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A98-16371

AIAA 98-0515A Conservative Algorithm for ExchangingAerodynamic and Elastodynamic Data inAeroelastic SystemsCharbel Farhat and Michel LesoinneUniversity of ColoradoBoulder, COPatrick LeTallecINRIA RocquencourtFrance

36th Aerospace SciencesMeeting & Exhibit

January 12-15, 1998 / Reno, NV

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A Conservative Algorithm for ExchangingAerodynamic and Elastodynamic Data in Aeroelastic

Systems

Charbel Farhat * Michel LesoinneWd Patrick LeTallec

Department of Aerospace Engineering and Sciences

and Center for Aerospace Structures

University of Colorado at Boulder

Boulder, CO 80309-0429, U.S.A.

Abstract

The prediction of many fluid/structure interactionphenomena requires solving simultaneously the cou-pled fluid and structural equations of equilibriumwith an appropriate set of interface boundary con-ditions. In this paper, we consider the realistic situa-tion where the fluid and structure subproblems havedifferent resolution requirements and their compu-tational domains have non matching discrete in-terfaces, and address the proper discretization ofthe governing interface boundary conditions. Wepresent and overview new and common algorithmsfor converting the fluid pressure and stress fields atthe fluid/structure interface into a structural load,and for transferring the structural motion to thefluid system. We discuss the merits of these algo-rithms in terms of conservation properties and solu-tion accuracy, and distinguish between theoreticallyimportant and practically significant issues. We val-idate our claims and illustrate our conclusions withseveral transient aeroelastic simulations.

*AIAA Associate FellowfAIAA Student MemberCopyright © 1997 by C. FarhatPublished by the American Institute of Aeronautics andAstronautics, Inc. with permission

1 Introduction

The numerical simulation of fluid/structure interac-tion phenomena arises in many scientific and engi-neering applications including, to name only a few,blood flow circulation, parachute dynamics, airfoiloscillations, flutter prediction, fighter tail buffeting,gate sliding, and a large class of acoustics problems.When in such applications some of the fluid domainboundaries undergo a motion with a large ampli-tude, it becomes necessary to solve the flow equa-tions on a moving and possibly deforming grid. Sucha grid is often referred to in the computational aero-dynamics literature as a dynamic mesh.

Several approaches have been proposed in the pastfor solving fluid/structure interaction problems onmoving and deforming meshes, among which we notethe two closely related Arbitrary Lagrangian Eule-rian (ALE) [1,2] and dynamic mesh [3] methods. Inthe most general case, all of these methods can beused to formulate the fluid/structure problem of in-terest as a three-field problem: the fluid, the struc-ture, and the dynamic mesh that is often representedby a pseudo-structural system. For example, in thecase of the ALE method, a fluid/structure interac-tion problem can be described by the following cou-

American Institute of Aeronautics and Astronautics

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pled partial differential equations

JV,.(7:(W)-xW) =

PS aV= - div(E : c(us}) = b

f - div(E : e(x}} = g(us)

(1)The first of Eqs. (1) is the ALE nondimensional con-servative form of the Navier-Stokes equations anddescribes viscous flows on dynamic meshes. Here, tdenotes time, a dot designates a derivative with re-spect to time, x(t] denotes the time-dependent po-sition or displacement of a fluid grid point (depend-ing on the context of the sentence and the equa-tion), £ its position in a reference configuration,J = det(dxId£), W is the fluid state vector using theconservative variables, and T and 1Z, denote respec-tively the convective and diffusive fluxes. The sec-ond of Eqs. (1) is the classical elastodynamic equa-tion where us denotes the displacement field of thestructure and ps its density, e and E denote respec-tively the strain tensor and the tensor of elasticities,and 6 represents the body forces acting on the givenstructure. This equation can be replaced by anotherone describing a nonlinear behavior of the structurewithout affecting the issues raised and resolved inthis paper. Finally, the third of Eqs. (1) governsthe dynamics of the fluid moving grid. It is similarto the elastodynamic equation because the dynamicmesh is viewed here a pseudo-structural system. Abar notation is used to indicate that p is a fictitiousdensity, and E is a fictitious tensor of elasticities [4].Usually, the fluid mesh motion is triggered by thedisplacement or vibration of the structure, which isrepresented here by g(u$)- The various Dirichletand Neumann boundary conditions intrinsic to eachof the fluid and structure problems are omitted forsimplicity.

