(c.2) Updatetheuidowvariableswiththenewboundaryconditions.(d) Updatethestructuralvariableswiththeloadstransferredbytheuid.(e) Repeatsteps(c)and(d)untiltheobjectivesofthesimulationarereached.Kinematics boundary conditions, taken into account the structural motion, must be applied tothe uiddomain. Therefore, at the uidstructure interface the followingconditionmust besatised:xn1iCFtDtxiCEi 1; 2; 3 35where xn1ijCFandtDtxijCEare the updated coordinates for the instant t Dtat the uid and solidinterfaces CFand CE, respectively. For viscous uids the same velocity components are prescribedforthemesh,theuidandthestructureattheinterface.Therefore,wn1iCF vn1iCFtDt_ uiCEi 1; 2; 3 36Fornon-viscousuidsthecorrespondingprescribedboundaryconditionsare:wn1iCF ni vn1iCF ni tDt_ uiCE nii 1; 2; 3 37where ni is the normal vector with respect to the interface surface at time t Dt. Eq. (37) is appliedtoeachnodelocatedintheinterfacesurface. Fig. 3.Algorithmforuidstructureinteractionwithsubcycles.258 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)2492735.Numericalapplications5.1.SupersonicinviscidowoveraatplateInthisexampleasupersonicinviscidowoveraatplate,whichisclampedonbothends,asshown in Fig. 4, is analyzed. The plate length, width and thickness are 0.5 m, 0.2 m and 1.35 mm,respectively. The density of the uidis qF 0:339 kg/m3, while the plate has the followingproperties:specicmass qE 2710kg/m3,Youngsmodulus E 77:28GPaandPoissonsratiom 0:33.Inthisproblem,avalueequalto5.0wasadoptedforCC,denedinEq.(6).Initiallytheplateissubjecttothesamepressureonbothsides.Itisassumedthatthepressureon the inferior face of the plate falls 0.1% instantaneously and remains in this condition for 4 ls,originating displacements in the plate and disturbances in the ow. After this period, this pressurexyzv2=0.0v2=0.0v.n=0v3=0.0, vi, Fig. 4. Geometry and boundary conditions in the uid domain for the example of a supersonic inviscid ow over a atplate.Fig. 5.Generalviewoftheniteelementmeshfortheuiddomain.P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 2590 0.02 0.04 0.06 0.08 0.1-0.002-0.0015-0.001-0.000500.00050.0010.00150.002Mach = 1.9Mach = 2.0Mach = 2.02Mach = 2.05displacement (m)in x=0.35mtime (s)Fig. 6.Displacementsoftheplatein x 0:35malongthetime,forMach 1.9,2.0,2.02and2.05.0 0.02 0.04 0.06 0.08 0.1-0.002-0.0015-0.001-0.000500.00050.0010.00150.002Mach = 2.0Rifai et al., 1999displacement (m)in x=0.35mtime (s)Fig. 7. Comparison of the displacements in x 0:35 m obtained by the present work for Mach2.0 and by Rifai et al.[4]forMach1.98.260 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273returns to its initial value of 28 kPa. The solution of the linear aeroelastic problem indicates a limitof stability with a critical Mach number equal to 2.0 using the piston theory [23]; above this value,continuousgrowthofoscillationsamplitudesareobservedandinstabilityoftheplateoccurs.Thedomaindiscretization isaccomplished suchthattheuid tetrahedral facesadjacent totheplate are coincident with the triangles belonging to the structure, although this fact would not benecessary for the solution scheme. 13303 nodes and 38915 tetrahedral elements for the mesh of theuiddomainand506nodesand508triangularelementsforthestructural meshareused. Thisinviscidowhasatwo-dimensionalbehavior,onlyonelayerofelementsisadoptedtodiscretizethedomaininthedirectionoftheaxisz, transversetotheow. Ageneral viewofthemeshisshowninFig.5.TheboundaryconditionsimposedtotheuidareshowninFig.4,whileintheplateareimposedthefollowingconditions: u v w hx hy hz 0:0intheclampedendsand w hx hy 0:0intheplaneofsymmetry xy.Fig. 8.PressuredistributionforMach 2.0at t 0:07s.Fig. 9.CongurationsoftheclampedplateforMach 2.0at t 0:03,0.07and0.1s.P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 261Inarstanalysis,simulationsareaccomplishedwithMachnumbersaroundthecriticalvaluepredictedbytheanalytical solutionusingthepistontheory[22]. Resultsofthedisplacementofthe plate in x 0:35 m along the time are shown in Fig. 6 for Mach numbers 1.9, 2.0, 2.02 and2.05. Theseresultswereobtainedusingtoanalyzetheplatetimestepsequal to1.8 107sfor0 0.02 0.04 0.06 0.08 0.1-0.02-0.015-0.01-0.00500.0050.010.0150.02LinearNon lineardisplacement (m)in