Sponsor
PZ and GM thank the Swiss National Science Foundation funding the project “Immersed Methods for Fluid-Structure-Contact-Interaction Simulations and Complex Geometries” (nr.: 215627), and the Platform for Advanced Scientific Computing (PASC) Program funding the project “XSES-FSI: toward eXtreme scale Semi-Structured discretizations for Fluid-Structure Interaction.” PZ thanks USI’s “Fondo Istituzionale per la Ricerca.” AN and PV acknowledge the support from the U.S. NSF RTG grant DMS-2136228. RK and PV thank the Swiss National Science Foundation for Scientific Exchanges funding 229884.
Published In
Journal of Computational Physics
Document Type
Article
Publication Date
1-15-2027
Subjects
Augmented Lagrangian multigrid, Obstacle problems, Linear elasticity, Finite element method
Abstract
Efficient and robust algorithms are essential for solving large-scale contact problems involving complex obstacles. Monotone multigrid (MMG) methods provide a solid foundation with optimal computational complexity, while penalty and augmented Lagrangian formulations offer remarkable flexibility for handling over-constrained systems and fuzzy contact conditions. Among these, the augmented Lagrangian method enforces contact constraints with high accuracy and remains competitive with non-smooth techniques such as the semi-smooth Newton method. We combine the efficiency of multigrid methods with the adaptability of augmented Lagrangian formulations, introducing the Matrix-free Multigrid Augmented Lagrangian (MaMAL) method for contact problems. The approach inherits the optimal complexity of multigrid solvers while retaining the simplicity and flexibility of penalty-based techniques. MaMAL employs multigrid strategies to enforce constraints accurately while supporting matrix-free operators, enabling simulations exceeding 600 million degrees of freedom on a single NVIDIA GH200 GPU. It integrates nonlinear smoothing and constraint-coarsening strategies, incorporating penalty terms directly into the multilevel preconditioner within an augmented Lagrangian iterative framework. This penalty-based approach is particularly effective for fuzzy or over-constrained problems, such as those encountered in fluid-structure-contact interactions. Numerical experiments demonstrate that MaMAL achieves convergence rates and accuracy comparable to state-of-the-art methods, and highlight the impact of algorithmic parameters, including the admissible range of the penalty parameter and the use of a linear line search strategy.
Rights
Copyright (c) 2026 The Authors Creative Commons License This work is licensed under a Creative Commons Attribution 4.0 International License.
Locate the Document
DOI
10.1016/j.jcp.2026.115531
Persistent Identifier
https://archives.pdx.edu/ds/psu/45337
Publisher
Elsevier BV
Citation Details
Marchi, G., Nelson, A., Kothari, H., Vassilevski, P. S., Krause, R., & Zulian, P. (2027). Matrix-free multigrid method for obstacle problems in linear elasticity. Journal of Computational Physics, 569, 115531.