Towards automatic selection of direct vs. iterative solvers for cloud-based finite element analysis

Abstract

The new trend in engineering is to solve complex computational problems in the cloud using high performance computing (HPC) services provided by different vendors. In this paper, we compare performances of direct vs. iterative linear equation solvers to help with the development of job schedulers that can automatically choose the best solver type and tune them (e.g. precondition the matrices) according to job characteristics and workload conditions seen in the HPC cloud services. As a proof of concept, we use three classical elasticity problems, namely a cantilever beam, Lame problem and the stress concentration factor (SCF), whose analytical solutions are well-known. We mesh these linear problems with increasing granularities, which leads to various matrix sizes; the largest having one billion non-zero elements. Detailed finite element analyses using an IBM HPC cluster are executed. We first use the multi-frontal parallel, sparse direct solver MUMPS and evaluate its performance with Cholesky and LU decompositions of the generated matrices with respect to memory usage, and multi-core, multi-node execution performances. As for the iterative solver, we use the PETSc library and carry out studies with several Krylov subspace methods (CG, BiCG, GMRES) and preconditioner combinations (BJacobi, SOR, ASM, None). Finally, we compare and contrast the direct and iterative solver results in order to find the most suitable algorithm for varying cases obtained from numerical modelling of these three-dimensional linear elasticity problems.