Semi-Smooth Newton Method for Solving 2D Contact Problems with Tresca and Coulomb Friction

Kristina Motyckova,Radek Kucera

doi:10.15598/aeee.v11i3.801

Abstract

The contribution deals with contact problems for two elastic bodies with friction. After the description of the problem we present its discretization based on linear or bilinear finite elements. The semi--smooth Newton method is used to find the solution, from which we derive active sets algorithms. Finally, we arrive at the globally convergent dual implementation of the algorithms in terms of the Langrange multipliers for the Tresca problem. Numerical experiments conclude the paper.

Highlights

Nowadays, solving contact problems with friction counts among very challenging tasks in mechanics and is of crucial importance in various practical applications
This paper proposes effective solvers for contact problems with friction using the semi–smooth Newton method
Let us consider the mapping Ψ : L2+(Γ1c) → L2+(Γ1c) defined by: Ψ(g) = −F σν(g), where σν(g) := σν(u(g)) is the normal contact stress corresponding to the solution of the contact problem with Tresca friction with given g

Summary

Introduction

Nowadays, solving contact problems with friction counts among very challenging tasks in mechanics and is of crucial importance in various practical applications. As an example we can state biological processes, the design of machines and transportation systems, and metal forming. This is a strong motivation for the development of methods allowing a reliable and fast simulation, i.e., to implement robust numerical solvers. Newton–type methods for solving contact problems with friction were used already in 1991, see [1]. Semi–smooth Newton method in finite dimensions appears already in 1993, see [22], and more recently in infinite dimensions. The concept of the slant differentiability was introduced; see [2] and the references given therein. This paper proposes effective solvers for contact problems with friction using the semi–smooth Newton method. For any non-empty set of indices I and a matrix M ∈ Rm×n, we denote by M I a submatrix of M with the rows given by I

Formulation

Discrete Problems

Algorithms

Non–Symmetric Case

Implementation

Conclusion