System reliability analysis with small failure probability based on active learning Kriging model and multimodal adaptive importance sampling

System reliability analysis with small failure probability is investigated in this paper. Because multiple failure modes exist, the system performance function has multiple failure regions and multiple most probable points (MPPs). This paper reports an innovative method combining active learning Kriging (ALK) model with multimodal adaptive important sampling (MAIS). In each iteration of the proposed method, MPPs on a so-called surrogate limit state surface (LSS) of the system are explored, important samples are generated, optimal training points are chosen, the Kriging models are updated, and the surrogate LSS is refined. After several iterations, the surrogate LSS will converge to the true LSS. A recently proposed evolutionary multimodal optimization algorithm is adapted to obtain all the potential MPPs on the surrogate LSS, and a filtering technique is introduced to exclude improper solutions. In this way, the unbiasedness of our method is guaranteed. To avoid approximating the unimportant components, the training points are only chosen from the important samples located in the truncated candidate region (TCR). The proposed method is termed as ALK-MAIS-TCR. The accuracy and efficiency of ALK-MAIS-TCR are demonstrated by four complicated case studies.