Clearly, the first and third, and the second andthird of Eqs. (1) are directly coupled. If UF denotesthe ALE displacement field of the fluid and p its pres-sure field, <T5 and crp the structure stress tensor andthe fluid viscous stress tensor, F the fluid/structureinterface boundary (wet boundary of the structure),and n the normal at a point to F, the fluid and struc-ture equations are usually coupled by imposing that

as- n = —p

us = up

on F

on F(2)

The first of these two interface boundary conditionsstates that the tractions on the wet surface of thestructure are in equilibrium with those on the fluidside of F. The second of Eqs. (2) expresses thecompatibility between the displacement fields of thestructure and the fluid at the fluid/structure inter-face. For inviscid flows, this second equation is re-placed by the slip wall boundary condition

up.n — on F (3)Of course, the structure and dynamic mesh motionsare also coupled by the continuity conditions

x = us

dx_ _ dusdt ~ dt

on F

onF(4)

It should be noted that the three-way coupled for-mulation presented here is tailored for aeroelasticsimulations. However, it also covers hydroelastic andstructural acoustic vibrations [5] as well as a largeclass of linear and nonlinear fluid/structure interac-tion problems as special instances.

When the computational domains of the fluid andstructure problems have matching discrete inter-faces, the discretization of Eqs. (2-4) is straight-forward. However in most realistic applications, thefluid and structure meshes are incompatible alongthe fluid/structure interface, either because theyhave been designed by different analysts, or becausethe fluid and structure problems have different reso-lution requirements. For example in aeroelastic com-putations, the fluid grid is typically finer than thestructure mesh, as illustrated in Fig. 1. In suchcases, the proper discretization and enforcement ofEqs. (2-4) raise a few issues that are dealt with inthis paper. These issues relate primarily to the con-servation of momentum and energy [6], and to theoptimality of the global numerical solution. Theyare addressed in this paper in the context of par-titioned procedures for the solution of the govern-ing fluid/structure equations. However, the discus-sions, methodologies, and conclusions offered herehold also for monolithic solution schemes.

In the sequel, we denote by Ff and FS the discretefluid/structure interfaces on the fluid and structuresides, respectively. In general, F^ and FS are nonmatching and do not coincide with F (Fig. 2).


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Structure

Figure 1: Fluid and structure meshes with nonmatching interfaces

Figure 2: Continuum and discrete fluid/structureinterfaces

2 Evaluation of the load induced bythe fluid on the structure

First, we address the problem of discretizing andenforcing the equilibrium on the fluid/structure in-terface boundary of the tractions on the wet sur-face of the structure and the tractions on the wallboundary of the fluid (first of Eqs. (2)). For illustra-tion purposes, we use the example depicted in Fig.1. We consider the case where the structural prob-lem is solved by the finite element method, becausethis approximation method has been dominating thefield of computational structural mechanics since atleast 1975. We do not make any assumption aboutthe discretization method used for solving the flowproblem.

2.1 Consistent interpolation basedmethods

Let e denote a finite element of the structural modelthat is in contact with the wet surface of the struc-ture, and fig G TS the geometrical support of thewet part of this element. The finite element rep-resentation of the nodal loads induced by the fluidpressure and stress fields on the structure element eis given by

•(«) _ Ni(-pn + , ds (5)

where n denotes here the normal to £1$ , and AT,-is the finite element shape function associated withnode i of element e. Most if not all finite elementstructural analysis codes adopt a quadrature rule forevaluating the integral in Eq. (5), and therefore com-pute the approximate nodal loads

f l S ) =g=ng

).n) (6)9=1

where wg is the weight of the Gauss point Xg, and ngis the number of Gauss points used for approximat-

*(«) (<=)ing f> '. Hence, in order to compute /•' J, a struc-tural analysis code needs the values of the fluid pres-sure and stress fields at the Gauss points of the ele-ments on the wet surface of the structure. This taskis simplified if in a first step, every Gauss point of awet structural element is properly paired with a fluidcell or a fluid element as shown in Fig. 3. Fast algo-rithms for pairing the interface boundary elements of


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two unstructured meshes in view of fluid/structurecomputations can be found in [7,8].

