Added-mass effect in the design of partitioned algorithms ... · y dep ends on the structure densit...

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THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Added-mass effect in the design ofpartitioned algorithms forfluid-structure problems

Paola Causin — Jean-Frédéric Gerbeau — Fabio Nobile

N° 5084

Janvier 2004

Unité de recherche INRIA RocquencourtDomaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

Added-mass e�ect in the design of

partitioned algorithms for

�uid-structure problems

Paola Causin� , Jean-Frédéric Gerbeau� , Fabio Nobiley

Thème 4 � Simulation et optimisationde systèmes complexes

Projets BANG

Rapport de recherche n° 5084 � Janvier 2004 � 28 pages

Abstract: The aim of this work is to provide a mathematical contribution to explain thenumerical instabilities encountered under certain combinations of physical parameters in thesimulation of �uid-structure interaction (FSI) when using loosely coupled time advancingschemes. It is also shown how the same combinations of parameters lead, in the case ofstrongly coupled schemes, to problems that demand a greater computational e�ort to besolved, requiring for example a high number of subiterations. The application that wehave in mind is FSI simulation for blood �ow in large human arteries, but the discussionapplies as well to several FSI problems in which an incompressible �uid interacts with a thinelastic structure. To carry out the mathematical analysis, we consider a simpli�ed modelrepresenting the interaction between a potential �uid and a linear elastic thin tube. Despiteits simplicity, this model reproduces propagation phenomena and takes into account theadded-mass e�ect of the �uid on the structure, which is known to be source of numericaldi�culties. This allows to draw conclusions that apply to more realistic problems, as well.

Key-words: �uid-structure interaction; added-mass e�ect; numerical stability; partionedalgorithms

� Projet BANG, INRIA Rocquencourt, Francey ICES, University of Texas at Austin, USA

E�et de masse ajoutée dans l'étude d'algorithmes

partitionnés en interaction �uide-structure

Résumé : L'objet de cette étude est de proposer une explication mathématique à des insta-bilités numériques apparaissant dans certaines situations lors de simulations en interaction�uide-structure avec des schémas faiblement couplés. On explique également pourquoi, dansles mêmes situations, des schémas fortement couplés peuvent nécessiter un très grand nombrede sous-itérations. L'application visée est l'interaction �uide-structure dans les grosses ar-tères, mais la discussion s'étend à d'autres situations où un �uide incompressible interagitavec une structure élastique. Pour e�ectuer l'analyse mathématique, on considère un modèlesimpli�é décrivant l'interaction entre un �uide potentiel et un tube élastique mince. Mal-gré sa simplicité, ce modèle reproduit des phénomènes de propagations réalistes et prenden compte l'e�et de masse ajoutée, qui une source connue de di�cultés numériques. Lesconclusions obtenues sur ce modèle corroborent des observations faites sur des modèles pluscomplexes.

Mots-clés : interaction �uide-structure; masse ajoutée; stabilité numérique; algorithmespartitionnés

Added-mass e�ect in the design of partitioned algorithms for �uid-structure problems 3

1 Introduction

In this work we focus our attention on the numerical simulation of an incompressible �uidinteracting with a thin elastic structure. The target application is the simulation of blood�ow in large arteries, although the analysis presented here applies to a much wider class of�uid-structure interaction (FSI) problems.

Among the most used numerical techniques in this context are the so-called �partitioned�time marching algorithms, which are based on subsequent solutions of the �uid and structuresubproblems and allow one to reuse existing computational codes (see e.g. [21, 20, 17, 5, 4,11, 12])

Although loosely coupled (explicit) algorithms are successfully used in aeroelasticity (see[5] and the references therein), it is common experience in other applications [19, 10, 16]that they feature instabilities under certain choices of the physical parameters, typicallywhen the densities of the �uid and the structure are comparable or when the domain hasa slender shape, irrespectively of the choice of the time step. Conversely, strongly coupled(implicit) algorithms feature convergence problems under the same conditions. These issuesare still not well understood and, in particular, very few mathematical explanations andmathematically formulated stability or convergence conditions are available, so far, alsobecause of the high complexity and nonlinearity of FSI problems.

The goal of this work is to provide a tool for the study of the stability and/or the conver-gence analysis of partitioned coupled algorithms, which is based on a �toy FSI model� thatrepresents the interaction between a potential �uid and a linear elastic thin tube. We do notclaim that this model is relevant to properly describe complex situations, like �uid-structureinteraction in arteries since, for example, it does not include nonlinearities and dissipationphenomena. Nevertheless, it retains important physical features common to more complexmodels: in particular, it reproduces propagation phenomena and takes into account theadded-mass e�ect of the �uid on the structure, which is known to induce numerical di�-culties [10]. This model problem is simple enough to perform mathematical and numericalstudies but, at the same time, complex enough to mimic more realistic situations, at leastin the case of incompressible �uids. As it will be shown in Sections 5 and 6, starting fromthis simple FSI model, we are able to derive stability and convergence conditions that are inexcellent agreement with the numerical observations collected in much more complex situa-tions. Moreover, this toy FSI model, and the type of analysis proposed in this paper, maybe helpful also in devising new and, possibly, more e�cient coupled algorithms.

The work is organized as follows. In Sect. 2 we give some details on empirical observationsand results concerning the stability of partitioned algorithms. In Sect. 3 we present thesimpli�ed �uid-structure model on which we are going to carry out our analysis. In Sect. 4we introduce the functional setting and we present the FSI problem as a structure equationmodi�ed by the introduction of an added-mass operator that represents the e�ect of the�uid on the structure itself. In Sect. 5 and 6 we consider di�erent explicit and implicitschemes to advance in time the interface FSI problem and we propose mathematical criteria

RR n° 5084

4 P. Causin, J.-F. Gerbeau, F. Nobile

to individuate range of values of the physical parameters leading to numerical instabilities.In Sect. 7 we brie�y address the space discretization of the FSI problem and we presentnumerical results that validate the mathematical analysis of the preceding section. Finally, inSect. 8 we use scaling arguments to reproduce and extend to a wider spectrum of situations,even if in a more qualitative way, the results of Sect. 5 and 6.

