Obtaining the statistical moments of system responses remains one of the main topics of stochastic analysis. This paper presents a new adaptive point estimate method (PEM) based on the exact dimension reduction method in terms of vector. Firstly, the rigorous and exact dimension reduction method in terms of vector is derived theoretically. Secondly, by introducing the nonnormal-to-normal transformation, the original variables are transformed into independent standard normal variables which are classified into several sub-vectors based on the delineation of the cross terms, and then the response moments can be described by an explicit function of the moments of multiple component functions. Thirdly, by combining with the two different approaches for estimating the moments of component sub-vector function, an adaptive PEM based on the exact dimension reduction method in terms of vector, which comprises two sub-methods, is proposed. Finally, several examples illustrate the accuracy and efficiency of the proposed method. (C) 2019 Elsevier Ltd. All rights reserved.

In reliability analysis of geotechnical engineering, the performance function is usually implicit and has strong nonlinearity. However, practical moment methods such as JC method and SORM are mainly applicable to the explicit performance function. In this paper, an improved fourth-moment method for reliability analysis of geotechnical engineering is presented by combining the efficient point estimate method for probability moments with the high order moments method for reliability analysis. Firstly, by introducing independent transformation and linear transformation, the performance function becomes a function of the reference variables, and then an efficient estimate method for the first four moments of performance function is established, which combines the univariate dimension-reduction method for multivariable function with the method for the points and weights of reference variables. Secondly, based on the combination of the estimated probability moments estimation and the cubic normal transformation, an improved fourth-moment method to assess reliability of geotechnical engineering is proposed. Finally, two numerical cases are investigated to verify the proposed point estimate method for probability moments, and results indicate that the proposed estimate method is efficient and accurate, and a classical geotechnical engineering case is used to illustrate the easy-implementation, efficiency and accuracy of the proposed fourth-moment method for reliability analysis of geotechnical engineering.

Point estimate method (PEM) is an efficient approach for stochastic system analysis. For most of PEMs, the precision depends on the number of nodes which is always determined subjectively and empirically. In this work, two more objective PEMs are proposed. One is the direct iterative point estimate method (DIPEM), in which all moments from different nodes are compared until the results converge. The other is the adaptive iterative point estimate method (AIPEM), in which the nonlinear degree of function is deduced, then the number of nodes is determined rationally and the moments are obtained. Several numerical cases are analyzed to verify the proposed methods. Results indicate that the efficiency of existing PEMs is the best, but the results may be inaccurate, especially for higher statistical moments. The precision of DIPEM and AIPEM are always high enough, and AIPEM is more efficient relatively, more practical.

Purpose The purpose of this paper is to find an accurate, efficient and easy-to-implement point estimate method (PEM) for the statistical moments of random systems. Design/methodology/approach First, by the theoretical and numerical analysis, the approximate reference variables for the frequently used nine types of random variables are obtained; then by combining with the dimension-reduction method (DRM), a new method which consists of four sub-methods is proposed; and finally, several examples are investigated to verify the characteristics of the proposed method. Findings Two types of reference variables for the frequently used nine types of variables are proposed, and four sub-methods for estimating the moments of responses are presented by combining with the univariate and bivariate DRM. Research limitations/implications In this paper, the number of nodes of one-dimensional integrals is determined subjectively and empirically; therefore, determining the number of nodes rationally is still a challenge. Originality/value Through the linear transformation, the optimal reference variables of random variables are presented, and a PEM based on the linear transformation is proposed which is efficient and easy to implement. By the numerical method, the quasi-optimal reference variables are given, which is the basis of the proposed PEM based on the quasi-optimal reference variables, together with high efficiency and ease of implementation.