Probability density estimation of polynomial chaos and its application in structural reliability analysis

Polynomial chaos expansion (PCE) is a widely used approach for establishing the surrogate model of a time-consuming performance function for the convenience of uncertainty quantification of a stochastic structure. However, it remains difficult to calculate the probability density function (PDF) of the PCE accurately for general cases, though the PDF, as a complete representation of a random variable, is often required in some uncertainty problems. To address this problem, this paper proposes a semi-analytical method to compute the PDF of a PCE. This method derives the closed-form solutions of characteristic functions (CFs) of the first- and second-order PCEs, while an equivalent parabolization technique is proposed to provide the approximate solutions of CFs of higher-order PCEs. Then, the PDF of the PCE can be obtained by the Fourier transform of the resulting CF. Three numerical examples are investigated to demonstrate the accuracy, applicability, and efficiency of the proposed method for probability density estimation of PCE in structural reliability analysis.