2 Motivations

In order to motivate the main results of this article, we recall some empirical observationsmade on a basic FSI test case proposed in previous studies (see [6, 7, 16]). The goal of thistest case is to simulate, in a very idealized framework, the mechanical interaction betweenblood and arterial wall. The geometry at rest is a cylinder. The �uid is described by theincompressible Navier-Stokes equations in Arbitrary Lagrangian Eulerian formulation. Thestructure is described either by a 1D generalized string model (for 2D simulations) or by anonlinear shell model in large displacements regime (for 3D simulations). An overpressureis applied at the inlet of the �uid for a short duration of time. Although the �uid is incom-pressible, there is a �nite velocity propagation of the overpressure due the �uid-structurecoupling. Fig 1 shows the propagation phenomenon in 2D. The 3D case gives analogousresults (see [7]). All the details regarding this test case can be found in the above citedreferences.

Figure 1: Propagation of a pressure wave in a portion of artery (2D), t=6, 10 and 14 ms.

We �rst tried explicit (or �loosely�) coupled methods, i.e. roughly speaking, algorithmswhich do not ensure exact balance of energy, but which only require one �uid and onestructure resolution per time step.

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It was observed that these algorithms exhibit numerical instabilities

(R1) for a given geometry, as soon as the density of the structure is lower than a certainthreshold;

(R2) for a given structure density, as soon as the length of the domain is greater than acertain threshold.

We then tried �strongly� coupled methods, i.e. we ensure at each time step an exact balanceof energy by sub-iterating several times between the �uid and the structure. When thesubiterations consist of a relaxed �xed-point method, it was observed that an increasingamount of relaxation is needed when

(R3) the density of the structure decreases;

(R4) the length of the domain increases.

All the details concerning these observations (values of parameters, algorithms, experiments,etc.) can be found in [16, Chap.4].

The fact that the numerical stability depends on the structure density has a clear phys-ical interpretation. This is not the same for the dependence on the geometry, which is quiteamazing: since the main physical phenomenon is a wave propagation with a �nite veloc-ity, it is surprising that the length of the domain modi�es the stability of the algorithm(independently of the space and time steps).

It is quite di�cult to explain these observations on the original fully nonlinear equations.That is why we propose in the next section a simpli�ed model which exhibits an analogousbehavior and which is simple enough to be analyzed in detail.

Moreover, the simpli�ed model is also a good candidate for testing new approaches.As an example, we consider in Sect. 6.2 a possible alternative to the Dirichlet/Neumanniterations classically used in �uid-structure simulations.

3 A simpli�ed �uid-structure problem

We consider a rectangular domain F � R2 whose boundary is split into �1F , �2F , �

3F and

� (see Fig. 2). The part � corresponds to the �uid-structure interface. In this simpli�edmodel, the domain S occupied by the structure is such that S = �. We set �F = �1F [�

2F .

We denote by n the unit outward normal vector on @F

3.1 The structural problem

In the domain S we use a structural model derived from the theory of linear elasticity fora cylindrical tube with small thickness, under the assumption of membrane deformations.The reference con�guration is a cylindrical surface of base circle radius R that is supposed tomove only radially, the longitudinal and angular displacements being neglected. Intersecting

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S

�1F F

�2F

�3F

�

n

x

y

L

R

Figure 2: Schematic representation of the computational domain (the dashed line representsthe symmetry axis).

the cylindrical tube with the plane � = �� we obtain the following generalized string model:�nd the displacement � = �(t; x) such that8<

: %shs@2�

@t2+ a� � b

@2�

@x2= f in (0; T )� S ;

� = 0 on (0; T )� @s;

(1)

where hs is the thickness of the structure and %s its mass per unit volume, a = Ehs=R2(1� �2),

E being the Young modulus and � the Poisson coe�cient, b = �TGhs, G being the shearstress modulus, �T the Timoshenko shear correction factor, and f the external forcing termcoming from the �uid (whose expression will be made precise in the following). Equation (1)must be supplied with initial conditions

�(0; x) = �0(x);@�

@t(0; x) = _�0(x) in S ; (2)

and boundary conditions �(t; 0) = �(t; L) = 0;8t 2 (0; T ).

3.2 The �uid problem

For the �uid, we use a linear incompressible inviscid model. Moreover, the deformation � ofthe structure is assumed to be very small, so that the �uid domain F can be considered�xed. Thus, the �uid problem reads:

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�nd the �uid velocity u = u(t; x; y) and pressure p = p(t; x; y) such that8>>>>>>>>><>>>>>>>>>:

%f@u

@t+rp = 0 in (0; T )�F ;

divu = 0 in (0; T )�F ;

p = p(t) on �1F [ �2F ;

u � n = 0 on �3F ;

u � n = w on �;

(3)

where %f is the �uid mass per unit volume and where p and w are given functions (we referto Fig. 2 for the notations). System (3) may be conveniently reformulated as follows:8>>>>>>><

>>>>>>>:

�4p = 0 in F ;

p = p(t) on �1F [ �2F ;

@p

@n= 0 on �3F ;

@p

@n= �%f

@w

@ton �:

(4)

3.3 Fluid-structure interaction

The �uid and structure systems are coupled by the following conditions8<: u � n = w =

@�

@ton �;

f = p on �:

(5)

The �rst equation is the kinematic compatibility condition, where we have used the geomet-rical assumption that F does not change with time. The second condition in (5) transfersthe pressure load from the �uid to the structure as a volume force for this latter, this beingin the simpli�ed model the only contribution coming from the �uid.

