A Newton method using exact jacobians for solving fluid–structure coupling

Computers and Structures 83 (2005) 127–142

www.elsevier.com/locate/compstruc

A Newton method using exact jacobians for solvingfluid–structure coupling

Miguel Angel Fernandez a,*, Marwan Moubachir b

a Ecole Polytechnique Federale de Lausanne, IACS, 1015 Lausanne, Switzerlandb Institut National de Recherche en Informatique et en Automatique, OPALE, BP 93, 2004 route de Lucioles,

06902 Sophia-Antipolis, France

Accepted 6 April 2004

Available online 23 November 2004

Abstract

This paper aims at introducing a partitioned Newton based method for solving nonlinear coupled systems arising in

the numerical approximation of fluid–structure interaction problems. We provide a method which characteristic lies in

the use of exact cross jacobians evaluation involving the shape derivative of the fluid state with respect to solid motion

perturbations. Numerical tests based on an implementation inside a 3D fluid–structure interaction code show how the

exactness of the cross jacobians computation guarantee the overall convergence of the Newton�s loop.� 2004 Elsevier Ltd. All rights reserved.

Keywords: Fluid–structure interaction; Navier–Stokes equations; ALE formulation; Full coupled schemes; Newton methods; Shape

sensitivity analysis; Haemodynamics

1. Introduction

Nowadays, Computational Fluid-Structure Dynam-

ics (CFSD) spreads out in every engineering field, from

aeroelasticity to bio-mechanics problems (see, for in-

stance, [28,9,11,24,16,27,25,31,18,34,32,38]). One issue

arising in the numerical approximation of these nonlin-

ear coupled systems, is the definition of coupling algo-

rithms based on specific solvers involving efficient

discretization for each of the solid and fluid subsystems,

that may guarantee accurate and fast convergence of the

overall system. This issue is particularly difficult to face

0045-7949/$ - see front matter � 2004 Elsevier Ltd. All rights reserv

doi:10.1016/j.compstruc.2004.04.021

* Corresponding author. Present address: INRIA Project

REO, Rocquencourt BP 105, 78153 Le Chesnay Cedex, France.

Tel.: +33 1 3963 5470; fax: +33 1 3963 5882.

E-mail address: [email protected] (M.A.

Fernandez).

when the structure is light, namely, when the fluid and

the solid densities are of the same order, as it happens

in haemodynamics for example. Indeed, in this case,

numerical experiments show that only fully coupled

schemes can ensure stability of the resulting method

(see [24,30,7,18,25,26]). Thus, at each time step, the rule

is to solve a coupled highly nonlinear system using effi-

cient methods that may preserve, inside inner loops,

the fluid-structure subsystem splitting.

Standard and simple strategies to solve these nonlin-

ear problems are fixed-point based methods [4,1]. Unfor-

tunately, these methods are very expensive (even if

several acceleration techniques may improve their effi-

ciency [27,7]) and may fail to converge [25–27,7,18]. Re-

cent advances in this topic suggest the use of Newton

based methods for a fast convergence towards the solu-

tion of the nonlinear coupled system [1,25,26,34,18,

2,38,21]. These methods rely on the evaluation of the

ed.

mailto:[email protected]

Fig. 1. Geometric configurations.

128 M. �A. Fern�andez, M. Moubachir / Computers and Structures 83 (2005) 127–142

jacobians associated to the fluid-solid coupled state

equations. More precisely, the critical step consists in

the evaluation of the cross jacobians [34], e.g. the sensi-

tivity of the fluid state with respect to solid motions. Up

to now, this difficulties have been overcome either by

using finite difference approximations [25,26,34,21], or

by replacing the tangent operator of the coupled system

by a simpler operator [34,2,38,18,6,8,19]. In both cases,

such approximations may deteriorate or avoid the over-

all convergence [34]. In this paper, we provide an explicit

expression of these cross jacobians, using shape sensitiv-

ity calculus [33], in the case of an incompressible Newto-

nian fluid coupled with a nonlinear elastic solid under

large displacements. These expressions are then imple-

mented inside a 3D finite element (FE) library [12] and

we show on some model cases the superiority of using

exact jacobians against approximated versions of the

Newton�s method.

The remainder of the paper is organized as follows.

In Section 2, we introduce a general fluid-solid coupled

system and describe its associated mathematical model.

In Section 3, the resulting coupled set of equations is

used to build a coupled weak variational formulation.

The latter is discretized in time using a fully coupled

scheme in Section 4, where, the resulting nonlinear cou-

pled system is turned into an abstract form. In order to

solve this abstract coupled system, in Section 5 we de-

scribe the main steps of the Newton�s method in terms

of fluid and solid operators and its derivatives. The

expressions of such derivatives are obtained, in Section

6, using general shape derivative calculus results that

are recalled in Appendix A. These expressions have been

already announced in [14,15] as brief notes. In Section 7,

we detail the major steps of the Newton�s algorithm ap-

plied to the coupled fluid-solid problem. The numerical

results are presented in Section 8, showing the relevance

of our approach. Finally, we give some conclusions and

draw some lines for future works.

Fig. 2. The map x.

2. Mechanical problem

Let us first describe a general nonlinear fluid-struc-

ture system in large displacements. We consider a

mechanical system occupying a moving domain X(t). It

consists of a deformable structure Xs(t) (vessel wall,

pipe-line, . . .) surrounding a fluid under motion (blood,

oil, . . .) in the complement Xf(t) of Xs(t) in X(t) (see Fig.

1). The problem is to determinate the time evolution of

the configuration X(t), as well as the velocity and Cau-

chy stress tensor within the fluid and the structure.

The latter being governed by the classic conservation

laws of the continuum mechanics, endowed with appro-

priate constitutive laws.

The evolution of X(t) (see e.g. [9,22,24]), can be de-

scribed through a smooth and injective mapping

x : X0 � Rþ ! R3;

which maps any point x0 of a given fixed (reference) con-

figuration X0 ¼def Xf0 [ Xs

0 into its corresponding image

x(x0, t), inside the present configuration X(t) (see Fig.

