AMG with Filtering: An Efficient Preconditioner for Interior Point Methods in Large-Scale Contact Mechanics Optimization
Socratis Petrides, Tucker Hartland, Tzanio Kolev, Chak Shing Lee, Michael Puso, Jerome Solberg, Eric B. Chin, Jingyi Wang, Cosmin Petra
TL;DR
AMGF introduces a scalable preconditioner that augments algebraic multigrid with a filtering subspace correction to address ill-conditioning in Newton-based interior-point methods for large-scale contact mechanics. By focusing the correction on the small contact-subspace associated with the contact interface and reducing the rest of the system with AMG, AMGF achieves bounded, mesh-insensitive performance and robust convergence in both linear and nonlinear, large-scale problems. Theoretical analysis shows a near-constant bound on the preconditioned system's condition number, and numerical experiments on two-block, ironing, and beam-sphere benchmarks confirm that AMGF outperforms standard AMG in the IP setting, especially under mesh refinement and tightening contact constraints. The approach is presented as broadly applicable to problems where solver performance deteriorates due to a low-dimensional subspace, making Newton-based IP methods more tractable in challenging engineering simulations.
Abstract
Large-scale contact mechanics simulations are crucial in many engineering fields such as structural design and manufacturing. In the frictionless case, contact can be modeled by minimizing an energy functional; however, these problems are often nonlinear, non-convex, and increasingly difficult to solve as mesh resolution increases. In this work, we employ a Newton-based interior-point (IP) filter line-search method; an effective approach for large-scale constrained optimization. While this method converges rapidly, each iteration requires solving a large saddle-point linear system that becomes ill-conditioned as the optimization process converges, largely due to IP treatment of the contact constraints. Such ill-conditioning can hinder solver scalability and increase iteration counts with mesh refinement. To address this, we introduce a novel preconditioner, AMG with Filtering (AMGF), tailored to the Schur complement of the saddle-point system. Building on the classical algebraic multigrid (AMG) solver, commonly used for elasticity, we augment it with a specialized subspace correction that filters near null space components introduced by contact interface constraints. Through theoretical analysis and numerical experiments on a range of linear and nonlinear contact problems, we demonstrate that the proposed solver achieves mesh independent convergence and maintains robustness against the ill-conditioning that notoriously plagues IP methods. These results indicate that AMGF makes contact mechanics simulations more tractable and broadens the applicability of Newton-based IP methods in challenging engineering scenarios. More broadly, AMGF is well suited for problems, optimization or otherwise, where solver performance is limited by a problematic low-dimensional subspace. This makes the method widely applicable beyond contact mechanics and constrained optimization.