x=0.35mtime (s)Fig. 10. Comparison of the displacements in x 0:35 m between the geometrically linear and non-linear solutions withMach2.3.Fig. 11.Descriptionoftheproblemtostudytheinteractionbetweenairowandaexiblemembrane.262 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273Mach1.9and1.75 107sforMach2.0,2.02and2.05.ItisobservedthatthecriticalMachnumberobtainedhereispracticallyequaltothevaluepredictedbythesolutionusingthepistontheory (Mach2.0). In Fig. 7 these results are compared with those obtained by Rifai et al. [4] (inthisworkacritical Machnumberequal to1.98wasfound). Itisnoticedthat, althoughalittlediscrepancy stays among these results, the amplitudes are very close. The pressure distribution inthedomainforaMachnumber 2.0attheinstant t 0:07sisshowninFig.8.Fig.9presentstheplatecongurationsandvaluesofverticaldisplacementsat t 0:03,0.07and0.1s.Acomparisonbetweenthelinearandthenon-linearbehaviouroftheplateisperformedforMachnumber 2.3andatimestepDt 1:6 107s, whosedisplacementsinx 0:35mareshowninFig. 10. Thegeometricallylinearsolution, aswell asthesolutionbasedinthepistontheory [22], presents an exponential growth of the oscillations, while the geometrically non-linearsolutionpresentslimiteddisplacement amplitudes. Thedierencebetweenthetwosolutionsisattributedtothecouplingbetweenmembraneandbendingeectsinthepanel whennon-linearelasticity due to nite displacements is considered. This phenomenon increases the eectivestinessoftheplate,modifyingthebehaviourandtheresponseofthesystem.Fig. 12.Finiteelementmesh.(a)Generalview.(b)Detailoftheregionclosetothemembrane.Table 1AirandmembranepropertiesMembraneproperties Value Airproperties ValueYoungsmodulus, E 3.333 108N/m2Soundspeed, C1345m/sPoissonsratio, m 0.0 Kinematicviscosity, l 17.9 106Pa sSpecicmass, qE1000kg/m3Specicmass, qF1.21kg/m3P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 263Asthestructureofthisexamplepresentsrelativelysmall displacements, theanalysismaybecarried outusingthe simpliedmeshmovement algorithmdescribedpreviouslyinSection 2.3.Itis obtained the same results for the displacements along the time, in simulations with Mach2.20,usingthecompleteandsimpliedmeshmovement algorithms. Thespeed-upobtainedwiththesimpliedalgorithminrelationtothecompletealgorithmis1.83, evenwiththefall ofMops(from888to670inaCRAYT-94computer). The reasonof this result is the dierence ofcomputationalcostthatthemeshmovementalgorithmrepresentswithrespecttothetotalCPUtime. While the complete algorithmrepresents 45.82%of the total CPUtime, the simpliedalgorithmrepresentsonly0.74%.Fig. 13.Pressuredistributionattheinstants:(a)1.5s,(b)3.0s,(c)6.0s,(d)9.0s,(e)10.5sand(f)12.0s.264 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)2492735.2.AirowpastaninatedmembraneTheinteractionbetweenairow(consideredasanincompressibleow)andaexiblemem-braneis analyzed. Theproblemis describedinFig. 11. Themembranehas asemi-cylindricalformat with diameter dand thickness h equal to 20 m and 3.0 mm, respectively, and is clamped atbothends.Theexternalboundary oftheuiddomainisasemi-cylinderwithradius Randwidth Lintheperpendicular direction to the plane where the ow take place equal to 200 and 1 m, respectively.At the external boundary a velocity prole, which is a function of the distance with respect to theground level, is prescribed. This velocity is referred to the velocity at 10 m above the ground level,Fig. 14.Velocityvectorsattheinstants:(a)1.5s,(b)3.0s,(c)6.0s,(d)9.0s,(e)10.5sand(f)12.0s.P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 265whichvarieswithtimeasshowninFig.11.Areferencepressure p 0isprescribedinthepointlocated in the external boundary condition, 200 m above the ground level. The air and membranepropertiesaregiveninTable1. Theinternal pressurep0whichkeepsthemembraneinated, istakenequalto60%oftheairstagnationpressureatavelocityof28m/s(p0 285:0Pa).Duetotheexistenceof airinsidethespacecoveredbythemembrane, adampingmatrixisincludedtoanalyzethestructure.ARayleighviscousdampingisadopted,whichmeansthatC aM bK 38Fig. 15.Meshcongurationsintheuiddomainat:(a)1.5s,(b)3.0s,(c)6.0s,(d)9.0s,(e)10.5sand(f)12.0s.266 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273where C, M and K are the damping, mass