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Introduction

Applied mathematician

## Publications

Publications (63)

This chapter introduces geometric notations and concepts related to curves and surfaces moving in a flow. It defines level set methods as alternatives to Lagrangian methods for the implicit tracking of these surfaces, with illustrations from classical examples in image processing and fluid mechanics. Finally, it discusses stability issues related t...

This chapter describes penalization methods to describe rigid solids interacting with an incompressible fluid. Extensions to the case of solids, such as swimmers, also subject to deformations not resulting from the action of the fluid are also discussed.

This chapter regroups technical elements of differential calculus, gives the proof of certain results used in the book and provides elements on classical finite difference methods for the approximation of transport equations.

This chapter gives a number of definitions and mathematical results relating to the notions of trajectories in a regular velocity field. It then recalls certain principles of conservation of continuum mechanics.

This chapter generalizes the previous one to the case of general elastic solids, starting with a complete description of hyperelastic media in Lagrangian and Eulerian formulations. It provides a number of illustrations, in both the compressible and incompressible case.

This chapter defines a first class of level set methods for the interaction of a membrane or an elastic curve, with or without shear elasticity, with a fluid, both in the case of compressible and incompressible flows. It discusses time discretization methods from the point of view of stability and gives examples of algorithms and codes for these sy...

This chapter deals with contacts between solids and their modeling by level set methods. It describes a fast algorithm allowing the efficient calculation of interactions between a large number of objects.

In this paper we present a novel algorithm for simulating geometrical flows, and in particular the Willmore flow, with conservation of volume and area. The idea is to adapt the class of diffusion-redistanciation algorithms to the Willmore flow in both two and three dimensions. These algorithms rely on alternating diffusions of the signed distance f...

In this paper we present a novel algorithm for simulating geometrical flows, and in particular the Willmore flow, with conservation of volume and area. The idea is to adapt the class of diffusion-redistanciation algorithms to the Willmore flow in both two and three dimensions. These algorithms rely on alternating diffusions of the signed distance f...

An efficient method to capture an arbitrary number of fluid/structure interfaces in a level-set framework is built, following ideas introduced for contour capturing in image analysis. Using only three label maps and two distance functions we succeed in locating and evolving the bodies independently in the whole domain and get the distance between t...

20th CEMRACS Summer School: contributions

Benamou and Brenier formulation of Monge transportation problem (Numer. Math. 84:375-393, 2000) has proven to be of great interest in image processing to compute warpings and distances between pair of images (SIAM J. Math. Analysis, 35:61-97, 2003). One requirement for the algorithm to work is to interpolate densities of same mass. In most applicat...

Optimal transportation theory is a powerful tool to deal with image interpolation. This was first investigated by Benamou and Brenier \cite{BB00} where an algorithm based on the minimization of a kinetic energy under a conservation of mass constraint was devised. By structure, this algorithm does not preserve image regions along the optimal interpo...

In this paper we present a novel semi-implicit time-discretization of the level set method introduced in [8] for fluid-structure interaction problems. The idea stems form a linear stability analysis derived on a simplified one-dimensional problem. The semi-implicit scheme relies on a simple filter operating as a post-processing on the level set fun...

This work deals with the resolution of the optimal transport problem between 2D images in the fluid mechanics framework of Benamou and Brenier formulation [1], which numerical resolution is still challenging even for medium-sized images. We develop a method using the Helmholtz-Hodge decomposition [2] in order to enforce the divergence-free constrai...

We introduce an Eulerian model for the coupling of a fluid governed by the Navier-Stokes equations, with an immersed interface endowed with full membrane elasticity (i.e. including shear effects). We show numerical evidences of its ability to account for large displacements/shear in a relatively simple way, avoiding some drawbacks of Lagrangian rep...

We propose a level-set model of phase change and apply it to the study of the Leidenfrost effect. The new ingredients used in this model are twofold: first we enforce by penalization the droplet temperature to the saturation temperature in order to ensure a correct mass transfer at interface, and second we propose a careful differentiation of the c...

In this work three branches of Immersed Boundary Methods (IBM) are described and validated for incompressible aerodynamics and fluid-structure interactions. These three approaches are: Cut Cell method, Vortex-Penalization method and Forcing method. The first two techniques are validated for external bluff-body flow around a circular obstacle. The l...

Benamou and Brenier formulation of Monge transportation problem has proven to be of great interest in image processing to compute warpings and distances between pair of images. In some applications, however, the built-in minimization of kinetic energy does not give satisfactory results. In particular cases where some specific regions represent phys...

Phospholipidic membranes and vesicles constitute a basic element in real biological functions. Vesicles are viewed as a model system to mimic basic viscoelastic behaviors of some cells, like red blood cells. Phase field and level-set models are powerful tools to tackle dynamics of membranes and their coupling to the flow. These two methods are some...

We present and analyze a penalization method wich extends the the method of [1] to the case of a rigid body moving freely in an incompressible fluid. The fluid-solid system is viewed as a single variable density flow with an interface captured by a level set method. The solid velocity is computed by averaging at avery time the flow velocity in the...

We describe in this paper two applications of Eulerian level set methods to fluid-structure problems arising in biophysics. The first one is concerned with three-dimensional equilibrium shapes of phospholipidic vesicles. This is a complex problem, which can be recast as the minimization of the curvature energy of an immersed elastic membrane, under...

We present and analyze a penalization method wich extends the the method of [1] to the case of a rigid body moving freely in an incompressible fluid. The fluid-solid system is viewed as a single variable density flow with an interface captured by a level set method. The solid velocity is computed by averaging at avery time the flow velocity in the...

