Projection methods for stochastic dynamic systems: A frequency domain approach

A collection of hybrid projection approaches are proposed for approximating the response of stochastic partial differential equations which describe structural dynamic systems. In this study, an optimal basis for the approximation of the response of a stochastically parametrised structural dynamic system has been computed from its generalised eigenmodes. By applying appropriate approximations in conjunction with a reduced set of modal basis functions, a collection of hybrid projection methods are obtained. These methods have been further improved by the implementation of a sample based Galerkin error minimisation approach. In total six methods are presented and compared for numerical accuracy and computational efficiency. Expressions for the lower order statistical moments of the hybrid projection methods have been derived and discussed. The proposed methods have been implemented to solve two numerical examples: the bending of a Euler-Bernoulli cantilever beam and the bending of a Kirchhoff-Love plate where both structures have stochastic elastic parameters. The response and accuracy of the proposed methods are subsequently discussed and compared with the benchmark solution obtained using an expensive Monte Carlo method.