Fluid-Structure_Interaction/1903996163/files/00000___743cbde1a0ed249906070abafa1174af.pdf

Fluid-Structure_Interaction/1903996163/files/00001___3209d3ebfb8f3a2d54722fa443eca058.pdfFluid-StructureInteraction

Fluid-Structure_Interaction/1903996163/files/00002___90524b976018ac7efb3d0cf0d0430316.pdf

Fluid-Structure_Interaction/1903996163/files/00003___44a9da5d3e3232a67cfec90023ff83ea.pdfI N N O V A T I V E T E C H N O L O G Y S E R I E S

Fluid-StructureInteraction

edited byAlain Deruieux

London and Sterling, VA

Fluid-Structure_Interaction/1903996163/files/00004___05224873504af9053fb77ab78e62e9d7.pdfFirst published in 2000 by Hermes Science Publications, ParisFirst published in Great Britain and the United States in 2003 by Kogan Page Science, animprint of Kogan Page LimitedDerived from Revue europeenne des elements finis. Fluid-Structure Interaction, Vol. 9, no.6-7.

Apart from any fair dealing for the purposes of research or private study, or criticism orreview, as permitted under the Copyright. Designs and Patents Act 1988, this publication mayonly be reproduced, stored or transmitted, in any form or by any means, with the priorpermission in writing of the publishers, or in the case of reprographic reproduction inaccordance with the terms and licences issued by the CLA. Enquiries concerning reproductionoutside these terms should be sent to the publishers at the undermentioned addresses:

120 Pentonville Road 22883 Quicksilver DriveLondon Nl 9JN Sterling VA 20166-2012UK USAwww.koganpagescience.com

Hermes Science Publishing Limited, 2000 Kogan Page Limited, 2003

The right of Alain Dervieux to be identified as the editor of this work has been asserted byhim in accordance with the Copyright, Designs and Patents Act 1988.

ISBN 1 903996163

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library.

Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynnwww. biddies.co.uk

Fluid-Structure_Interaction/1903996163/files/00005___1da63b9672dc1b92c48e7a15ae9f9006.pdfContents

ForewordAlain Dervieux vii

1. Fluid-Structure Interaction: A Theoretical Point of ViewCeline Grandmont and Yvon Maday 1

2. Design of Efficient Partitioned Procedures for the Transient Solution ofAeroelastic ProblemsSerge Piperno and Charbel Farhat 23

3. Deriving Adequate Formulations for Fluid-Structure InteractionProblems: From ALE to TranspirationThierry Fanion, Miguel Fernandez and Patrick le Tallec 51

4. Sensitivity Analysis and Control in an Elastic CAD-Free Frameworkfor Multi-Model ConfigurationsBijan Mohammadi 81

5. Numerical Study of the Aeroelastic Stability of an OverexpandedRocket NozzleEmmanuel Lefrancois, Gouri Dhatt and Dany Vandromme 99

6. Fully Coupled Fluid-Structure Algorithms for Aeroelasticity andForced Vibration Induced Flutter: Applications to a CompressorCascadePenelope Leyland, Frederic Blom, Volker Carstens and Tiana Tefy 137

Fluid-Structure_Interaction/1903996163/files/00006___62fcd23819eb8b36df9e918ddba9049e.pdf7. Interaction between a Pulsating Flow and a Perforated MembraneRaphael Lardat, Romauld Carpentier, Alain Dervieux, Bruno Koobus,Eric Schall, Charbel Farhat, Jean-Fran9ois Guery and Patrick Delia Pieta 179

8. Analysis of a Possible Coupling in a Thrust InverterRaphael Lardat, Alain Dervieux, Bruno Koobus, Eric Schall andCharbel Farhat 193

9. Aeroelastic Coupling between a Thin Divergent and High PressureJetsEric Schall, Raphael Lardat, Alain Dervieux, Bruno Koobus andCharbel Farhat 211

Index 229

vi Fluid-Structure Interaction

Fluid-Structure_Interaction/1903996163/files/00007___2af81d910c1f959189815c7275bc1723.pdfForeword

This publication is devoted to mathematical and numerical models for fluid-structureinteraction. Fluid-structure interaction concerns the study of mechanical systemsinvolving a fluid and a structure that have a mechanical influence on each other.

This publication has two special foci - compressible fluids and unsteady models.

Although we are restricting our attention to unsteady aeroelasticity, we cannot claim thatthe subject is new. It seems to me, however, that, at the beginning of the new millenium,aeroelasticity is a meeting point between the well developed theories of unsteadyaeroelasticity and recently developed and still immature methods for unsteady flowcalculations. Indeed, if the 1980s were Euler model years in CFD, unsteady Navier-Stokes is the challenge for the years 1995-2005.

In the example of aircraft flutter analysis in the transonic range, simplified models donot yield accurate enough predictions of critical parameters, and there is a need for morecomplex models. Improved predictions can be obtained using Euler models; newdecisive tools are expected from the use of Navier-Stokes models.

A second frontier is the analysis of flow involving vortices. Some of them can bepredicted using an unsteady Euler model, most of them make Navier-Stokes mandatory.

It is now clear that in the next few years that progress in aeroelasticity will be heavilydependant on the improvement in simulation tools for unsteady turbulent flows,particularly those involving large vortices.

The coming together of coupling methods, with the emergence of URANS or LES inmoving geometries could fulfi l l expectations. Indeed, as our technological societydevelops larger machines with greater demands for safety, environmental considerationsand energy savings, this leads to a demand for accurate studies in aeroelasticity.

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The study of numerical problems for fluid-structure coupling has shown that many non-trivial questions were and are still open. This is why this publication is devoted to amajor extent to theoretical issues, with the first paper proposing a review of recentresults in energy estimates and the existence of continuous solutions. With the recentdesign of efficient numerical strategies as the result of theoretical advances, some of thecontributions published here are devoted to such theoretical advances.

The last three contributions describe recent applications of a 3D unsteady numericalmodel.

Our thanks are due to the contributors and to Hermes Science for providing theopportunity to gather so well-focussed a collection of papers on this stimulating topic.

Alain DervieuxINRIA, Sophia-Antipolis

Fluid-Structure_Interaction/1903996163/files/00009___3bbdf4c2e7a5b90ff65baeabaca498ee.pdfChapter 1

Fluid-Structure Interaction:A Theoretical Point of View

Celine GrandmontCeremade, Universite Paris Dauphine, France

Yvon MadayLaboratoire d'Analyse Numerique, Universite Pierre et Marie Curie, Paris; andLaboratoire ASCI, Universite Paris Sud, France

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Fluid-Structure_Interaction/1903996163/files/00011___bb4a98bdfe304620af01d899154d038e.pdfA Theoretical Point of View 3

1. Introduction

This paper deals with the mathematical and numerical analysis of problems dealingwith unsteady fluid-structure interaction phenomena. These phenomena are of majorimportance for aerospace, mechanical or biomedical applications, and thus have beenstudied by many authors over the past few years from different points of view (theory,numerical analysis and simulations). The problem is to describe the evolution of aviscous fluid coupled with a moving structure. The fluid can be compressible or in-compressible, and the structure can be rigid or elastic. Several conditions determine thecoupling between the two media at their interface. First, the kinematic condition statesthat the fluid velocity and the structure velocity are equal. The second coupling condi-tion traduces the action-reaction principle. The interaction is not reduced to these onlytransmission conditions since, in most interesting cases where the deformations of thestructure are large enough, one can not neglect the variation of the fluid domain. Wethus have to solve a problem defined (at least for the fluid part) over a time dependentdomain. We focus here on the analysis and numerical analysis of the fluid-structureinteraction problems in the case where the deformation of the fluid domain is actuallypart of the unknown. We refer to [MOR 92] for a study of the vibrations of coupledproblems in which the domain is fixed.

Section 2 is devoted to a general presentation of the problem. Energy estimatesare formally derived. Then we expose existence of weak solutions (section 3) andexistence of strong solutions (section 4). Finally, in section 6, we give some results ofthe numerical analysis of the discretizations.

2. Standard mathematical formulation

In order to simplify the presentation we assume that the flow is viscous and ho-mogeneous, incompressible or compressible and that its behaviour is described by theNavier-Stokes equations. We denote by v its constant viscosity, pf its density. Forthe structure, we can consider several cases: rigid bodies immersed in fluid, full 3Delasticity or hyperelastic models. But we can also consider asymptotic models suchas plates, shells, beams that are used when the thickness or the section of the elas-tic media is small with respect to its other dimensions. We can even consider thatthe displacement of the structure is a linear combinaison of a finite number of elasticeigenmodes. This latter modelization and the case of rigid bodies actually reduces thestructure equations to ordinary differential equation (o.d.e). In the other cases, the par-tial differential equations (p.d.e) describing the structure are classically set in a fixeddomain ns. The domain ns is, in general, the reference configuration of the body(that will be assumed to coincide with its initial state for the sake of simplicity). Thebehaviour of the structure is described by the displacement d of each point x of thereference configuration. Consequently, each point x 6 ns is, at time t, at the positionx(t) = x + d(x, t). On the contrary, the fluid equations are set in Eulerian coordi-nates and are thus defined in an unknown domain n f ( t ) depending on the structuredisplacement d. All the unknowns linked to the fluid part are thus evaluated at each

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point of the physical domain, at time t. The incompressible Navier-Stokes equationsare: Find (u,p) in ns() such that

where u denotes the fluid velocity, p its pressure and ff a given exterior force. Theseequations are completed using initial data

and by boundary conditions. These boundary conditions are of two different types.We have to distinguish the part of a n f ( t ) which is not in contact with the structurefrom the fluid-structure interface that we will denote by r(t). On rf = anf (t) \ r(),we consider standard boundary conditions and in order to simplify the presentation,we will suppose that they are of Dirichlet type:

On r(t), coupling conditions have to be considered and will be detailed later on.The compressible isentropic Navier Stokes model is: Find (u, pf) in n f ( t ) such

that

where y is a real number > 2 and is the parameter of the pressure law. Again, theseequations are completed by initial data

and by boundary conditions.

