ITERATIVE SOLUTION OF LARGE THREE-DIMENSIONAL BEM ELASTOSTATIC ANALYSES USING THE GMRES TECHNIQUE

摘要

The GMRES (Generalized Minimal RESidual) iterative method is receiving increased attention as a solver for the large dense and unstructured matrices generated by boundary element elastostatic analyses. Existing published results are predominantly for two-dimensional problems, of only medium size. When these methods are applied to large three-dimensional problems, which actually do require efficient iterative methods for practical solution, they fail. This failure is exacerbated by the use of the pre-conditioning otherwise desirable in such problems. The cause of the failure is identified as being in the orthogonalization process, and is demonstrated by the divergence of the true residual, and the residual calculated during the GMRES algorithm. It is shown that double precision arithmetic is required for only the small fraction of the work comprising the orthogonalization process, and exploitation of this largely removes the penalties associated with the use of double precision. Additionally, it is shown that full re-orthogonalization can be employed to overcome the lack of convergence, extending the applicability of the GMRES to significantly larger problems. The approach is demonstrated by solving three-dimensional problems comprising 4000 and 5000 equations.