The Lagrange-Galerkin method for fluid-structure interaction problems
© San Martín et al.; licensee Springer. 2013
Received: 30 June 2013
Accepted: 10 September 2013
Published: 19 November 2013
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.
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 ). The presented methods are generalizations of the numerical scheme introduced in , 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 [3–9].
For the numerical treatment of convection term in the Navier-Stokes equations, we discretize the material derivative along trajectories (see ) combined with the Lagrange-Galerkin mixed finite element approximation of Navier-Stokes equations in a velocity/pressure formulation studied in . In , 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  (one-dimensional problem). For the ALE method applied to interaction problems, we may cite  in the case of the unsteady Stokes equations in a time dependent domain and  for a two-dimensional problem describing the motion of a rigid body in a viscous fluid. In [2, 16], 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 [17, 18], 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 ).
2 Setting of the problem
2.1 Notation and hypothesis
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 will be supposed to be known and to satisfy several assumptions as in , which will be recalled in the sequel.
(H1) For every , the mapping is a diffeomorphism from onto , where . Moreover, for every , the mapping is of class and .
for every , and .
where stands for the Jacobian matrix of .
(H3) for all .
(H4) for all , where we denote by the vector for .
Conditions (H3), (H4) correspond to the so-called self-propelling conditions which are natural requirements for understanding swimming viewed as a self-propelled phenomena.
Remark 2.1 In the case (rigid solid), all hypotheses (H1)-(H4) are satisfied, and the undulatory velocity field is equal to zero.
Let Ω be an open bounded set in representing the domain occupied by the solid-fluid system. Recalling that is the domain occupied by the solid at instant t, we have that the fluid fills, at instant t, the domain .
where Id is the identity matrix in . Moreover, for and we denote by the unit normal to oriented towards the solid. Recall that the mass M and the moment of inertia of the solid at instant t are defined by (11) and (12).
because the undulatory velocity field w is equal to zero.
2.3 Weak formulation
where denotes the function defined by for .
where is the strain rate tensor defined by (19).
The extended vector belongs to . In the remaining part of this paper, the solution u and p of (13)-(22) will be extended as above.
the Jacobian matrix of the transformation .
and that u and p are extended to Ω as above.
for a.e. .
More details on the existence and uniqueness of the solution and the complete proof of this result could be found in .
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).
Moreover, taking in definition (28) of , where η is defined in (40), we have that .
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.
where for any .
where is extended by zero outside of Ω.
for all .
Let us now state our first main result concerning the convergence of the semi-discrete scheme (46)-(47).
4 Fully discrete formulation and second main result
Let us recall an approximation property of the projection on (see ).
, , , where .
We denote by the orthogonal projection from onto , i.e., for any , then the projection is such that for all .
where is the projection of the initial condition on defined in (54).
where is defined by the identity (60) below.
The characteristic function Π is defined by (50).
We also denote by the union of all triangles such that all their vertices are contained in . The triangles of are then split into the following four categories (see Figure 1):
is the subset of formed by all triangles such that .
is the subset formed by all triangles such that .
is the subset formed by all triangles such that and .
where is the -projection of on .
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  for the case of rigid body and in the forthcoming paper  if the structure is deformable.
Let us mention that in order to get an approximation of first order in time (i.e., in Theorem 4.2), we have to choose . In this case, the corresponding condition on h becomes which is similar to the one obtained in [, Theorem 3.2], where the densities of the fluid and of the solid 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 , where convergence is obtained under condition and , where h and Δt are chosen such that and (with h and Δt small enough). We also mention  for an ALE scheme applied to Stokes equations in a time-dependent domain, where the authors obtain an error estimate of order under condition .
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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