Numerical 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 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 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. [1] Ph. Angot, C.-H. Bruneau and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows, Numer. Math. 81: 497--520 (1999)Comment: 23 page

Eulerian models and algorithms for unbalanced optimal transport

International audienceBenamou 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 applications to image interpolation, this is a serious limitation. Existing approaches to overcome this caveat are reviewed, and discussed. Due to the mix between transport and $L^2$ interpolation, these models can produce instantaneous motion at finite range. In this paper we propose new methods, parameter-free, for interpolating unbalanced densities. One of our motivations is the application to interpolation of growing tumor images

Multi-physics Optimal Transportation and Image Interpolation

International audienceOptimal 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 interpolation path, and it is actually not very difficult to exhibit test cases where the algorithm produces a path of images where high density regions split at the beginning before merging back at its end. However, in some applications to image interpolation this behaviour is not physically realistic. Hence, this paper aims at studying how some physics can be added to the optimal transportation theory, how to construct algorithms to compute solutions to the corresponding optimization problems and how to apply the proposed methods to image interpolation

Primal-dual formulation of the Dynamic Optimal Transport using Helmholtz-Hodge decomposition

This work deals with the resolution of the dynamic optimal transport (OT) problem between 1D or 2D images in the fluid mechanics framework of Benamou-Brenier [6]. The numerical resolution of this dynamic formulation of OT, despite the successful application of proximal methods [36] is still computationally demanding. This is partly due to a space-time Laplace operator to be solved at each iteration, to project back to a divergence free space. In this paper, we develop a method using the Helmholtz-Hodge decomposition [23] in order to enforce the divergence-free constraint throughout the iterations. We prove that the functional we consider has better convexity properties on the set of constraints. In particular we explain that in 1D+time, this formulation is equivalent to the resolution of a minimal surface equation. We then adapt the first order primal-dual algorithm for convex problems of Chambolle and Pock [12] to solve this new problem, leading to an algorithm easy to implement. Besides, numerical experiments demonstrate that this algorithm is faster than state of the art methods for dynamic optimal transport [36] and efficient with real-sized images

Comparison between advected-field and level-set methods in the study of vesicle dynamics

International audiencePhospholipidic 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 somewhat similar, but to date no bridge between them has been made. This is a first focus of this paper. Furthermore, a constitutive viscoelastic law is derived for the composite fluid: the ambient fluid and the membranes. We present two different approaches to deal with the membrane local incompressibility, and point out differences. Some numerical results following from the level-set approach are presented

Axisymmetric Level Set model of Leidenfrost effect

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 capillary interface with respect to a moving interface with phase change. We perform some numerical tests in the axisymmetric case and show that our numerical method, while not avoiding well known numerical caveats of diffuse interface methods, behave quite well in the limit of numerical interface width going to zero in comparison to an analytical formula

Une méthode level set semi-implicite pour les écoulements multiphasiques et l'interaction fluide-structure

International audienceIn 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 function. It applies to multiphase flows driven by surface tension as well as to fluid-structure interaction problems. The semi-implicit scheme avoids the stability constraints that explicit scheme need to satisfy and reduces significantly the computational cost. It is validated through comparisons with the original explicit scheme and refinement studies on two and three-dimensional membranes

Primal-dual formulation of the Dynamic Optimal Transport using Helmholtz-Hodge decomposition

This work deals with the resolution of the dynamic optimal transport (OT) problem between 1D or 2D images in the fluid mechanics framework of Benamou-Brenier [6]. The numerical resolution of this dynamic formulation of OT, despite the successful application of proximal methods [36] is still computationally demanding. This is partly due to a space-time Laplace operator to be solved at each iteration, to project back to a divergence free space. In this paper, we develop a method using the Helmholtz-Hodge decomposition [23] in order to enforce the divergence-free constraint throughout the iterations. We prove that the functional we consider has better convexity properties on the set of constraints. In particular we explain that in 1D+time, this formulation is equivalent to the resolution of a minimal surface equation. We then adapt the first order primal-dual algorithm for convex problems of Chambolle and Pock [12] to solve this new problem, leading to an algorithm easy to implement. Besides, numerical experiments demonstrate that this algorithm is faster than state of the art methods for dynamic optimal transport [36] and efficient with real-sized images