Before specifying the proper boundary conditions over the interface r() for bothmodelizations, we have to specify the different models for the structure we consider:3D linearised elasticity, asymptotic model (many situations can be thought about, likebeams, plate or shells equations...) or the equations of rigid body motion or even finitedimensional modal decomposition.

First, we consider the 3D linearised elasticity. The constitutive law of the elasticmedia is supposed to be Hooke's law. The Lame constants of the media are \s andus and ps is its density, fs denotes the volumic force applied on the structure. Theequations can be written as follows

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where the stress tensor a is given by

and e(d) represents the linearised strain tensor and is equal to

Initial conditions have to be added, for instance

Boundary conditions on the part of the structure rs that is not in contact with theflow have also to be added (for instance homogeneous Dirichlet boundary conditions).Next, boundary conditions on the interface have to be stated. Denoting by r the partof the boundary of ns which corresponds to the fluid-structure interface, we write thatr and r ( t ) represent the same entity.

and

Next, we introduce the kinematic condition at the interface: on F(t) the fluid sticksto the structure, and thus the velocities of the fluid part and of the structure part areequal. Since the fluid is supposed to be viscous (in the case of an invicid flow only thecontinuity of the normal component of the velocities is required):

The other boundary condition corresponds to the action-reaction principle and can bewritten as follows:

where T F (u ,p ) stands for the expression of the normal component of the fluid stresstensor af (af = 2vD(u) pI in the incompressible case and 07 = 2vD(u) +(A divu pf)I in the compressible case with D(u) denoting the symmetric part ofVu) written in the reference configuration.

If the structure under consideration possesses a small section or thickness, then,asymptotic models can be proposed to describe its behaviour. The coupling condi-tions are expressed again by the equality of the velocities at the interface and, presentin the right hand side of the plate, beam or shell equations, since a forcing term ap-pears that comes from the stress applied by the fluid to the structure. Let us consider,

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for instance, a plate of thickness 2e and of average surface w. The elastic medium issupposed to be isotropic and homogeneous. E denotes its Young modulus, v itsPoisson ratio and ps its volumic mass. With the previous notations we have

We only present here the equations satisfied by the transverse displacement of theplate d3 = d3(1, 2). The equations can be written as follows using the Einsteinconvention, and considering that the Greek indices belong to {1,2}:

where fs denotes a volumic force and g+, g- surfacic forces applied respectively onw x {e} and w x {e}. The plate is moreover supposed to be clamped on aw x[e, +e]. The longitudinal displacement is given by

Let us express the coupling conditions between this plate and the viscous flow onthe interface which is supposed to be r = w x {e} . We have, for all x E r,

Furthermore, the structure is submitted to a surfacing force coming from the fluid,and thus g- depends on the fluid stress tensor written in the reference configuration.More details about plate, beam, shell equations can be found in [CIA 90], [DES 86],[BER94].

The third case that has been considered is the rigid body motion or the reducedbasis motion where the structure is deformable with displacements that are written as alinear combination of a finite number of modal functions associated with the continuouselastic operator. The first coupling condition is the equality of the velocities at theinterface. The second coupling condition appears in the right hand side of the structureequations which are now o.d.e with respect to the position of the center of gravityXG and the rotation angle vector 0 and the coefficients of the modal decomposition.Considering a solid sphere immersed in a fluid, we have the Newton equations

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where ns(t) is the rigid region at time t and m and J denote respectively the massof the rigid body and its inertia momentum. The displacement of the structure can berecovered, writing

2.1. Energy estimates

First, we write the coupled problem in a variational formulation, assuming that thesolution of the problem exists and is sufficiently smooth to justify all the manipula-tions. Multiplying the fluid equations [1] by a divergence free function v, the trace ofwhich over a n f ( t ) \ r() is equal to zero, and then integrating over af(t), it becomes,after integration by parts

For the structure part - for instance in the case of 3D linearised elasticity - we alsomultiply equation [8] by a test function b, satisfying homogeneous Dirichlet boundaryconditions over ans \ r, then integrate over ns. After integrating by parts, we obtain

with

If we choose the test functions such as

then adding [20] to [19] and taking into account the coupling condition [14], we havefor any b and v satisfying [21]

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Similarly, starting from equations [5], [6], we deduce for any b and v satisfying [21]

NOTE The same kind of variational formulation can be obtained for the plateequations. Concerning the rigid body, it is slightly different and we can obtain aglobal weak formulation where the unknown is the global velocity, obtained byextending the velocity of the fluid by a rigid body velocity in the domain occupied bythe structure at time t.

NOTE In all the cases, the test functions of the coupled problem depend on time(the problem is set in a non cylindrical domain). Moreover, the test functions dependon the solution of the problem, which is not standard.

Let us derive energy estimates. We choose (u, ad/at) as test functions in [22]. Notethat they are admissible test functions, in particular they satisfy [21]. This leads to

Considering the incompressibility [2], the convection term becomes, after integrationby parts,

where n is the outer unit normal vector of nf(t) and w is the velocity of each pointu2

of the interface r(t). Using it with $ = - we derive (we recall that pf is a constant2

the fluid is homogeneous)

We recall the Reynolds formula

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Thus taking into account these two equalities, [24] yields

recalling the ellipticity of the bilinear form a over HQ rs ( n s ) (c.f [CIA 86]), an energy

estimate can then be deduced if the solution of the coupled system [1], [2], [8], [13],[14] exists:

In case of compressible flows, we obtain, in a similar way from [23], if the solutionof the coupled system [5], [6], [8], [13], [14] exists:

The same type of energy estimate can also be obtained (at least formally) for thetwo other models presented before. Such estimates are the first step to prove theexistence of weak solutions.NOTE As mentioned in [ERR 94], the convection term seems to be necessary,in most cases, in order to obtain energy estimates without imposing supplemen-tary conditions on the data (small enough data, small time....) when dealing withunknown time dependent domains. Besides the theoretical analysis, this fact is impor-tant to point out, especially for the numerical simulation of fluid-structure interaction.Indeed, as a starting point, the Stokes problem is often the first step of the implemen-tation of the fluid discrete problem:the coupling of the Stokes model with the structureinteraction may be unstable regardless of the presence of bugs, indeed the correspond-ing continuous problem may not be stable either to start with, as was exhibited in[ERR 94]. To simulate the coupling it is thus important to consider the full, nonlinear,Navier-Stokes problem.

Let us now present some of the theoretical results that can be found with such mod-els. In a first section we review the results dealing with the existence of weak solutions,then look at the question of existence of strong solutions.

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3. Weak solution

3.1. Elasticity

For the time being, as far as we are aware, no result is available on the fullinteraction problem: Navier-Stokes coupled with the 3D elasticity in time dependentdomain. Nevertheless, we refer to [GRA OOa] for an existence result on a similar butsteady state problem.

For the time being, the rigid body case seems to be the most accessible one.

Note however that, in the case of weak perturbations of the interface, one canassume that the fluid occupies a fixed domain: nf(t) = nf. Numerically, whenthis assumption is made, other interface boundary conditions are often considered inorder to take into account the interface motion: these are the transpiration techniques(see [BAR 94], [FAN 00]). From a theoretical point of view, J.L. Lions [LIO 69]proves the existence of a unique weak solution "a la Leray" using the Galerkin methodfor the coupled problem: Navier-Stokes coupled with the linearised elasticity. Sincenf(t) = nf, the convection term does not disappear when the energy estimates arederived. Consequently, the interface condition [13] is modified in order to obtain aboundary of the solution. If we want to keep [13], existence of weak solutions can beproven considering the Stokes equations instead of the Navier-Stokes equations.

3.2. Rigid Body

Several papers treat the model of rigid bodies immersed in a viscous, incom-pressible or compressible flow from the theoretical point of view and answer the ques-tion of the existence of weak solutions. We consider three of them, where quite thesame weak formulation is used but where the techniques to prove the solvability aredifferent. The unknown is the global velocity u equal to the fluid velocity in nf (t)and to the rigid bodies velocity in ns(t) ( h A (x XG)). Let us denote by

at atp the global density: p = p f 1n f ( t ) + P s 1 s ( t ) where 1E denotes the characteristicfunction of a given part E. In view of the conservation of mass [2] or [6], p satisfies alinear transport equation

Now, we consider a test function w that is rigid in n s ( t ) and divergence free in theincompressible case, which is the case exposed below. The fact that w is rigid inns(t) can be written as follows ln s(t)D(w) = 0. Let us introduce the space of testfunctions

Fluid-Structure_Interaction/1903996163/files/00019___a0376ef4aa440139c602695c8cd5aad3.pdfNote that the space of test functions depends on the continuous solutions. Startingfrom equations [2], [16], [17], one can derive the following variational formulationfor all w E V and a.e. t,

A Theoretical Point of View 11

where f fs in the structure part and f = ff in the fluid part.NOTE Such a global formulation can be used for the numerical simulation ofparticles in flow, and enables one to employ fixed mesh. The fictitious domain methodrelies also on such a formulation. We refer to [GLO 94a], [GLO 94b] for more detailson the fictitious domain method.

