Reliability analysis of structures using probability density evolution method and stochastic spectral embedding surrogate model

This study presents an efficient reliability analysis method using probability density evolution method (PDEM) and stochastic spectral embedding (SSE) based surrogate model. The PDEM is used to estimate the structural response's probability density function (PDF). The PDEM is derived based on the principle of probability conservation where generalized density evolution equations (GDEEs) are decoupled from the physical system. The GDEEs are solved using finite difference method coupled with total variation diminishing, in which a set of representative points of random parameters are generated using the generalized F-discrepancy scheme. To obtain satisfactory accuracy of the numerical solution, representative points are needed, which becomes computationally expensive for complex structures. To reduce the computation burden, the SSE is used, which approximates the original response surface. The SSE is a class of supervised machine learning algorithm where it is trained by few observations and enables output prediction as spectral representation. This is achieved by minimizing the residual using domain decomposition technique. To illustrate the proposed SSE-based PDEM, three numerical examples are investigated, including the reliability analysis of four-branch problem and shear building frame subjected to ground acceleration, and the reliability-based design optimization of a moment-resisting frame coupled with the nonlinear energy sink with negative stiffness and sliding friction. Numerical results show that the proposed SSE-based PDEM can estimate failure probability using a very small number of representative points without compromising accuracy compared with Monte Carlo simulation, which leads to a reduction in computational costs.