MPI solver for 3D elasticity problems. (English) Zbl 1180.74055
Summary: 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 techniques, displacement decomposition and circulant block-factorization, are highly parallelizable.
A parallel algorithm is developed 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 MPI parallel FEM code is developed. Numerical tests for real-life engineering problems of the geomechanics in geosciences on a number of modern parallel computers are presented. The reported speed-up and parallel efficiency well illustrate the parallel features of the proposed method and its implementation.
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 techniques, displacement decomposition and circulant block-factorization, are highly parallelizable.
A parallel algorithm is developed 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 MPI parallel FEM code is developed. Numerical tests for real-life engineering problems of the geomechanics in geosciences on a number of modern parallel computers are presented. The reported speed-up and parallel efficiency well illustrate the parallel features of the proposed method and its implementation.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74B15 | Equations linearized about a deformed state (small deformations superposed on large) |
| 74L10 | Soil and rock mechanics |
| 65Y05 | Parallel numerical computation |
Software:
MPIReferences:
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| [3] | Blaheta, R., Displacement decomposition—incomplete factorization preconditioning techniques for linear elasticity problems, Num. Lin. Alg. Appl., 1, 107-128 (1994) · Zbl 0837.65021 |
| [5] | Lirkov, I.; Margenov, S.; Vassilevski, P. S., Circulant block-factorization preconditioners for elliptic problems, Computing, 53, 1, 59-74 (1994) · Zbl 0810.65039 |
| [6] | Lirkov, I.; Margenov, S.; Zikatanov, L., Circulant block-factorization preconditioning of anisotropic elliptic problems, Computing, 58, 3, 245-258 (1997) · Zbl 0879.65022 |
| [8] | Walker, D.; Dongara, J., MPI: a standard message passing interface, Supercomputer, 63, 56-68 (1996) |
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