A robust adaptive algebraic multigrid linear solver for structural mechanics

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size linear systems, especially when accurate results are sought for derived variables, like stress or deformation fields. Such a task represents the most time-consuming kernel, and motivates the development of robust and efficient linear solvers for these applications. On the one hand, direct solvers are robust and easy to use, but their computational complexity in the best scenario is superlinear, which limits applicability according to the problem size. On the other hand, iterative solvers, in particular those based on algebraic multigrid (AMG) preconditioners, can reach up to linear complexity, but require more knowledge from the user for an efficient setup, and convergence is not always guaranteed, especially in ill-conditioned problems. In this work, we present a novel AMG method specifically tailored for ill-conditioned structural problems. It is characterized by an adaptive factored sparse approximate inverse (aFSAI) method as smoother, an improved least-squared based prolongation (DPLS) and a method for uncovering the near-null space that takes advantage of an already existing approximation. The resulting linear solver has been applied in the solution of challenging linear systems arising from real-world linear elastic structural problems. Numerical experiments prove the efficiency and robustness of the method and show how, in several cases, the proposed algorithm outperforms state-of-the-art AMG linear solvers. Even more important, the results show how the proposed method gives good results even assuming a default setup, making it fully adoptable also for non-expert users and commercial software.