2). We set xf ¼def xjXf0and xs ¼def xjXs

0. For x0 2 Xs

0, xs(x0, t)

represents the position at time tP 0 of the material

point x0 inside the solid domain. This corresponds to

the classical lagrangian flow. In particular, the solid dis-

placement ds(x0, t) reads

dsðx0; tÞ ¼def xsðx0; tÞ x0; x0 2 Xs0:

This implies that the configuration velocity matches

the solid velocity inside the solid domain. Conversely,

the map xf ¼ IXf0þ df can be defined from the interface

displacement dsjCw0as an arbitrary extension over the do-

main Xf0, namely,

df ¼ ExtðdsjCw0Þ:

This explains the use of the terminology Arbitrary

Lagrangian Eulerian (ALE) formulation for the result-

ing equations [9,22,24]. In practice, we can choose an

harmonic extension operator [24,30,7,18], which means

that df solves the following elliptic problem:

jDdf ¼ 0; in Xf0;

df ¼ ds; on Cw0 ;

(

where j > 0 is a given ‘‘diffusion’’ coefficient, that might

depend on ds. Other alternative extension approaches

can be found, for instance, in [3,37].

Therefore, the fluid domain can be parameterized as

follows:

M. �A. Fern�andez, M. Moubachir / Computers and Structures 83 (2005) 127–142 129

XfðtÞ ¼ Xfðdfð�; tÞÞ ¼ ðIXf0þ dfÞðXf

0Þ; ð1Þ

and its corresponding velocity (the so-called ALE veloc-

ity) by

wðdfÞ ¼def odf

ot:

We deal with a Newtonian viscous, homogeneous

fluid under incompressible flow with density q and

kinetic viscosity l. Its behavior is described by its

Eulerian velocity u and pressure p. The constitutive

law for the Cauchy stress tensor is given by the following

expression:

rðu; pÞ ¼ pI þ l½ruþ ðruÞT�:

The elastic solid under large displacements is de-

scribed by its displacement ds and its stress tensor S (sec-

ond Piola–Kirchhoff tensor). The field S is related to ds

through an appropriate constitutive law, S = S(ds) (see

[5,23,20]). Its boundary is divided into three disjoint

parts Cd0 [ Cn

0 [ Cw0 . We impose respectively a homoge-

neous Dirichlet boundary condition on Cd0 and a homo-

geneous Neumann boundary condition on Cn0 .

The coupling between the solid and the fluid is realized

through standard boundary conditions at the fluid-struc-

ture interface Cw0 , namely, the kinematic continuity of the

velocity and the kinetic continuity of the stress:

u ¼ wðdfÞ;F ðdsÞSðdsÞn0 ¼ JðdfÞrðu; pÞF ðdfÞTn0;

ð2Þ

with n0 being the unit outward normal vector to Xf0, and

F ðdsÞ ¼def I þr0ds; F ðdfÞ ¼def I þr0d

f ;

JðdsÞ ¼def det F ðdsÞ; JðdfÞ ¼def det F ðdfÞ;

where F(ds) and F(df) stand, respectively, for the defor-

mation gradient inside the solid and the fluid domains.

Finally, the fluid-structure coupled state (u,p,df;ds)

satisfies the following strong coupled system (involving

an ALE fluid formulation),

qoJðdf Þuot jx0

þ div ½qu� ðu wðdfÞÞ rðu; pÞ� ¼ 0;

in XfðtÞ;divu ¼ 0; in XfðtÞ;u ¼ wðdfÞ; on Cw

0 ;

rðu; pÞn ¼ g; on CinðtÞ [ CoutðtÞ;df ¼ ExtðdsjCw

0Þ; in Xf

0;

q0o2ds

ot2 div0ðF ðdsÞSðdsÞÞ ¼ 0; in Xs0;

F ðdsÞSðdsÞn0 ¼ JðdfÞrðu; pÞF ðdfÞTn0; on Cw0 ;

ds ¼ 0; on Cd0 ;

F ðdsÞSðdsÞn0 ¼ 0; on Cn0 ;

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

ð3Þ

with q0 the solid density, n the unit outward normal vec-

tor to Xf(t), given by

n ¼ F TðdfÞn0kF TðdfÞn0k

;

and g standing for the external forces acting on the fluid.

For the sake of simpleness, no body forces are consid-

ered. In the sequel, we shall set Cinout ¼def Cin [ Cout.

Remark 1. Let notice that, in (3), the fluid domain Xf(t)

is directly related to the unknown df through expression

(1).

Remark 2. A variant to bypass the fluid ALE formu-

lation in (3) is to use a fully Eulerian fluid description

and then to discretise using espace-time finite elements,

as in [35,36,34].

3. Continuous weak formulation

Problem (3) can be reformulated in a weak varia-

tional form using appropriate test functions (see

[17,24]), performing integration by parts and taking into

account the boundary and interface conditions. This

leads to the following coupled weak formulation (see

[30,24]), involving a fluid and a solid weak-formulations:

Find df : Xf0 ! R3, u : XfðtÞ ! R3, p : XfðtÞ ! R and

ds : Xs0 ! R3, satisfying the following coupled nonlinear

subproblems:

(1) The fluid weak-formulation:

qd

dt

ZXf ðtÞ

u � vf dxþ qZ

Xf ðtÞdiv ½u� ðu wðdfÞÞ� � vf dx

þZ

Xf ðtÞrðu; pÞ : rvf dx

ZCinoutðtÞ

gðtÞ � vf da

Z

Xf ðtÞqdivudxþ

ZCwðtÞðu wðdfÞÞ � nda

þZ

Xf0

df Ext dsjCw0

� �� kdx0 ¼ 0;