and stiness matrices, respectively. Considering a linearbehaviour, adoptingdampingratiosequal ton1 1:0%andn2 2:0%fortherstandsecondmodes, respectively, andcalculatingthecorrespondingcircularfrequenciesx1andx2thecoef-cient aandbcanbe determinedbythe procedure describedbyBathe [16]. For this case,a 0:003529and b 0:443162areobtained.Although this case is essentially a two-dimensional problemand beamelements for thestructuremaybeused, athree-dimensional codeisemployed, wherethemembranemodel wasbuilt with triangular at elements and the uid domain was discretized with tetrahedral elements.Intheperpendiculardirectiontotheow,onlyonelayerofelementsisused.The non-structured nite element mesh in the uid domain has 20 550 tetrahedral elements and7097nodes, asindicatedinFig. 12. Thestructural meshhas124nodesand124triangularele-ments (these elements are coincident withthe faces of the tetrahedral elements at the uidstructureinterface).Theprescribedboundaryconditionsonthelateralendsare w hx hy 0fortheplateand v3 0fortheuiddomain.As Reynolds number in this problem is relatively high (Re 1:9 107), if it is determined withareferencevelocityequal to28m/sandareferencelengthequal totheradiusofthestructure(10m),asimplealgebraicturbulencemodelisused.TheReynoldsstressesaremodeledusingageneralizationof thePrandtl mixinglengthhypothesis[24]. Thismodel wasalsoemployedbyMittal andTezduyar[25], andconsistsintheadditionofaeddyviscositylTtothemolecularkinematicviscosity l.Theeddyviscosityisgivenbythefollowingexpression:lT Kl212ovioxj ovjoxi_ _:ovioxj ovjoxi_ _39whereK 0:41istheVonKarmanconstant andl istheshortest distancebetweenthepointwhere lTwillbedeterminedandtheclosestwalltothispoint.Atime interval DtF 1:25 103s is adopted for the uid domain, while a time stepDtS 1:25 104sisused toanalyzethemembrane. Pressuredistributions,velocityvectorsandFig. 16.Geometryandboundaryconditionsforthepanelvibrationinducedbyvortexsheddingoriginatedbyarigidxedrectangularprism.P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 267mesh congurations in the uid domain are shown in Figs. 13, 14 and 15, respectively for t 1:5,3.0, 6.0, 9.0, 10.5and12.0s. Anoverpressureontheleft-handsideoftheeldandanunder-pressure over the middle andthe right-handside of the eldare showninFig. 13. The re-bouncingof themembrane, whileavortexisbeinggenerateddownstreamof thestructure, isTable 2FluidandpanelpropertiesProperties ValueYoungsmodulus, E 2.5MN/cm2Poissonsratio, m 0.35Specicmass, qE0.1g/cm3Soundspeed, C134 500cm/sSpecicmass, qF1.18 103g/cm3Kinematicviscosity, l 1.82 104g/cmsFig. 17.Finiteelementmeshfortheuiddomain.(a)Generalview.(b)Detailintheregionclosetothepanel.268 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273shown in Fig.14 atrst 3s. Within thesubsequent 3s, ata constant free-stream velocity,nearlysteadyconditionsareattained. Whenthefreestreamvelocityincreasesfromv10 14to28m/s(t 69s), themembranemovesagaintotherightpushedbytheairowandthevortexfor-mation downstream, behind the membrane, becomes pronounced. Following this initial phase thefree-streamairvelocityremainsconstantandtheperiodicdevelopmentanddetachmentofvor-ticesincorrespondencewiththemembranestructuremovementisindicatedinFig.14.Itisob-served in Fig. 15 the distortion of the nite element mesh at dierent times and a goodperformanceofthemeshmovementalgorithmwasobtained.tempo (s)displacement(cm)displacement(cm)0 1 2 3-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5Present workWall and Ramm, 1998tempo (s)0 1 2 3-1.5-1-0.500.511.5Present workWall and Ramm, 1998time (s) time (s)(a)(b)Fig. 18. Comparison of results obtained in this work and those obtained by Wall and Ramm [17]. (a) Displacements ofthepanelattheend.(b)Displacementsofthepanelatthemidspan.P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 269MembranecongurationsobtainedinthepresentworkareverysimilartothosepresentedbyArgyrisetal. [26]. However,pressuredistributionsandvelocityvectorshereshowvorticesgen-erated downstream of the structure that was not well captured by the above mentioned reference,probablybecausetheniteelement meshwas poorlyrened(takingintoaccount thecharac-teristicsofthisproblem).5.3.PanelvibrationinducedbyvortexsheddingoriginatedbyaxedrectangularprismAexible panelis clampedinaxed rigidrectangular prism,whichgenerates vortex sheddinginducingvibrations onthe deformable structure. This problemis describedinFig. 