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

Abstract

ABASTRACTThe 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 distributed memory computer cluster. Through alternatively performing global equilibrium computation and local relaxation, our proposed algorithm will reach the specific accuracy requirement in a few of iterative steps. Moreover, each set/slave-processor majorly communicate with its nearest neighbors, and the transferring data between sets/slave-processors and master is always far below the set-neighbor communication. The corresponding algorithm for implicit finite element analysis has been implemented based on MPI library, and a super large 2-dimension square system of triangle-lattice truss structure under random static loads is simulated with over one billion degrees of freedom and up to 2001 processors on "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 application in other implicit simulations.

A Hybrid Molecular Dynamics/Atomic-Scale Finite Element Method for Quasi-Static Atomistic Simulations at Finite Temperature

“…That is because in pure MD, larger systems need longer physical relaxation time, and in each time step the time consumption is on order-M When the number of atoms is large, MD can be parallelized to use thousands of CPU/cores to simulate large or complex molecular systems. We have developed a hierarchical parallel algorithm for the solution of superlarge-scale sparse linear equations [45]. It has good parallel efficiency and can deal with more than 1 x 10 unknowns in implicit FEM.…”

“…When the number of atoms is large, MD can be parallelized to use thousands CPU/cores to simulate large or complex molecular systems. We have developed a hierarchical parallel algorithm for the solution of super large-scale sparse linear equations (Xu R et al, 2013). It has good parallel efficiency and can deal with more than one billion unknowns in implicit FEM.…”

Broadband transmission losses and time dispersion maps from time-domain numerical simulations in ocean acoustics

“…When used in conjunction with the spectral-element method in the time domain, the implementation of the proposed way of computing TL or time dispersion maps thus has the additional advantage that, contrary to full-wave methods in the frequency domain, the time-domain spectral-element method does not exhibit decreasing performance when increasing the number of processor cores used to perform the calculations. Matrix system solvers (linear solvers) are needed when solving the wave equation in the frequency domain, and their known performance scaling issues on large machines above a thousand processor cores or so (Xu et al, 2013), which is not that high by current high-performance computing standards, implies that some large problems are numerically difficult to handle in the frequency domain, even on the current largest supercomputers.…”