1687-2770-2013-2461687-2770Research <p>The Lagrange-Galerkin method for fluid-structure interaction problems</p> San MartínJorgejorge@dim.uchile.cl ScheidJean-Françoisscheid@iecn.u-nancy.fr SmarandaLoredanasmaranda@dim.uchile.cl

Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile and Centro de Modelamiento Matemático, UMR 2071 CNRS-UChile, Casilla 170/3-Correo 3, Santiago, Chile

Institut Elie Cartan de Nancy UMR 7502, Université de Lorraine, CNRS, INRIA, B.P. 239, 54506, Vandoeuvre-lès-Nancy Cedex, France

Department of Mathematics and Computer Science, Faculty of Mathematics and Computer Science, University of Piteşti, Str. Târgu din Vale nr. 1, Piteşti, 110040, Romania

Boundary Value Problems1687-2770201320131246http://www.boundaryvalueproblems.com/content/2013/1/24610.1186/1687-2770-2013-246
30620131092013191120132013San Martín et al.; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider a Lagrange-Galerkin scheme to approximate a two-dimensional fluid-structure interaction problem. The equations of the system are the Navier-Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the solid. We are interested in studying numerical schemes based on the use of the characteristics method for rigid and deformable solids. The schemes are based on a global weak formulation involving only terms defined on the whole fluid-solid domain. Convergence results are stated for both semi and fully discrete schemes. This article reviews known results for rigid solid along with some new results on deformable structure yet to be published.

1 Introduction

In this article, we present a modified characteristics method for the discretization of the equations modelling the motion of a solid immersed in a cavity filled by a viscous incompressible fluid. We are interested in rigid and deformable solids modelling some particulate flows in the case of rigid solid and the swimming of slender, neutrally buoyant fish, for the deformable structure (see 1). The presented methods are generalizations of the numerical scheme introduced in 2, where the solid immersed in the fluid is rigid and has the same density with the fluid.

The fluid-structure interaction problem that we study is characterized by the strong coupling between the nonlinear equations of the fluid and those of the structure, as well as the fact that the equations of the fluid are written in a variable domain in time, which depends on the displacement of the structure. From the numerical point of view, in this kind of problems it is necessary to solve equations on moving domains. For this reason, in recent years various authors have proposed a number of different techniques 3456789.

For the numerical treatment of convection term in the Navier-Stokes equations, we discretize the material derivative along trajectories (see 10) combined with the Lagrange-Galerkin mixed finite element approximation of Navier-Stokes equations in a velocity/pressure formulation studied in 11. In 12, the convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations is done.

The numerical analysis of some time decoupling algorithms in the case, where the deformation of the structure induces an evolution in the fluid domain has been developed in 13 (one-dimensional problem). For the ALE method applied to interaction problems, we may cite 14 in the case of the unsteady Stokes equations in a time dependent domain and 15 for a two-dimensional problem describing the motion of a rigid body in a viscous fluid. In 216, the authors have introduced a convergent numerical method based on finite elements with a fixed mesh for a two-dimensional fluid-rigid body problem, where the densities of the fluid and the solid are equal. In 1718, we have introduced crucial modifications on the characteristic function, and we have proposed a convergent numerical scheme for a two dimensional fluid-rigid body problem where the densities of the fluid and the solid are different. In this paper, we go further, and we present a new characteristic function which gives us convergent algorithms for the simulation of aquatic organisms (for the existence and regularity of the solution in this kind of interactions, see 19).

2 Setting of the problem

2.1 Notation and hypothesis

Let us now introduce some notation following paper 19, where the existence and uniqueness for the solution of similar problem are treated. We denote by S0 the domain occupied by the solid in a reference configuration. We assume that S0 is an open connected set with C boundary, and we choose a system of coordinates with the origin at the mass center of S0. For the deformable solid, we suppose that the motion is given by a smooth mapping

X:S0×[0,)R2,

which satisfies

X(y,t)=ξ(t)+Rθ(t)X(y,t)yS0,t0,

where for every t0, ξ(t) is the trajectory of the mass center, and θ(t) represents the angle giving the orientation of the solid. Rθ denotes the matrix associated to the rotation of angle θ. In (1), X denotes an appropriate smooth mapping, representing the undulatory deformation of the creature. The rigid solid is obtained by considering the map

X=I,

where I denotes the identity map.

