Parallel implementation of the finite element method on shared memory multiprocessors

Abstract

The work presented in this thesis concerns parallel methods for finite element analysis. The research has been funded by British Gas and some of the presented material involves work on their software. Practical problems involving the finite element method can use a large amount of processing power and the execution times can be very large. It is consequently important to investigate the possibilities for the parallel implementation of the method. The research has been carried out on an Encore Multimax, a shared memory multiprocessor with 14 identical CPU's. We firstly experimented on autoparallelising a large British Gas finite element program (GASP4) using Encore's parallelising Fortran compiler (epf). The par- allel program generated by epj proved not to be efficient. The main reasons are the complexity of the code and small grain parallelism. Since the program is hard to analyse for the compiler at high levels, only small grain parallelism has been inserted automatically into the code. This involves a great deal of low level syn- chronisations which produce large overheads and cause inefficiency. A detailed analysis of the autoparallelised code has been made with a view to determining the reasons for the inefficiency. Suggestions have also been made about writing programs such that they are suitable for efficient autoparallelisation. The finite element method consists of the assembly of a stiffness matrix and the solution of a set of simultaneous linear equations. A sparse representation of the stiffness matrix has been used to allow experimentation on large problems. Parallel assembly techniques for the sparse representation have been developed. Some of these methods have proved to be very efficient giving speed ups that are near ideal. For the solution phase, we have used the preconditioned conjugate gradient method (PCG). An incomplete LU factorization ofthe stiffness matrix with no fill- in (ILU(O)) has been found to be an effective preconditioner. The factors can be obtained at a low cost. We have parallelised all the steps of the PCG method. The main bottleneck is the triangular solves (preconditioning operations) at each step. Two parallel methods of triangular solution have been implemented. One is based on level scheduling (row-oriented parallelism) and the other is a new approach called independent columns (column-oriented parallelism). The algorithms have been tested for row and red-black orderings of the nodal unknowns in the finite element meshes considered. The best speed ups obtained are 7.29 (on 12 processors) for level scheduling and 7.11 (on 12 processors) for independent columns. Red-black ordering gives rise to better parallel performance than row ordering in general. An analysis of methods for the improvement of the parallel efficiency has been made.