Rg. Ghanem et Rm. Kruger, NUMERICAL-SOLUTION OF SPECTRAL STOCHASTIC FINITE-ELEMENT SYSTEMS, Computer methods in applied mechanics and engineering, 129(3), 1996, pp. 289-303
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. Two efficient solution procedures are pr
esented that dramatically reduce the amount of computations involved i
n numerically solving these problems. A brief review is first provided
of the underlying spectral approach which highlights the peculiar str
ucture of the matrices generated and how their properties are related
to both the level of approximation involved as well as to the converge
nce behavior of the proposed solution procedure. The differences betwe
en these matrices from their deterministic finite element counterparts
are illustrated. An iterative solution scheme is proposed, which util
izes their specific properties for efficient memory management and enh
anced convergence behavior. Results from numerical tests are presented
. Comparisons with standard algorithms illustrate the efficiency of th
e proposed algorithm. The second solution procedure presented in this
paper is based on hierarchical basis concepts. Results from numerical
tests are again provided, and the limitations of this approach are ass
essed. The performance of both proposed algorithms indicates that the
linear algebraic systems from the underlying SSFEM formulation can be
solved with considerably less effort in memory and computation time th
an their size suggests. Furthermore, the data structures and the hiera
rchical concept introduced in this study are found to have great poten
tial for the future development of adaptive procedures in stochastic F
EM.