The fact that this problem is a weak form of the original coupled frame is standardwith regard to the fluid domain, indeed, we recover [1] by using a test function w = 0on nS(t). Let us assume now enough regularity of the solutions, then we first observethat over n s ( t ) , the functions u and w correspond to a rigid motion and thus have theform

where b and are arbitrary functions of t and a(t) = *dt usm ^rst C 0 menb = 0, we obtain [16] and [17].

In [DES 99], [DES OOa], [CON 99] and [HOF 99] this weak formulation (or aquite similar one) is introduced. There are mainly three non standard features in thesetype of problems:

1. The Navier-Stokes equations are set in a non cylindrical domain, so classicalGalerkin method does not apply.

2. The test functions depend on the solution.3. The convection term requires compactness results for the velocity.

In [DES 99], [DES OOa] the existence of weak solutions for all T > 0 is proven,assuming that there is no collision (i.e. the body does not touch the exterior boundaryan or if there are several particles they do not enter in collision). In the incompressiblecase, the velocity belongs to L2((0, T), H^(n)) x L((0, T), L2(n)) and the densitybelongs to Loc((0,T),L00(n)), provided that u0 L2(n) , f E L2((0,T) x n)and P0 E L(n). In the compressible isentropic case the velocity is such that^/pu E L((0,T),L2(n)) and u E L2((0,T),H01(n)) and the density belongs toI((0, T), L r ( n ) ) , provided suitable assumptions on the data. The proof is based onSchauder fixed point theorem and compactness results.Stepl

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The motion of the rigid body is supposed to be given, together with a global ve-locity v satisfying a nonlinear constraint (which we denote with (*)) stating that v cor-responds to the rigid body motion in the given rigid body region. This velocity isregularized in space and time through an operator RE that preserves the constraint (*).Then the original problem is linearised using this regularized velocity vE = RE (v).For E fixed, the problem is now linear and the test functions depend on vE. The equa-tions are next written, thanks to a change of variables, in Lagrangian coordinates. AGalerkin method is then used to solve them. The basis functions considered are oftwo types: first the eigenfunctions of a Stokes-like problem defined in the initial fluiddomain, where the coefficients depend on vE, and a basis of the set of rigid motionextended in the fluid part. At this step, the assumption that there are no collision isrequired.Step 2

Knowing that the previous linearized problem is well posed, we obtain a new ve-locity uE, that gives us a new rigid body motion. The Schauder fixed point theorem isapplied on the mapping S that associates with v, S(v) = uE (which has been actuallyslightly modified in order to satisfy the nonlinear constraint (*)). The compactness isobtained because of regularity results on uE.Step 3

Next, one has to pass to the limit when E goes to zero, using an a priori energybound that states that /pE uE E L((0,T),L2(n)) and uE 6 L2((0,T),H01(n))(and also in the compressible case that pE L((0, T), L r(n))). One can then passto the limit in the mass equation thanks to Di Perna-Lions compactness results for lin-ear transport equations [DIP 89] (the case of isentropic compressible flow is slightlymore complicated). Next, one has to study the momentum equation. At this step, com-pactness results on the velocity are required: it is shown that the velocity is compactin L2((0, T) x n), thanks to Riesz-Frechet-Kolmogorov compactness theorem in Lp(see [BRE 83], p. 72). In order to pass to the limit in the weak equation, consideringw E V, a test function wE VE is built such that wE converges in the good spacesthrough w. At this step the assumption that no collision occurs is needed again.

Note that more recently, M. Tucknak has proven in [TUC 00] a compactness resultstating that the weak limit of any weakly convergent sequence of solution of a fluid-structure interaction problem is still a solution of such a problem using basic theoryon Navier-Stokes, thus simplifying the approach based on the use of renormalizedsolutions involved in the previous approach.

These types of methods can be extended to tackle a structure described by a fi-nite number of modal functions (see [DES OOb]). It seems difficult to apply them tomore general structure models since one needs space and time regularity of the fluid-structure interface, and 3D elastic models do not provide the required regularity.

The next articles deal with the incompressible model.In [CON 99], the same result is proven but for only one rigid body. The problem

is written in a new system of coordinates moving together with the body (thus making

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difficult the generalization of the approach to more than one rigid body). This toolhad been already used by D. Serre in [SER 87] where he studies a ball in a viscousfluid that occupies the whole space R3. So the ball is then fixed inside a domainn(t) whose exterior boundary is moving (and over which a homogeneous Dirichletboundary condition is set). They embed this moving domain inside a fixed larger oneE and extend the functions defined over n(t) by 0 over the complementary set notedas n c ( t ) . As in [FUJ 70] a penalizing term is added to the translated-rotated equation[29] corresponding to a mass term on the velocity over n c ( t ) , so that, in the limit,the velocity will be zero over &(t). This allows one to suppress the difficulty inducedby the time dependent domain. A standard Faedo-Galerkin method is used to prove thatthe penalized problem is well posed and the a priori bounds on the solution are thenused to pass to the limit following the same compactness arguments as the ones usedin [FUJ 70].

In [HOF 99], the problem of the motion of a solid body ns (t) in a bounded domainfilled with a viscous incompressible fluid is adressed. The approach is again to writethe problem over the whole domain but the equation over p is replaced by an equa-tion over 0 = 1 n t ( t ) that belongs to the space Char(Q) the class of characteristicfunctions of subsets of Q. The problem reads as follows: find (u, 0) with

such that [29] is satisfied together with

holding for any n E C l ( Q ) , n ( T ) = 0.The analysis of problem [29], [32] then proceeds by using a penalized method

that consists in adding the term -D(v}D(v} in [29], since the rigid body motionbelongs to the kernel of D(.). An additional ingredient is required that consists ofregularizing the equations by adding the term 6VAvVAu. It is not hard to check thatthis resulting problem is well posed, providing a solution vE,s and that the followinga priori estimate holds

The proof proceeds by passing to the limit first in E, second in 5 using a generalizedAubin compactness theorem. In their paper, the authors also investigate the behaviourof the body near the wall and prove that if the body comes to the wall, its speed mustvanish, which proves the limit of the modelization of the phenomenon in this extremesituation by the Navier-Stokes equation.

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3.3. Asymptotic models

By asymptotic models we mean models of plates, beams or shells. In [FLO 00],the authors study a three-dimensional problem where a plate interacts with a linearcompressible fluid. The plate equations are the ones described previously, with an ad-ditional term which represents the rotation inertia (= A^^2-). This term regularizesthe plate equation. The existence of weak solutions (in the energy spaces) is obtainedthanks to a fixed point procedure (Kakutani fixed point theorem), provided that thedata are small enough. The fluid equations are studied for a given geometry of thefluid domain. In order to deal with the non cylindrical domain an elliptic regular-ization of the equation is performed. These tools have been used in [SAL 85], whereNavier-Stokes equations defined in a given time dependent geometry were considered.The structure equations are studied for a given forcing (coming from the regularizedfluid equations). These are linear equations defined in a fixed domain, so there is noparticular difficulty. Then the problem is recoupled thanks to a fixed point procedure.The final step is the convergence of the regularized problem through the real one.We refer also to [FLO 99], where the same problem in 2D is treated, and where theexistence of a smooth solution is proven.

4. Strong solution

In [GRA 00b], we study the existence of strong solutions for rigid bodies im-mersed in a viscous incompressible fluid contained in a bounded domain. To handlethe problem of the time dependent domain the Navier-Stokes equations are written inLagrangian coordinates, and thus the new unknowns are the Lagrangian velocity andpressure (still denoted by u and p). The space K^ (n ) in which we search the solutionis defined as follows: we set for T > 0,

The main result is the following:

Let r be a real number, 1 < T < 3/2. We assume that u0 H r+1(n f(0)), ff andfs are sufficiently smooth and that the mass and the momentum of inertia of the bodyare sufficiently large, then there exists a time T1 > 0 depending on the data (nf(0),||u0||Hr+1(nf ((o))), ff ... ) such that the Problem has a unique solution with u

The proof is based on several fixed point procedures. We study, in a first step, afluid problem with a given velocity over ans. For such equations we prove that thereexists a smooth solution with the help of a fixed point theorem (contraction mappingprinciple). The ideas are the same that one can find in the papers [ALL 83], [ALL 87],[BEA 81], [SOL 77], [SOL 88a] where the authors have studied the solvability of theNavier-Stokes equations with free surface in bounded or unbounded domains. Theapproach is the following: the equations are rewritten in Lagrangian coordinates (this

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change of variable enables one to solve the difficulties linked to the time dependent do-main) and it is shown that solutions for the initial value problem exist locally in time,in smooth functions spaces K r t(n). This first step enables us, for a given velocityof the rigid body, to define a fluid velocity and a fluid pressure. This gives us thefluid constraints. The next step is to solve the structure equations for given exteriorforces (coming from the fluid constraints). A new velocity of the interface is derived.The problem is recoupled thanks to a fixed point procedure. It is at this step that weneed to add the constraint on the size of the mass and inertia momentum of the rigidbody. Nevertheless, this additional condition that seems unnatural, allows us to obtaina uniqueness statement on the solution of the coupled problem.

Theses results can also be extended to a deformable structure whose displacementis a linear combinaison of a finite number of modal functions associated to the contin-uous elastic operator. The equations that described the evolution of the structure areo.d.e. and admit sufficiently smooth solutions in order to apply the same techniquesdescribed above (see [GRA OOc]).