16usingFig. 19.Meshcongurationsat(a) t 3:765sand(b) t 3:85899s.270 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273dimensionless values. The uid is considered as slightly compressible and Re 332. Although thiscase is essentially a two-dimensional problem, a three-dimensional code is used, where tetrahedralelementsareemployedfortheuiddomainandat plateelementsfortheexiblepanel. Theboundary conditions on the lateral ends are w hx hy 0 for the plate and v3 0 for the uiddomain. Fluidandpanel properties aregiveninTable2. Bearman[27] performedadetailedexperimental studyandclariedthe various regimes of the owencounteredbyvaryingthesplitterplatelengthforrangeofReynoldsnumber.A non-structured mesh with 19198 tetrahedral elements and 6544 nodes, with only one layer ofelementsintheperpendiculardirectiontotheow,isusedfortheuiddomainandisshowninFig. 17, whilethestructural meshhas 84triangular plateelements and84nodes. The uidstructureinteractiontakesplaceat bothsidesof thepanel (it meansthat therearetwouidstructureinterfaces).Fig. 20.Pressuredistributionsat(a) t 3:765sand(b) t 3:85899s.P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273 271Simulationof the dynamic behaviour of the panel was carriedout using a time intervalDt 3:0 105s. DisplacementatthefreeandatthemidspanareshowninFig. 18, andarecomparedwiththoseobtainedbyWall andRamm[28]. Theseauthorsuseatwo-dimensionalcode.Some discrepancies can be observed between the results of the present work and those obtainedbyWallandRamm[28].Reasonsofthesedierencesareverydiculttoestablishbecausebothworks use dierent schemes, dierent meshes and the structure was modeled with beam elements,whereasinthepresent workthinplateelementswereused. Meshcongurationsandpressuredistributionsat t 3:765and3.8589sareshowninFig.19andFig.20,respectively.6.ConclusionsApartitionedalgorithmforthesolutionofuidstructureinteractionproblemsusingatwo-stepTaylorGalerkinmethodfortheuidintheALEformulationandtheNewmarkimplicitschemeforthesolutionofthestructural dynamicequationsispresentedinthiswork. Thepo-tential of themethodwas shownthroughseveral examples of compressibleandslightlycom-pressible ows acting on exible structures (plates and shells) and good results were obtained. Thedierences between the dynamic responses of a linear and a non-linear system were studied in theexampleofasupersonicinviscidowactingonadeformableatplate.Itwasalsoveriedthatusingprocedures withsubcycles, the precisionof the results are reducedandthe number ofiterationstoobtaintheconvergenceofthesolutionoftheequilibriumequationsofthestructureincreases. But, itwasalsoobservedthatthedecreaseincomputerprocessingtimemaybecomeexpressive when a non-structured mesh withvery dierent sizeof elements are used. Themethodof meshmovement adoptedhere was showntobe ecient andthe complete andsimpliedalgorithms weretested. Thecompleteschemehas muchmoreaccuracy, howeverit requires aconsiderable CPU time, disadvantage that can be avoided using the simplied scheme, which hasreasonable precision for problems with small displacements. Finally, the good performance of thevectorized algorithms was veried. Speeds going from 670 to 888 Mops in a Cray T94 computerandwithaCPUtimeoftheorderof1.22.3 105s/Dt nodehavebeenobtained.AcknowledgementsTheauthorswishtothankCNPqandCAPESforthenancialsupport.References[1] Felker FF. Direct solution of two-dimensional NavierStokes equations for static aeroelasticity problems. AIAA J1993;31(1):14853.[2] BlomFJ. Amonolithical uidstructureinteractionalgorithmappliedtothepistonproblem. ComputMethodsApplMechEng1998;167:36991.[3] Soria A, Casadei F. Arbitrary LagrangianEulerian multicomponent compressible owwith uidstructureinteraction.IntJNumerMethodsFluids1997;25:126384.272 P.R.F.Teixeira,A.M.Awruch/Computers&Fluids34(2005)249273[4] Rifai SMet al. Multiphysics simulationof ow-inducedvibrations andaeroelasticity onparallel computingplatforms.ComputMethodsApplMechEng1999;174:393417.[5] Farhat C, Lesoinne M, Maman N. 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