Semismooth Newton and Augmented Lagrangian Methods for a Simplified Friction Problem

Georg Stadler

doi:10.1137/s1052623403420833

Abstract

In 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.

Highlights

This paper is devoted to the convergence analysis of iterative algorithms for the solution of mechanical problems involving friction
Since we focus on higher-order methods, and further since for (P∗) the iterates of the algorithms presented in section 4 are not contained in spaces of square integrable functions, we introduce a regularization procedure for (P∗) that allows the statement and analysis of our algorithms in infinite dimensional Hilbert spaces and will be shown to be closely related to augmented Lagrangians
Due to the local superlinear convergence of (PDAS), the auxiliary problem is solved in very few iterations, as can be seen in Table 6.4, where for γ = 102 and γ = 104 we report on the number of iterations #iterP D required by (PDAS)

Summary

Introduction

This paper is devoted to the convergence analysis of iterative algorithms for the solution of mechanical problems involving friction. It is an iterative algorithm which uses the current variables λk, ξk for (P∗γ) to predict new active sets Ak−+1, Ak++1 for the constrained optimization problem (P∗γ), whereas this prediction is motivated from expressing the complementarity condition in the form (3.2c). Note that the norm gap required for Newton differentiability of the max- and min-function results from directly exploiting the smoothing property of the operator C.

Results

Conclusion