Computational Procedures in SFEM
This paper addresses the issues involved in
solving systems of linear equations which arise in the context of the spectral
stochastic finite element (SSFEM) formulation. A brief review of the underlying spectral
approach is provided which highlights the peculiar structure of the
matrices generated and how their properties are related to both the
level of approximation involved as well as the convergence behavior of the
proposed solution procedure. The differences of these matrices
from their deterministic finite element counterparts are illustrated.
An iterative solution scheme is proposed, which utilizes their specific
properties for efficiency memory management and enhanced convergence behavior. Further, it is
shown, that the data dependencies within the algorithm suggest that
implementation with a high level of parallelism is readily implementable.
Results from numerical tests are presented. Comparisons with standard
algorithms illustrate the efficiency of the proposed algorithm.
In addition to straightforward iterative procedures,
an alternative method, based on hierarchical concepts, is presented.
Results from numerical tests are again provided, and the limitations of this approach
are assessed. The performance of both algorithms indicates that
the linear systems
from the underlying SSFEM formulation can be solved with considerably
less effort in memory and computation time than their size suggests.
Furthermore, the data structures and the
hierarchical concept introduced in this study are
found to have great potential for the future development of adaptive procedures
in stochastic FEM.