MSC (2000) 49J52, 70F40, 90C25This paper presents a new formulation of the dynamical Coulomb friction problem in finite dimension with discretized time. The novelty of our approach is to capture and treat directly the friction model as a parametric quadratic optimization problem with second-order cone constraints coupled with a fixed point equation. This intrinsic formulation allows a simple existence proof under reasonable assumptions, as well as a variety of solution algorithms. We study mechanical interpretations of these assumptions, showing in particular that they are actually necessary and sufficient for a basic example similar to the so-called "paradox of Painlevé". Finally, we present some implementations and experiments to illustrate the practical aspect of our work.

We focus on the challenging problem of simulating thin elastic rods in contact, in the presence of friction. Most previous approaches in computer graphics rely on a linear complementarity formulation for handling contact in a stable way, and approximate Coulombs's friction law for making the problem tractable. In contrast, following the seminal work by Alart and Curnier in contact mechanics, we simultaneously model contact and exact Coulomb friction as a zero finding problem of a nonsmooth function. A semi-implicit time-stepping scheme is then employed to discretize the dynamics of rods constrained by frictional contact: this leads to a set of linear equations subject to an equality constraint involving a non-differentiable function. To solve this one-step problem we introduce a simple and practical nonsmooth Newton algorithm, which proves to be reasonably efficient and robust for systems that are not over-constrained. We show that our method is able to finely capture the subtle effects that occur when thin elastic rods with various geometries enter into contact, such as stick-slip instabilities in free configurations, entangling curls, resting contacts in braid-like structures, or the formation of tight knots under large constraints. Our method can be viewed as a first step towards the accurate modeling of dynamic fibrous materials.

To cite this version:Florent Cadoux. Computing deep facet-defining disjunctive cuts for mixed-integer programming.[ Abstract: The problem of separation is to find an affine hyperplane, or "cut", that lies between the origin O and a given closed convex set Q in a Euclidean space. We focus on cuts which are deep for the Euclidean distance, and facet-defining. The existence of a unique deepest cut is shown and cases when it is decomposable as a combination of facet-defining cuts are characterized using the reverse polar set. When Q is a split polyhedron, a new description of the reverse polar is given. A theoretical successive projections algorithm is proposed that could be used to compute deep facet-defining split cuts.Key-words: integer programming, separation, cut generation, disjunctive cut, convex analysis, reverse polar * florent.cadoux@inria.frCoupes disjonctives profondes exposant des facettes pour la programmation entière générale Résumé : Le problème de séparation consisteà trouver un hyperplan affine, ou "coupe", situé entre l'origine O et un ensemble convexe fermé Q dans un espace euclidien. On s'intéresse aux coupes profondes au sens de la distance euclidienne, et qui exposent une facette. L'existence d'une unique coupe de profondeur maximale est prouvée, et les cas où elle peutêtre décomposée en combinaison de coupes exposant une facette sont caractérisés grâce au polaire inverse de Q. Quand Q est un polyèdre disjonctif, une nouvelle description du polaire inverse est donnée. Un algorithme théorique de projections successives est proposé, qui pourraitêtre utilisé pour calculer des coupes profondes exposant une facette.

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