8ðvf ; q; n; kÞ 2 V fðtÞ: ð4Þ

(2) The solid weak-formulation:ZXs0

q0

o2ds

ot2� vsdx0 þ

ZXs0

F ðdsÞSðdsÞ : rvs dx0

þ qd

dt

ZXf ðtÞ

u �RðvsjCw0Þdx

þ qZ

Xf ðtÞdiv ½u� ðu wðdfÞÞ� �RðvsjCw

0Þdx

þZ

Xf ðtÞrðu; pÞ : rRðvsjCw

0Þdx ¼ 0; 8vs 2 V s: ð5Þ

Here, the space of solid test functions is given by

V s ¼def vs 2 H 1ðXs0Þ

3jvs ¼ 0 on Cd0

n o;


and the space of fluid test functions Vf(t) is defined using

the ALE parameterization xf (see [17]),

V fðtÞ ¼def vf ¼ v � ðxfÞ1 j v 2 H 1ðXf0Þ

3; v ¼ 0 on Cw0

n o� q ¼ q � ðxfÞ1 j q 2 L2ðXf

0Þn o� n ¼ n � ðxfÞ1jCwðtÞ j n 2 L2ðCw

0 Þ3

n o� L2ðXf

0Þ3:

In addition, we denote by R a given linear continu-

ous lift operator

R : H12ðCw

0 Þ

! vf ¼ v � ðxfÞ1 j v 2 H 1ðXf0Þ

3; v ¼ 0 on Cinout

n o:

Remark 3. Notice that the interface coupling condition

(2)2 is implicitly treated in (5). Indeed, the last three

terms of Eq. (5) represent nothing but the residual of

the fluid momentum equations in (4). This furnishes a

weak description of the fluid load at the interface

(see [24]).

4. Time semi-discretized weak formulation

The weak coupled formulation (4) and (5) is now

semi-discretized in time. Numerical experiments (see

[30,24–26,18,7]) show that only fully coupled schemes

(e.g. the fluid geometry and the interface coupling condi-

tions are implicitly treated) ensure stability and allow to

solve effectively problems in which both the fluid and

structural densities are of the same order. This arises,

for instance, in haemodynamics applications (see

[30,16,7,18,19]).

4.1. An implicit coupling scheme

We use an implicit Euler scheme for the ALE Na-

vier–Stokes equations, with a semi-implicit treatment

of the nonlinear convective term. Furthermore we use

a mid-point rule for the structural equation (see

[24,16,18,7]). Thus, for n = 0,1, . . . , the semi-discretized

in time problem writes: Given (un,df,n; ds,n,ws,n), find

ðunþ1; pnþ1; df;nþ1; ds;nþ1Þ;

satisfying the following coupled nonlinear subproblems:

(1) Fluid time semi-discretized weak-formulation:

qDt

ZXf ðtnþ1Þ

unþ1 � vfdx qDt

ZXf ðtnÞ

un � vf dx

þ qZ

Xf ðtnþ1Þdiv unþ1 � ðun wgðdf ;nþ1ÞÞ

� �� vfdx

þZ

Xf ðtnþ1Þrðunþ1; pnþ1Þ : rvfdx

Z

Cinoutðtnþ1Þgðtnþ1Þ � vfda

ZXf ðtnþ1Þ

qdivunþ1dx

þZ

Cwðtnþ1Þunþ1 wgðdf;nþ1Þ �

� nda

þZ

Xf0

df ;nþ1 Ext ds;nþ1jCw

0

�� kdx0 ¼ 0;

8ðvf ; q; n; kÞ 2 V fðtnþ1Þ; ð6Þ

with the notation wgðdf ;nþ1Þ ¼def 1Dt df ;nþ1 df ;n �

:(2) Solid time semi-discretized weak-formulation:

2

ðDtÞ2Z

Xs0

q0ds;nþ1 � vs dx0

2

ðDtÞ2Z

Xs0

q0ðds;n þ Dtws;nÞ � vs dx0

þ 1

2

ZXs0

F ðds;nþ1ÞSðds;nþ1Þ : rvs dx0

þ 1

2

ZXs0

F ðds;nÞSðds;nÞ : rvs dx0

þ qDt

ZXf ðtnþ1Þ

unþ1 �RðvsjCw0Þdx

qDt

ZXf ðtnÞ

un �RðvsjCw0Þdx

þ qZ

Xf ðtnþ1Þdiv unþ1 � ðun wgðdf;nþ1ÞÞ

� ��RðvsjCw

0Þdx

þZ

Xf ðtnþ1Þrðunþ1; pnþ1Þ : rRðvsjCw

0Þdx ¼ 0; 8vs 2 V s;

ð7Þwith ws;nþ1 ¼ 2

Dt ds;nþ1 ds;n �

ws;n:

4.2. Abstract formulation

In order to use general methods to solve the coupled-

system (6) and (7), it is useful to turn it into an abstract

form. Let Uf · Us be the space of fluid and solid states.

We introduce the following operators:

(1) Fluid operator:

: U f � U s ! ðV fÞ0

ðuf ; usÞ 7! ðuf ; usÞ;

with uf ¼def ðu; p; dfÞ and us ¼def ds, whose action on fluid

test functions (in Vf(tn+1)) is defined as follows,

h ðu; p; df ; dsÞ; ðvf ; q; n; kÞi

¼def qDt

ZXf ðdf Þ

u � vf dx qDt

ZXf ðtnÞ

un � vf dx

þ qZ

Xf ðdf Þdiv u� ðun wgðdfÞÞ

� �� vf dx

þZ

Xf ðdf Þrðu; pÞ : rvf dx

ZCinoutðdf Þ

gðtnþ1Þ � vf da

Z

Xf ðdf Þqdivudxþ

ZCwðdf Þ

u wgðdfÞ �

� nda

þZ

Xf0

df Ext dsjCw

0

�� kdx0; ð8Þ

with wgðdfÞ ¼ 1Dt df df ;n �

.


