Scalable Hierarchical Parallel Algorithm for the Solution of Super Large-Scale Sparse Linear Equations

Abstract

The parallel linear equations solver capable of effectively using 1000+ processors becomes the bottleneck of large-scale implicit engineering simulations. In this paper, we present a new hierarchical parallel master-slave-structural iterative algorithm for the solution of super large-scale sparse linear equations in a distributed memory computer cluster. Through alternatively performing global equilibrium computation and local relaxation, the specific accuracy requirement can be met in a few iterations. Moreover, each set/slave-processor majorly communicates with its nearest neighbors, and the transferring data between sets/slave-processors and the master-processor is always far below the communication between neighboring sets/slave-processors. The corresponding algorithm for implicit finite element analysis has been implemented based on the MPI library, and a super large 2-dimension square system of triangle-lattice truss structure under randomly distributed loadings is simulated with over 1 × 109 degrees of freedom (DOF) on up to 2001 processors of the “Exploration 100” cluster in Tsinghua University. The numerical experiments demonstrate that this algorithm has excellent parallel efficiency and high scalability, and it may have broad applications in other implicit simulations.