Using (5)1, condition (4)4 may be conveniently written as

@p

@n= �%f

@2�

@t2on �: (6)

4 Mathematical preliminaries

4.1 Functional setting

We introduce the functional spaces

V = H10 (S);

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Q = fq 2 H1(F ); qj�1F[�2F = 0g:

We denote by (�; �) the L2(F ) or L2(S) inner products and we de�ne the bilinearforms:

aF (p; q) =

ZF

rp � rq dx dy 8p; q 2 H1(F );

aS(�; �) =

ZS

a � � dx+

ZS

b@�

@x

@�

@xdx 8�; � 2 H1(S):

In all this Section, the Hilbert space V will be endowed with the scalar product de�ned byaS(�; �). We make the following regularity assumptions on the boundary data for the �uid

p 2 C(0;1;H1=2(�F )); (7)

and on the initial data for the structure

�0 2 V; _�0 2 V: (8)

In the following, we will make use of the notation q� to indicate the restriction of the functionq : F ! R to the interface �.

A variational formulation of the �uid-structure problem introduced in the previous sec-tion is:�nd (p; �) 2 C(0;1;H1(F ))�C(0;1;V ) such that (2) is satis�ed, p(t)j�1F[�2F = p(t); 8t 2

(0;1) and for all (q; �) 2 Q� V :8>><>>:

aF (p; q) = �%f

Z�

@2�

@t2q�

%shs@2�

@t2; �

�+ aS(�; �) = (pj�; �):

(9)

In order to write problem (9) in a more compact form, some additional notation is needed.We introduce the following operators: for �; � 2 V , we de�ne

< L�; � >= aS(�; �) (10)

and for ` 2 L2(S), we denote by T` the element of V such that

< L(T`); � >= (`; �): (11)

Using the above de�nitions, the structure problem reads:�nd � such that

%shs@2�

@t2+ L� = p�: (12)

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Next, for any w 2 H�1=2(�), we denote by Rw the element of Q solution of the followingproblem 8>>>>>>><

>>>>>>>:

��Rw = 0 in F ;

Rw = 0 on �1F [ �2F ;

@Rw

@n= 0 on �3F ;

@Rw

@n= w on �:

(13)

We de�ne the operator MA : H�1=2(�)! H1=2(�) by

MAw = Rwj�: (14)

Note that, for w; v 2 L2(�),

<MAw; v >= aF (Rw;Rv):

We gather in the following proposition some standard results ([3, 8]):

Proposition 1

1. The operator T : L2(S)! V is compact self-adjoint and positive on V ;

2. the operator MA : H�1=2(�)! H1=2(�) is continuous;

3. the operator MA is compact, self-adjoint and positive on L2(�).

Let us now introduce an arbitrary continuous extension operator EF : H1=2(�F ) !H1(F ). By de�nition, for q 2 H1=2(�F ), EF qj�F = q and kEF qkH1(F ) � CkqkH1=2(�F ).

We de�ne p� 2 C(0;1;H1(F )) as the solution to8>>>>>>><>>>>>>>:

��p� = 4EFp in F ;

p� = 0 on �1F [ �2F ;

@p�

@n= �

@EF p

@non �3F ;

@p�

@n= �

@EF p

@non �:

(15)

Using equations (13) and (15), the solution to (4) is given by

p = p� +EF p� %fR@2�

@t2: (16)

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Then, de�ning

pext = p�j� +EF pj�; (17)

and using (14), we have

p� = pext � %fMA

@2�

@t2: (18)

Let I denote the identity operator. Plugging (18) into (12), we see that (9) can be writtenin the following form:�nd � such that

(%shsI + %fMA)@2�

@t2+ L� = pext; (19)

the pressure in the �uid being given by (16). The following result holds.

Proposition 2 Under assumptions (7) and (8) on the data, there exists a unique solution

� 2 C([0;1);V ) to equation (19) satisfying the initial conditions (2). Moreover,@�

@t2

C([0;1);L2(�)).

Sketch of the proof.This proposition is just a variant of a result proved in [8]. The main ingredient is to

notice that the operator:A = T (%shsI + %fMA)

is compact, self-adjoint and positive on V (endowed with the scalar product aS(�; �), whichstraightforwardly comes from Proposition 1 (recall (11)). Consequently, there exists a com-plete orthogonal sequence in V made by the eigenfunctions of A. The existence of thesolution to (19) can then be obtained in a standard way, by using this basis in the Faedo-Galerkin method for example. �

Remark 1 Equation (19) looks like the structure equation (12) except for the extra operator%fMA in front of the second order time derivative. This operator represents the interactionof the �uid on the structure and acts as an extra mass (hence the name of �added-mass�e�ect). In particular, the presence of the �uid changes the natural vibration frequencies ofthe structure. We refer the interested reader to [9, 13, 14].

4.2 Spectrum of the added-mass operator

In order to carry out the subsequent mathematical analysis, it is convenient to dispose ofan estimate of the maximum eigenvalue ofMA, that we will denote by �max. Note that theinverse of �max is the smallest eigenvalue of the standard Steklov-Poincaré operator [18].Note also that �max is a purely geometric quantity.

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When dealing with a generic geometry, a closed expression for �max cannot be found.The procedure that can be followed then is the analysis proposed in [2] (see also [1]), adaptedto a single subdomain problem. The indication provided by such an estimate is that theoverall e�ect of an increase of the domain diameter is an increase of the largest eigenvalueof MA. Nevertheless, it may be di�cult to sharply separate the e�ects of the increase ofdiameter from the e�ects of the increase of the aspect ratio.

