Performance-portable multigrid solvers for semi- and block-structured grids in computational solid mechanics

• Applicants: Dr.-Ing. Matthias Mayr; Professor Dr.-Ing. Alexander Popp
• Subject Area: Mechanics ; Applied Mechanics, Statics and Dynamics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 560547102

Computational solid mechanics (CSM) is a long-standing field of research, where generations of scientists have developed mechanical theories, mathematical descriptions, constitutive models, discretization schemes and other numerical tools to study and analyze complex systems in science, engineering, and biomedicine. One particular success story, the Finite Element Method (FEM), began more than 80 years ago and has since conquered all application areas of CSM. On the other hand, computing technology - in particular the use of distributed and parallel computing or high-performance computing - has transformed simulation technology, allowing larger and more sophisticated models to be solved on laptops, workstations, and supercomputers. However, they often require specialized methods and algorithms to take full advantage of the available computing power. So far, existing simulation frameworks have been enriched with many models and capabilities to address today's predictive needs in CSM, however they often cannot exploit the full power of modern parallel computers. In contrast to conventional approaches, this project pursues a non-invasive approach to remove the most restrictive bottleneck in the implicit treatment of CSM problems, namely the linear solver, by improving its efficiency without requiring extensive code modifications. Recognizing prevalent mesh generation workflows for FEM and isogeometric analysis (IGA), which often result in semi- or block-structured grids to represent a complex geometry by a mesh with localized grid structure, the construction and execution of multigrid preconditioners can exploit the grid structure to accelerate the performance by at least a factor of 10. To achieve such speed-ups for FEM and IGA schemes and make them available to the entire CSM community, this project will develop performance-portable algebraic multigrid preconditioners for semi- and block-structured grids. First, the performance of the recently proposed Region-MG framework will be improved through specialized computational kernels tailored to structured grids to speed-up the triple-matrix-product during multigrid setup. These will then be used to devise multi-level block preconditioners for semi- and block-structured grids arising from FEM and IGA discretizations with a particular focus on saddle-point systems in case of multiple subdomains with non-matching mesh interfaces. Innovatively, this project will design algebraic coarsening strategies for IGA, coarsening either the polynomial degree or the mesh. After completion, a performance-portable and scalable multigrid preconditioning framework will be available to the CSM community. Through the interface to the iterative linear solver, it will be usable in existing well-matured application codes without having to modify the existing code base, hence delivering huge computational performance on today’s and tomorrow’s computing hardware in a non-invasive manner.

• DFG Programme: Research Grants