Anchored analysis of variance Petrov–Galerkin projection schemes for linear stochastic structural dynamics

Abstract

In this paper, we propose anchored functional analysis of variance Petrov–Galerkin (AAPG) projection schemes, originally developed in the context of parabolic stochastic partial differential equations (Audouze C, Nair PB. 2014 Comput. Methods Appl. Mech. Eng. 276, 362–395. ( doi:10.1016/j.cma.2014.02.023 )) for solving a class of problems in linear stochastic structural dynamics. We consider the semi-discrete form of the governing equations in the time-domain and the proposed formulation involves approximating the dynamic response using a Hoeffding functional analysis of variance decomposition. Subsequently, we design a set of test functions for a stochastic Petrov–Galerkin projection scheme that enables the original high-dimensional problem to be decomposed into a sequence of decoupled low-dimensional subproblems that can be solved independently of each other. Numerical results are presented to demonstrate the efficiency and accuracy of AAPG projection schemes and comparisons are made to approximations obtained using Monte Carlo simulation, generalized polynomial chaos-based stochastic Galerkin projection schemes and the generalized spectral decomposition method. The results obtained suggest that the proposed approach holds significant potential for alleviating the curse of dimensionality encountered in tackling high-dimensional problems in stochastic structural dynamics with a large number of spatial and stochastic degrees of freedom.