Forward and inverse structural uncertainty propagations under stochastic variables with arbitrary probability distributions

In this study, a general frame of the forward and inverse structural uncertainty propagations (UPs) based on the dimension reduction (DR) method and the derivative lambda probability density function (λ-PDF) is proposed to handle arbitrary probability distribution. For the forward UP, a DR method is applied to decompose a multivariable system into multiple univariate subsystems and a derivative λ-PDF is adopted to transform the arbitrary probability distribution of each subsystem. Then, the statistical moments and a fitting region are mathematically derived to analyze the fitting ability of the derivative λ-PDF. According to whether the kurtosis-skewness point lies in or out the fitting region, two different strategies combining the Gauss-Gegenbauer quadrature are proposed to implement the forward propagation. Compared with the conventional methods, the proposed method has advantages in higher accuracy, stability and efficiency. For the inverse propagation, because the unknown variable may be arbitrary distribution, the general frame based on the derivative λ-PDF and the Gauss-Gegenbauer quadrature are utilized to convert the uncertainty propagation into multiple deterministic calculations. Based on this, optimization method is adopted to accurately obtain the statistical moments and PDFs of the unknown stochastic variables. Five examples are provided to verify the accuracy and efficiency of the proposed general frame for the forward and inverse UPs.