Cross-entropy-based directional importance sampling with von Mises-Fisher mixture model for reliability analysis

In structural reliability analysis, the estimation of rare failure probability is a significant challenge. Directional importance sampling may address this challenge when a quasi-optimal sampling density is chosen. Directional importance sampling involves sampling on the unit hypersphere and one-dimensional reliability analysis. This paper develops a cross-entropy-based directional importance sampling (CE-DIS) for reliability analysis. The von Mises-Fisher mixture (vMFM) model is selected as sampling density to generate direction vectors on the unit hypersphere. The updating rules based on the cross-entropy method are derived for the vMFM model to find a quasi-optimal sampling density. In addition, the DBSCAN algorithm is used to choose the number of distributions in the vMFM model in case of no prior information of failure domain, and the intermediate failure event is introduced in case of not enough samples to update the distribution parameters. Several benchmark numerical examples and engineering applications are used to test the performance of the proposed CE-DIS method. The results demonstrate that the CE-DIS method significantly improves the computational efficiency compared to original directional sampling and other CE-based importance sampling methods. In addition, the CE-DIS method can be applied to different types of reliability problems, e.g., multiple design points, multiple failure modes and rare failure probability.