On Domain Decomposition Algorithms for Contact Problems with Tresca Friction

Abstract

Development of numerical methods for the solution of contact problems is a challenging task whose difficulty lies in the non-linear conditions for non-penetration and friction. Recently, many authors proposed to use various numerical algorithms combined with multigrid or domain decomposition techniques; see, e.g., the primal-dual active set algorithm [8], the non-smooth multiscale method [10], or the augmented Lagrangian based algorithm [3]. Another alternative consists in the formulation of suitable iterations solving the elasticity equations for each sub-body separately with certain boundary conditions [5]. In [1], the authors proposed a Dirichlet-Neumann algorithm which takes into account the natural interface for frictionless contact problems. Another improvement has led to a Neumann–Neumann algorithm in which they added two Neumann sub-problems in order to ensure the continuity of normal stresses [2]. Later, various numerical implementations of this approach was given in [7, 9].