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The Lagrange-Galerkin method for fluid-structure interaction problems
Boundary Value Problemsvolume 2013, Article number: 246 (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
Let us now introduce some notation following paper , where the existence and uniqueness for the solution of similar problem are treated. We denote by the domain occupied by the solid in a reference configuration. We assume that is an open connected set with boundary, and we choose a system of coordinates with the origin at the mass center of . For the deformable solid, we suppose that the motion is given by a smooth mapping
where for every , is the trajectory of the mass center, and represents the angle giving the orientation of the solid. denotes the matrix associated to the rotation of angle θ. In (1), denotes an appropriate smooth mapping, representing the undulatory deformation of the creature. The rigid solid is obtained by considering the map
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 we denote by the inverse of , i.e., the diffeomorphism satisfying
for every , and .
(H2) The total volume of the creature is preserved, i.e.,
Denote by the undulatory velocity of creature, written as a vector field on , i.e.,
Let be the density field of the solid in the reference configuration , and let be the density field of . The mass conservation principle applied to the whole body gives
whereas the local form of the conservation of mass yields
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.
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,
From (1), it follows that the region occupied by the creature at time t is given by
Moreover, by differentiating equation (1) with respect to t, it follows that the Eulerian velocity field of the solid is given for every by
The Eulerian density field of the body is given by
with 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 , are as usually given by
Let us notice that from (4), (10) and (11), we have that
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 .
With the notation above, the full system describing the self-propelled motion of the creature can be written as
In the system above, and 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
the stress tensor field σ is defined by
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).
System (13)-(20) is completed by the initial conditions
Remark 2.2 In the case of rigid solid, equation (16) becomes
because the undulatory velocity field w is equal to zero.
2.3 Weak formulation
Let , be two functions such that for all . In the sequel, we define and . Moreover, if no confusion is possible, we define
Let be a mapping such that for every , the function is a -diffeomorphism from ℱ onto and such that the derivatives
exist and are continuous. The existence of such a function is due, in particular, to the fact that for all t (see ). We can now define the following functions spaces:
where denotes the function defined by for .
In order to introduce the weak formulation, we first define some additional functions spaces. For every , let be an arbitrary position of the creature at time t, such that . We denote
where is the strain rate tensor defined by (19).
Let be a solution of (13)-(22). The vector velocity field u and the pressure p can be extended to Ω by setting
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.
We also need to extend the density field of the creature (defined in (10)) to the whole domain Ω by setting
By a slight variation of the argument in Ladyzhenskaya [, p.27], it can be shown that for every , there exists a continuous function such that, for every , the map is on and such that the function is of class for every and
For every , let be an arbitrary position of the creature at time t, such that . We then define by
Then the function Λ satisfies
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, [10, 11]) the characteristic function is defined as the solution of the initial value problem
It is well known that the material derivative of the velocity field u at instant satisfies:
Remark 2.3 By using a classical result of Liouville (see, for instance, [, p.251]), if
are such that for any , we have and
then we get
where we have denoted by
the Jacobian matrix of the transformation .
In order to give the global weak formulation of our problem, we need to introduce the bilinear forms
Proposition 2.4 Assume that
and that u and p are extended to Ω as above.
Then is the solution of (13)-(22) if and only if for all , , , and satisfies
for a.e. .
More details on the existence and uniqueness of the solution and the complete proof of this result could be found in .
In the remainder of the paper, we suppose that f and satisfy
where and , are given as initial data in (22). Let us also assume that the corresponding solution of problem (13)-(22) satisfies the following regularity properties:
Moreover, we suppose that there exists a nonempty open connected subset of Ω such that for any , we have
Using this notation, we assume that
Remark 2.5 The hypotheses (37) and (39) imply the existence of such that
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 into subintervals with , where N is a positive integer and . Let be the approximation of the solution of (34)-(35) at time (remark that and are functions defined on the whole domain Ω). We denote
and we consider the functions
Remark 3.1 Combining the regularity properties of and , it follows that
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.
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
The second step consists in computing the global velocity field and the global pressure field . To this end, we look for and such that for all , and for all , we have
where for any .
In the equations above, the approximate characteristic is given by
for all , where is the solution of the problem
where is extended by zero outside of Ω.
For all , and , function corresponds to the characteristic function of the extended undulatory velocity , defined by
Remark 3.2 Let us note that for any , equation (50) has the explicit solution
Then for any , since
we obtain that the initial condition in (49) is
Moreover, since , we have and
In particular, for we have that
for all .
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 , f and satisfy (36), and the exact solution of problem (13)-(22) satisfies hypotheses (37)-(39). Then there exist a constant depending on T and a constant independent of T such that for all , the solution of the time-discretization problem (46)-(47) satisfies
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 and defined in (41), (23) and (24). To this end, for any discretization parameter , we consider a quasi-uniform triangulation of the domain Ω. Suppose that Ω is a bounded convex domain with a polygonal boundary. We denote by the finite elements space associated with for the velocity field and by the -finite elements space for the pressure, that is,
Then, we define the following finite elements spaces for a conform approximation of the fluid-solid system:
Let us recall an approximation property of the projection on (see ).
Lemma 4.1 Suppose that and . Then there exists a unique couple in such that
In addition, if we suppose that and , then there exists a positive constant C, independent of h, such that
In order to define the approximate characteristics, let us denote by the -finite elements space associated with the triangulation , and we introduce the space
, , , where .
We denote by the orthogonal projection from onto , i.e., for any , then the projection is such that for all .
Let N be a positive integer. We denote and for all . For , we define
where is the projection of the initial condition on defined in (54).
Assume that the approximate solution of (13)-(18) at time is known. We describe below the numerical scheme allowing to determinate the approximate solution at . First, we compute and by
where is defined by the identity (60) below.
We consider the approximated characteristic function defined as the solution of the following ordinary differential equation:
where is extended by zero outside of Ω and
The characteristic function Π is defined by (50).
Finally, we define
In the sequel, we shall split the mesh into the union of 4 different types of triangle subsets. We first introduce as the union of all triangles intersecting the solid , i.e.,
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 .
We introduce the approximated density function as follows:
With these notations, we introduce the following mixed variational fully discrete formulation: Find , such that
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.
Theorem 4.2 Let Ω be a convex domain with a polygonal boundary. Suppose that f and satisfy the conditions from (36), and that is a solution of (13)-(18) satisfying regularity properties (37)-(39). Let and be two fixed constants. Then there exist two positive constants K and , independent of h and Δt such that for all and for all , we have
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 .
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San Martín, J, Scheid, JF, Smaranda, L: The Lagrange-Galerkin method for a fluid-deformable system (2013, submitted)
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.
The authors declare that they have no competing interests.
All authors have contributed equally to this research work. All authors read and approved the final manuscript.