5. Summary

Let us summarize the different techniques and strategies used in those type ofproblems. The question of time dependent domain can be solved as follows:

- Considering the fluid equations written in Lagrangian coordinates;- Penalization techniques;- Elliptic regularization of the equations.

We have also seen that one can try to find a global formulation of the fluid-structure in-teraction problem or look at the fluid equations and the structure equations separatelyand used a fixed point procedure to recoupled the problem.

Numerically, the dependence in time of the fluid domain requires also special pro-cedures and can be treated thanks to different techniques. One has to deal with movingmesh or to develop special formulations in order to work on a fixed mesh. Neverthe-less, in the latter case one has to find a way to track the moving interface. For flowsin time dependent domains, the ALE (Arbitrary Lagrangian Eulerian) formulation isoften used. It consists in working on a moving mesh whose motion is determined by amesh velocity whose only constraint is to be equal to the fluid velocity at the interface.We refer to [HUG 81], [DOE 82] for more details. Other approaches have also beenproposed among which we note the space-time formulations [TEZ 92].

We have already mentioned the fictitious domain techniques that has been used inthe case of a fluid interacting with rigid bodies (to simulate sedimentation phenom-ena). The mesh is fixed and a global formulation on the global velocity is used for thefluid-structure problem. A Lagrange multiplier is introduced in order to enforce therigid motion in the rigid region. One can also refer to the level set techniques used in adifferent context: modelization of the evolution of nonmiscible flows, where a global

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formulation is used and solved on a fixed mesh and where the interface is trackedthanks to a level set function satisfying a transport equation [CHA 96].

Nevertheless, in most cases, one has a fluid code able to solve the fluid equations(on a moving mesh) and a different structure code that treats the structure part. Thequestion is then how to couple these two codes in order to obtain efficient algorithms;the advantage is that the fluid or structure models can be easily modified in such a"black-box" approach. In the next section, we focus on the numerical problems en-countered for fluid-structure interaction simulation and more particularly: the timediscretization and the spatial discretization.

6. Numerical analysis

6.1. Time discretization

In order to propose a time discretization of the model, we have to choose betweendifferent strategies for the treatment of the interaction. Many different schemes fromfully implicit to fully explicit can be investigated. When the structure and the fluid arerepresented by models having the same dimension, there are numerous strategies thatcan be thought about.

The first one is a complete implicit treatment where at each time step, the geom-etry, the interaction and the fluid and structural data (velocities and stresses) are allbalanced. This highly nonlinear algorithm has to be solved iteratively. It is naturallystable but may require a special procedure (relaxation) to ensure the convergence ateach iteration [MOU 96]. In this article, the authors prove -on a linear fluid-structureinteraction problem- that an iterative process, based on the separated resolution of afluid problem and a structure problem converges through the coupled problem, pro-vided a relaxation procedure is used.

A less implicit scheme is the one where we treat explicitly the behaviour of thegeometry while retaining the coupling implicit. Thus, knowing the approximationsof velocity, displacement and geometry at the time nAt, we begin to extrapolate thegeometry at the time (n + l)At. We then have to discretize the fluid part and thestructure part and we can propose to solve (iteratively again) the fluid and the struc-ture equations so that the interface conditions (velocities and stresses) are balanced attime (n + 1)A. Another possibility is to make the coupling more explicit by solvingat each time step the fluid and the structure parts only once and independently. Thisleads to the so-called staggered or partitioned strategies. We can compute the fluidequation with Dirichlet boundary conditions (provided from the previous time steps)and then the structure equation with Neumann boundary conditions (obtained from therecently computed fluid motion) or the opposite, first the fluid with Neumann bound-ary conditions and then structure with Dirichlet boundary conditions or first structurewith either Dirichlet or Neumann boundary conditions. We can also compute first thestructure part and the fluid part with the same possibilities for the interface conditions.In the case of a 3D fluid interacting with a plate, a shell or a structure modelled by

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modal functions then the only possible staggered strategy is to treat the fluid partwith Dirichlet boundary conditions and compute the displacement knowing the ap-plied fluid efforts. There are numerous variations for all those strategies dependingon the choice of the time integrators for the fluid and the structure part, on the choiceof the evaluation of the load applied by the fluid to the structure.... One can alsothink of prediction-correction procedures. Here, we review a few articles that discussthe efficiency and stability properties of some of those strategies.

In [PIP 95], the authors introduce a criterion that ensures the stability of the nu-merical solution. This criterion expresses the energy balance at the fluid-structure in-terface. Next, they build and study a few staggered procedures applied to a 1D linearizedfluid-structure interaction problem (an Euler compressible flow interacting with a pis-ton). The basic and popular staggered algorithm denoted by CSS (Conventional SerialStaggered) they consider is the following: (1) predict the motion of the interface, (2)update the fluid mesh, (3) compute the fluid part with a given velocity, (4) computethe new force applied to the structure, (5) advance the structural system. In partic-ular, they prove that the basic CSS method completed with a well-chosen correctionprocedure provides an unconditionally stable and time-accurate scheme. Their the-oretical linear analysis is confirmed by numerical simulations on a two-dimensionalaeroelastic problem.

In part II of the previous article [PIP 99], the authors develop a new criterionthat predicts the performance of the considered partitioned procedure. They considera three field formulation - the fluid, the structure and the dynamic of the mesh- todescribe the fluid-structure interaction problem. They estimate the energy that is in-troduced at the interface of the two media by the various staggered schemes, assumingthat the structure and the pressure induced by the flow are vibrating with constant am-plitudes at the same frequency but assuming that they are not in phase. This givesan estimate of the created energy with respect to the time step. They validated theirapproach on two-dimensional and three-dimensional aeroelastic applications (super-sonic and transonic flow/ panel and wing).

In [GRA 98b], three different strategies are studied, applied to a one-dimensionalnonlinear problem where a modified Burger equation in an unknown time dependentdomain (written in ALE formulation) is coupled with a wave equation. In all thosestrategies the geometry is predicted. In the first one the interface conditions are treatedimplicitly, in the two others the treatment of the interface conditions is explicit. Thefirst explicit strategy consists in solving the fluid part with Neumann boundary con-dition and then the structure part with Dirichlet boundary condition, and the secondone corresponds to the so-called CSS procedure. The considered schemes are first or-der in time. We prove, provided that the time step and the data are sufficiently small,that the implicit strategy is stable and convergent with a rate of convergence of At3/4.Concerning the explicit schemes, the first one is stable with a stability constant thatcan explode as time increases. Moreover if the equations are also discretized in spaceusing a finite element discretization, then the scheme is stable under CFL-like condi-tions. Under more constraining CFL-like conditions, stability estimates are derived for

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the second explicit scheme but we are not able to derive stability without consideringspace discretization. These latter results seem to confirm the well-known limitation ofCSS procedure.

6.2. Spatial discretization

We now consider the question of spatial discretization of the fluid-structure inter-action problem [22]. In most of the applications and because one has -in most ofthe cases- at its disposal a fluid code and a structure code built independently, thefluid mesh and the structure mesh may be non-matching or incompatible (even if themeshes match, the partial differential equations describing the two media may requiredifferent types of discrete functions). In this framework, the question is how to couplethese two codes, how to transfer the information at the interface of the two media, ina reliable and energy-consistant way. In most of the cases the structure solver is basedon a finite element discretization, and we will denote by hs the associated spatial meshsize. Concerning the fluid one can consider either finite element discretization or finitevolume discretization. At the fluid-structure interface one has to introduce the equalityof the velocities and the load balance. Nevertheless, at the discrete level, when the dis-cretizations are incompatible, this can not be done in a strong way. The weak equality(all the quantities are now discrete ones) can be written as follows:

where Q and P are two linear operators (possibly depending on time if dealing withmoving boundary) and as (resp. af) represents the numerical structure (resp. fluid)stress tensor. The question is how to define Q and P in order to be conservative?

If Q (depending for instance on the fluid discretization) is fixed the balance ofthe fluid and structure virtual works requires that P = QT (see [GRA 98c] and[GRA 98a]. This is what is underlined in [FAR 98] where the authors present a conser-vative algorithm where the operator Q is the fluid interpolation operator and comparethe choice P = QT to the non conservative interpolation based methods where Q andP are both interpolation operators. Thanks to some numerical simulations, they showthat these non-conforming methods can be accurate in some cases (when the fluidand the structure interface share the same geometrical support) and that the conform-ing methods are accurate in all the considered cases. Considering finite element dis-cretizations for the structure part they also discuss the accuracy of the mortar elementapproach introduce by Bernadi, Maday, Patera [BE 90]. This method is conserva-tive and (as we shall see) mathematically optimal, in the sense that the error estimatefor the fluid-structure interaction problem obtained when considering incompatiblemeshes is the same as the one obtained when the meshes match. Nevertheless, itcan be noted that, considering P1 finite element discretization for the fluid part, and if

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the fluid mesh size hf is chosen of the order of h2s then the conservative interpolationmethod is also accurate.

