An augmented integral method for probability distribution evaluation of performance functions

Abstract

The paper proposes an efficient augmented integral method for probability distribution evaluation of performance functions. In the proposed method, the performance function is augmented by adding an auxiliary random variable, whose probability density function (PDF) and cumulative distribution function (CDF) are formulated as the integrations of the original performance function with respect to basic random variables. The optimal auxiliary random variable is determined to provide an accurate estimation of the integrations by investigating the geometric properties of integrands and a distribution parameter optimization approach based on moment analysis. According to the convolution formula, the relationship between the PDFs of the augmented performance function and the original performance function is clarified. Then, the PDF of the original performance function is calculated by solving an unconstrained optimization problem that is established using the convolution formula. Finally, four numerical examples are investigated to demonstrate the efficiency and accuracy of the proposed method for structural reliability analysis. The results indicate that the proposed method can evaluate the probability distribution of performance functions accurately and efficiently, even when the performance functions are strongly nonlinear and implicit.