In the case of the simple geometry of Fig. 2, �max can be computed analytically. To dothis, we consider the following problem:�nd u = u(x; y) such that 8>>>>>>>><

>>>>>>>>:

�4u = 0 in F ;

u = 0 on �1F [ �2F ;

@u

@y= 0 on �3F ;

@u

@y= g on �;

(20)

where g = g(x) is a prescribed function such that g(0) = g(L) = 0. Expressing g as the

superposition g =Xk�1

gk sin

�k�

Lx

�, allows us to obtain the solution to (20)

u(x; y) =Xk�1

gkL

k�sin

�k�x

L

� ch

�k�y

L

�

sh

�k�R

L

� :

Recalling de�nition (14), we can write

MAg = ujy=R =Xk�1

gkL

k�th

�k�R

L

� sin

�k�x

L

�:

Finding the eigenvalue �n; n = 1; 2; : : : ; ofMA associated to the eigenvector g = gn sin�n�Lx�

amounts to solving the eigenvalue problem

MAg = �ng;

which yields

�n =L

n�th

�n�R

L

� :

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The largest eigenvalue �max, corresponding to n = 1, is thus

�max =L

�th

��R

L

� : (21)

Figure 3 shows the value of �max for di�erent values of L and R.

0 5 10 15 20 250

50

100

150

200

250

R=.25

R=.5

R=1

R=2

L

µmax

Figure 3: Largest eigenvalue of MA given by (21) as a function of the domain length L andheight R.

5 Stability of explicit time-marching schemes

We present in this section a stability analysis of an explicit time-marching scheme for thetemporal discretization of the FSI problem presented in Sect. 3. The space discretization willbe addressed in the next section. Moreover, for the sake of simplicity and under the scalingconsiderations presented in Sect. 8 for the problem at hand, we assume that the coe�cientb in (1) is zero. The di�erential structural operator de�ned in (10) therefore reduces toL� = a�. The results obtained in the next sections generalize easily to the case b 6= 0.

By explicit time marching schemes we mean time discretization algorithms of the coupledFSI problem (1), (2), (3), (5) that allow to solve only once (or just a few times) the �uid andthe structure equations within each time step. They can be typically obtained by combining

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an explicit algorithm for one of the subsystems (either �uid or structure) with an implicitone for the other subsystem.

Our goal is to show that those kind of algorithms might be unconditionally unstable incertain cases, depending on the relative mass density of the structure and the �uid and onsome geometric properties of the domain.

As a prototype of an explicit algorithm, we consider the one obtained by employinga Leap-Frog scheme for the structure and an Implicit Euler scheme for the �uid (LF-IEscheme).

Denoting by �t the time step, the time-discrete system that we consider is the following:8>>>>>>><>>>>>>>:

%fun � u

n�1

�t+rpn = 0 in F ;

divun = 0 in F ;

pn = p(tn) on �1F [ �2F ;

un � n = 0 on �3F ;

(22)

and

un � n =

�n � �n�1

�ton �; (23)

%shs�n+1 � 2�n + �n�1

�t2+ a�n = pn in S : (24)

Observe that, given the wall displacement �n at time tn, the �uid equation (22) with bound-ary condition (23) allows to compute the �uid velocity un and pressure pn. With this latter,we can now solve the structure equation (24) and get the new wall displacement �n+1 attime tn+1. Hence, this coupling algorithm is explicit.

We now analyze its stability. We proceed as in Sect. 4.1 and we reduce the coupledproblem to a single structure equation with added-mass. We have:8>>>>>>>><

>>>>>>>>:

�pn = 0 in @pn

@n= �%f

(un � un�1) � n

�t= �%f

�n � 2�n�1 + �n�2

�t2on �

@pn

@n= 0 on �3F

pn = p(tn) on �F :

Hence, with the notation introduced in (17), we have

pn��= pnext � %fMA

�n � 2�n�1 + �n�2

�t2;

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and

%shs�n+1 � 2�n + �n�1

�t2+ %fMA

�n � 2�n�1 + �n�2

�t2+ a�n = pnext on S : (25)

The following result holds

Proposition 3 Let �max be the largest eigenvalue of the operator MA. Then, the scheme(25), and hence the coupled explicit algorithm (22)-(24), is unconditionally unstable if

%shs%f�max

< 1 (26)

Proof. From the compactness of the operator MA (see Proposition 1), we know thatthere exists an orthonormal Hilbert basis of L2(�), made of eigenvectors fzig of MA. Wethus expand the solution � and the right-hand side pext on this basis: �n =

Pi �

ni zi and

pnext =P

i(pnext)i zi. Each Fourier coe�cient �i of the solution satis�es

%shs�n+1i � 2�ni + �n�1i

�t2+ %f�i

�ni � 2�n�1i + �n�2i

�t2+ a�ni = (pnext)i ;

�i being the eigenvalue associated to zi.By direct calculation, the characteristic polynomial �(s) 2 P3 of the previous di�erence

equation satis�es:

�(�1) = �1 and �(�1) = a+4

�t2(%f�i � %shs): (27)

It follows that, if %f�i > %shs for some i, then �(�1) > 0 and there exists a root s� of �(s)such that s� < �1. In particular, we can conclude that, if %f�max > %shs, the di�erenceequation (25) is unstable irrespectively of the time step. �

Remark 2 Observe that the �instability� condition (26) con�rms empirical observations(R1) and (R2) of Sect. 2. Indeed, this condition is more and more restrictive as %shs=%fdecreases and as �max increases, the connection of this latter with the geometry being shownin (21). Namely, the more F becomes a slender geometry (that is when, for a �xed radiusR, L increases or when, for a �xed length L, R decreases), the larger �max becomes (seealso Fig. 3).