Figure 3: Gauss-point — fluid cell (or element) pair-ing

Once a Gauss point Xg 6 FS has been paired witha fluid cell C,- £ I> or an element fi^ £ TF, p(Xg)and (TF(Xg) can be computed by a number of ap-proximation methods [9]. However, if such a compu-tational strategy is selected for enforcing equilibriumat the fluid/structure interface, we favor computingp(Xg) and <jp(Xg) using the discretization methodthat is intrinsic to the flow solver. For example,if the flow solver is finite volume or finite elementbased, we favor interpolating p(Xg) and <Tp(Xg) in-side using the same shape functions as thoseemployed by the fluid solution method in the ab-sence of a structure problem. In the sequel, we referto such an interpolation scheme as a consistent in-terpolation method.

The global system defined by the union of the fluidand structure subsystems being a closed system, itfollows that at any time t, the reaction of the struc-ture is equal to the action of the fluid, and the energyreleased (except for eventual structural damping) orabsorbed by the structure is equal to the energygained or released by the fluid. Therefore, it is de-sirable that the fluid and structure loads computedrespectively on IV and FS also verify this prop-erty. Clearly, if the fluid and structure meshes havenon matching discrete interfaces, and/or the fluidand structure solvers employ different discretizationmethods — for example, different shape functions— a consistent interpolation method such as theone described earlier does not guarantee that thesum of the discrete loads /,- on the wet surface

of the structure are exactly equal to the sum of thefluid loads computed on IV. For this reason, most ifnot all interpolation based algorithms designed forconverting the fluid pressure and stress fields at thefluid/structure interface into a structural load arestrictly speaking non conservative. However, thisdoes not mean that they are necessarily unreliableor inaccurate, as demonstrated below.

In order to highlight the good performance of awell designed consistent interpolation method forcomputing the load induced by a fluid on thewet surface of a structure, we consider the two-dimensional simulation of the transient aeroelasticresponse of a flat panel with infinite aspect ratio ina supersonic stream. More specifically, we considerthe case where the free-stream Mach number is setto MOQ = 1.98, which corresponds to a free-streamvelocity that is slightly below the flutter speed ofthis panel [2]. The target panel is damped at bothends. It has a length L = 0.5m, a uniform thick-ness h = 1.35 x 10~3 m, a Young modulusE = 7.728 x 1010 N/m2, a Poisson ration = 0.33,and a density p = 2710 Kg/m3. It is representedhere by a finite element model with 10 Euler beam el-ements. Hence, this structural mesh is coarse enoughto reveal any potential deficiency of an interpola-tion based method. Yet, this coarse finite elementstructural model is fine enough to reproduce the firstthree fundamental modes of the panel with less than2 % error. This is an important remark becauseat MOO = 1-98, the aeroelastic response of the flatpanel is dominated by its first two bending modes.The flow above the panel is assumed to be inviscid(Euler equations), and its computational domain isdiscretized into 4163 vertices and 8016 triangular el-ements. Hence, for this problem, IV is ten timesfiner than FS. The software package Matcher [7] isused for pairing the non matching fluid and struc-ture interfaces (Fig. 4), and the consistent interpo-lation method described by Eq. (6) is employed withtwo Gauss points per element for computing the fi-nite element nodal loads acting on the structure. Inparticular, the effect of the structural rotations aretaken into account both for computing the momentsinduced by the fluid on the structure, and for trans-ferring the structural vibrations to the fluid system.

First, the shape of the panel is set to the firstbending mode, and a steady-state flow is computedabove the slightly deformed panel. Then, the de-formed shape is used as an initial displacement,and the transient aeroelastic response of the cou-


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pled fluid/structure system is computed by solvingEqs. (1).

77

7

7

Figure 4: Partial view of the non matching discreteinterfaces

Clearly, the results reported in Fig. 5 and Fig. 6show that the consistent interpolation scheme sum-marized by Eq. (6) generates surface structuralloads that are in perfect equilibrium with the fluidtractions (aerodynamic lift). The work of the struc-tural tractions acting on F5 is also shown to beequal to the work of the fluid tractions acting onTp at all time steps. Given the coarse mesh reso-lution of the structural finite element model, theseare rather amazingly conservative numerical results.They demonstrate the fact that a numerical algo-rithm can be non exactly conservative, and yet pro-duce numerically conservative results.