Throughout this paper, the deformable body will be called the creature or sometimes just the body in particular, when considering the rigid body case.

In the remaining part of this work, the functions ξ, θ are unknowns to be determined from the governing equations below, whereas the undulatory motion X will be supposed to be known and to satisfy several assumptions as in 19, which will be recalled in the sequel.

(H1) For every t0, the mapping yX(y,t) is a C diffeomorphism from S0¯ onto S¯(t), where S(t)=X(S0,t). Moreover, for every yS0, the mapping tX(y,t) is of class C and X(y,0)=y.

For every t0 we denote by Y the inverse of X, i.e., the diffeomorphism satisfying

X(Y(x,t),t)=x,Y(X(y,t),t)=y

for every xS(t), t0 and yS0.

(H2) The total volume of the creature is preserved, i.e.,

S(t)dx=S0dyt0.

Denote by w the undulatory velocity of creature, written as a vector field on S(t), i.e.,

w(x,t)=Xt(y,t)|y=Y(x,t)xS(t),t0.

Let ρS,0 be the density field of the solid in the reference configuration S0, and let ρS(t) be the density field of S(t). The mass conservation principle applied to the whole body gives

S(t)ρS(x,t)dx=S0ρS,0(y)dyt0,

whereas the local form of the conservation of mass yields

ρS(x,t)=ρS,0(Y(x,t))det(X)(Y(x,t))t0,xS(t),

where X stands for the Jacobian matrix of X(,t).

(H3) S(t)ρS(x,t)w(x,t)dx=0 for all t0.

(H4) S(t)ρS(x,t)xw(x,t)dx=0 for all t0, where we denote by x the vector x=(x2x1) for x=(x1x2).

Conditions (H3), (H4) correspond to the so-called self-propelling conditions which are natural requirements for understanding swimming viewed as a self-propelled phenomena.

In particular, hypotheses (H1) and (H3) imply that the position of the center of mass of the creature is not affected by the undulatory motion, that is,

S0ρS,0(y)X(y,t)dy=0t0.

From (1), it follows that the region occupied by the creature at time t is given by

S(ξ(t),θ(t),t)=Rθ(t)S(t)+ξ(t)t0.

Moreover, by differentiating equation (1) with respect to t, it follows that the Eulerian velocity field of the solid is given for every t0 by

uS(x,t)=ξ(t)+θ(t)(xξ(t))+w(x,t)xS(ξ(t),θ(t),t),

where

w(x,t)=Rθ(t)w(Rθ(t)(xξ(t)),t)xS(ξ(t),θ(t),t).

The Eulerian density field of the body is given by

ρS(x,t)=ρS(Rθ(t)(xξ(t)),t)t0,xS(ξ(t),θ(t),t),

with ρS given by (5). The mass M of the body and its moment of inertia with respect to an axis orthogonal to the plane of the motion and passing by the mass center of S(ξ(t),θ(t),t), are as usually given by

M=S(ξ(t),θ(t),t)ρS(x,t)dx.

I(t)=S(ξ(t),θ(t),t)ρS(x,t)|xξ(t)|2dx.

Let us notice that from (4), (10) and (11), we have that

M=S0ρS,0(y)dy.

Remark 2.1 In the case X=I (rigid solid), all hypotheses (H1)-(H4) are satisfied, and the undulatory velocity field w is equal to zero.

2.2 Equations

Let Ω be an open bounded set in R2 representing the domain occupied by the solid-fluid system. Recalling that S(ξ(t),θ(t),t) is the domain occupied by the solid at instant t, we have that the fluid fills, at instant t, the domain F(ξ(t),θ(t),t)=ΩS(ξ(t),θ(t),t)¯.

With the notation above, the full system describing the self-propelled motion of the creature can be written as

ρF(ut+(u)u)μΔu+p=ρFf,xF(ξ(t),θ(t),t),t(0,T),

divu=0,xF(ξ(t),θ(t),t),t(0,T),

u=0,xΩ,t(0,T),

u(x,t)=ξ(t)+θ(t)(xξ(t))+w(x,t),xS(ξ(t),θ(t),t),t(0,T),

Mξ(t)=S(ξ(t),θ(t),t)σndΓ+S(ξ(t),θ(t),t)ρS(x,t)f(x,t)dx,t(0,T),

(Iθ)(t)=S(ξ(t),θ(t),t)(xξ(t))σndΓ(Iθ)(t)=+S(ξ(t),θ(t),t)ρS(x,t)(xξ(t))f(x,t)dx,t(0,T).

In the system above, ρF>0 and μ>0 stand for the density and the viscosity of the fluid, which are supposed to be constant, u is the Eulerian velocity field of the fluid, and p denotes the pressure field of the fluid. A prime stands for the derivation operator with respect to time. By using the classical notation

D(u)=12((u)+(u)T),

the stress tensor field σ is defined by

σ(u,p)=2μD(u)pId,

where Id is the identity matrix in M2(R). Moreover, for t[0,T] and xS(ξ(t),θ(t),t) we denote by n(x,t) the unit normal to S(ξ(t),θ(t),t) oriented towards the solid. Recall that the mass M and the moment of inertia I(t) of the solid at instant t are defined by (11) and (12).

System (13)-(20) is completed by the initial conditions

u(x,0)=u0(x),xF(ξ(0),θ(0),0),

ξ(0)=ξ0,θ(0)=θ0,ξ(0)=ξ1,θ(0)=ω0.

Remark 2.2 In the case of rigid solid, equation (16) becomes

u(x,t)=ξ(t)+θ(t)(xξ(t)),xS(ξ(t),θ(t),t),t[0,T],

because the undulatory velocity field w is equal to zero.

2.3 Weak formulation

Let ξH2((0,T);R2), θH2((0,T);R) be two functions such that S(ξ(t),θ(t),t)¯Ω for all t[0,T]. In the sequel, we define F=F(ξ(0),θ(0),0) and S=S(ξ(0),θ(0),0). Moreover, if no confusion is possible, we define

S(t)=S(ξ(t),θ(t),t),F(t)=F(ξ(t),θ(t),t)=ΩS(t)¯.

Let Ψ:R2×[0,T]R2 be a mapping such that for every t[0,T], the function Ψ(,t) is a C-diffeomorphism from ℱ onto F(ξ(t),θ(t),t) and such that the derivatives

i+k1+k2Ψtiy1k1y2k2,i1,k10,k20

exist and are continuous. The existence of such a function is due, in particular, to the fact that dist(S(t),Ω)>0 for all t (see 19). We can now define the following functions spaces:

L2(0,T;H2(F(t))2)={uuΨL2(0,T;H2(F)2)},H1(0,T;L2(F(t))2)={uuΨH1(0,T;L2(F)2)},C([0,T];H1(F(t))2)={uuΨC([0,T];H1(F)2)},L2(0,T;H1(F(t)))={ppΨL2(0,T;H1(F))},

where vΨ denotes the function defined by vΨ(y,t)=v(Ψ(y,t),t) for (y,t)F×(0,T).

In order to introduce the weak formulation, we first define some additional functions spaces. For every t0, let (ξ,θ) be an arbitrary position of the creature at time t, such that S(ξ,θ,t)¯Ω. We denote

K(ξ,θ,t)={uH01(Ω)2:D(u)=0 in S(ξ,θ,t)},

M(ξ,θ,t)={pL2(Ω):Ωpdx=0 and p=0 in S(ξ,θ,t)},

where D(u) is the strain rate tensor defined by (19).

Let (u,p,ξ,θ) be a solution of (13)-(22). The vector velocity field u and the pressure p can be extended to Ω by setting

u(x,t)=ξ(t)+θ(t)(xξ(t))+w(x,t)if xS(ξ(t),θ(t),t),

p(x,t)=0if xS(ξ(t),θ(t),t).

The extended vector u(,t) belongs to H01(Ω)2. In the remaining part of this paper, the solution u and p of (13)-(22) will be extended as above.

We also need to extend the density field ρS of the creature (defined in (10)) to the whole domain Ω by setting

ρ(x,t)={ρFfor all xF(ξ(t),θ(t),t),ρS(x,t)for all xS(ξ(t),θ(t),t),t0.

