A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics

Abstract

A new, generalized, multivariate dimension‐reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of an N‐dimensional response function into at most S‐dimensional functions, where S≪N; an approximation of response moments by moments of input random variables; and a moment‐based quadrature rule for numerical integration. A new theorem is presented, which provides a convenient means to represent the Taylor series up to a specific dimension without involving any partial derivatives. A complete proof of the theorem is given using two lemmas, also proved in this paper. The proposed method requires neither the calculation of partial derivatives of response, as in commonly used Taylor expansion/perturbation methods, nor the inversion of random matrices, as in the Neumann expansion method. Eight numerical examples involving elementary mathematical functions and solid‐mechanics problems illustrate the proposed method. Results indicate that the multivariate dimension‐reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods, such as first‐ and second‐order Taylor expansion methods, statistically equivalent solutions, quasi‐Monte Carlo simulation, and the fully symmetric interpolatory rule. While the accuracy of the dimension‐reduction method is comparable to that of the fourth‐order Neumann expansion method, a comparison of CPU time suggests that the former is computationally far more efficient than the latter. Copyright © 2004 John Wiley & Sons, Ltd.