We perform a linear stability analysis of an elementary 1D model obtained from a Level Set formulation of the coupling between an immersed elastic interface and the surrounding fluid. Despite the striking simplicity of the studied model, relevant instability regimes are obtained, unifying results obtained in previous studies (Brackbill et al, 1992,...

In this work, we consider an optimization problem described on a surface. The approach is illustrated on the problem of finding a closed curve whose arclength is as small as possible while the area enclosed by the curve is fixed. This problem exemplifies a class of optimization and inverse problems that arise in diverse applications. In our approac...

This paper is devoted to Eulerian models for incompressible fluid-structure systems. These models are primarily derived for computational purposes as they allow to simulate in a rather straightforward way complex 3D systems. We first analyze the level set model of immersed membranes proposed in [Cottet and Maitre, Math. Models Methods Appl. Sci. 16...

An Eulerian approach is presented for generic fluid-structure coupling of an elastic body with an incompressible fluid. We consider the coupling as a multiphysics problem where fluid-solid interfaces are captured by a level-set method. The main features of the method are its simplicity, and its natural control of mass and energy. We are indeed able...

An eulerian approach for fluid-structure coupling is presented for generic fluid-structure coupling an elastic body with an incompressible fluid. Fluid-solid interfaces are captured by a level-set method. The main features of the method are its simplicity, and its natural control of mass and energy. We are indeed able to prove an energy equation wh...

Revised (Day Month Year) Communicated by (xxxxxxxxxx) This paper is devoted to the derivation and the validation of a level set method for fluid-structure interaction problems with immersed surfaces. The test case of a pressurized membrane is used to compare our method to Peskin's Immersed Boundary methods in the two-dimensional case and to demonst...

We propose a level set formulation of the immersed boundary method for fluid–structure problems in two and three dimensions. We prove that the resulting model verifies an energy estimate. To cite this article: G.-H. Cottet, E. Maitre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

We consider a nonlinear counterpart of a compactness lemma of
Simon (1987), which arises naturally in the study of doubly
nonlinear equations of elliptic-parabolic type. This
paper was motivated by previous results of Simon (1987), recently
sharpened by Amann (2000), in the linear setting, and by a
nonlinear compactness argument of Alt and Luckhaus...

In this note we establish the existence of minimizer of a nonconvex energy functional. This functional is an energy of deformation of a woven fabric subject only to his own weight and xed on a part of its boundary. A typical example is the case of a tablecloth on a table. We make the hypothesis that the fabric is inextensible in the direction of th...

This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the deg...

In this Note we establish the existence of minimizer of a nonconvex energy functional. This functional is an energy of deformation of a woven fabric subject only to his own weight and fixed on a part of its boundary. A typical example is the case of a tablecloth on a table. We make the hypothesis that the fabric is inextensible in the direction of...

Pseudo-monotonicity seems to be a good notion to deal with convergence in nonlinear terms of partial differential equations. J.-L. Lions [16] used two different definitions of pseudo-monotonicity for elliptic and parabolic problems, and derived associated existence results. Nonlinear elliptic-parabolic equations are intermediate equations for which...

We consider a nonlinear counterpart of a compactness lemma of J. Simon, which arises naturally in the study of doubly nonlinear equations of elliptic-parabolic type. Our work was motivated by previous results J. Simon, recently sharpened by H. Amann, in the linear setting, and by a nonlinear compactness argument of H.W. Alt and S. Luckhaus.

We consider a nonlinear counterpart of a compactness lemma of J. Simon [1], which arises naturally in the study of doubly nonlinear equations of elliptic-parabolic type. Our work was motivated by previous results J. Simon [1], recently sharpened by H. Amann [2], in the linear setting, and by a nonlinear compactness argument of H.W. Alt and S. Luckh...

. Relying on the splitting of the collision operator introduced by [6] [1], we prove theoretical convergence for an innite dimensional adaptation of the minimal residual algorithm for Boltzmann transport equation in dimension two. Then we compare this solver with known ones from a numerical point of view. 1.

In the last years, several schemes have been proposed to solve transport equation in slab geometry [1,2] and [3]. In this work we present a new algorithm based on a splitting of the collision operator according to the characteristics of the transport operator and an infinite dimensional adaptation of the minimal residual algorithm. The theoretical...

Ces dernières années, de nouveaux algorithmes pour la résolution de l'équation de transport des neutrons ont été introduits et analysés dans [1,2] et [3]. Notre travail porte sur l'introduction d'un nouvel algorithme basé sur la décomposition de l'opérateur de collision tenant compte des lignes caractéristiques de l'opérateur de transport et l'adap...

During the filling stage of an injection moulding process, which consists in casting a melt polymer in order to manufacture
plastic pieces, the free interface between polymer and air has to be precisely described. We set this interface as a zero
level set of an unknown function. This function satisfies a transport equation with boundary conditions,...

This work has been draw out of an industrial problem dealing with injection moulding of thermo-plastic. We focussed our attention on the filling stage of the injection process, and on the localisation of melt polymer front. The present work is thus divided in two parts : => The mathematical and numerical study of the pressure equation which is the...

Pseudomonotonicity seems to be the good notion to deal with convergence in nonlinear terms of partial differential equations. Lions [1] used two different definitions of pseudomonotonicity for elliptic and parabolic problems, and derived associated existence results. Nonlinear elliptic-parabolic equations are intermediate equations for which an int...

Mes recherches ont porté sur l'analyse des équations à double non linéarité, le transport neutronique et la mécanique des textiles, et plus récemment sur la méthode Level Set et ses applications au couplage fluide-structure, notamment dans le domaine biomécanique.