An improved adaptive bivariate dimension-reduction method for efficient statistical moment and reliability evaluations

Abstract

Statistical moment estimations of the performance function with the aim of balancing accuracy and efficiency still remains a challenge for moment-based structural reliability analysis. In this paper, an improved adaptive bivariate dimension-reduction method in terms of vectors (i-VBDRM) is proposed for efficient statistical moments and reliability evaluations. In the proposed method, the delineation of cross terms of two-dimensional functions involved in the bivariate dimension-reduction method (BDRM) is first implemented, where the random variables are classified into several sub-vectors after the non-normal to normal transformation. Then, the explicit expressions of the moments of multiple component sub-vector functions are formulated, where a novel cubature formula is proposed and the Gauss-Hermite quadrature is employed to evaluate the involved two- and one-dimensional Gaussian-weighted integrals. In that regard, the first-four central moments can be calculated with accuracy and efficiency. Then, the probability density function of the performance function is rebuilt by a flexible distribution model called the shifted generalized lognormal distribution based on the first-four central moments evaluated by the proposed method. To demonstrate the efficacy of the proposed method, four numerical examples are presented, where some other forms of BDRM, univariate dimension-reduction method and the crude Monte Carlo simulation are performed for comparisons. The results show that the proposed method can keep the trade-off of precision and efficiency for both the statistical moment assessments and structural reliability analysis.