Semismooth Newton and Augmented Lagrangian Methods for a Simplified Friction Problem

International audienceIn this paper a simplified friction problem and iterative second-order algorithms for its solution are analyzed in infinite dimensional function spaces. Motivated from the dual formulation, a primal-dual active set strategy and a semismooth Newton method for a regularized problem as well as an augmented Lagrangian method for the original problem are presented and their close relation is analyzed. Local as well as global convergence results are given. By means of numerical tests, we discuss among others convergence properties, the dependence on the mesh, and the role of the regularization and illustrate the efficiency of the proposed methodologies. 1. Introduction. This paper is devoted to the convergence analysis of iterative algorithms for the solution of mechanical problems involving friction. As a model problem we consider a simplified friction problem that can be stated as the minimization of the nondifferentiable functional J(y) :

Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

International audienceThe 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas

Optimization and Applications

Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research

Numerical simulation of two-dimensional Bingham fluid flow by semismooth Newton methods

AbstractThis paper is devoted to the numerical simulation of two-dimensional stationary Bingham fluid flow by semismooth Newton methods. We analyze the modeling variational inequality of the second kind, considering both Dirichlet and stress-free boundary conditions. A family of Tikhonov regularized problems is proposed and the convergence of the regularized solutions to the original one is verified. By using Fenchel’s duality, optimality systems which characterize the original and regularized solutions are obtained. The regularized optimality systems are discretized using a finite element method with (cross-grid P1)–Q0 elements for the velocity and pressure, respectively. A semismooth Newton algorithm is proposed in order to solve the discretized optimality systems. Using an additional relaxation, a descent direction is constructed from each semismooth Newton iteration. Local superlinear convergence of the method is also proved. Finally, we perform numerical experiments in order to investigate the behavior and efficiency of the method

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces

For a Banach space X of R^M-valued functions on a Lipschitz domain, let K(X) ⊂ X be a closed convex set arising from pointwise constraints on the value of the function, its gradient or its divergence, respectively. The main result of the paper establishes, under certain conditions, the density of K(X_0) in K(X_1) where X_0 is densely and continuously embedded in X_1. The proof is constructive, utilizes the theory of mollifiers and can be applied to Sobolev spaces such as H (div,Ω) and W1,p(Ω), in particular. It is also shown that such a density result cannot be expected in general.Peer Reviewe

A primal-dual active set algorithm for three-dimensional contact problems with coulomb friction

International audienceIn this paper, efficient algorithms for contact problems with Tresca and Coulomb friction in three dimensions are presented and analyzed. The numerical approximation is based on mortar methods for nonconforming meshes with dual Lagrange multipliers. Using a nonsmooth com-plementarity function for the three-dimensional friction conditions, a primal-dual active set algorithm is derived. The method determines active contact and friction nodes and, at the same time, resolves the additional nonlinearity originating from sliding nodes. No regularization and no penalization are applied, and superlinear convergence can be observed locally. In combination with a multigrid method, it defines a robust and fast strategy for contact problems with Tresca or Coulomb friction. The efficiency and flexibility of the method is illustrated by several numerical examples