By a slight variation of the argument in Ladyzhenskaya [20, p.27], it can be shown that for every δ>0, there exists a continuous function (x,t)Λ(x,t) such that, for every t0, the map xΛ(x,t) is C on R2S(t) and such that the function tΛ(x,t) is of class C for every xR2S(t)¯ and

{divΛ=0in R2S(t)¯,t(0,T),Λ(x,t)=0if dist(x,S(t))δ>0,t(0,T),Λ(x,t)=w(x,t)if xS(t),t(0,T).

For every t0, let (ξ,θ) be an arbitrary position of the creature at time t, such that S(ξ,θ,t)¯Ω . We then define Λ(,t;ξ,θ) by

Λ(x,t;ξ,θ)=RθΛ(Rθ(xξ),t)xR2.

Then the function Λ satisfies

{divΛ=0in R2S(ξ,θ,t)¯,Λ=0on Ω,Λ=win S(ξ,θ,t).

An important ingredient of the numerical method we use is given by the characteristic functions whose level lines are the integral curves of the velocity field. More precisely (see, for instance, 1011) the characteristic function ψ:[0,T]2×ΩΩ is defined as the solution of the initial value problem

{ddtψ(t;s,x)=u(ψ(t;s,x),t)t[0,T],ψ(s;s,x)=x.

It is well known that the material derivative Dtu=u/t+(u)u of the velocity field u at instant t0 satisfies:

Dtu(x,t0)=ddt[u(ψ(t;t0,x),t)]|t=t0.

Remark 2.3 By using a classical result of Liouville (see, for instance, [21, p.251]), if

ξH2(0,T)2,θH2(0,T),uC([0,T];H01(Ω)2)

are such that for any t[0,T], we have S(ξ(t),θ(t),t)Ω and

divu=0in F(ξ(t),θ(t),t),

then we get

detJψ(t,s,x)=1xF(ξ(t),θ(t),t),

where we have denoted by

Jψ=(ψiyj)i,j

the Jacobian matrix of the transformation yψ(y).

In order to give the global weak formulation of our problem, we need to introduce the bilinear forms

a:H01(Ω)2×H01(Ω)2R,b:H01(Ω)2×L2(Ω)R,

defined by

a(u,v)=2μΩD(u):D(v)dxu,vH01(Ω)2,b(u,q)=Ω(divu)qdxuH01(Ω)2,qL2(Ω).

Proposition 2.4 Assume that

uL2(0,T;H2(F(t))2)H1(0,T;L2(F(t))2)C([0,T];H1(F(t))2),pL2(0,T;H1(F(t))),ξH2(0,T)2,θH2(0,T),

and that u and p are extended to Ω as above.

Then (u,p,ξ,θ) is the solution of (13)-(22) if and only if for all t[0,T], u(,t)Λ(,t;ξ(t),θ(t))K(ξ(t),θ(t),t), p(,t)M(ξ(t),θ(t),t), and (u,p) satisfies

(ρDtu,v)+a(u,v)+b(v,p)=(ρf(t),v)vK(ξ(t),θ(t),t),

b(u,q)=0qM(ξ(t),θ(t),t),

for a.e. t(0,T).

More details on the existence and uniqueness of the solution and the complete proof of this result could be found in 19.

In the remainder of the paper, we suppose that f and u0 satisfy

fC([0,T];H1(Ω)2),u0H2(F)2,div(u0)=0in F,u0=0on Ω,u0(x)=ξ1+ω0(xξ0)+Rθ0w(Rθ0(xξ0),0)for xS,

where ξ0,ξ1R2 and θ0, ω0R are given as initial data in (22). Let us also assume that the corresponding solution (u,p,ξ,θ) of problem (13)-(22) satisfies the following regularity properties:

{uC([0,T];H2(F(t))2)H1(0,T;L2(F(t))2),Dt2uL2(0,T;L2(F(t))2),uC([0,T];C0,1(Ω¯)2),pC([0,T];H1(F(t))),ξH3(0,T)2,θH3(0,T).

Moreover, we suppose that there exists a nonempty open connected subset Ω0 of Ω such that for any ξ1,ξ2Ω0, we have

S(ξ1+λ(ξ2ξ1),θ,t)Ωλ[0,1],θ[0,2π],t[0,T].

Using this notation, we assume that

ξ(t)Ω0,dist(S(t),Ω)>0t[0,T].

Remark 2.5 The hypotheses (37) and (39) imply the existence of η>0 such that

dist(ξ(t),Ω0)>3η,dist(S(t),Ω)>3ηt[0,T].

3 Time discretization and first main result

In this section, based on a weak form of the governing equations, we describe a method for the time discretization of (13)-(22).

Let us first divide the time interval [0,T] into subintervals [tk,tk+1] with tk+1tk=Δt=TN, where N is a positive integer and k{1,,N}. Let (uk,pk,ξk,θk) be the approximation of the solution of (34)-(35) at time t=tk (remark that uk and pk are functions defined on the whole domain Ω). We denote

Kk=K(ξk,θk,tk),Mk=M(ξk,θk,tk),Sk=S(ξk,θk,tk),Fk=F(ξk,θk,tk),

and we consider the functions

Λk(x)=RθkΛ(Rθk(xξk),tk)xR2,ρk(x)={ρS(Rθk(xξk),tk)if xSk,ρFif xFk.

Remark 3.1 Combining the regularity properties of Λ and w, it follows that

ΛkC0,1(Ω)2.

Moreover, taking δ<η in definition (28) of Λ, where η is defined in (40), we have that Λk+KkH01(Ω)2.

Now, let us describe the numerical scheme for approximating the solutions of (13)-(22). This procedure is based on the weak form derived in Proposition 2.4.

The first step of our scheme consists in computing the new position of the mass center and the new orientation of the creature by setting

ξk+1=ξk+Δtuk(ξk),

θk+1=θk+ΔtI(tk)Skρk(uk(x)uk(ξk))(xξk)dx.

The second step consists in computing the global velocity field uk+1 and the global pressure field pk+1. To this end, we look for uk+1Λk+1+Kk+1 and pk+1Mk+1 such that for all vKk+1, and for all qMk+1, we have

(ρk+1uk+1ukX¯ukkΔt,v)+a(uk+1,v)+b(v,pk+1)=(ρk+1fk+1,v),

b(uk+1,q)=0,

where fk+1(x)=f(x,tk+1) for any xΩ.

In the equations above, the approximate characteristic is given by

X¯ukk(x)=χk(tk;tk+1,x),

for all xΩ, where χk is the solution of the problem

{ddtχk(t;tk+1,x)=uk˜(χk(t;tk+1,x)),t[tk,tk+1],χk(tk+1;tk+1,x)=ξk+RθkΠ(tk;tk+1,Rθk+1(xξk+1)),

with

uk˜(z)=uk(z)uk(ξk)θk+1θkΔt(zξk)Λk(z)zR2,

where uk(z) is extended by zero outside of Ω.

For all s[0,T], and zR2, function Π(;s,z) corresponds to the characteristic function of the extended undulatory velocity Λ, defined by

{ddtΠ(t;s,z)=Λ(Π(t;s,z),t),t[0,T],Π(s;s,z)=z.

Remark 3.2 Let us note that for any zS(tk+1), equation (50) has the explicit solution

Π(t;tk+1,z)=X(Y(z,tk+1),t)S(t).

Then for any xS(ξk+1,θk+1,tk+1), since

Rθk+1(xξk+1)S(tk+1),

we obtain that the initial condition in (49) is

χk(tk+1;tk+1,x)=ξk+RθkX(Y(Rθk+1(xξk+1),tk+1),tk).

Moreover, since z=χk(tk+1;tk+1,x)S(ξk,θk,tk), we have uk˜(z)=0 and

χk(t;tk+1,x)=ξk+RθkX(Y(Rθk+1(xξk+1),tk+1),tk)t[tk,tk+1].

In particular, for t=tk we have that

X¯ukk(x)=ξk+RθkX(Y(Rθk+1(xξk+1),tk+1),tk)

for all xS(ξk+1,θk+1,tk+1).

It is easy to see that for any k{1,,N}, equations (46)-(47) represent a mixed formulation of a well-posed Navier-Stokes-type system, so that our scheme is well defined.

Let us now state our first main result concerning the convergence of the semi-discrete scheme (46)-(47).

Theorem 3.3 Suppose that Ω is an open smooth bounded domain in R2, f and u0 satisfy (36), and the exact solution (u,p,ξ,θ) of problem (13)-(22) satisfies hypotheses (37)-(39). Then there exist a constant K>0 depending on T and a constant δ>0 independent of T such that for all 0<Δtδ, the solution (uk,pk,ξk,θk)k{1,,N} of the time-discretization problem (46)-(47) satisfies

sup1kN(u(tk)ukL2(Ω)2+|ξ(tk)ξk|+|θ(tk)θk|)KΔt.

