Sur la résolution du problème de frottement tridimensionnel : Formulations et comparaisons des méthodes numériques

In this report, we review several formulations of the discrete frictional contact problemthat arises in space and time discretized mechanical systems with unilateral contact andthree-dimensional Coulomb’s friction. Most of these formulations are well–known concepts in theoptimization community, or more generally, in the mathematical programming community. Tocite a few, the discrete frictional contact problem can be formulated as variational inequalities,generalized or semi–smooth equations, second–order cone complementarity problems, or as optimizationproblems such as quadratic programming problems over second-order cones. Thanks tothese multiple formulations, various numerical methods emerge naturally for solving the problem.We review the main numerical techniques that are well-known in the literature and we also proposenew applications of methods such as the fixed point and extra-gradient methods with self-adaptivestep rules for variational inequalities or the proximal point algorithm for generalized equations.All these numerical techniques are compared over a large set of test examples using performanceprofiles. One of the main conclusion is that there is no universal solver. Nevertheless, we are ableto give some hints to choose a solver with respect to the main characteristics of the set of testsDans ce rapport, plusieurs formulations du problème discret de contact frottant qui apparaîtdans les systèmes mécaniques avec du contact unilatéral et du frottement de Coulomb, sont présentées.La plupart de ces formulations sont des objets bien connus dans la communauté de l’optimisation, etplus généralement, de la programmation mathématique. Pour en citer quelques uns, le problème decontact frottant peut être formulé comme une inégalité variationnelle, comme une équation non-régulièreou semi–lisse, comme un problème de complémentarité sur des cônes, ou encore comme des problèmesd’optimisation par exemple des problèmes quadratiques sur des cônes du second ordre. Grâce à cesmultiples formulations, de nombreuses méthodes numériques de résolutions émergent naturellement. Ondétaille dans ce rapport les principales techniques numériques bien connues dans la littérature et nousproposons aussi des nouvelles méthodes comme les méthodes de point fixe et d’extra-gradient pour lesinégalités variationnelles avec une règle d’adaptation automatique du pas, ainsi que l’application del’algorithme du point optimal pour les équations généralisées. Toutes ces techniques sont comparées surun grand ensemble de problème–tests en utilisant des profils de performance. Une des conclusions est qu’iln’existe pas de méthode universelle. Néanmoins, on peut donner des conseils pour choisir une méthodeparticulière la mieux adaptée aux caractéristiques d’un problème donné

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The paper leads to a derivation of the latent variable proximal point (LVPP) algorithm: an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this work, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis