Multigrid for the dual formulation of the frictionless Signorini problem

• Published in: International journal for numerical methods in engineering 2023-05, Vol.124 (10), p.2367-2388
• Main Authors: Rovi, Gabriele, Kober, Bernhard, Starke, Gerhard, Krause, Rolf
• Format: Article
• Language: English
• Subjects: Algorithms; Basis functions; dual Signorini problem; Finite element method; Formability; incompressibility; multigrid; nonlinear; Parameters; Robin conditions; Sobolev space
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.7214

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 Hdiv$$ {\mathbf{H}}_{\mathrm{div}} $$. 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 α$$ \alpha $$. Numerical experiments show that multigrid performance is recovered if α$$ \alpha $$ is chosen properly. We propose an algorithm to dynamically update the parameter α$$ \alpha $$ during the multigrid process, in order to provide a near optimal value of α$$ \alpha $$.