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

Abstract

SUMMARYA 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.

Predicting Harmonic Distortion of Multiple Converters in a Power System

“…The dimension reduction [22,29,33,34] is an approximate technique for estimating the statistical moments of an output function. The technique involves an additive decomposition of an -dimensional function involvingdimensional integral into a series sum of -dimensional functions such that < .…”

“…The UDR method is briefly described below while a detailed mathematical derivation of the techniques can be found in [33].…”

Stochastic dynamic systems with complex‐valued eigensolutions

“…, x N } T ∈ R N , where subscript m = R (real) or I (imaginary). A dimensional decomposition of m (x) and / m (x), 966 S. RAHMAN described by [12][13][14][15][16][17][18][19][20] …”

“…Further generalizations to multi-cut HDMR [19] and monomial preconditioning (mp)-cut-HDMR [20] have also appeared. Recently, the author's group exploited this decomposition technique in calculating statistical moments of response [12] and reliability [13] of uncertain mechanical systems.…”

Uncertainty propagation analysis by an extended sparse grid technique

“…Some existing methods, namely, Monte Carlo simulation (MCS) [11][12][13], Taylor expansion-based method [14,15], reliability-based method [16][17][18], polynomial chaos expansion (PCE) [19][20][21] and numerical integration-based method [1,22,23] have already contributed to UP. The MCS method was first used in simulating a neutron chain reaction.…”