A multi-fidelity integration rule for statistical moments and failure probability evaluations

This paper presents a multi-fidelity integration rule for statistical moments and failure probability evaluations. The contribution-degree analysis is first conducted for dividing the random inputs as relatively important and unimportant ones, where the multi-dimensional Gaussian-weighted integral respect to moments estimation can be separated into two lower-dimensional integrals in an additive form. A flexible spherical-radial cubature rule is derived to evaluate the integral consisting of important random variables, where the free parameter is optimally determined via a moment-matching strategy. A low-degree spherical-radial cubature rule, whose algebraic degree of accuracy is between 3 and 5, is then applied to estimate the integral related to unimportant variables. In this regard, a multi-fidelity integration rule, where different numerical schemes are employed, is established accordingly for estimating the statistical moments of the limit state function, which can ensure the balance of precision and efficiency. The maximum entropy method is then applied to obtain the entire probability distribution of the limit state function based on the statistical moments, where the failure probability can be straightforwardly assessed. The efficiency and accuracy of the proposed method are demonstrated through five numerical examples for both the statistical moments and failure probability evaluations.