Algebraic multigrid methods for saddle point systems arising from mortar contact formulations

TL;DR

This paper introduces a new algebraic multigrid method tailored for nonlinear saddle point contact problems using mortar finite elements, featuring an interface aggregation strategy and saddle point smoothers, demonstrating robustness and scalability.

Contribution

It develops and implements a novel interface aggregation strategy for saddle point systems in contact problems, enhancing multigrid efficiency and robustness.

Findings

Robustness demonstrated in complex dynamic contact problems.

Scalability up to 23.9 million unknowns on 480 MPI ranks.

Open-source implementation in Trilinos MueLu package.

Abstract

In this paper, a fully aggregation-based algebraic multigrid strategy is developed for nonlinear contact problems of saddle point type using a mortar finite element approach. While the idea of extending multigrid methods to saddle point systems can already be found, e.g., in the context of Stokes and Oseen equations in literature, the main contributions of this work are (i) the development and open-source implementation of an interface aggregation strategy specifically suited for generating Lagrange multiplier aggregates that are required for coupling structural equilibrium equations with contact constraints and (ii) a review of saddle point smoothers in the context of constrained interface problems. The new interface aggregation strategy perfectly fits into an aggregation-based multigrid framework and can easily be combined with segregated transfer operators, which allow to preserve the saddle point structure on the coarse levels. Further analysis provides insight into saddle point smoothers applied to contact problems, while numerical experiments illustrate the robustness of the new method. We have implemented the proposed algorithm within the MueLu package of the open-source Trilinos project. Numerical examples demonstrate the robustness of the proposed method in complex dynamic contact problems as well as its scalability up to 23.9 million unknowns on 480 MPI ranks.