For a numerical analysis, we refer to [GRA 98a], where error estimates are derivedfor a steady-state limit model interaction problem. On the one hand, for the fluid part,a two-dimensional second-order equation is considered discretized with a Pk finiteelement discretization. On the other hand, for the structure part, several cases can beconsidered: a 2D structure, a beam (modelized by two decoupled one-dimensionalequations: a fourth-order equation for the transverse displacement and a second-orderequation for the longitudinal displacement), or a structure modelized by a finite num-ber of modal functions. In this article, two different types of matching are considered:a pointwise matching, i.e. Q is equal to the Pk finite element interpolation operatorassociated to the fluid part, and an integral matching, i.e. the mortar element method.In all the cases, the error estimate will be of the form O(haf) + O(hBs), where (3 isoptimal in regard to the discretization associated to the structure part. The possiblelack of optimality will come from a. The following results hold ( with no assumptionmade on the relative size of hf with respect to hs)

1. For 2D structure modelized by second order elliptic equation then the standardconclusion holds (see [BE 90]) i.e.:

(a) the mortar method is optimal, Vk (a k),(b) the pointwise matching is not optimal (a = 1/2).2. For a 1D fourth-order equation :

(a) the mortar method is optimal, VK (a = k),(b) the pointwise matching is optimal for k < 2 (a = max(k, 2)).3. For a 1D second-order equation :

(a) the mortar method is optimal, Vk (a = k),(b) the pointwise matching is optimal for k < 1 (a = max(k, 1)).4. For a finite number of model functions (then the motion of the structure is

modelized by an o.d.e)(a) the mortar method is optimal, Vk (a = k),(b) the pointwise matching is also optimal, Vk.

As we see, the optimality of the pointwise matching is linked to the regularity of thedisplacement at the interface of the two media.

7. References

[ALL 83] G. ALLAIN, Un probleme de Navier-Stokes avec surface libre, These de troisiemecycle de l'Universite Paris VI, 1983.

[ALL 87] G. ALLAIN, Small-time Existence for the Navier-Stokes Equations with a FreeSurface , Appl. Math. Optim., 16, n 1, 1987.

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[BAR 94] C.BARDOS and O. PIRONNEAU, Petites perturbations et equations d'Euler pour1'aeroelasticite , M2AN,vol 28,4, 1994.

[BEA 81] J.T. BEALE, The Initial Value Problem for the Navier-Stokes Equation with a FreeSurface , Comm. on Pure and Applied Mathematics, vol XXXIV, 1981.

[BER 94] M. BERNADOU, Methodes d'elements finis pour les problemes de coques minces,RMA, Masson, 1994.

[BE 90] C. BERNARDI, Y. MADAY and A. PATERA, A New Non Conforming Approach toDomain Decomposition: the Motar Element Method , College de France Seminar, Pitman,H. Brezis, J.L. Lions, 1990.

[BRE 83] H. BREZIS, Analyse Fonctionnelle, Masson, Paris, 1983.[BRU 97] C.H. BRUNEAU, Calcul d'ecoulements incompressibles derriere des obstacles et

analyse des solutions transitoires , Actes du congres d'Analyse Numerique, 1997.[CHA 96] Y. C. CHANG, T. Y. Hou, B. MERRIMAN and S. OSHER, A Level Set Formulation

of Eulerian Interface Capturing Methods for Incompressible Fluid Flow , J. of Comp.Phys., 124, 1996.

[CIA 78] P.G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland,Amsterdam, New York NY, Oxford, 1978.

[CIA 86] P.G. CIARLET, Elasticite tridimensionnelle, Masson, Paris, 1986.[CIA 90] P.G. CIARLET, Plates and Junctions in Elastic Multi-Structures, RMA, Masson,

Paris, 1990.[CON 99] C. CONCA, J. SAN MARTIN and M. TUCSNAK, Analysis of a Fluid-Rigid Body

Problem , C. R. Acad. Sci. Paris Ser. I Math., 328, 1999.[DES 99] B. DESJARDINS, M.J. ESTEBAN, Existence of Weak Solutions for the Motion of

Rigid Bodies in a Viscous Fluid , Arch. in Rat. Mech. Anal., 146, 1999.[DES OOa] B. DESJARDINS and M.J.ESTEBAN, On Weak Solutions for Fluid-Rigid Structure

Interaction: Compressible and Incompressible Models , to appear in Comm. Pan. Diff. Eq.[DES OOb] B. DESJARDINS, M.J.ESTEBAN, C. GRANDMONT and P. LE TALLEC, Weak So-

lutions for a Fluid-Elastic Structure Interaction Model , submitted.[DES 86] P. DESTUYNDER, Une theorie asymptotique des plaques minces en elasticite

lineaire, RMA, Masson, 1986.[DIP 89] R. J. DIPERNA and J. L. LIONS, Ordinary Differential Equations, Transport Theory

and Sobolev Spaces , Invent. Math., 98, 1989.[DOE 82] J. DONEA, An arbitrary Lagrangian-Eulerian Finite Element Method for Transient

Fluid-Structure Interactions , Comp. Methods in Appl. Mech. and Eng., 33, 1982.[DUV 90] G. DUVAUT, Mecanique des milieux continus, Masson, Paris, Milan, Barcelone,

Mexico, 1990.[ERR 94] D. ERRATE, M. ESTEBAN and Y. MADAY, Couplage fluide structure, un modele

simplifie , C. R. Acad. Sci. Paris Ser. I Math., 318, 1994.[FAN 00] T. FANION, M. FERNANDEZ and P. Le Tallec, Deriving Adequate Formulations

for Fluid-Structure Interactions Problems: from ALE to Transpiration , Rev. EuropeenneElem. Finis, 2000.

[FAR 98] C. FARHAT, M. LESOINNE and P. LE TALLEC, Load and Motion Transfert Al-gorithms for Fluid/Structure Interaction Problems with Non-Matching Discrete Interfaces:

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Momentum and Energy Conservation, Optimal Discretization and Application to Aeroelas-ticity , Comp. Methods in Appl. Mech. and Eng., 157, 1998.

[FLO 99] F. FLORI and P. ORENGA, On a Nonlinear Fluid-Structure Interaction Problemdefined on a Domain Depending on Time , Non linear Analysis, 38, 1999.

[FLO 00] F. FLORI and P. ORENGA, Fluid-Structure Interaction: Analysis of a 3D Compress-ible Model , To appear in les Annales de L'lHP, 2000.

[FUJ 70] H. FUJITA and N. SAUER, On Existence of Weak Solutions of the Navier StokesEquations in Regions with Moving Boundaries , J. Fac. Sci. Univ. Tokyo, 17, 1970.

[GLO 97] R. GLOWINSKI, B. MAURY, Fluid-particle Flow: a Symmetric Formulation , C.R. Acad. Sci. Paris Ser. I Math. t. 324, 1997.

[GLO 94a] R. GLOWINSKI, T-W. PAN and J. PERIAUX, A Fictitious Domain Method forDirichlet Problem and Applications , Comp. Methods in Appl. Mech. and Eng. 1ll, 1994.

[GLO 94b] R. GLOWINSKI, T-W. PAN and J. PERIAUX, A Fictitious Domain Method for Ex-ternal Incompressible Viscous Flow Modeled by Navier-Stokes Equations , Comp. Meth-ods in Appl. Mech. and Eng. 112, 1994.

[GRA 98a] C. GRANDMONT and Y. MADAY, Nonconforming Grids for the Simulation ofFluid-Structure Interaction. Domain Decomposition Methods , 10 (Boulder, CO, 1997),262-270, Contemp. Math., 218, Amer. Math. Soc., Providence, RI, 1998.

[GRA 98b] C. GRANDMONT, V. GUIMET and Y. MADAY, Results about some DecouplingTechniques for the Approximation of the Unsteady Fluid-Structure Interaction , Enu-math97 Proceedings, published by World Scientific in Singapore, Oct. 1998.

[GRA 98c] C. GRANDMONT and Y. MADAY, Analyse et methodes numeriques pourla simulation de phenomenes d'interaction fluide-structure , ESAIM: Proceedings,http://www.emath.fr/proc/VoL3/,Vol. 3, 1998, 101-117.

[GRA 00a] C. GRANDMONT, Existence for a Three-dimensional Steady State Fluid-Structure Interaction Problem , Preprint,

[GRA OOb] C. GRANDMONT and Y. MADAY, Existence for a 3D Fluid-Structure InteractionProblem , M2AN Math. Model. Numer. Anal. , Vol 34, 2000.

[GRA 00c] C. GRANDMONT, Y. MADAY and P. METIER, Existence de solutions regulieresd'un probleme de couplage fluide structure instationnaire , Submitted. to C. R. Acad. Sci.Paris, 2000.

[HOF 99] K. H. HOFFMANN and V. N. STAROVOITOV, On a Motion of a Solid Body in aViscous Fluid. Two Dimensional Case , Adv. Math. Sci. Appl., 9, 1999.

[HUG 81] T.J.R. HUGHES, W. LIu, T.K. ZIMMERMAN, Lagrangian-Eulerian Finite ElementFormulation for Incompressible Viscous Flows , Comp. Methods in Appl. Mech. and Eng.,29, 1981.

[INO 77] A. INOUE and M. WAKIMOTO, On Existence of Solutions of the Navier-StokesEquations in a Time Dependent Domain , J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24,1977.

[LES 93] M. LESSOINE and C. FARHAT, Stability Analysis of Dynamic Mesches for Tran-sient Aeroelastic Computatitions , 11th AIAA Computational Fluid Dynamics Conference,Orlando, Florida, July 6-9, 1993.

[LIO 69] J.L. LIONS, Quelques methodes de resolution des problemes aux limites nonlineaires, Dunod, Paris, 1969.

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[LIO 96] P.L. LIONS, Mathematical topics in fluid mechanics. Vol. 1 and Vol. 2, Oxford Lec-ture Series in Mathematics and its Applications, 3, Clarendon Press, Oxford UniversityPress, New York, 1996.

