Bertrand Maury

Bertrand Maury, Juliette Venel

The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: We first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people; The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here the underlying mathematical framework, and we explain how recent results by J.F. Edmond and L. Thibault on the sweeping process by uniformly prox-regular sets can be adapted to handle this situation in terms of well-posedness. We propose a numerical scheme for this contact dynamics model, based on a prediction-correction algorithm. Numerical illustrations are finally presented and discussed.

Julien Dambrine, Nicolas Meunier, Bertrand Maury et al.

This paper deals with a class of macroscopic models for cell migration in a saturated medium for two-species mixtures. Those species tend to achieve some motion according to a desired velocity, and congestion forces them to adapt their velocity. This adaptation is modelled by a correction velocity which is chosen minimal in a least-square sense. We are especially interested in two situations: a single active species moves in a passive matrix (cell migration) with a given desired velocity, and a closed-loop Keller-Segel type model, where the desired velocity is the gradient of a self-emitted chemoattractant. We propose a theoretical framework for the open-loop model (desired velocities are defined as gradients of given functions) based on a formulation in the form of a gradient flow in the Wasserstein space. We propose a numerical strategy to discretize the model, and illustrate its behaviour in the case of a prescribed velocity, and for the saturated Keller-Segel model.

Benoit Fabrèges, Bertrand Maury

We are interested in the finite element solution of elliptic problems with a right-hand side of the single layer distribution type. Such problems arise when one aims at accounting for a physical hypersurface (or line, for bi-dimensional problem), but also in the context of fictitious domain methods, when one aims at accounting for the presence of an inclusion in a domain (in that case the support of the distribution is the boundary of the inclusion). The most popular way to handle numerically the single layer distribution in the finite element context is to spread it out by a regularization technique. An alternative approach consists in approximating the single layer distribution by a combination of Dirac masses. As the Dirac mass in the right hand side does not make sense at the continuous level, this approach raises particular issues. The object of the present paper is to give a theoretical background to this approach. We present a rigorous numerical analysis of this approximation, and we present two examples of application of the main result of this paper. The first one is a Poisson problem with a single layer distribution as a right-hand side and the second one is another Poisson problem where the single layer distribution is the Lagrange multiplier used to enforce a Dirichlet boundary condition on the boundary of an inclusion in the domain. Theoretical analysis is supplemented by numerical experiments in the last section.