(2) Solid operator:

: U f � U s ! ðV sÞ0

ðuf ; usÞ 7! ðuf ; usÞ;

whose action on solid test functions is defined as follows,

h ðu; p; df ; dsÞ; vsi

¼def 2

ðDtÞ2Z

Xs0

q0ds � vs dx0

2

ðDtÞ2Z

Xs0

q0ðds;n þ Dtws;nÞ � vs dx0

þ 1

2

ZXs0

F ðdsÞSðdsÞ : rvsdx0

þ 1

2

ZXs0

F ðds;nÞSðds;nÞ : rvsdx0 þ h ðu; p; dfÞ; vsi;

ð9Þ

where the fluid load transfer operator : U f ! ðV sÞ0 isdefined by

h ðu; p; dfÞ; vsi ¼def h ðu; p; df ; dsÞ; ðRðvsjCw0Þ; 0; 0; 0Þi

¼ qDt

ZXf ðdf Þ

u �RðvsjCw0Þdx q

Dt

ZXf ðtnÞ

un �RðvsjCw0Þdx

þ qZ

Xf ðdf Þdiv ½u� ðun wgðdfÞÞ� �RðvsjCw

0Þdx

þZ

Xf ðdf Þrðu; pÞ : rRðvsjCw

0Þdx: ð10Þ

Therefore, using the above definitions, the coupled

problem (6) and (7) reads: For n = 0,1, . . . , find

ðunþ1; pnþ1; df ;nþ1; ds;nþ1Þ;

solution of the following nonlinear coupled system:

h ðunþ1; pnþ1; df ;nþ1; ds;nþ1Þ; ðvf ; q; n; kÞi ¼ 0;

8ðvf ; q; n; kÞ 2 V fðtnþ1Þ;h ðunþ1; pnþ1; df ;nþ1; ds;nþ1Þ; vsi ¼ 0; 8vs 2 V s;

that is,

ðunþ1; pnþ1; df ;nþ1; ds;nþ1Þ ¼ 0;

ðunþ1; pnþ1; df ;nþ1; ds;nþ1Þ ¼ 0:

To simplify the notation, we shall only consider one

time step so that we drop the upper index n + l. Thus,

the above coupled problem writes: find uf ¼def ðu; p; dfÞand us ¼def ds such that

ðuf ; usÞ ¼ 0;

ðuf ; usÞ ¼ 0:ð11Þ

In the sequel, we shall assume that the methods and

software which have been developed to solve separately

each subsystem (the fluid and the structure) will con-

tinue to be used, Hence, it is useful to turn problem

(11) into an abstract form in terms of the fluid and struc-

ture solvers. To this purpose, we introduce the fluid sol-

ver operator F : U s ! U f , defined by

ðFðusÞ; usÞ ¼ 0; 8us 2 U s; ð12Þ

and the solid solver operator S : U f ! U s, defined by

ðuf ;SðufÞÞ ¼ 0; 8uf 2 U f : ð13Þ

Therefore, using the above definitions, the coupled

nonlinear problem (11) can be reduced to: find us 2 Us

solution of the following nonlinear equation

ðusÞ ¼def us S �FðusÞ ¼ 0; ð14Þ

see [18]. If (uf,us) solves (11) then us satisfies (14) and,

conversely, if us is a solution of (14) then ðuf ¼FðusÞ; usÞ solves (11).

In the next section, we describe the main steps of the

Newton�s algorithm when applied to the above nonlin-

ear problem.

5. The Newton�s algorithm

Standard forms for solving the nonlinear problem

(14) are fixed-points based iterations, see [30,27,

38,18,7]. Unfortunately, these methods usually show

poor convergence properties and may fail to converge

(see [25–27,7,18]). In order to speed up the convergence,

it is useful to use Newton-Raphson based methods (see

[25,26,2,38,34,18,21]).

The Newton�s method applied to the nonlinear Eq.

(14) reads:

(1) Choose �us 2 U s

(2) Do until convergence

(a) Evaluate the fluid operator �uf ¼Fð�usÞ

(b) Evaluate the solid operator us ¼Sð�ufÞ

(c) Evaluate the residual ð�usÞ ¼ �us us

(d) Solve

Dus ð�usÞ½ �dus ¼ ð�usÞ

(e) Update rule: �us �us þ dus.

Step 2 (b) of this algorithm can be carried out by

using an iterative free matrix method as GMRES (see

[25,26,34,18]). In this case, we only need to evaluate sev-

eral times the operator ½Dus ð�usÞ� against solid state per-

turbations z. In particular, we have


By differentiating the Eqs. (12) and (13), we are able

to split the above expression in terms of the derivatives

of the fluid an solid operators (8) and (9), defined in

the previous Section:

(i) Solve the fluid tangent sub-pro-

blem:

Duf ð�uf ; �usÞ� �

duf ¼ Dus ð�uf ; �usÞ� �

z; ð15Þ

(ii) Solve the solid tangent sub-pro-

blem:

Dus ð�uf ; usÞ� �

dz ¼ Duf ð�uf ; usÞ� �

duf ;

(iii) Evaluate

Dus ð�usÞ½ �z ¼ z dz:

The main difficulty here (see [25,26,34,18,21]) relies

on the fluid tangent subproblem (15). Indeed, as we shall

see in the next section, this problem involves the evalu-

ation of the following cross-derivative of the fluid

operator:

Ddf ð�u; �p; �df; �d

sÞddf ;

which corresponds to the directional derivative with re-

spect to fluid-domain perturbations. In previous works,

the evaluations of these cross-jacobians were performed

using finite difference approximations, that only require

state operators evaluations [25,26,34,21]. However, the

lack of a priori rules for selecting optimal finite differ-

ence infinitesimal steps, leads to nonconsistent jacobians

and a reduction of the overall convergence speed [18]. In

the following section, we will show how to avoid these

approximations by establishing the exact expression of

the above linearized sub-systems.

6. Weak state operators derivatives

In this section, we present the differentiation of the

fluid and solid operators with respect to the fluid and

solid state variables. We will use shape derivative calcu-

lus results in order to perform the differentiation of inte-

gral terms with respect to their supports.

6.1. Fluid operator derivatives

We are able to obtain the expressions of the action of

the derivative of the fluid weak state operator with re-

spect to the state variables uf = (u,p,df) and us ¼def ds; atpoint ð�u; �p; �df ; �dsÞ in the direction (du,dp,ddf,dds). We

can split the operator derivative in two terms:

Dðu;p;df Þ ð�u; �p; �df; �d

sÞðdu; dp; ddfÞ¼ Dðu;pÞ ð�u; �p; �d

f; �d

sÞðdu; dpÞ þ Ddf ð�u; �p; �df; �d

sÞddf :ð16Þ

The first term Dðu;pÞ ð�u; �p; �df; �d

sÞðdu; dpÞ is a classical

Frechet derivative, in the sense we do not need to per-

turb the support of the integrals inside the operator.