[MOR 92] H. MORAND and R.OHAYON, Interaction fluide structure, RMA, Masson, 1992.[MOU 96] J. MOURO, P. LE TALLEC, Structure en grands deplacement couplees a des flu-

ides en mouvement , Rapport INRIA N2961, 1996.[PIP 95] S. PIPERNO, C. FARH AT and B. LARROUTUROU, Partioned Procedures for the Tran-

sient Solution of Coupled Aerolelastic Problems. Part I: Model Problem, Theory and Two-dimensional Applications , Comp. Methods in Appl. Mech. and Eng., 124, 1995.

[PIP 99] S. PIPERNO and C. FARHAT, Partitioned Procedures for the Transient Solution ofCoupled Aerolelastic Problems. Part II: Energy Transfert Analysis and Three-dimensionalApplications , Preprint, 1999.

[SAL 85] R. SALVI, On the Existence of a Weak Solution of a Non-Linear Mixed Problemfor the Navier-Stokes Equations in a Time Dependent Domain , J. Fac. Sci. Univ. TokyoSect. IA Math., vol 32, 1985.

[SER 87] D. SERRE, Chute libre d'un solide dans un fluide visqueux incompressible: Exis-tence , Japan J. Appl. Math., 4, 1987.

[SOL 77] V.A. SOLONNIKOV, Solvability of a Problem on the Motion of a Viscous Incom-pressible Fluid Bounded by a Free Surface , Math. USSR Izvestiya, 11, N 6, 1977.

[SOL 88a] V.A. SOLONNIKOV, On the Transient Motion of an Isolated Volume of ViscousIncompressible Fluid , Math. USSR Izvestiya, 31, N 2, 1988.

[SOL 88b] V.A. SOLONNIKOV, Unsteady motion of a finite mass of fluid, bounded by a freesurface , J. Soviet Math. 40, 1988.

[TEM 77] R. TEMAM, Navier-Stokes Equations, North-Holland Publishing Company, 1977.[TEZ 92] T. TEZDUYAR, M. BEHR and J. LIou, A New Strategy for Finite Element Com-

putations Involving Moving Boundaries and Interfaces. The Deforming Spatial/Space-TimeProcedure: I. The Concept and the Preliminary Numerical Tests Comput. Methods Appl.Mech. Engrg., 94, 1992.

[TUC 00] M. TUCSNAK, Weak Stability of the Solution of a Fluid-rigid Body Problem ,Preprint, 2000.

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Design of Efficient PartitionedProcedures for the TransientSolution of Aeroelastic Problems

Serge PipernoCERMICS - INRIA, Sophia Antipolis, France

Charbel FarhatDepartment of Aerospace Engineering Sciences and Center for AerospaceStructures, University of Colorado at Boulder, USA

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1. Introduction

Wing flutter, fighter tail buffeting and flow induced pipe vibrations are examplesof fluid-structure interaction phenomena that are of great concern to aerospace, mech-anical, and civil engineering. Since the underlying fluid system is represented by anon-linear model (i.e. the Euler or Navier-Stokes equations), these problems are refer-red to as non-linear transient aeroelastic problems.

A non-linear transient aeroelastic problem where the fluid domain boundaries un-dergo a motion with a large amplitude can be formulated as a three-field problem(fluid, structure, and pseudo-structural dynamic mesh) governed by the following cou-pled semi-discrete equations [LES 93], [FAR 95b], [PIP 95]:

where a dot denotes a time-derivative, x is the displacement or position vector of themoving fluid grid points (depending on the context), W is the fluid state vector, u isthe structural displacement vector, M, D, and K denote respectively the finite ele-ment mass, damping, and stiffness matrices of the structure, fext is the vector of exter-nal forces acting on the structure, x is the grid displacement vector, M, D, and K arefictitious mass, damping, and stiffness matrices associated with the moving fluid gridand constructed to control its motion, A results from the finite element/volume discre-tization of the fluid equations, Fc = F xW is the vector of Arbitrary LagrangianEulerian (ALE) convective fluxes, F denotes the vector of convective fluxes and Rthe vector of diffusive fluxes. The first two equations of [1] express the equilibrium ofthe fluid and structure subsystems, respectively. The third equation is a mathematicalrepresentation of a fluid dynamic mesh (note that M = D = 0 includes as particu-lar cases the spring analogy and continuum mechanics based mesh motion schemesadvocated by many investigators [BAT 90], [FAR 98a]).

For complex structural systems, the simultaneous solution of [1] by a monolithicscheme (the three fields are combined in a single numerical formulation) is in gen-eral computationally challenging, mathematically and economically suboptimal, andsoftware-wise unmanageable.

Alternatively, equations [1] can be solved by a partitioned procedure where thefluid and structure subproblems are time-discretized by different methods tailored totheir different mathematical models, and the resulting discrete equations can be sol-ved by a "staggered", or "segregated", or "time-lagged" algorithm (see for example[FAR 95b], [PIP 95], [STR 90], [MOU 96], [GUP 96]). Such a strategy simplifies ex-plicit/implicit treatment, subcycling, load balancing, software modularity, and soft-

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ware replacements as better mathematical models and methods emerge in the fluidand/or structure disciplines. The basic, most popular staggered algorithm, referredto in this paper as the Conventional Serial Staggered (CSS) procedure, goes as fol-lows:

(a) transfer the motion of the wet structural boundary to the fluid system,(b) update the position of the moving fluid mesh accordingly,(c) advance the fluid system and compute new pressure and fluid stress fields,(d) convert the new fluid pressure and stress fields into a structural load,(e) advance the structural system under the flow induced load.

Such a staggered procedure, which can be described as a loosely coupled solution al-gorithm, can also be equipped with a subcycling strategy where the fluid and structuresubsystems are advanced using different time-steps AtF and Ats [PIP 95]. Usually,one has AF < Ats.

Unfortunately, it is well-known that the time-accuracy of the CSS procedure is ingeneral at least one order lower than that of its underlying flow and structure time-integrators, and its stability limit can be much more restrictive than that of the flowand/or structure solvers. For this reason, several ad-hoc strategies have been publi-shed in the literature for improving the time-accuracy and stability properties of theCSS procedure. Most of them consist essentially in inserting some type of predic-tor/corrector iterations within each cycle of the CSS procedure, in order to compensatefor the time-lag between the fluid and structure solvers [STR 90], [PRA 94].

The mathematical analysis of the time-accuracy and numerical stability of parti-tioned procedures constructed for the solution of equations [1] has been at center ofmany works, including those of the authors, for years. However, because the depend-ence of the structure equations of equilibrium on the motion of the fluid dynamicmesh is implicit rather than explicit, and the fluid equations of motion can be stronglynon-linear, our previous investigations were limited to a one-dimensional aeroelasticmodel problem [PIP 95]. This analysis yielded guidelines for exchanging aerodyna-mic and elastodynamic data (possibly in the presence of subcycling) in a manner thatpreserves the unconditional stability and order of time-accuracy of a given partitio-ned procedure. We were able to apply some but not all of these ideas to the realisticproblem represented by equations [1] [PIP 97]. However, the formal analysis of stag-gered algorithms remains a formidable challenge, because of the same reasons thatpreviously incited us to consider a representative model problem.

For this reason, we sum up in this paper our previous contributions, including avery promising criterion for assessing the suitability of a given partitioned procedurefor the solution of the non-linear transient aeroelastic equations [1], and we give newperspectives for the construction of more efficient partitioned procedures, based onthe designed criterion [PIP 99a]. This criterion is essentially based on the evaluationof the energy that is numerically - and hence, artificially - created at the fluid-structureinterface by the staggering process and typically dictates the choice of the predictor,and/or the time-discretization of the transfer of the fluid pressure and stress fields to

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the structure subsystem. Since it is very effective at discriminating between staggeredalgorithms as well as improving them, we apply this criterion to the analysis of newpartitioned procedures constructed with enhanced parallel features, in the case of two-and three-dimensional, transonic and supersonic, wing and panel flutter problems.

2. An energy-based analysis

The global system defined by the union of the fluid and structure subsystems beinga closed system, it follows that at each time t, the reaction of the structure is equal tothe action of the fluid and the works performed by the structural forces and by the fluidpressure and stress fields at the fluid-structure interface rf/s must be opposite.

In [LES 95], [LES 96], [PIP 97], it was argued and shown that the loss in time-accuracy and numerical stability induced by staggering can be traced to a lack ofconservation of the momentum and energy at the fluid-structure interface. It was alsoargued in [PIP 97] that the time-accuracy and stability properties of a given partitionedprocedure can be improved by controlling the unbalance of energy at rp / s .

In view of the above remarks, we have proposed a framework for analyzing par-titioned procedures designed for the solution of equations [ 1 ] that is based on theevaluation of the works performed at the fluid-structure interface. More specifically,given the fluid and structure responses predicted by a staggered algorithm at eachtime tn - where n designates the n-th time-station - we propose to evaluate a parti-tion procedure by assessing the order of the difference between the work performedby the fluid pressure and stress fields, and that performed by the structural forces, atthe fluid-structure interface, as a function of the computational time-step. Not onlycan such a criterion discriminate between different partitioned procedures designedfor the solution of non-linear transient aeroelastic problems, but it can also suggesta conservative time-discretization of the pressure and stress fields transmitted by thefluid to the structure. A similar idea was exploited in [FAR 98b] to semidiscretize (inspace) the exchange of aerodynamic data between the fluid and the structure acrossnon-matching discrete fluid and structure interfaces.