The complete proof of this result could be found in 18 for the case of rigid solid and in the forthcoming paper 22 for the deformable structure.

4 Fully discrete formulation and second main result

In order to discretize problem (46)-(47) with respect to the space variable, we introduce two families of finite element spaces which approximate spaces Kk and Mk defined in (41), (23) and (24). To this end, for any discretization parameter h(0,1), we consider a quasi-uniform triangulation Th of the domain Ω. Suppose that Ω is a bounded convex domain with a polygonal boundary. We denote by Wh the P1+bubble finite elements space associated with Th for the velocity field and by Eh the P1-finite elements space for the pressure, that is,

Wh={vC(Ω¯)2:TTh,v|TP1+bubble(T)},Eh={qC(Ω¯):TTh,q|TP1}.

Then, we define the following finite elements spaces for a conform approximation of the fluid-solid system:

Kh(ξ,θ,t)=WhK(ξ,θ,t)ξΩ0,θ[0,2π],t[0,T],Mh(ξ,θ,t)=EhM(ξ,θ,t)ξΩ0,θ[0,2π],t[0,T].

Let us recall an approximation property of the projection on Kh(ξ,θ,t)×Mh(ξ,θ,t) (see 2).

Lemma 4.1 Suppose that VK(ξ,θ,t) and PM(ξ,θ,t). Then there exists a unique couple (V¯h,P¯h) in Kh(ξ,θ,t)×Mh(ξ,θ,t) such that

{a(VV¯h,v)+b(v,PP¯h)=0vKh(ξ,θ,t),b(VV¯h,q)=0qMh(ξ,θ,t).

In addition, if we suppose that V|ΩS(ξ,θ,t)H2(ΩS(ξ,θ,t))2 and P|ΩS(ξ,θ,t)H1(ΩS(ξ,θ,t)), then there exists a positive constant C, independent of h, such that

VV¯hL2(Ω)2Ch.

In order to define the approximate characteristics, let us denote by Fh the P2-finite elements space associated with the triangulation Th, and we introduce the space

Rh(ξ,θ,t)={ϕh:ϕhFh,ϕh=0 on Ω}K(ξ,θ,t),

ξΩ0, θ[0,2π], t[0,T], where ϕh=(ϕhyϕhx).

We denote by P(ξ,θ,t) the orthogonal projection from L2(Ω)2 onto Rh(ξ,θ,t), i.e., for any uL2(Ω)2, then the projection P(ξ,θ,t)uRh(ξ,θ,t) is such that (uP(ξ,θ,t)u,rh)=0 for all rhRh(ξ,θ,t).

Let N be a positive integer. We denote Δt=T/N and tk=kΔt for all k{0,,N}. For k=0, we define

uh0()=u(,0)¯,ξh0=ξ0andθh0=θ0,

where (u(,0)¯,p(,0)¯)Kh(ξ0,θ0,0)×Mh(ξ0,θ0,0) is the projection of the initial condition (u(,0),p(,0)) on Kh(ξ0,θ0,0)×Mh(ξ0,θ0,0) defined in (54).

Assume that the approximate solution (uhk,phk,ξhk,θhk) of (13)-(18) at time t=tk is known. We describe below the numerical scheme allowing to determinate the approximate solution (uhk+1,phk+1,ξhk+1,θhk+1) at t=tk+1. First, we compute ξhk+1R2 and θhk+1R by

ξhk+1=ξhk+uhk(ξhk)Δt,

θhk+1=θhk+ΔtI(tk)Skρhk(uhk(x)uhk(ξhk))(xξhk)dx,

where ρhk is defined by the identity (60) below.

We consider the approximated characteristic function χhk defined as the solution of the following ordinary differential equation:

{ddtχhk(t;tk+1,x)=P(ξhk,θhk,tk)uhk˜(χhk(t;tk+1,x)),t[tk,tk+1],χhk(tk+1;tk+1,x)=ξhk+RθhkΠ(tk;tk+1,Rθhk+1(xξhk+1)),

with

uhk˜(z)=uhk(z)uhk(ξhk)θhk+1θhkΔt(zξhk)Λhk(z)zR2,

where uhk(z) is extended by zero outside of Ω and

Λhk(x)=RθhkΛ(Rθhk(xξhk),tk)xR2.

