Emmanuel Maitre

A LEVEL SET METHOD FOR FLUID-STRUCTURE INTERACTIONS WITH IMMERSED SURFACES

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 Peskins Immersed Boundary methods in the two-dimensional case and to demonstrate its capabilities for three-dimensional flows. The method in particular exhibits appealing mass and energy conservation properties.

Applications of level set methods in computational biophysics

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 a constant area constraint. The second deals with isolated cardiomyocyte contraction. This problem corresponds to a generic incompressible fluid-structure coupling between an elastic body and a fluid. By the choice of these two quite different situations, we aim to bring evidence that Eulerian methods provide efficient and flexible computational tools in biophysics applications.

A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations

The case where B is an unbounded linear operator was considered 1rst by Bardos and Br7 ezis [4]. In the nonlinear case Raviart [20], Grange and Mignot [13], DiBenedetto and Showalter [11] proved existence results assuming that A and B are at least monotone operators, and B is compact. Alt and Luckhaus [1] investigated the case of a non-compact operator B, assuming A is strongly monotone; their work was extended more recently by Ka? cur [14], Filo and Ka? cur [12] or Zadrzy7 nska and ZajB aczkowski [26]. Berm7 udez et al. [7] devoted their work to the case where B is compact and strongly monotone and A is pseudo-monotone. We are interested in the same case excepted that B is no longer assumed to be strongly monotone, thus the equation may

Convergence Analysis of a Penalization Method for the Three-Dimensional Motion of a Rigid Body in an Incompressible Viscous Fluid

We present and analyze a penalization method which extends the the method of [Ph. Angot, C.-H. Bruneau, and P. Fabrie, Numer. Math., 81 (1999), pp. 487-520] 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. The interface is captured by a color function satisfying a transport equation. The solid velocity is computed by averaging at every time the flow velocity in the solid phase. This velocity is used to penalize the flow velocity at the fluid-solid interface and to move the interface. Numerical illustrations are provided to illustrate our convergence result. A discussion of our result in the light of existing existence results is also given.

Level set methods for optimization problems involving geometry and constraints II. Optimization over a fixed surface

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 approach, we assume that the surface is given parametrically. A level set formulation for the curve is developed in the surface parameter space. We show how to obtain a formal gradient for the optimization objective, and derive a gradient-type algorithm which minimizes the objective while respecting the constraint. The algorithm is a projection method which has a PDE interpretation. We demonstrate and verify the method in numerical examples.

Transport equation with boundary conditions for free surface localization

Summary. 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, where the velocity field has few regularity properties.In a first part, we obtain existence and uniqueness result for these equations, under weaker regularity assumptions than C. Bardos [Bar70], and C. Bardos, Y. Leroux and J.C. Nedelec [BLN79] in previous articles, but stronger assumptions than R.J. DiPerna and P.L. Lions [DL89b] who studied the case without boundary condition. We also study some regularity properties of the interface.A second part is devoted to an application to injection molding of melt polymer. We give a numerical experiment which shows that our method leads to an accurate localization of interface, which is robust, since it easily handles changes of topology of the free interface, as bubble formation or fusion of two fronts of melt polymer.

Optimal Transport using Helmholtz-Hodge Decomposition and First-Order Primal-Dual Algorithms

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 constraint throughout the iterations. We then show how to use a first order primal-dual algorithm for convex problems of Chambolle and Pock [3] to solve the obtained problem, leading to a new algorithm easy to implement. Besides, numerical experiments demonstrate that this algorithm is faster than state of the art methods and efficient with real-sized images.