In order to simplify our analysis, we consider only a generic point on the fluid-structure interface, and a surrounding patch of unit length in two dimensions, and unitarea in three dimensions. We denote by P the pressure at this point, and consider thecase of an inviscid flow. The extension of our analysis to a viscous flow is straight-forward. We apply our analysis framework to the investigation of several staggeredalgorithms: serial or parallel, collocated or non-collocated. Because the aeroelasticstructural response is usually dominated by low frequencies, we consider exclusivelyan implicit structural time-integrator. More specifically, we consider only the mid-point rule because of the popularity of this scheme in production structural codes.Even though for aeroelastic applications we also recommend an implicit scheme asa flow time-integrator, we consider both explicit and implicit flow solvers because ofthe popularity of both approaches in the computational fluid dynamics community.

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2.1. Analysis of serial collocated partitioned procedures

2.1.1. The generalized conventional serial staggered (GCSS) procedureFirst, we consider the popular CSS procedure as summarized by Farhat et al

[FAR 96]. This partitioned procedure is collocated - that is, it evaluates the fluid andthe structure subsystems at the same time-stations. It is serial, or sequential, i.e. thefluid and the structure are advanced in time successively, but not simultaneously.

We generalize this method to incorporate a prediction of the displacement of thestructure and an evaluation of the flow induced structural load after the fluid subsystemhas been advanced from one time-step to the next one. Each cycle [tn, tn+l] of theGCSS procedure goes as follows:Step 1. Predicts the structural displacement at time tn+l

where a0 and a1 are two real constants. The prediction [2] is first-order time-accurateif a0 = 1, and second-order time-accurate if a0 = 1 and a1 = 1/2. Then, transfersthe motion of the wet boundary of the structure to the fluid.Step 2. Updates the position of the fluid grid xn+1 to match on rf/s the position thatthe structure would have if it were advanced by the predicted displacement un+1P.Then, time-integrates the fluid subsystem from tn to tn+1 = tn 4 + Ats using a fluidtime-step AtF < Ats. If AtF = Ats, subcycles the flow solver.Step 3. Transfers a fluid pressure field Psn+1 to the structure, and computes the corre-sponding flow induced structural load f^g- In the case where the fluid and structuremeshes have non-conforming discrete interfaces, a conservative algorithm for compu-ting the spatial distribution of f^g is recommended [FAR 98b].Step 4. Time-integrates the structure subsystem from tn to tn+l = tn + Ats.

Note that when the flow solver is subcycled, the velocity of the fluid moving gridis held constant constant and equal to x = (xn+1 - x n}/&t s , in order to satisfy thegeometric conservation law (GCL) [LES 95], [LES 96]. Also, Psn+l is "a" pressurefield sent to the structure and computed after the fluid subsystem has been advancedfrom tn to tn+l. Most importantly, Psn+l is not necessarily the pressure field Pn+lwhich is computed by the flow solver at time-station tn+l. Hence, a specific expressionof Psn+1 defines a particular instance of the generalized CSS procedure, and can varyfrom one staggered solution algorithm to another.

2.1.2. Energy balance at the fluid-structure interfaceAs stated earlier, we assume in this paper that the structural subsystem is always

time-integrated by the midpoint rule, but we do not make any assumption for thechoice of the flow time-integrator. For any given unsteady flow solver, the energytransferred during the time-interval [tn, tn+l] from the fluid to the structure through

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a patch of rF/S of unit length/area, as viewed by the fluid, can be evaluated as

where the superscript T designates the transpose operation, fF denotes the pressureforces exerted by the fluid on rF/S, and PFn+1 is a vector of nodal pressures whoseexact expression depends on the time-integrator used by the flow solver. More spec-ifically, PFn+l depends on the fluid pressure values that are used by the flow solverto compute the fluxes across the interface boundary rF/S, when advancing the fluidstate vector from tn to tn+1. Hence, PFn+1 can take any of the following values

For one-dimensional aeroelastic problems where the interface rF/S reduces to asingle point, formulas [4] are exact. For two- and three-dimensional problems, we useit to estimate the energy transferred from the fluid to the structure, as viewed by thefluid.

Next, we turn our attention to the energy transferred from the fluid to the struc-ture, as viewed by the structure. First, we note that the displacement, velocity, andacceleration of the structure computed by the midpoint rule satisfy

Hence, if the structural energy is defined as ES =1/2uT Mu + 1/2uTKu, it followsfrom equations [5] that the variation of ES during a time-step Ats is given by

The first right hand side term of [6] represents the energy transferred from the fluidto the structure, as viewed by the structure. Hence, this energy can be written as

From equations [3] and [7], it follows that the GCSS procedure conserves energyat the fluid-structure interface if and only if AEFn+1 + AESn+1 = 0, i.e.

explicit forward -Euler Scheme:

implicit backward-Euler scheme:

second-order time-accurate solvers:

subcycling with

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This relationship underscores the importance of formulating computational aero-elasticity problems as three-field problems (the fluid, the structure, the fluid dynamicmesh), especially for the analysis of their solution by partitioned procedures.

In general, the predictor [2] - and for that matter any other predictor - does notguess exactly the position of the structure at tn+1; therefore

Furthermore, although Psn+1 is a free parameter of the generalized CSS proce-dure, it cannot be easily constructed to enforce the conservation equation [8], becauseun+1 is computed by a partitioned procedure only after Psn+1 has been evaluated.For all these reasons, we conclude that the general family of collocated partitionedprocedures, and more specifically the GCSS procedure, cannot conserve energy at thefluid-structure interface rF/S. However, three key algorithmic components can be ca-refully adjusted for controlling the unbalance of energy at rF/S, and reducing it asmuch as possible:

- The motion scheme of the dynamic fluid mesh;- The structural predictor;- The time-discretization of the transfer during [tn, tn+1] of the aerodynamic data

from the fluid to the structure, i.e. the construction of the pressure field PSn+1.

2.1.3. Framework for characterizing a partitioned procedureHere, we specify our framework of analysis and illustrate it for one particular

instance of the GCSS procedure. Our objective is to estimate the amount of energyunbalance created at the fluid-structure interface rF/s by a staggering process duringa relatively long period of time. To this end, we make the following assumptions:

- The only external forces applied to the structure are those corresponding to theflow pressure on TF/s;

- The structure is vibrating with a constant amplitude U0 and circular frequency(w: u(t) = U0 cos(wt);

- The pressure induced by the flow on rF/s is also vibrating at the same circularfrequency w, but with a phase difference denoted by u: P(t) = P0 cos(wt + u). Thisphase difference can be linked to well-known added mass, damping, and stiffnesseffects of a fluid loading. We define two scalar parameters k and d as follows:

- The fluid and structure subsystems are advanced in time using fixed time-stepsAtF and Ats, respectively.

To illustrate our analysis framework, we consider first a specific instance of theGCSS procedure where the prediction step is as trivial as xn+1 = un+1u = un

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(a0 = a1 = 0), Psn+1 is constructed as Psn+l = Pn+l - i.e. the pressure field sentto the structure is the most recent pressure field computed by the flow solver - andthe fluid subsystem is time-integrated by the explicit forward-Euler scheme (whichimplies PFn+l = Pn), with AtF = Ats = At. Let

At each time-step, the structure and fluid variables un and Pn computed bythis collocated partitioned procedure can be viewed as approximations of u(tn) andP(tn), where tn = nAt. From equations [3] and [10], it follows that

The energy AEpTw transferred during the time-interval [0, N x Tw] from thefluid to the structure through a patch of r F / S of unit length/area, as viewed by thefluid, can be estimated by summing AEFn from n = 0 to n = INH/h = NTw/At.Noting that for large values of N/h

we deduce from equations [12-13] that &ENFTw =En2NH/h = 0AEnF ~ NH 6EF with6Ef = [k (cos(h) 1) d s i n ( h ) ] / h . Assuming h = wAt PFn+1 = Pn) and PSn+1/2 = Pn+1, the resultingCSS partitioned procedure becomes at least fourth-order energy-accurate. However,this specific instance of the CSS method does not conserve well the momentum at thefluid-structure interface (momentum variations of the fluid and the structure duringa time-step are not accurately opposite), and becomes less energy-accurate when thefluid system is subcycled.* First-order prediction. Next, we consider the first-order prediction obtained bysetting a0 = 1 and Q1 = 0 in [2]. In that case, several CSS algorithms becomesecond-order energy-accurate. The optimal one (using a second-order time-accurateflow integrator and PSn+1 given by [18d] is such that 6E = -5dh2/12 + O(h3).* Second-order prediction. The second-order prediction of the structural displace-ment obtained with a0 1 and Q1 = 1/2 leads to third-order energy-accurate CSSprocedures if and only if P5n+1/2 = PFn+1, where PFn+1 depends on the flowtime-integrator [18a-18d]. Independently of the flow time-integrator, momentum isconserved at r F / s (momentum variations of the fluid and the structure during a time-step are numerically opposite) and 6E 5kh3/12 -I- O(h4).

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2.2. Analysis of a serial non-collocated partitioned procedure

In [LES 95], [LES 96], [FAR 96], [LES 98], Lesoinne and Farhat have shown thatcollocated partitioned procedures can never satisfy simultaneously both displacementsand velocities continuity equations on the fluid-structure interface, while verifyingthe Geometrical Conservation Law (GCL) in the fluid domain [THO 79], [LES 95],[LES 96]. Because of staggering, continuity of displacements at rF/s is almost neversatisfied. Also, the continuity of velocities at rF/s, the satisfaction of the GCL, andthe use of the trapezoidal rule for the structure are incompatible.

