A stratified beta-sphere sampling method combined with important sampling and active learning for rare event analysis

Accurate and efficient estimation of small failure probability subjected to high-dimensional and multiple failure domains is still a challenging task in structural reliability engineering. In this paper, we propose a stratified beta-spheres sampling method (SBSS) to tackle this task. Initially, the whole support space of random input variables is divided into a series of subdomains by using multiple specified beta-spheres, which is a hypersphere centered in the origin in standard normal space, then, the corresponding samples truncated by beta-spheres are generated explicitly and efficiently. Based on the truncated samples, the real failure probability can be estimated by the sum of failure probabilities of these subdomains. Next, we discuss and demonstrate the unbiasedness of the estimation of failure probability. The proposed method stands out for inheriting the advantages of Monte Carlo simulation (MCS) for highly nonlinear, high-dimensional problems, and problems with multiple failure domains, while overcoming the disadvantages of MCS for rare event. Furthermore, the SBSS method equipped with importance sampling technique (SBSS-IS) is also proposed to improve the robustness of estimation. Additionally, we combine the proposed SBSS and SBSS-IS methods with GPR model and active learning strategy so as to further substantially reduce the computational cost under the desired requirement of estimated accuracy. Finally, the superiorities of the proposed methods are demonstrated by six examples with different problem settings.