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

Abstract

In this article, 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, for example, 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.

Scalable computational kernels for mortar finite element methods

“…Since this paper is concerned with the efficient evaluation of the mortar terms on parallel computing clusters, we will detail the discretization of all mortar-related terms in Section 2.2. However, to keep the focus tight and concise, we refer to the extensive literature for any further details on the finite element formulation and discretization [31,48,49,56,57,76,80], the solution of the nonlinear problem via active set strategies [37,43,45,44,55], as well as for details on the structure of the arising linear systems of equations and efficient solvers thereof [2,14,67,69,70,72,71,77].…”

“…Depending on the specific details of the discretization, the resulting linear system might exhibit saddle-point structure. Efficient preconditioners to be used in conjunction with Krylov solvers are available in literature [2,14,67,69,70,72,71,77] and, thus, are not in the scope of this paper. We rather focus on the cost of evaluating all mortar-related terms.…”

Dynamic load balancing for large‐scale mortar contact formulations

“…By using a nonlinear complementarity function, we can merge the search for the active contact set into a semi-smooth Newton method to solve for all nonlinearities within a single iterative scheme. To solve the linear system in each Newton step, we rely on specialized multi-level preconditioners for mortar-based contact formulations [6,7].…”

A scalable preconditioning framework for stabilized contact mechanics with hydraulically active fractures

“…Algebraic multigrid (AMG, [72]) is one of the most effective multilevel approaches and consists of the complementary use of: (i) a smoother that reduces high frequency errors, (ii) a coarse grid correction that reduces low frequency errors, and (iii) restriction and interpolation operators, to move from one grid to another. Starting from the original works, e.g., [73], a wide range of multigrid approaches has appeared in the literature, extending the applicability of this method, originally designed for elliptic PDEs, to both non-symmetric [74,75] and block matrices [76][77][78][79][80][81]. Nonetheless, robustness and efficiency is still an open issue for AMG whenever used as a black-box tool in problems with these algebraic properties.…”

“…The Jacobian matrix arising from the model considered herein is a non-symmetric 3 × 3 block matrix and, despite the available studies for similar problems, none of them can be straightforwardly and effectively applied to our case. In the context of geomechanical simulations, only a few studies on 2 × 2 block Jacobian systems [80,82,83] are found by the authors.…”