Sample-based and sample-aggregated based Galerkin projection schemes for structural dynamics

A comparative study of two new Galerkin projection schemes to compute the response of discretised stochastic partial differential equations is presented for discretised structures subjected to static and dynamic loads. By applying an eigen-decomposition of a discretised system, the response of a discretised system can be expressed with a reduced basis of eigen-components. Computational reduction is subsequently achieved by approximating the random eigensolutions, and by only including dominant terms. Two novel error minimisation techniques have been proposed in order to lower the error introduced by the approximations and the truncations: (a) Sample-based Galerkin projection scheme, (b) Sample-aggregated based Galerkin projection scheme. These have been applied through introducing unknown multiplicative scalars into the expressions of the response. The proposed methods are applied to analyse the bending of a cantilever beam with stochastic parameters undergoing both a static and a dynamic load. For the static case the response is real, however the response for the case of a dynamic loading is complex and frequency-dependent. The results obtained through the proposed approaches are compared with those obtained by utilising a direct Monte Carlo approach.