École normale supérieure de Cachan

About: École normale supérieure de Cachan is a education organization based out in Cachan, Île-de-France, France. It is known for research contribution in the topics: Decidability & Nonlinear system. The organization has 2717 authors who have published 5585 publications receiving 175925 citations.

Multi-time-step explicit–implicit method for non-linear structural dynamics

Abstract: We present a method with domain decomposition to solve time-dependent non-linear problems. This method enables arbitrary numeric schemes of the Newmark family to be coupled with different time steps in each subdomain: this coupling is achieved by prescribing continuity of velocities at the interface. We are more specifically interested in the coupling of implicit/explicit numeric schemes taking into account material and geometric non-linearities. The interfaces are modelled using a dual Schur formulation where the Lagrange multipliers represent the interfacial forces. Unlike the continuous formulation, the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements, velocities and accelerations at the interfaces. We show that, within the framework of the Newmark family of numeric schemes, continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged. To prove this stability, we use an energy method, i.e. a global method over the whole time interval, in order to verify the algorithms properties. Then, we propose to extend this to non-linear situations in the following cases: implicit linear/explicit non-linear, explicit non-linear/explicit non-linear and implicit non-linear/explicit non-linear. Finally, we present some examples showing the feasibility of the method. Copyright © 2001 John Wiley & Sons, Ltd.

On the trend to global equilibrium in spatially inhomogeneous entropy‐dissipating systems: The linear Fokker‐Planck equation

Abstract: We study the long-time behavior of kinetic equations in which transport and spatial confinement (in an exterior potential or in a box) are associated with a (degenerate) collision operator acting only in the velocity variable. We expose a general method, based on logarithmic Sobolev inequalities and the entropy, to overcome the well-known problem, due to the degeneracy in the position variable, of the existence of infinitely many local equilibria. This method requires that the solution be somewhat smooth. In this paper, we apply it to the linear Fokker-Planck equation and prove decay to equilibrium faster than O(t−1/ϵ) for all ϵ > 0. © 2001 John Wiley & Sons, Inc.

Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey

Abstract: The goal of this paper is to describe the state of the art on the question of global existence of solutions to reaction-diffusion systems for which two main properties hold: on one hand, the positivity of the solutions is preserved for all time; on the other hand, the total mass of the components is uniformly controlled in time. This uniform control on the mass (or – in mathematical terms- on the L1-norm of the solution) suggests that no blow up should occur in finite time. It turns out that the situation is not so simple. This explains why so many partial results in different directions are found in the literature on this topic, and why also the general question of global existence is still open, while lots of systems arise in applications with these two natural properties. We recall here the main positive and negative results on global existence, together with many references, a description of the still open problems and a few new results as well.

Three dimensional image correlation from X-Ray computed tomography of solid foam

Abstract: A new methodology is proposed to estimate 3D displacement fields from pairs of images obtained from X-ray computed microtomography (XCMT). Contrary to local approaches, a global approach is followed herein that evaluates continuous displacement fields. Although any displacement basis may be considered, the procedure is specialized to finite element shape functions. The method is illustrated with the analysis of a compression test on a polypropylene solid foam (independently studied in a companion paper). A good stability of the measured displacement field is obtained for cubic element sizes ranging from 16 to 6 voxels.