Figure 5: Aerodynamic lifting force vs. structuralsurface force

As illustrated above, conservativity is not in gen-eral a major problem for the consistent interpolationmethod presented herein. However, we can point thefollowing drawback of such an approach. In aeroelas-ticity, the structure is often represented by a "fish-bone" (Fig. 1, left) or "wing-box" (Fig. 1, right)

Figure 6: Energy gained/released by the fluid vs.energy released/gained by the structure

equivalent model for which the geometric discrepan-cies between I> and FS are rather large. In suchcases, the lifting force can still be accurately com-puted by Eq. (6) if a larger than usual number ofGauss points is employed for computing the finite el-ement nodal loads, but the drag forces may never becaptured. For example in the case of an inviscid flow,the consistent interpolation method described herepredicts a zero drag force for the fish-bone structuralmodel shown at the left of Fig. 1, while the integra-tion of the pressure forces on Tp may not necessarilyyield a zero drag force.

Figure 1: Fish-bone and wing-box structural models

2.2 A new conservative method2.2.1 The general algorithmOne obvious solution for enforcing exactly theequilibrium between the fluid tractions computedon TF and the structure tractions computed onFS independently from any discretization issue, isto compute the tractions on both sides of thefluid/structure interface using the discretizationmethod and mesh of the same field, either the fluid,or the structure. For example, if the fluid (structure)grid is finer than the structure (fluid) mesh alongthe fluid/structure interface, we propose to computethe surface forces and moments induced by the fluidon the structure using the discretization method ofthe fluid (structure) and the geometrical support Tp(Fs). Such a computational strategy will guarantee


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that the momentum of all the loads acting on thefluid/structure interface will always be equal to zero.In order to ensure that the total energy of these in-terface loads will also be equal to zero at all timesteps, we propose the following computational algo-rithm for evaluating the forces and moments inducedby the fluid on the structure.

Let UF and us denote two fluid and structureadmissible virtual displacement fields. By admis-sible, we mean that the traces of UF and us on thefluid/structure interface satisfy in the viscous case

UF = us

and in the inviscid case

UF -n = us-

on T

on T

(7)

(8)

Whichever approximation method is chosen for en-forcing compatibility on F between the virtual —or real — displacement fields of the fluid and thestructure, its outcome can be formulated as

UFj = (9)

where V.FJ is the discrete value of UF at the fluidpoint j, ws; is the discrete value of 115 at the struc-ture node i, and is and Cjj are constants that de-pend on the chosen method of approximation. Ofcourse in the inviscid case, Eq. (9) holds only forthe components of UFJ and us; that are normal tothe interface boundaries Tp and FS, respectively.

Consider a virtual displacement field UF that iszero on each degree of freedom in the flow domainexcept on those on the boundary Tp • Whicheverdiscretization method is chosen for solving the flowproblem, UF can be expressed as

uF = (10)

where Dj is some function with a local or global sup-port on Tp • The virtual work of the fluid tractionsacting on Tp can be written as

SWF = /r (— pn ds

where $j has the physical meaning of a numericalpressure flux and is given by

j = / (—pn + crp-n)Dj dsJrF

(12)

Substituting Eq. (9) into Eq. (11) leads to

swF = =(13)

Finally, noting that the virtual work of the finiteelement structure forces and moments acting on FScan be written as

(14)

and that energy is conserved at the fluid/structureinterface if SWF = SWS, we conclude from Eqs.(13-14) that

3=3 F(15)

As anticipated in the beginning of this section, thenew expression of /j does not depend on the dis-cretization method of the structure. The term inthe first bracket in Eq. (15) depends exclusively onthe discretization method chosen for solving the flowproblem, and the term in the second bracket de-pends only on the approximation method selectedfor enforcing the compatibility on F between the dis-placement fields of the fluid and the structure. Thisconcludes the description of our general conserva-tive algorithm for converting the fluid pressure andstress fields into a structural load. For illustrationpurposes, we specify next this algorithm to the casewhere the fluid/structure displacement compatibil-ity conditions are discretized by the finite elementmethod.