Remark 3 In view of (27), it appears that even when %shs=%f�max > 1 (that is when, forexample, the structure is much denser than the �uid), instabilities may occur if the positiveparameter a is large enough ( e.g. when the structure is characterized by a large Youngmodulus). Nevertheless, in such a case, the scheme can be stabilized by suitably decreasingthe time step. This situation will be addressed more in detail in Sect. 8.

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6 Implicit time-marching schemes

Proposition 3 suggests that explicit coupling algorithms might not work for �uid-structureinteraction problems in certain conditions. An obvious remedy consists in switching toimplicit couplings. As a prototype of implicit coupling algorithms we choose here the oneobtained by combining the Implicit Euler scheme for the �uid with the �rst order backwarddi�erence scheme for the structure (IE-BDF scheme). The time-discrete problem reads:8>>>>>><

>>>>>>:

%fun+1 � u

n

�t+rpn+1 = 0 in F ;

divun+1 = 0 in F ;

pn+1 = p(tn+1) on �1F [ �2F ;

un+1 � n = 0 on �3F ;

(28)

and

un+1 � n =

�n+1 � �n

�ton �; (29)

%shs�n+1 � 2�n + �n�1

�t2+ a�n+1 = pn+1 on S : (30)

This problem corresponds to the following discrete added-mass problem for the structure

(%shs + %fMA)�n+1 � 2�n + �n�1

�t2+ a�n+1 = pn+1ext in S : (31)

It is straightforward to show that problem (31), and hence the coupled problem (28)-(30),is stable for any choice of the time step �t.

In this simple example, at each time step, the coupled problem (28)-(30) turns intoa linear system in the unknowns (un+1; pn+1; �n+1) (after spatial discretization) and canbe assembled and solved quite easily. Yet, in more realistic (and complex) situations, theproblem might be non linear (non-linearities in the �uid and/or structure equations andpossibly large displacements of the structure). Then, in order to solve (28)-(30) at each timestep, it is often convenient to consider iterative methods; this approach allow to decouplethe �uid from the structure step and, ultimately, to use already available computer codes.

In the next two subsections we will analyze two possible iterative methods that we will callDirichlet/Neumann and Neumann/Dirichlet, borrowing the terminology from correspondingDomain Decomposition algorithms [18]. By Dirichlet/Neumann (D-N) method we mean thatat each iteration we solve the �uid equations with respect to primitive variables (u; p) subjectto Dirichlet boundary conditions at the interface (imposed displacements or velocities) andthe structure equations subject to Neumann boundary conditions (imposed loads). TheNeumann/Dirichlet (N-D) method is carried out with a procedure dual to the one abovedescribed.

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6.1 Dirichlet/Neumann subiterations

At each time step, we use the following algorithm to solve (28)-(30):given an initial guess �n+10 , we solve for each k = 1; 2; : : :

1. Fluid step: �nd (uk ; pk) s.t.8>>>>>>>>>><>>>>>>>>>>:

%fuk � u

n

�t+rpk = 0 in F ;

divuk = 0 in F ;

pk = p(tn+1) on �1F [ �2F ;

uk � n = 0 on �3F ;

uk � n =�k�1 � �n

�ton �:

2. Structure step: �nd ~�k s.t.

%shs~�k � 2�n + �n�1

�t2+ a~�k = pk in S :

3. Relaxation step:

�k = !~�k + (1� !)�k�1:

4. Convergence test:

� if jj�k � �k�1jj < tol then set �n+1 = �k, un+1 = uk and pn+1 = pk

� else set k = k + 1 and go to step 1.

We can eliminate the unknowns uk and pk, using the same technique employed before.We obtain in this case:

p�;k = �%fMA

��k�1 � 2�n + �n�1

�t2

�+ pn+1ext :

Keeping in mind that ~�k =�k!�

1� !

!�k�1, the previous algorithm corresponds to the

following iterative method to solve the added-mass problem (31):

%shs�t2

��k!� 2�n + �n�1

�+ a

�k!

=1� !

!

�%shs�t2

+ a

��k�1

� %fMA

��k�1 � 2�n + �n�1

�t2

�+ pn+1ext ; k = 1; 2; : : :

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or, equivalently, the �xed-point iteration

1

!

�%shs�t2

+ a

��k =

�1� !

!

�%shs�t2

+ a

��%fMA

�t2

��k�1 + g(�n; �n+1; pn+1ext ): (32)

Concerning the convergence of the �xed point iteration (32), we state the following

Proposition 4 The D-N iterative method converges to the solution to (28)-(30) if and onlyif

0 < ! <2(%shs + a�t2)

%shs + %f�max + a�t2(33)

Proof. The solution �n+1 to (28)-(30) is the �xed point of the iterative method (32) anda necessary and su�cient condition for the convergence of this latter is��(1� !)(%shs + a�t2)� !%f�i

%shs + a�t2

�� < 1:

A straightforward calculation leads to (33). �

Remark 4 This result con�rms the empirical observations (R3) and (R4) of Sect. 2 (seealso Remark 2).

Remark 5 In the limit �t! 0, condition (33) reads

0 < ! <2

1 + %f�max=(%shs):

Observe that, in this case, whenever the explicit algorithm diverges (%f�max > %shs), theD-N iterative method needs a relaxation parameter strictly smaller than 1 to converge.

6.2 Neumann/Dirichlet subiterations

The aim of this Section is to investigate a possible alternative to the standard Dirich-let/Neumann iterations. We consider the following scheme:

given an initial guess �n+10 , we solve for each k = 1; 2; : : :

1. Structure step: compute the load �k s.t.

�k = %shs�k�1 � 2�n + �n�1

�t2+ a�k�1 � pn+1ext in S :

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2. Fluid step: �nd (uk ; pk) s.t.

