Shifted Penalty Multigrid Method for Obstacle Problems in Elasticity
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High-performance computing is essential for efficiently solving large-scale contact problems. Simulating such phenomena at engineering scale is often limited by computational resources, making it crucial to design algorithms that fully exploit modern hardware like multi-core CPUs and GPUs. Iterative solvers and preconditioners play a central role in this efficiency. Monotone Multigrid (MMG) methods offer optimal complexity and robustness. In parallel, Penalty and Augmented Lagrangian methods handle over constrained and fuzzy constraints effectively. Among these, the Shifted-Penalty method is notable for accurately enforcing constraints while remaining competitive with non-smooth techniques like the semi-smooth Newton method. To combine the optimal complexity of MMG with the flexibility of shifted-penalty methods, we introduce the Shifted-Penalty Multigrid (SPMG) method. Designed from the ground up for GPU architectures, SPMG unifies nonlinear smoothing with constraint-aware multigrid strategies. Our implementation uses matrix-free differential operators and memory-efficient semi-structured meshes to discretize elasticity equations. We present the SPMG algorithm with a focus on nonlinear smoothing and constraint coarsening. We evaluate performance on the Grace-Hopper superchip of the CSCS Alps supercomputer. Emphasis is placed on single-node GPU performance, kernel design, and convergence behavior in simple contact scenarios. Finally, we demonstrate SPMG’s scalability on large-scale problems with hundreds of millions of degrees of freedom.