2.2.2 A special caseA natural but not necessarily mathematically opti-mal approximation method for enforcing the secondof Eqs. (2) or Eq. (3) is a consistent finite element

Is based interpolation method. It is natural because,as stated earlier, the structural problem is almost al-ways solved by the finite element method, and there-

(11) fore the structural displacement field Ug inside the


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wet region tig' 6 FS of an element e is often given

«s, (16)

where ie denotes the total number of wet nodes be-longing to element e, and AT,- is as before the finiteelement shape function associated with node i of el-ement e. Hence, in the presence of fluid and struc-ture meshes with non matching discrete interfaces,the second of Eqs. (2) or Eq. (3) can be discretizedby

a) pairing each fluid grid point Sj on IV with theclosest wet structural element tig £ T$ (seeFig. 8 and [7]).

b) determining the natural coordinates Xj in tigof the fluid point Sj (or its projection onto tig ).

c) interpolating up inside tig using the sameshape functions Nj as in Eq. (14) to obtain

UF, = UF(SS) =

j e e rs(17)

From Eqs. (9,17) it follows that for a finite ele-ment approximation of the fluid/structure displace-ment compatibility conditions, cy,- = Ni(Xj), andtherefore

(18)

Figure 8: Fluid grid point—wet structural elementpairing

Given that the shape functions of a finite element» = >'e

satisfy ^ AT,- = 1, the reader can easily verify that: = 1

the nodal loads expressed in Eq. (18) satisfy

1=1 S

(19)

which is an alternative proof of the conservationproperties of our new algorithm.

For example, consider the simple fluid/structureinterface depicted in Fig. 9, where T$ has one struc-tural finite element with two nodes, and Tp has fourfluid cells and four vertices. Using Eq. (18), the fi-nite element nodal loads induced by the fluid on thestructure are

h =

and the sum of the loads acting on the structure is

X3) + N2(X3))$a + $4

which, once again, illustrates the conservation prop-erties of our new load computation algorithm.

Using Eq. (17) and the specific form (18) of theconservative algorithm presented in this paper, wehave recomputed the transient aeroelastic responseof the supersonic panel problem introduced in Sec-tion 2.1. The obtained time histories of the lift coef-ficient and the fluid/structure energy exchange arereported in Fig. 10 and Fig. 11, respectively, andcontrasted with those obtained with the non conser-vative load evaluation algorithm discussed in Section2.1. These figures show that both conservative andnon conservative load transfer schemes produce ex-actly the same aeroelastic results.


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Figure 9: Application example of the new conserva-tive algorithm

3 Transfer of the structural motionto the fluid system

When the structure moves and possibly deforms, itsmotion is transferred to the fluid system by the sec-ond of Eqs. (2) or Eq. (3), and Eqs. (4). Be-cause the fluid and structure meshes have usuallynon matching interfaces, several interpolation pro-cedures have been [10] and continue [9] to be de-veloped for exchanging elastodynamic data betweenthese two mechanical systems. As stated in Sec-tion 2.2.1, all these interpolation methods can becast into a general form where the fluid displace-ment field is expressed in terms of the structuraldisplacement field as follows

UF, = , » e r5 (22)

SUPERSONIC PANEL -U

Figure 10: Time history of the lift coefficient

Figure 11: Time history of the fluid/structure en-ergy exchange

where UFJ is the discrete value of the fluid displace-ment field at the fluid point j, us, is the discretevalue of the structure displacement field at the struc-ture node i, and is and Cji are constants that dependon the interpolation algorithm.

Alternatively, a mortar method can be used forthat purpose [11], but mortar methods are in gen-eral computationally expensive for aeroelastic appli-cations.

4 Applications

Here, we consider the prediction of the transientaeroelastic response of the ARW-2 [12] wing in atransonic airstream. We model the spars, ribs, skin,hinges, control surfaces, and discrete masses of thiscomposite aeroelastic research wing by suitable finiteelements as recommended in [12]. More specifically,our detailed finite element structural model (Fig.14) contains 308 bar elements, 304 beam elements,806 triangular membrane elements, 338 triangularshell elements, 456 nodes, and 2556 active degreesof freedom. The wet surface of this structural modelcorresponds to the lower and upper skins and con-tains a total number of 830 triangular elements (Fig.12).

In order to construct a fluid mesh that is rea-sonably fine for Euler flow computations, we re-discretize the upper and lower skins of the ARW-2 into a total number of 3879 triangles, and useGHS3D [13] to generate a three-dimensional un-structured fluid mesh with 27872 vertices and 159073


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Figure 12: Detailed finite element structural modelof the ARW-2 (top skin is removed for clarity)

Figure 14: Partial view of the fluid surface mesh ofthe ARW-2 wing

tetrahedra. Note that our fluid surface mesh whichhas 3879 triangles is more than 4 times finer than thestructure surface mesh (Fig. 13). We use Matcher[7] for pairing the non matching fluid and structureinterfaces.