Thus, we have

hDðu;pÞ ð�u; �p; �df; �d

sÞðdu; dpÞ; ðvf ; q; n; kÞi

¼ qDt

ZXf ð�df Þ

du � vf dx

þ qZ

Xf ð�df Þdiv ½du� ðun wgð�d

fÞÞ� � vf dx

þZ

Xf ð�df Þrðdu; dpÞ : rvf dx

ZXf ð�df Þ

qdivdudx

þZ

Cwð�df Þdu � nda: ð17Þ

The second term, Ddf ð�u; �p; �df; �d

sÞddf , requires the

derivation with respect to df of Eulerian integrals over

Xf(df) and its boundary. Actually, this kind of derivatives

can be viewed as shape derivatives, see for instance [33].

Some elements on shape derivative calculus are given in

Appendix A and allow to establish the following identity,

hDdf ðu; p; �df; �d

sÞddf ; ðvf ; q; n; kÞi

¼ qDt

ZXf ðdf ÞðdivddfÞu � vf dx

þZ

Xf ðdf Þqdivfu� ðun wgðd

fÞÞ½Idivddf

ðrddfÞT�g � vfdx qDt

ZXf ðdf Þ

div ðu� ddfÞ � vf dx

þZ

Xf ðdf Þrðu; pÞ½Idivddf ðrddfÞT� : rvf dx

Z

Xf ðdf Þl½rurddf þ ðrddfÞTðruÞT� : rvf dx

Z

Cinoutðdf ÞðgðddfÞ � nÞgðtnþ1Þ � vfda

Z

Xf ðdf Þqdivfu½Idivddf ðrddfÞT�gdx

þZ

Xf0

ddf � kdx0 þZ

Cwðdf Þðu wgðd

fÞÞðgðddfÞ � nÞ�

ddf

Dt

�� nda: ð18Þ

Here, we used the notation gðddfÞ ¼def ½IdivddfðrddfÞT�n, which represents the first order variation of

the surface vector nda (see, for instance, [13,10]). The deriv-

ative of with respect to the solid variable is given by,

hDds ðu; p; df; d

sÞdds; ðvf ; q; n; kÞi

¼ Z

Xf0

Ext0ðdsÞdds � kdx0: ð19Þ

with the notation Ext0ðdsÞdds ¼def DdsExtðdsÞdds.


6.2. Solid operator derivatives

We proceed in the same fashion for the solid operator

differentiation. We first get the classical Frechet

derivatives,

hDds ðu; p; df; dsÞdds; vsi ¼ 2

ðDtÞ2Z

Xs0

q0dds � vsdx0

þ 1

2

ZXs0

F ðdsÞdS þrddsSðdsÞ� �

: rvsdx0; ð20Þ

where dS ¼def DdsSðdsÞdds.

Again we can split the derivative with respect to the

fluid state in two terms:

Dðu;p;df Þ ðu; p; df; d

sÞðdu; dp; ddfÞ

¼ Dðu;pÞ ðu; p; df; d

sÞðdu; dpÞ

þ Ddf ðu; p; df; d

sÞddf : ð21Þ

The first term Dðu;pÞ ðu; p; df; d

sÞðdu; dpÞ is a classical

Frechet derivative. Thus, we have

hDðu;pÞ ðu; p; df; d

sÞðdu; dpÞ; vsi

¼ qDt

ZXf ðdf Þ

du �R vsjCw0

� �dx

þ qZ

Xf ðdf Þdiv du� ðun wgðd

fÞh i

�RðvsjCw0Þ

þZ

Xf ðdf Þrðdu; dpÞ : rRðvsjCw

0Þdx: ð22Þ

Using the definition (9), the derivative of the solid

operator with respect to df corresponds to the derivative

of the fluid loads operator with respect to df, i.e.,

hDdf ðu; p; df ; dsÞddf ; vsi ¼ hDdf ðu; p; dfÞddf ; vsi

¼ hDdf ðu; p; df; d

sÞddf ;ðRðvsjCw

0Þ; 0; 0; 0Þi:

Then using (18), we get

hDdf ðu; p; df; d

sÞddf ; vsi

¼ qDt

ZXf ðdf ÞðdivddfÞu �R vsjCw

0

� �dx

þZ

Xf ðdf Þqdivfu� ðun wgðd

fÞÞ½I divddf

ðrddfÞT�g �R vsjCw0

� �dx

qDt

ZXf ðdf Þ

div ðu� ddfÞ �R vsjCw0

� �dx

þZ

Xf ðdf Þrðu; pÞ I divddf ðrddfÞT

h i: rR vsjCw

0

� �dx

Z

Xf ðdf Þl rurddf þ ðrddfÞTðruÞTh i

: rR vsjCw0

� �dx

ð23Þ

7. Detailed Newton�s algorithm sub-steps

Let us recall the following notations:

z ¼def z; uf ¼def ðu; p; dfÞ; us ¼def ds;

us ¼def ds; duf ¼def ðdu; dp; ddfÞ; dz ¼def dz:

Thus, from (16), it follows that step 2(d) (i) consists

in solving for (du,dp,ddf) the following linear variational

problem

Dðu;pÞ ðu; p; df; d

sÞðdu; dpÞ þ Ddf ðu; p; df; d

sÞddf

¼ DdS ðu; p; df; dsÞz: ð24Þ

Using the expressions of the fluid operator deriva-

tives provided in (17)–(19) and by taking (0, 0, 0, k) as

test function in (24) we obtain that

ddf ¼ Ext0ðdsÞz; in Xf0:

On the other hand, taking the quadruplet (0, 0, n, 0)

as test functions in (24), we obtain the following bound-

ary condition,

du ¼ ddf

Dt ðu wgðd

fÞÞðgðddfÞ � nÞ on CwðdfÞ: ð25Þ

Moreover, since by definition uf ¼FðusÞ, it followsthat u wgðd

fÞ ¼ 0 on CwðdfÞ so that (25) reduces to

du ¼ ddf

Dt; on CwðdfÞ: ð26Þ

Finally, using (17)–(19) and choosing the quadruplet

(0, 0, n, 0) as test functions in (24), we obtain the follow-

ing variational expression satisfied by (du,dp),

hDðu;pÞ ðu; p; df; d

sÞðdu; dpÞ; ðvf ; q; 0; 0Þi

¼ hDdf ðu; p; df; d

sÞddf ; ðvf ; q; 0; 0Þi; ð27Þ

Actually, the last expression corresponds to the weak

formulation associated to the following PDE involving

the linearized ALE Navier–Stokes equations:

qDt duþ div qdu� ðun wgðd

fÞÞ rðdu; dpÞh i

¼ qDt ðdivddfÞu div qu� ðun wgðd

fÞÞhn

rðu; pÞiIdivddf ðrddfÞTh io

þ qDt divðu� ddfÞ div l

hrurddf

nþðrddfÞTðruÞT

io; in XfðdÞ;

divdu ¼ div u I divddf ðrddfÞTh in o

; in XfðdfÞ;

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:endowed with the following inflow – outflow condition

on CinoutðdfÞ:


rðdu; dpÞn ¼ gðtnþ1ÞðgðddfÞ � nÞ rðu; pÞgðddfÞ

þ l rurddf þ ðrddfÞTðruÞTh i

n:

In short, taking into account the above consider-

ations, step 2(d) (i) can be carried out by computing

ddf ¼ Ext0ðdsÞz and by solving the following linearized

Navier–Stokes problem for (du,dp):



¼ qDt ðdivddfÞu div qu� ðun wgðd

fÞÞhn

rðu; pÞiIdivddf ðrddfÞTh io

þ qDt divðu� ddfÞ div l

hrurddf

nþðrddfÞTðruÞT

io; in XfðdÞ;

divdu ¼ div u Idivddf ðrddfÞTh in o

;

in XfðdfÞ;du ¼ ddf

Dt ; on CwðdfÞ;rðdu; dpÞn ¼ gðtnþ1ÞðgðddfÞ � nÞ rðu; pÞgðddfÞ

þl rurddf þ ðrddfÞTðruÞTh i

n;

on CinoutðdfÞ:

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

ð28Þ

Finally, using the expressions of the solid operator

derivatives provided in (20)–(23), we are able to identify

the structure of step 2(d) (ii). It consists in solving for the

perturbed solid state dz the following problem:

2

ðDtÞ2Z

Xs0

q0dz:vsdx0 þ

1

2

ZXs0

ðF ðdsÞdSðzÞ

þ rdzSðdsÞÞ : rvsdx0

¼ qDt

ZXf ð�df Þ

du �RðvsjCw0Þdx

þ qZ

Xf ð�df Þdiv½du� ðun wgðd

fÞ� �RðvsjCw0Þ

þZ

Xf ð �df Þrðdu; dpÞ : rRðvsjCw

0Þdx

þ qDt

ZXf ð �df ÞðdivddfÞ�u �RðvsjCw

0Þdx

þZ

Xf ð �df Þqdivf�u� ðun wgð �dfÞÞ½I divddf

ðrdd fÞT�g �RðvsjCw0Þdx

qDt

ZXf ð�df Þ

divð�u� ddfÞ �RðvsjCw0Þdx

þZ

Xf ð �df Þrð�u; �pÞ½I divddf ðrddfÞT� : rRðvsjCw

0Þdx

Z

Xf ð �df Þl½r�urddf þ ðrddfÞTðr�uÞT� : DRðvsjCw

0Þdx;

8vs 2 V s:

Let notice, that the right hand side of this problem is

nothing but the residual of the linearized fluid momen-

tum Eq. (28)1 involved in step 2(d) (i).

An approximate fluid tangent problem can be simply

derived from exact expression (28) by neglecting the

shape derivative terms (see [2,38,34,6,8]), yielding



¼ 0;

in XfðdfÞ;

divdu ¼ 0; in XfðdfÞ;

du ¼ ddf

Dt ; on CwðdfÞ;

rðdu; dpÞn ¼ 0; on CinoutðdfÞ;

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

ð29Þ

Still, by neglecting here the convective and diffusive

terms we get the following approximate fluid tangent

problem:

qDt duþrdp ¼ 0; in XfðdfÞ;

divdu ¼ 0; in XfðdfÞ;

du ¼ ddf

Dt ; on CwðdfÞ;

dpn ¼ 0; on CinoutðdfÞ;

8>>>>>>><>>>>>>>:that was proposed in [18] (see also [8,19]).

8. Numerical results

In this section we compare the exact Newton�s algo-rithm, whose jacobian evaluations have been described

in Section 7, with two former approaches:

• fixed-point iterations combined with Aitken�s acceler-ation techniques (FP-Aitken) [27,18],

• a quasi-Newton�s algorithm, whose approximate jac-

obians evaluations are carried out using the simpli-

fied fluid tangent problem (29), i.e., without taking

into account the shape derivatives of the fluid opera-

tor (see [2,38,34,6,8]).

As we shall see, in the sequel, the expected superiority

of the exact jacobian approach is validated by the

numerical experiments, mainly when using moderate

time steps.

We consider a fluid-structure problem arising in the

modelling of the blood flow in large arteries: it consists

of a thin elastic vessel conveying an incompressible vis-

cous fluid (see [7,6,30,16,18]). The solid is described by

the linear elasticity equations (Saint Venant–Kirchhof

material), while the fluid is described by the incompress-

ible Navier–Stokes equations. We consider two different

geometries:


(1) a straight cylinder of radius R0 = 0.5cm and length

L = 5cm,

(2) a curved cylinder of radius R0 = 0.5cm with curva-

ture ratio 0.25cm1.

The surrounding structure has a thickness of

h = 0.1cm. The physical parameters are the following

(see [16,18]):

Fig. 3. Pressure wave propaga

Fig. 4. Pressure wave propag

• Fluid: viscosity l = 0.03Poise, density q = 1g/cm3,

• Solid: density q0 = 1.2g/cm3, Young modulus

E = 3 · 106dynes/cm2 and Poisson ratio m = 0.3.