Given that both continuity equations are desirable in order not to introduce any pa-rasitic discontinuity at rF / S where the interaction between the fluid and the structureoccurs, and violating the GCL restricts severely the time-step of the flow-integratorand/or the coupling time-step Ats [FAR 95b], [GUI 99], Lesoinne and Farhat haveproposed to resolve this dilemma by non-collocating the staggered procedure. Theiradvocated algorithm, named the Improved Serial Staggered (ISS) procedure, is builtas a leap-frog scheme (and then it is basically sequential) where the fluid subsystem isalways computed at half time-stations (..., tn-1/2, tn+1/2, tn+ 3/2, ...), while the struc-ture subsystem is always computed at full time-stations (..., tn, tn+l, tn+2, ...). TheISS staggered procedure can be summarized as follows [LES 98]:

-Predicts the structural displacement at time tn+1/2: un+1p/2 = un + ^ ts/2un;- Updates the position of the fluid grid xn+1/2 to match on Tp/s the predicted

position un+1p/2. Then, time-integrates the fluid subsystem from tn-1/2 to tn+1/2 =tn-1/2 + Ats using a fluid time-step ATF < AtS . If AtF = Ats, subcycles theflow solver;

- Transfers a fluid pressure field Psn+l to the structure, and computes the corres-ponding flow induced structural load fn+1F/S. Then, time-integrates the structure sub-system from tn to tn+1 = tn + Ats using the midpoint rule.

Using this non-collocated partitioned method, the expression of the energy AEFtransferred during the time-interval [tn, tn+1] from the fluid to the structure througha patch of rF/s of unit length/area, as viewed by the fluid, has to be re-adjusted,because unlike the structural subsystem, the fluid subsystem is advanced from tn-1/2to tn+1/2). AEF can be computed as

Depending on the flow solver with which the ISS procedure is equipped, PFn+1/2can take several values corresponding to a forward-Euler, backward-Euler, second-order implicit, and highly subcycled flow time-integrator, respectively. The value for6EF can be computed accordingly. Depending on analogous choices for Psn+l, es-

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timates for 6Es can be obtained, as in [17-refDESB]. From tedious developments on6E for many combinations, we obtain that* Second-order energy-accurate instances of the ISS procedure can be obtainedusing a flow time-integrator where PFU+1/2 = Pn-1/2, and setting psn+1/2 to anyof the possible pressure fields deriving from the fluid solvers considered, or using asubcycled flow time-integrator and setting psn+1/2 = pn+1/2.In the latter case, 6Ebecomes 6E = -dh2/12 + O(h3).* A third-order energy-accurate instance of the ISS method can be obtained bychoosing a flow time-integrator where PFn+1/2= (Pn - 1 / 2 + Pn+1/2)/2 (even whenthe flow solver is significantly subcycled), and setting Psn+1/2 = pn+1/2. This cor-responds to the original ISS procedure proposed by Lesoinne and Farhat in [LES 95],[LES 96], [FAR 96], [LES 98]

2.3. Construction of new serial partitioned procedures

The analysis presented in this paper gives us the basic tools to construct new par-titioned procedures. We can get through many combinations of fluid and structuralsolvers, as well as choices for the structural predictor and fluid pressure fields trans-mitted. We have already noticed in Section 2.1.6 that, if a predictor [2] with Q0 = 1and Q1 = 1/2 is used and if we set Psn+1/2 = PFn+1, where PFn+1 depends on theflow time-integrator, then the partitioned procedure obtained is third-order energy-accurate with 6E = 5kh3/12 + O(h4), independently of the flow time-integrator. Ouranalysis tells us also, after tedious calculations that* if a predictor [2] with a0 = 3/2 and a1 = 5/4 is used, whereas Psn+l = PFn+1,where PFn+l depends on the flow time-integrator, then the partitioned procedureobtained is again third-order energy-accurate with 6E = 33kh3/24 + O(h4), inde-pendently of the flow time-integrator.* finally, a linear combination of the results reported above yields the amazing re-sult that: if a predictor [2] with a0 = 18/23 and a1 = 4/23 is used and if we setPsn+1 = 56/23 PFn+1 - 33/23Psn, where PFn+1 depends on the flow time-integrator,then the partitioned procedure obtained is at least fourth-order energy-accurate(6E = O(h4)), independently of the flow time-integrator.

However, the last two families of partitioned procedures do not satisfy one of themost important properties of procedures of the first family mentioned above, whichis the conservative exchange of momentum between the fluid and the structure sub-systems at each time-step. This momentum unbalance, although the energy error ac-cumulated at the fluid-structure interface is controlled and small, can lead in someundamped cases to high-frequency numerical instabilities (errors in the structural pre-diction and in momentum conservation are both present).

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2.4. Construction of parallel collocated partitioned procedures

The collocated and non-collocated partitioned procedures presented in Sections2.1,2.2 and 2.3 were all constructed as serial partitioned procedures, in the sense thatthe fluid and structural subsystems have to be advanced in time successively, and notsimultaneously at each coupled time-step. More precisely, the fluid time-integrationor the structural time-integration requires an information (coming from the other sub-system) which is not available at the beginning of the coupled time-step.

In this section, we propose new partitioned procedures for the transient solution ofthe aeroelastic problem [1], which are inherently parallel, i.e. the fluid and the structurecan be time-integrated simultaneously inside each coupled time-step. Such procedurescan reduce the total simulation time when the computational cost of the structuralanalyzer is comparable to that of the fluid one, for example, when the structure is acomplete aircraft configuration and geometrical non-linearities must be accounted for.

2.4.1. A new family of parallel partitioned procedures deriving form the GCSSWe first propose the following parallel version of the GCSS procedure, which can

be sketched asFluid Step 1. Predicts the structural displacement at time tn+l using formula [2].Note that this prediction does not require any further structural information than thosealready known and exchanged before the beginning of the current coupled time-step.Fluid Step 2. Updates the position of the fluid grid xn+l to match on rF/s the posi-tion that the structure would have if it were advanced by the predicted displacementu

n+1P. Then, time-integrates the fluid subsystem from tn to tn+1 using a fluid time-

step AF < AtS. If AtF = Ats, subcycles the flow solver.Structural Step 1. Computes a fluid pressure field Psn+1, depending only on theavailable fluid field history on [ t n - l , tn], and computes the corresponding flow indu-ced structural load.Structural Step 2. Time-integrates the structure subsystem from tn to tn+l.

Note that this procedure is actually parallel, in the sense that fluid and structuralsteps can be performed simultaneously, and therefore computational load balancingcan be done on all fluid and structural subdomains. For this particular partitionedprocedure, the energy based analysis leads to the following general results: in thefamily of parallel partitioned procedures defined above, third-order and even fourth-order energy accurate procedures can be constructed. More precisely,* if a predictor [2] with &Q0 = Q1 = 2 is used and if we set PSn+1/2 = PFn, wherePFn depends on the flow time-integrator, then the partitioned procedure obtained isthird-order energy-accurate with 6E = 7kh3/3 + O(h4), independently of the flowtime-integrator.* if a predictor [2] with a0 = 5/2 and a1 = 13/4 is used and if we set Psn+l = PFn,where PFn depends on the flow time-integrator, then the partitioned procedure obtain-

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ed is again third-order energy-accurate with 6E = 109kh3/24 + 0(h4), independ-ently of the flow time-integrator.* finally, a linear combination of the results reported above yields the amazing resultthat if a predictor [2] with a0 = 246/109 and a1 = 288/109 is used and if we setpsn+l = 162/109PFn - 53/109Psn, where PFn depends on the flow time-integrator, thenthe partitioned procedure is at least fourth-order energy-accurate (6E = O(h4)),in1dependently of the flow time-integrator.

Again, this spectacular result tells us that accumulation of artificial energy, crea-ted at the fluid-structure interface, can be very well controlled. However, some high-frequency instabilities can appear in partitioned procedures, because of important butcompensating errors in momentum conservation and in the structural prediction.

2.4.2. Other parallel partitioned procedures

In the previous section, we have shown how to construct families of parallel par-titioned procedures, which are quite general, because the coupling algorithm can bedesigned such that its energy accuracy is independent of the fluid flow solver chosen.However, many more parallel partitioned procedures can be constructed, which areactually third-order energy-accurate for a second-order backward difference schemeflow solver.

Among dozens of third-order energy-accurate partitioned procedures, let us citethe particular instance of the parallel procedure described in the previous section, witha0 = 3/2 and Psn+1/2 = pn. Note that, for this procedure, the fluid pressure fieldand the corresponding flow induced structural load are not directly derived from PFn.The energy based analysis leads to the following results: if a1 = 1, then the procedureis third-order energy-accurate with 6E = 7kh3/8 + O(h4); if a1 = 5/12, then theprocedure is second-order energy-accurate with 6E = 7dh2/12 + O(h4). Numericaltests involving these procedures will be presented in the sequel.

3. Applications and numerical results

We present here some numerical validation of our energy-based criterion, and inparticular of the underlying assumptions stated in Section 2.1.2 and Section 2.1.3.We consider the simulation of the non-linear transient aeroelastic response of twostructures: a flat panel with infinite aspect ratio in a critical supersonic air streamin two space dimensions, and the AGARD Wing 445.6 in a transonic air stream inthree space dimensions. The complete flutter analysis of both of these problems usingcollocated and non-collocated partitioned procedures can be found, among others,in [FA