First, we set the free-stream Mach number toMOO = 0.8, freeze the structural model in itsfirst bending mode shape, and compute a steady-state flow around this deformed configuration of theARW-2 wing (Fig. 15).

Figure 13: Finite element discretization of the skinof the ARW-2 wing Figure 15: Mach isovalues for a perturbation along

the first bending mode

Next, we consider the deformed configuration out-lined above as an initial displacement field, and


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seek to predict the transient behavior of the coupledfluid/structure system triggered by this initial per-turbation. We perform two numerical simulations.In the first one, we compute the forces induced bythe fluid on the structure by the non conservativeconsistent interpolation method discussed in Section2.1, with three Gauss points per wet structural ele-ment. In the second simulation, we compute theseloads by the conservative scheme presented in Sec-tion 2.2. In both cases, we use a consistent finiteelement interpolation method (Eq. (17)) to enforcethe compatibility on the interface boundary F of thefluid and structure displacement fields.

Figure 17: Energy gained/released by the fluid vs.energy released/gained by the structure (consistentinterpolation method)

•FnkI UWa_Fofc.- ——

Figure 16: Aerodynamic lifting force vs. structuralsurface force (consistent interpolation method)

The results reported in Fig. 16 and Fig. 17 clearlyshow that, for this ARW-2 transonic problem withstrong shocks, the consistent interpolation method isnot conservative. This is unlike the case of the shock-free supersonic panel problem (see Section 2.1) forwhich this interpolation based method produces nu-merically conservative results. However, the max-imum relative error in momentum conservation is3.7%, and the maximum relative error in energy con-servation is 5.0%. These are rather small errors thatcan be eliminated if a larger number of Gauss pointsare employed in Eq. (6) for computing the finite el-ement nodal forces.

On the other hand, the results reported in Fig. 18and Fig. 19 highlight the conservation properties ofthe new method proposed in this paper for comput-ing the loads induced by the fluid on the structure.

Finally, we note that despite its lack of conser-vation properties, the consistent interpolation basedmethod predicts a lift history for this ARW-2 wingthat differs at most by 1% from that computed withthe new conservative scheme (Fig. 20).

Figure 18: Aerodynamic lifting force vs. structuralsurface force (new conservative method)

Figure 19: Energy gained/released by the fluid vs.energy released/gained by the structure (new con-servative method)

Figure 20: Aerodynamic lifting force: new conserva-tive method v.s. interpolation method

10


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5 Conclusions

In this paper, we have addressed the problem of dis-cretizing a class of fluid/structure interface condi-tions, and exchanging aerodynamic and elastody-namic data between a flow solver and a structuralanalyzer. More specifically, we have considered therealistic situation where the fluid and structure sub-problems have different resolution requirements andtheir computational domains have non matching dis-crete interfaces. We have over vie wed a family ofinterpolation methods and presented a new conser-vative algorithm for computing the loads inducedby the fluid on the structure. We have shown thateven though they are not conservative, interpolationbased methods can be accurate and reliable whenthe fluid and structure interfaces share the same ge-ometrical support, but not necessarily the same dis-cretization points. When they do not, as in fish-bone and wing-box aeroelastic models, interpolationbased methods can be inaccurate, but the proposedconservative algorithm is accurate, robust, and reli-able.

References

[1] J. Donea, An arbitrary Lagrangian-Eulerian fi-nite element method for transient fluid-structure in-teractions, Comput. Meths. Appl. Mech. Engrg.33 (1982) 689-723.[2] C. Farhat, M. Lesoinne and N. Maman, Mixedexplicit/implicit time integration of coupled aeroe-lastic problems: three-field formulation, geometricconservation and distributed solution, Internat. J.Numer. Meths. Fluids 21 (1995) 807-835.

[3] J. T. Batina, Unsteady Euler airfoil solutions us-ing unstructured dynamic meshes, AIAA Paper No.89-0115, AIAA 27th Aerospace Sciences Meeting,Reno, Nevada, January 9-12, 1989.

[4] C. Farhat, High performance simulation ofcoupled nonlinear transient aeroelastic problems,AGARD Report R-807, Special Course on ParallelComputing in CFD (1'Aerodynamique numerique etle calcul en parallele), North Atlantic Treaty Orga-nization (NATO), October 1995.

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