Both systems, the fluid and the structure, are initially

at rest. The structure is clamped at the inlet and the out-

let. An over pressure of 1.3332 · 104 dynes/cm2

(10mmHg) is imposed on the inlet boundary Cin during

tion (straight cylinder).

ation (curved cylinder).

136 M. �A. Fern�andez, M. Moubachir / Compute

3 · 103 s. The fluid equations are discretized using

P1bubble=P1 finite elements, whereas for the solid equa-

tions we use P1 finite elements.

A pressure wave propagation is observed in both

configurations. Figs. 3 and 4 show the fluid pressure at

time t = 0.0025, 0.005, 0.0075, 0.01s with time step

Dt = 104 s. These results are similar to those provided

Fig. 5. Solid domain deformed con

Fig. 6. Solid domain deformed con

in [18,16]. In Figs. 5 and 6 the corresponding solid de-

formed configurations (half section) are displayed.

Remark 4. The boundary data imposed on the inlet

and outlet boundaries [16,18,19] do not have any

physiological meaning. Let us notice that the typical

period of a heart beat is about 1 second [29]. Providing

rs and Structures 83 (2005) 127–142

figuration (straight cylinder).

figuration (curved cylinder).

0

5

10

15

20

25

30

35

40

45

50

0 0.005 0.01 0.015 0.02 0.025 0.03

num

ber

of it

erat

ions

time

Newtonquasi-Newton

FP-Aitken

Fig. 7. Straight cylinder Dt = 104 s.

0

10

20

30

40

50

60

0 0.005 0.01 0.015 0.02 0.025 0.03

num

ber

of it

erat

ions

time

Newtonquasi-Newton

FP-Aitken

Fig. 8. Curved cylinder Dt = 104 s.

Table 1

Dimensionless CPU time (straight cylinder)

Algorithm CPU time

FP-Aitken 1.00

Quasi-Newton 0.56

Newton 0.52

Table 2

Dimensionless CPU time (curved cylinder)

Algorithm CPU time

FP-Aitken 1.00

Quasi-Newton 0.55

Newton 0.60


realistic physiological simulations of the interaction

between the blood and the arterial wall, lies outside

the scope of this paper. This will be the purpose of a

forthcoming work. In such cases, the use of moderate

time steps will be crucial to perform over several cardiac

beats.


For comparison purposes, we give in Figs. 7 and 8

the number of iterations of either method at each time

level, for Dt = 104 s. The superior convergence proper-

ties of both Newton�s algorithms are evident. However,

the cost of each Newton iteration is higher than the cost

of a fixed-point iteration. Therefore, the previous prom-

ising convergence behavior does not necessarily imply an

overall reduction of the computational cost. Hence, in

order to compare the computational cost of these algo-

0

5

10

15

20

25

0 0.005 0.01 0.015

num

ber

of it

erat

ions

tim

Fig. 9. Straight cylin

0

5

10

15

20

25

0 0.005 0.01 0.01

num

ber

of it

erat

ions

ti

Fig. 10. Curved cylin

rithms, we furnish in Tables 1 and 2 the elapsed CPU

time (dimensionless) for either algorithm after 300 time

steps of length Dt = 104 s. We can notice that both

Newton�s algorithms are (in our implementation) about

2 times faster than the fixed-point algorithm. This com-

putational gain can be also recovered (from Figs. 7 and

8) by simply estimating the cost of each Newton itera-

tion in terms of the number of fluid and solid solvers

calls. Each Newton iteration requires:

0.02 0.025 0.03

e

Newtonquasi-Newton

FP-Aitken

der Dt = 103 s.

5 0.02 0.025 0.03

me

Newtonquasi-Newton

FP-Aitken

der Dt = 103 s.


• a fluid solver evaluation,

• a solid solver evaluation,

• about 9 inner GMRES iterations (with a tolerance of

104).

At this point, it is important to remark that each call

of the fluid solver updates the fluid mesh and the corre-

sponding FE fluid matrices, while the tangent fluid sol-

ver keeps the same FE matrix during the GMRES

process. As a result, and since we are dealing with linear

solvers, the cost of each Newton iteration is approxi-

mately 7 times the cost of a fluid solver evaluation, plus

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0 20 40

resi

dual

numbe

Fig. 11. Straight cylin

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0 20 40

resi

dual

numbe

Fig. 12. Curved cylin

10 times the cost of a solid solver evaluation. This ex-

plains the observed overall computational cost reduc-

tion. Obviously, this performance rises much more

when the fluid and solid solvers are nonlinear.

Now, let us address in detail the comparison between

the exact Newton and quasi-Newton algorithms. Obvi-

ously, the GMRES iterations in the exact version are

more expensive than in the quasi version. Indeed, the ex-

act Newton�s algorithm involves the computation of a

shape derivative which is required by the fluid tangent

problem (28). However, this is usually compensated by

a reduced number of Newton iterations. For instance,

60 80 100

r of iterations

Newtonquasi-Newton

der Dt = 103 s.

60 80 100

r of iterations

Newtonquasi-Newton

der Dt = 103 s.


Figs. 7 and 8 and Tables 1 and 2 show that, for

Dt = 104 s, these algorithms exhibit a comparable

behavior.

The superiority of the exact Newton�s algorithm can

be simply illustrated by increasing the time step. Indeed,

in Figs. 9 and 10, we specify the number of iterations of

either method at each time level, with time step

Dt = 103 s. The fixed-point and quasi-Newton algo-

rithms fail to converge after two time steps: the allowed

maximum number of iterations is in both cases reached,

whereas the exact Newton�s method converges and re-

quires a low (almost constant) number of iterations.

This illustrates the expected good convergence proper-

ties of the Newton�s method using exact jacobians eval-

uations. Figs. 11 and 12 show the evolution of the

residual during the iteration process in both Newton�salgorithms at the third time step. We may observe that

the quasi-Newton�s algorithm is unable to reduce the

residual after 100 iterations, while the exact Newton

only requires 3 iterations to reach the convergence

threshold.

Let us notice that the initial guess used in our compu-

tations, for starting each Newton�s loop, is based on a

extrapolation of the displacement of the previous time

steps (see, for instance, [24,30,18]). Consequently, larger

time steps lead to worse initial guess and then to a higher

number of iterations. The variations of the fluid domain

become thus more relevant and, therefore, the shape

derivatives (18) cannot be neglected in (28).

