Parallel Performance of an MPI Solver for 3D Elasticity Problems

The numerical solution of 3D linear elasticity equations is considered. The problem is described by a coupled system of second order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements. The Preconditioned Conjugate Gradient iterative method is used for solving of the large-scale linear algebraic systems arising after the Finite Element Method (FEM) discretization of the problem. Displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block-diagonal part of the original matrix. Then circulant block-factorization is used for preconditioning of the obtained block-diagonal matrix. Both preconditioning techniques, displacement decomposition and circulant block-factorization, are highly parallelizable. A parallel algorithm is invented for the proposed preconditioner. The theoretical analysis of the execution time shows that the algorithm is highly efficient for coarse-grain parallel computer systems. A portable parallel FEM code based on MPI is developed. Numerical tests for real-life engineering problems in computational geomechanics are performed on a number of modern parallel computers: Cray T3E, Sunfire 6800, and Beowulf cluster. The reported speed-up and parallel efficiency well illustrate the parallel features of the proposed method and its implementation.