An importance sampling reliability method combining Kriging and Gaussian Mixture Model through ring subregion strategy for multiple failure modes

The limit state function for multiple failure modes has multiple failure regions, and Monte Carlo (MC) method usually requires a large amount of calculation costs, especially for small failure probability problems. In this paper, active learning Kriging model combined with Gaussian Mixture Model (GMM) is used to establish a novel general reliability analysis method for multiple failure modes based on importance sampling (IS) and ring subregion strategy. The proposed method is proceeded in the standard normal space. Firstly, a concentric circle is set up with the coordinate origin as the center. By expanding the radius of the concentric circle, the whole sample space is divided into multiple ring subregions. The candidate sample points are generated in each ring subregion, and the active learning Kriging model is built to recognize failure regions and obtain failure samples. Next, the convergence criterion of concentric circle radius expansion is defined to ensure that all the failure regions are identified. Failure samples in all subregions are extracted, and GMM is adopted to build an IS probability density function through expectation maximization (EM) algorithm. Finally, IS sample points are generated by the fitted GMM to update Kriging model, and the failure probability is calculated. The results show that: The ring subregion sampling method can effectively identify all the most probable failure regions. GMM can effectively approximate the optimal IS probability density function, and the generated IS sample points are distributed in all failure regions. The proposed method shows highly robustness and stability. Compared with the existing Kriging methods, the proposed method also shows highly efficiency and accuracy.