9. Conclusion

Using Newton based methods for solving the nonlin-

ear coupled systems arising in the numerical approxima-

tion of fluid-structure interaction problems, requires

evaluations of the jacobian associated to coupled non-

linear problem. Up to now, these evaluations were

approximated either using finite differences [25,26,

34,21], or by introducing simplified expressions [34,18,

6,8,19]. In any case, with a loss of some of the conver-

gence features of the Newton�s method. In this article,

we have proposed a new strategy consisting in imple-

menting linearized solvers in order to evaluate, in a con-

sistent way, the different jacobians involved in the

Newton�s algorithm. The main contribution of this

paper consists in establishing the expressions of these

jacobians, in the case of a coupled system consisting of

an incompressible Newtonian fluid interacting with a

nonlinear elastic solid under large displacements. On

the numerical view point, the Newton�s method has been

implemented in a 3D research code [12] in the case of a

linear elastic solid. Performing several numerical exper-

iments allowed us to show how the exactness level of

the jacobian evaluations influences drastically the con-

vergence properties of the Newton�s loop.

There exist several ways of improving the results pre-

sented in this paper. First we shall move from a linear to

a nonlinear elastic solid. Then we may address the tech-

nical point consisting in preconditioning the GMRES

iterations during the Newton�s loop [21,8]. We shall also

investigate the application of our approach in the case of

discretisations involving space-time finite elements [34].

Our underlying motivation is, in any case, to be able

to provide fast and reliable simulations of the mechani-

cal interaction between the blood and the arterial wall

[16,30,7,6,18,19], over several cardiac beats.

Acknowledgments

First author was supported by the Swiss National

Science Foundation (contract number 20-65110.01).

Second author was supported by the Semester Program

‘‘Mathematical Modelling of the Cardiovascular Sys-

tem’’ of the Bernoulli Center at the EPFL. This work

was partially supported by the Research Training Net-

work ‘‘Mathematical Modelling of the Cardiovascular

System’’ (Hae Model), contract HPRN-CT-2002-00270

of the European Community.

Appendix A. Shape derivative calculus

In this section we shall give some elements of the

shape derivative calculus used in Section 6 (see also

[13]), for establishing the expressions of the cross-deriv-

ative (18) of the fluid operator.

We consider a smooth domain X0 2 R3 and a one-to-

one mapping x 2 k;1 where

k;1 ¼def fxjx I 2 W k;1ðR3;R3Þ;x1 I 2 W k;1ðR3;R3Þg

with kP 1.

We introduce the transported domain X(x) = x(X0).

We introduce a scalar parameter s P 0 and a perturba-

tion mapping dx 2 sk,1 and we set Fs = $(x + sdx)and Js = det Fs. The following identities are easy to

verify:

Dx½J �dx ¼defd

dsJ sjs¼0 ¼ J divdx;

Dx½F 1�dx ¼ F 1rdx:

We introduce the unit outward normal to

CðxÞ ¼def oXðxÞ by

nðxÞ ¼ F Tn0jjF Tn0jj

;

where n0 is unit outward normal to C0 ¼def oX0. We have

the following identities

M. �A. Fern�andez, M. Moubachir / Compute

Dx

ZCðxÞ

da

" #dx ¼

ZCðxÞ

gðdxÞ � n da;

Dx

ZCðxÞ

n da

" #dx ¼

ZCðxÞ

gðdxÞ � da;

with gðdxÞ ¼def ½Idiv dx ðrdxÞT�n. We also have the fol-

lowing linearized Piola identity:

div½Idivx ðrxÞT� ¼ 0: ðA:1Þ

Now we consider a function uðxÞ : XðxÞ ! R3

depending on the mapping x. We set uðsÞ ¼defuðxþ sxÞ : Xðxþ sdxÞ ! R3:

Definition 1. We say that the function u(x) admits a

shape material derivative _uðx; dxÞ in the direction dx if

the following limit exists,

_uðx; dxÞ ¼def lims!0

uðsÞ � ðxþ sdxÞ uðxÞ � xs

:

In the sequel we shall need the following result,

whose proof can be found in [33, Section 2.31]

Dx

ZXðxÞ

u � vdx" #

dx ¼Z

XðxÞ½ _uðx; dxÞ � vþ u � _vðx; dxÞ

þ ðdivdxÞu � v�dx: ðA:2Þ

Let U(X0) be a given abstract functional space of

functions defined on X0, we define its transported image

through x as follows,

UðXðxÞÞ ¼def fu � x1; u 2 UðX0Þg:

Then we have the following result:

Lemma 2. For ((u, p), (v, q)) 2 U(X(x)) · V(X(x)) the

following identities hold true

Dx

ZXðxÞ

u � vdx" #

dx ¼Z

XðxÞðdivdxÞu � vdx; ðA:3Þ

Dx

ZXðxÞ

qdivudx

" #dx ¼

ZXðxÞ

qdivfu½Idivdx

ðrdxÞT�gdx; ðA:4Þ

Dx

ZXðxÞ

rðu; pÞ : rvdx

" #dx

¼Z

XðxÞrðu; pÞ½Idivdx ðrdxÞT� : rvdx

Z

XðxÞl½rurdxþ ðrdxÞTðruÞT� : rvdx: ðA:5Þ

Proof. In the special case where ((u,p), (v,q)) 2
U(X(x)) · V(X(x)), obviously we have
_uðx; dxÞ; _pðx; dxÞð Þ; _vðx; dxÞ; _qðx; dxÞð Þ ¼ ð0; 0Þ; ð0; 0Þð Þ;

then, using (A.2), we get

Dx

ZXðxÞ

u � vdx" #

dx ¼Z

XðxÞ½ðdivdxÞu � v�dx:

The next results follows using the following identities,ZXðxÞ

qdivudx ¼Z

X0

qdivfJ uF Tgdx0;

ZXðxÞru : rvdx ¼

ZX0

JruF 1 : rvF 1dx0;

and the definition r(u,p) = pI + l[$u + ($u)T]. h

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