The characteristic function Π is defined by (50).

Finally, we define

X¯hk(x)=χhk(tk;tk+1,x)xΩ.

In the sequel, we shall split the mesh into the union of 4 different types of triangle subsets. We first introduce Ah as the union of all triangles intersecting the solid S(ξhk,θhk,tk), i.e.,

Ah=TThTS(ξhk,θhk,tk)T.

We also denote by Qh the union of all triangles such that all their vertices are contained in Ah¯. The triangles of Th are then split into the following four categories (see Figure 1):

F1 is the subset of Th formed by all triangles TTh such that T¯S(ξhk,θhk,tk).

F2 is the subset formed by all triangles TThF1 such that T¯Qh¯.

F3 is the subset formed by all triangles TTh such that T¯Qh¯ and TQh¯.

F4=Th(F1F2F3).

<p>Figure 1</p>

In this figure, we see the splitting of the fixed triangulation related to the position of the solid at time t.

In this figure, we see the splitting of the fixed triangulation related to the position of the solid at time t.

We introduce the approximated density function ρhk as follows:

ρhk(x)={ρS(Rθhk(xξhk),tk)if xS(ξhk,θhk,tk),ρFif xΩS(ξhk,θhk,tk).

With these notations, we introduce the following mixed variational fully discrete formulation: Find uhk+1Λhk+1+Kh(ξhk+1,θhk+1,tk+1), phk+1Mh(ξhk+1,θhk+1,tk+1) such that

(ρhk+1uhk+1uhkX¯hkΔt,v)+a(uhk+1,v)+b(v,phk+1)=(ρhk+1fhk+1,v)vKh(ξhk+1,θhk+1,tk+1),

b(uhk+1,q)=0qMh(ξhk+1,θhk+1,tk+1),

where fhk+1 is the L2(Ω)2-projection of fk+1=f(tk+1) on (Eh)2.

Let us now state the second main result of this paper, which asserts the convergence of the fully-discrete scheme (61)-(62). The complete proof of this result could be found in 18 for the case of rigid body and in the forthcoming paper 22 if the structure is deformable.

Theorem 4.2 Let Ω be a convex domain with a polygonal boundary. Suppose that f and u0 satisfy the conditions from (36), and that (u,p,ξ,θ) is a solution of (13)-(18) satisfying regularity properties (37)-(39). Let C0>0 and 0<α1 be two fixed constants. Then there exist two positive constants K and τ, independent of h and Δt such that for all 0<Δtτ and for all hC0Δt1+α, we have

sup1kN(u(tk)uhkL2(Ω)2+|ξ(tk)ξhk|+|θ(tk)θhk|)KΔtα.

Let us mention that in order to get an approximation of first order in time (i.e., O(Δt) in Theorem 4.2), we have to choose α=1. In this case, the corresponding condition on h becomes hC0Δt2 which is similar to the one obtained in [2, Theorem 3.2], where the densities of the fluid ρF and of the solid ρS are equal.

Remark 4.3 Let us give some comments on the condition of h and Δt required for the convergence result in Theorem 4.2. First, we emphasize that the same type of condition appears in several works for approximation in a Lagrangian framework of the Navier-Stokes equations without any rigid body. We may cite 10, where convergence is obtained under condition hC0Δt and 11, where h and Δt are chosen such that h2CΔtC1hσ and σ>1/2 (with h and Δt small enough). We also mention 14 for an ALE scheme applied to Stokes equations in a time-dependent domain, where the authors obtain an error estimate of order O(Δt) under condition hCΔt3/4.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have contributed equally to this research work. All authors read and approved the final manuscript.

Acknowledgements

San Martín was partially supported by the Grant Fondecyt 1090239 and BASAL-CMM Project. Scheid gratefully acknowledges the Program ECOS-CONICYT (Scientific cooperation project between France and Chile) through the grant C07-E05. He was also partially supported by the ‘Agence Nationale de la Recherche’ (ANR), the Project CISIFS, the grant ANR-09-BLAN-0213-02. Smaranda was supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-RU-TE-2011-3-0059.

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