8>>>>>>>><>>>>>>>>:

%fuk � u

n

�t+rpk = 0 in F ;

divuk = 0 in F ;

pk = p(tn+1) on �1F [ �2F ;

uk � n = 0 on �3F ;

pk = �k on �:

3. Update structure: �nd ~�k s.t.

~�k = �tuk � n+ �n in S :

4. Relaxation step:

�k = !~�k + (1� !)�k�1:

5. Convergence test:

� if jj�k � �k�1jj < tol then set �n+1 = �k, un+1 = uk and pn+1 = pk

� else set k = k + 1 and go to step 1.

After elimination of uk we have:

MA�1�k =

@pk@n

= �%f(uk � u

n) � n

�t= �%f

~�k � 2�n + �n�1

�t2in S ;

and

%fMA

�t2

��k!� 2�n + �n�1

�=

%fMA

�t2(1� !)

!�k�1

� %shs�k�1 � 2�n + �n�1

�t2� a�k�1 + pn+1ext ; k = 1; 2; : : :

or, equivalently, as the �xed-point method

%fMA

!�t2�k =

�(1� !)

!

%fMA

�t2�%shs�t2

� a

��k�1 + g(�n; �n+1; pn+1ext ): (34)

Concerning the convergence of the �xed-point iteration (34), the following result holds.

Proposition 5 The N-D iterative method converges to the solution to (28)-(30) if and onlyif, for all i = 1; 2; : : : ,

0 < ! <2%f

%f + (%shs + a�t2)=�i(35)

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Proof. The solution �n+1 to (28)-(30) is the �xed point of the iterative method (34) anda necessary and su�cient condition for the convergence of this latter is��(1� !)%f�i � !(%shs + a�t2)

%f�i

�� < 1:

A straightforward calculation leads to (35). �

Remark 6 Notice that, at the continuous level, the in�mum of the eigenvalues of MA iszero. At the discrete level, �hmin = O(h), where h denotes the space discretization param-eter (see [18, 2]). Thus, the above result shows that the relaxation parameter ! for Neu-mann/Dirichlet iterations tends to zero with the space discretization parameter h. Therefore,such an algorithm seems not to be of practical use.

7 Numerical results

In the following we present a series of numerical tests carried out on a code implementing thetoy FSI problem and using the di�erent time-marching schemes discussed in the previoussections. The aim of these tests is to provide a numerical validation of the above theoreticalresults. Before presenting the results, we need to introduce some further notation.

7.1 Space discretization

Here we consider the space discretization of the �uid problem (4) written in terms of thepressure only. The space discretization of problem (3) in velocity and pressure would leadto analogous results.

Let Th be a regular triangulation of F made of quadrilaterals (denoted by K) and Jhbe a partition of s made of intervals (denoted by J), conforming to the nodes of Th lyingon �. For ease of presentation, we can suppose Th to be a regular Cartesian mesh suchthat its trace on � is made of equally spaced intervals. Moreover, we introduce the discretespaces

Qh = fqh 2 Q; qhjK 2 Q1(K); 8K 2 Thg;

Vh = f�h 2 V; �hjJ 2 P1(J); 8J 2 Jhg:(36)

Let us denote by NS = dim(Vh) the number of structure nodes internal to s and byNF = dim(Qh) the number of �uid nodes internal to F or lying on �.

Denoting by A the �uid sti�ness matrix, we have that the discrete �uid problem reads

AP = F; (37)

where P is the vector of the nodal values of ph. We can reorder the �uid nodes such that theinternal ones come �rst, i.e. I = f1; 2; :::; NF�NSg and � = fNF�NS+1; NFg. We indicate

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by PI the vector of the �rst NF �NS pressure values (pressure values at the interior of the�uid domain) and with P� the remaining NS boundary pressure values (pressure values onthe �uid-structure interface). The above decomposition in internal and interface nodes reads�

AII AI�

A�I A��

��PIP�

�=

�FIF�

�; (38)

where FI accounts for the external pressure at the entrance of the �uid domain and F�

is the discrete counterpart of the boundary term �%f

Z�

@2�

@t2q ds. Eliminating now the

internal unknowns in favor of the interface ones and solving the resulting equation, yieldsthe expression

P� = MA(F� �A�IA�1II FI); (39)

whereMA = (A��A�IA�1II AI�)

�1 is the discrete counterpart of the added-mass operator.Denoting byMS and KS the structure mass and sti�ness matrices, respectively, the discretestructure problem reads

%shsMS�D + (bKS + aMS)D = MSP�; (40)

where D is the vector of the nodal values of �h. Using (39) and observing that F� =�%fMS

�D, we eventually end up with the following discrete interface problem of dimen-sion NS

(%shsMS + %fMSMAMS) �D + (bKS + aMS)D = �MSMA(A�IA�1II FI ): (41)

We now apply to (41) the time discretization schemes discussed in the previous sections.The numerical tests are performed using geometrical and physical parameters pertinent tothe case of blood �ow in large arteries (see Tab. 4).

7.2 LF-EI time advancing scheme

Referring to the domain of Fig. 2, we set R = 1 cm and L = 6 cm and we take hs = 0:1 cm,%f = 1g=cm3, E = 0:75 � 106 dynes=cm2, � = 0:5. The �uid is initially at rest and an over-

pressure of 2 �104dynes=cm2(' 15mmHg) is imposed at the inlet for a duration of 0:005 s. A

discretization step hx = 0:15 cm in the x direction is considered. Evaluating the numericalvalue of �max, the limiting value of the structure density given by (26) turns out to be(hs%s)lim ' 3:98 g=cm2.

