Two-Level preconditioning method for solving saddle point systems in contact computation

TL;DR

This work addresses the challenge of solving saddle-point linear systems arising from Lagrange multiplier formulations of 2D tied contact problems. It introduces a two-level algebraic multigrid preconditioner that uses physics-based coarsening and two interpolation/smoothing strategies, yielding a coarse-grid operator equal to the Schur complement and SPD, enabling efficient coarse-grid solves with AMG. The method demonstrates rapid convergence (often within 30 iterations) and substantial time savings (often under 2 seconds for large problems) across multiple models, with favorable parallel scalability. These contributions provide a practical, scalable solver for mortar-based contact simulations and related saddle-point problems.

Abstract

In contact mechanics computation, the constraint conditions on the contact surfaces are typically enforced by the Lagrange multiplier method, resulting in a saddle point system. Given that the saddle point matrix is indefinite, solving these systems presents significant challenges. For a two-dimensional tied contact problem, an efficient two-level preconditioning method is developed. This method utilizes physical quantities for coarsening, introducing two types of interpolation operators and corresponding smoothing algorithms. Additionally, the constructed coarse grid operator exhibits symmetry and positive definiteness, adequately reflecting the contact constraints. Numerical results show the effectiveness of the method.

Figures (3)

• Figure 1: 2D two-body tied contact model.
• Figure 2: Tied contact example models.
• Figure 3: Comparison of results before and after approximation.

Theorems & Definitions (14)

• Proposition 3.1
• proof
• Proposition 3.2
• Lemma 3.1
• proof
• Proposition 3.3
• proof
• Theorem 3.4
• proof
• Proposition 3.5