Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers

International audienceThe purpose of this work is to present in a general framework the hybrid discretization of unilateral contact and friction conditions in elastostatics. A projection formulation is developed and used. An existence and uniqueness results for the solutions to the discretized problem is given in the general framework. Several numerical methods to solve the discretized problem are presented (Newton, SOR, fixed points, Uzawa) and compared in terms of the number of iterations and the robustness with respect to the value of the friction coefficient

Comparison of two approaches for the discretization of elastodynamic contact problems

International audienceThe purpose of this Note is to compare two approaches for the discretization of elastodynamic contact problems. First, we introduce an energy conserving method based on a standard midpoint scheme and a contact condition expressed in terms of velocity. The second approach consists in considering an equivalent distribution of the body mass so that the nodes on the contact boundary have no inertia. We prove that this method leads to an energy conservation for the space semi-discretized elastodynamic contact problem. Finally, some numerical results are presented in the two dimensional case

Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles

Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini’s conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam model the singular dynamic method introduced by Renard. A particular emphasis is given in the use of a restitution coefficient in the impact law. Finally, various numerical results are presented and energy conservation capabilities of the schemes are investigated

Combined sticking: a new approach for finite-amplitude Coulomb frictional contact

Engineering-level accuracy of discretization methods for frictional contact originates from precise representation of discontinuous frictional and normal interaction laws and precise discrete contact techniques. In terms of discontinuous behavior in the quasi-static case, two themes are of concern: the normal interaction (i.e. impact) and the jumps in tangential directions arising from high frictional values. In terms of normal behavior, we use a smoothed complementarity relation. For the tangential behavior, we propose a simple and effective algorithm, which is based a stick predictor followed by corrections to the tangential velocity. This allows problems with impact and stick-slip behavior to be solved with an implicit code based on Newton–Raphson iterations. Three worked examples are shown with comparisons with published results. An extension to node-to-face form in 3D is also presented

Problèmes de contact unilatéral avec frottement de Coulomb en élastostatique et élastodynamique. Etude mathématique et résolution numérique.

The modelling of problems of contact leads to serious difficulties: conceptual, mathematical and data processing difficulties much more complex than those coming from the linear structural mechanics. Motivated by the fundamental role that the contact plays in applications of computational mechanics, we are interested in problems of unilateral contact and friction (static and dynamic) in small deformations. This thesis is devoted to the study of certain formulations and methods to solve this problem and breaks up into two great parts. The first one is devoted to the presentation of the hybrid discretization of unilateral contact problem with Coulomb friction. A formulation with a projection is developed and an existence and uniqueness result is given for the discrete problem. Different methods of solution are presented (Newton, iterative method, fixed points, Uzawa) and are compared in terms of number of iteration and robustness with respect to the coefficient of friction. The second part relates to the elastodynamic contact problem. Several algorithms (θ-method, Newmark, midpoint) are presented. New strategies are considered (Paoli and Schatzman scheme, scheme with an equivalent contact condition, scheme with equivalent mass matrix) to overcome the difficulties met with the previous schemes. The last method allows us to have energy conserving problem and to prove an existence result of a Lipschitz continuous solution for the discrete elastodynamic contact problem. These results are validated by numerical results.La modélisation des problèmes de contact pose de sérieuses difficultés : conceptuelles, mathématiques et informatiques bien plus complexes que celles qui proviennent de la mécanique des structures linéaire classique. Motivés par le rôle fondamental que joue le contact dans les applications en calcul de structures, nous nous intéressons aux problèmes de contact unilatéral et frottement (statique et dynamique) en petites déformations. Cette thèse est consacrée à l'étude de certaines formulations et méthodes pour résoudre ce problème et se décompose en deux grandes parties. La première partie est consacrée à la présentation de la discrétisation hybride du problème de contact unilatéral avec frottement de Coulomb. Une formulation avec projection est étudiée et un résultat d'existence et d'unicité est donné pour le problème discret. Différentes méthodes de résolution sont présentées (Newton, méthode itérative, points fixes, Uzawa) et comparées en termes de nombre d'itérations et en termes de robustesse par rapport au coefficient de frottement. La deuxième partie concerne le problème de contact élastodynamique. Plusieurs schémas classiques d'intégration en temps (la θ-méthode, schéma de Newmark, point milieu) sont présentés dans cette partie. On donne aussi de nouvelles stratégies (schéma de Paoli et Schatzman, schéma avec la loi de contact équivalente, schéma avec la matrice de masse équivalente) pour venir à bout des difficultés rencontrées avec les schémas précédents. Cette dernière méthode nous permet de conserver l'énergie du problème et de montrer un résultat d'existence d'une solution lipschitzienne pour le problème de contact élastodynamique discret. Ces résultats sont validés par des simulations numériques

Numerical Methods for 3D Coulomb's Friction based on Nonsmooth Newton's Method and Nonlinear Complementarity Formulations

International audienc

Numerical Methods for 3D Coulomb's Friction based on Nonsmooth Newton's Method and Nonlinear Complementarity Formulations

International audienc

Problèmes de contact unilatéral avec frottement de Coulomb en élastostatique et élastodynamique (étude mathématique et résolution numérique)

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