We observe the behavior of scheme (25) in the time interval [0; Tmax = 3 s] for di�erentvalues of the structure surface density hs%s, ranging from below to above the theoreticallimiting value. The experimental results are collected in Tab. 1. In the greatest part of thesituations, the instability occurs fairly before reaching Tmax, and the smaller the structuraldensity is (below the critical value) the sooner this phenomenon manifests. We also observe

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hs%s [g=cm2] �t = 1e� 3 s �t = 1e� 4 s �t = 1e� 5 s

4:5stable

stable stable4:24:0

unstable

3:983:97

unstable unstable3:963:953:85

Table 1: Behavior of the LF-EI numerical scheme for di�erent values of the structure densityin the neighborhood of the value (hs%s)lim = 3:98g=cm2.

a slight dependence on the time step, and namely instability is reached earlier adopting acoarser time step. This e�ect is expected from the result of Proposition 3.

The same behavior displayed in Tab. 1 is observed when hx is varied (for example halved ordoubled). Varying in turn L, while keeping R �xed, leads to analogous results, con�rmingthat explicit numerical simulations of slender �uid-structure systems su�er from a morestringent lower limit for the structure surface density than bulky �uid-structure systems.Namely, in the case L = 2 cm we �nd (hs%s)lim ' 0:7 g=cm2, while for L = 10 cm we �nd(hs%s)lim ' 10:46 g=cm2.

Remark 7 The numerical tests are pretty adherent to Proposition 3. In particular, theycon�rm the in�uence of geometry and density ratio on the stability of explicit scheme. More-over, it must be observed that the value %s;lim corresponding to L = 6 is considerably greaterthan the physiological arterial wall density (see also Tab. 4).

Remark 8 The results of Tab. 1 are very close to those in [16, Table 4.1, page 145], theselatter corresponding to the full non-linear problem mentioned in Section 2, with the samechoice of parameters. This is a strong con�rmation that the reduced model investigated inthis paper captures the main source of instability appearing in the non-linear FSI problem ofinterest in hemodynamics.

7.3 Dirichlet-Neumann subiterations

We present the results of the numerical tests conducted using the iterative scheme presentedin Section 6.1. Considering again the geometry and the discretization parameters of Sect. 7.2,we set hs%s = 3:0 g=cm2, this value being well already in the domain of instability for theLF-EI scheme. From (33), we compute the upper limit value for the acceleration parameter!lim = [0:922; 0:861; 0:8603] for the �xed point method, using �t = [1e� 3; 1e� 4; 1e� 5] s,respectively. We set to 2000 the maximum number of iterations allowed to satisfy the

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! �t = 1e� 3 s �t = 1e� 4 s �t = 1e� 5 s0:75

stable

stable stable0:80:850:86

unstable unstable

0:870:880:890:910:922

unstable0:930:91

Table 2: Behavior of the iterative scheme (32) for di�erent values of the acceleration param-eter ! in the neighborhood of the limit !lim = [0:922; 0:861; 0:8603].

convergence test jj�k � �k�1jj < 1e � 6. The experimental results are collected in Tab. 2.The results are pretty in agreement with (33).

An analogous behavior is observed for di�erent values of the discretization parameter hxand for di�erent values of %s.

7.4 Neumann/Dirichlet subiterations

Finally, we present the results of the numerical tests conducted using the procedure (34).Again, we consider the same domain and the same physical parameters as before. Settinghs%s = 3:0 g=cm2 and using space steps hx = [L=20; L=40; L=60] from (35) we �nd that!lim = [0:0459; 0:0221; 0:0135], respectively. In Tab. 3 we show the results obtained witha time step �t = 10�4s and jj�k � �k�1jj < 1e � 6 (max number of iterations is 2000). Inparticular, we �nd con�rmation of the fact that in this case ! = !(h), this implying thatin the limit h ! 0, ! must also tend to zero! We also observe that, in order to observethe onset of instability as prescribed by the theoretical results, in some cases it should benecessary to integrate for a time interval longer than Tmax = 3 s. For coherence with theprevious results, we limit our analysis to this latter value.

8 Characterization of the numerical properties of parti-

tioned schemes through dimensional analysis

The analysis carried out in the preceding sections shows the dependence of the numericalstability of the solution of the added-mass problem on the values of the physical parameters.To highlight this fact, another possible way to proceed is proposed in [15, 10], where scaling

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! hx=L/20 hx=L/40 hx=L/600.058

unstable

unstable

unstable

0.0570.0560.0550.05

stable

0.0459

0.030.0250.024450.023

stable

0.0221

0.02150.020.0140.01390.01370.0135

stable0.01300.0125

Table 3: Behavior of the Neumann-Dirichlet scheme as the spatial discretization parametertends to zero.

arguments are used to provide indications about the in�uence of the physical parameters.This analysis applies to the linear case. In the present section we will pursue a similarapproach. This will allow, on the one hand, to put in the frame of [15, 10] the results ofthe present discussion, and on the other hand, to devise a more general point of view withrespect to the analysis of Sect. 6 - even if more qualitative - in presence for example ofa non-negligible second order term @2�=@x2. We observe that this latter is the situationthat actually corresponds to the one studied in [15, 10], where, di�erently from the presentanalysis, inertial terms were neglected.

We introduce the characteristic length and velocity scales Lm; Um and we de�ne thefollowing non-dimensional quantities

(ex; ey; ez)T =(x; y; z)T

Lm; eu =

u

Um; e� =

�

Lm

and

�t� =UmLm

�t; ep = p

%FU2m

:

In the case of blood �ow in large arteries in a human body, pertinent characteristicparameters are reported in Tab. 4.

