Optimisation et analyse convexe pour la dynamique non-régulière

Florent Cadoux 1
1 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : The aim of this work is to propose a new approach to the solution of 3D unilateral contact problems with Coulomb friction in solid mechanics. We consider dynamical systems composed of several bodies with a finite number of degrees of freedom: rigid bodies, or deformable bodies which are spatial approximations of continuous models. Friction between bodies is modelled using a classical formulation of Coulomb's law. After time discretization (or quasi-static approximation), we get at each time step a problem containing complementarity equations posed on a product of second order cones, plus other equations. Several methods have been proposed in the literature for different equivalent formulations of this problem, in particular by Moreau, Alart and Curnier, and De Saxcé. Considering the complementarity equations as optimality conditions (KKT) of an optimization problem, we propose a new equivalent reformulation as a parametric convex minimization problem coupled with a fixed point problem. Thanks to this viewpoint, we prove the existence of solutions under a quite mild assumption which can be checked in practice. Moreover, we can practically compute one of these solutions by solving numerically the fixed point equation. The performance of this approach is compared with existing methods.
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Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2009. Français
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Florent Cadoux. Optimisation et analyse convexe pour la dynamique non-régulière. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2009. Français. 〈tel-00440798v2〉

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