Multigrid Methods for Unilateral Contact Problems with Friction

In this paper, multigrid methods are tested on unilateral problems with friction. An optimal strategy is presented and efficiency of the solver is discussed on several examples.

Multigrid methods have been widely used in fluids mechanics when large numbers of degrees of freedom are involved. Usually the geometries are sufficiently simple to enable the generation of multiple overlapped meshes in an easy way (essentially in the context of finite difference methods). In nonlinear structure mechanics, the computational costs increase because of the treatment of nonlinearities and finite elements methods are dominant because of the complexity of the geometries. The present work investigates the ability of multigrid methods to reduce the computational times and analyzes the specific problems of formulation and implementation related to the treatment of nonlinearities in the context of finite element methods. This work is conducted on contact problems involving unilateral contact and friction between an elastic body and a rigid obstacle. The nonlinearities are stiff because the contact behavior laws are nonsmooth (the nonpenetration is characterized by the nonregularized Signorini conditions) and nondifferentiable because of the use of the nonregularized Coulomb law.

Key words: Unilateral contact, friction, multigrid.