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Parameter ValueReference length: Lm 6 cmReference radius: R 0.5 cmMean velocity: Um 10 cm/sBlood density: %f 1 g/cm3

Wall density: hs%s 1.1 g/cm2

Wall thickness: hs 0.1 cmShear modulus: kTG 2.5 � 105 dynes/cm2

Young modulus: E 0.75 � 106 dynes/cm2

Poisson coe�cient: � 0.5

Table 4: Physical parameter values for blood �ow in large arteries in human body.

From the time discretized form of equation (18)

p� = �%fMA�

�t2+ g; (42)

where g groups the forcing term pext and the terms coming form the previous time steps,we have that the adimensional added-mass operator reads

fMA =MA

Lmand MA

�

�t2=

U2m

�t2�fMAe�: (43)

The time discretized structure equation reads

SS� =

�%shs�t2

I + L

�� = p�; (44)

where we have introduced the symmetric and positive operator SS , for which we proposethe following scaling

eSSe� =�t2�Lm%shsU2

m

SS� = e� � �t2�kTG

%sU2m

@2e�@ez2 +

�t2�EL2m

R2(1� �2)%sU2m

e�; (45)

that yields

SS� =%shsU

2m

�t2�LmeSSe�: (46)

Remark 9 We observe that, with the present model, the ratio k1 =�t2�kTG

%sU2m

=�t2kTG

%sL2mmeasures the relative in�uence of the elastic shear terms and of the inertial terms, while the

ratio k2 =�t2�EL

2m

R2(1� �2)%sU2m

=�t2E

R2(1� �2)%smeasures the relative in�uence of the elastic

sti�ness terms and of the inertial terms.

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If we now gather (43) - (46), we obtain the adimensional form of the FSI problem

eSe� =

�eSS +%fLm%shs

fMA

� e� = eF : (47)

where eF accounts for known terms. Problem (47) is approximated in �nite dimension by

eShe�h = (eShS +%fLm%shs

fMhA)e�h = eFh; (48)

where the operator eSh is symmetric and positive de�nite. The iterative �xed point method(32), introduce in Sect. 6.1, reads in this context (for simplicity, we omit the h superscript):given e�k ; k � 0, compute 8<

: �e� = eS�1SeF �

%fLm%shs

eS�1SfMAe�k;

e�k+1 = (1� !)e�k + !�e�; (49)

that is

e�k+1 = ! eS�1SeF + [Id � !P ]e�k; (50)

where we have set

P = eS�1SeS =

�Id +

%fLm%shs

eS�1SfMA

�:

The choice ! = 1 means adopting a non-relaxed �xed-point algorithm to solve the coupledproblem.

As, in general, the matrix P is not symmetric, it is convenient to perform the substitutionb� = eS1=2S e�; this procedure leads to the form

b�k+1 = ! eS�1=2SeF +

�Id � !

�Id +

%fLm%shs

eS�1=2SfMA

eS�1=2S

��b�k; (51)

that corresponds to applying the gradient algorithm with �xed acceleration parameter tothe symmetric positive de�nite operator

bP =�Id +

%fLm%shs

eS�1=2SfMA

eS�1=2S

�:

Denoting by (A)max the extremal eigenvalues of the operatorA, a necessary and su�cientcondition (see [18], p. 38) for the convergence of the �xed point method with constantacceleration parameter ! is the following

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Proposition 6 The �xed point algorithm (49) converges linearly toward the solution of thelinearized implicit coupled problem at time tn+1 if and only if

0 < ! <2%shs=%fLm�

%shs%fLm

Id+ eS�1=2SfMA

eS�1=2S

�max

(52)

Remark 10 A �xed point algorithm applied to the solution of the coupled system requiresto use a smaller value of the relaxation parameter as the density of the �uid approaches thedensity of the structure, this limitation being not dependent on the choice of the time step.This qualitative result is in agreement with the analysis carried out in Sect. 6.1.

The above procedure allows as well to see what happens if we were instead in the case ofdominant sti�ness terms in the structure, that is if k2 >> 1. To do this, we should changethe scaling and in particular we need to introduce a characteristic sti�ness Em = E. Weperform accordingly the scaling of the structure operator, yielding

SS� =EmL

2m

R2(1� �2)eSSe� (53)

and thus

eSe� =

�eSS +%fU

2mR

2(1� �2)

�t2�EmL2mfMA

� e� = eF : (54)

Proceeding analogously as in the case of dominating inertial terms, we may write the fol-lowing relaxed �xed point method:given e�k ; k � 0, compute8<

: �e� = eS�1SeF �

%fU2mR

2(1� �2)

�t2�EmL2meS�1S

fMAe�k;e�k+1 = (1� !)e�k + !�e�; (55)

that is

e�k+1 = ! eS�1SeF + [Id � !P ]e�k; (56)

where we have set

P = eS�1SeS =

�Id +

%fU2mR

2(1� �2)

�t2�EmL2meS�1S

fMA

�:

Operating again the change of variables b� = eS1=2S e�, such that the corresponding operator b�is symmetric and positive de�nite, we have that the following proposition holds.

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Proposition 7 The �xed point algorithm (55) converges linearly toward the solution of thelinearized implicit coupled problem at time tn+1 if and only if

0 < ! <

2�t2�EmLm

%fU2mR

2(1� �2)��t2�EmLm

%fU2mR

2(1� �2)Id+ eS�1=2S

fMAeS�1=2S

�max

: (57)

Remark 11 This result is analogous to what found in [15, 10]. We observe in particularthat in this case a �xed point algorithm requires a smaller relaxation parameter for a softstructure or for a small time step (Em�t

2� small) or again as the �uid density increases.

Acknowledgments:

This work has been partially supported by the Research Training Network �Mathematical Mod-

elling of the Cardiovascular System� (HaeMOdel), contract HPRN-CT-2002-00270 of the European

Community.

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