An equivalent expectation evaluation method for approximating the probability distribution of performance functions

Abstract

This paper proposes an equivalent expectation evaluation method for approximating the probability distribution of performance functions. The innovation of the proposed method is to formulate the probability distribution of the performance function as an expectation regardless of the dimensionality and complexity of the problem. When a monotonic inverse function (MIF) of the performance function with respect to a certain random variable can be obtained, the probability distribution can be exactly formulated as an expectation related to the distribution of this variable and the MIF. While if such a MIF does not exist or cannot be easily obtained, by constructing two simple auxiliary functions whose MIF can be obtained, a generalized method is proposed to formulate the probability distribution of the performance function as an expectation related to the distribution of the auxiliary functions. Then, the Sobol sequence and point estimate method are employed to determine the expectation. The efficiency and accuracy of the proposed method are demonstrated through four numerical examples. The results show that the proposed method is applicable to approximating probability distributions of both static and dynamic reliability problems involving high-dimensional, nonlinear, or implicit performance functions.