Multigrid for the dual formulation of the frictionless Signorini problem

We examine the dual formulation of the frictionless Signorini problem for a deformable body in contact with a rigid obstacle. We discretize the problem by means of the finite element method. Since the dual formulation solves directly for the stress variable and is not affected by locking, it is very attractive for many engineering applications. However, it is hard to solve it efficiently, since many challenges arise. First, the stress belongs to the non-Sobolev space H-div. Second, the matrix block related to the stress is only semi-positive definite in the incompressible limit. Third, global equality constraints and box-constraints are enforced. In this paper, we propose a novel and optimal nonlinear multigrid method for the dual formulation of the Signorini problem, that works even in the incompressible limit. We opt for the combination of a truncation of the basis functions strategy and a nonlinear monolithic patch smoother with Robin conditions of parameter alpha. Numerical experiments show that multigrid performance is recovered if alpha is chosen properly. We propose an algorithm to dynamically update the parameter alpha during the multigrid process, in order to provide a near optimal value of alpha.

Automated summary

This paper examines the dual formulation of the frictionless Signorini problem for a deformable body in contact with a rigid obstacle and proposes a novel and optimal nonlinear multigrid method to solve it. The dual formulation directly solves for the stress variable and is not affected by locking, making it attractive for many engineering applications. However, solving it efficiently poses several challenges, including the stress belonging to the non-Sobolev space H-div, the semi-positive definiteness of the stress-related matrix block in the incompressible limit, and the enforcement of global equality constraints and box-constraints.