Solution techniques for sparse systems of equations arising in the finite element method.

摘要

Today, finite element method has become an integral part of many engineering designs, and the solution of sparse systems of equations is generally the most computationally intensive step in the solution of finite element problems. The direct solution of sparse linear systems of equations on vector and parallel architectures is the topic of this thesis. Several topics are described here which are all aimed at the solution of very large systems of equations using the multifrontal method, and are all intended to improve the efficiency of the solution step in terms of computation time, and other resources such as memory and secondary storage requirements.; Some background information is first presented on the finite element method and the solution techniques used to solve the sparse systems of equations arising in this field. An efficient method for solving very large systems of equations with very small memory requirements is described next. The results presented show that it is possible to implement an efficient out-of-core multifrontal solver. This will enable users to solve much larger problems on a given machine and to increase throughput in multiuser environments.; An ordering method suitable for indefinite matrices is presented next which is a variation on the minimum degree method. This method is found to be superior to the basic minimum degree ordering when applied to indefinite systems of equations which include Lagrange multipliers representing constraint equations.; A memory efficient method which enhances parallelism is described next. This method which is based on the Jess and Kees algorithm takes advantage of the finite element grid connectivity information to reduce the CPU time and memory requirements of the Jess and Kees algorithm.; An efficient method of solving systems of equations with many right hand side vectors is also presented. This method shows particular advantages in systems which consist of several smaller decoupled systems or systems whose inverse is also sparse. Finally, Concluding remarks and some recommendations on future work in the area is given.