Estimating augmented failure probability by an augmented importance sampling method based on vMFNM model

For the structure with uncertain distribution parameters of random inputs, the prior and posterior augmented failure probability (AFP) can be used to respectively evaluate the prior reliability level and the posterior one under newly observations. However, for the common engineering problem with small magnitude of the prior and posterior AFP and high-dimensional inputs, there is still a lack of efficient algorithm. Therefore, this paper proposes an augmented importance sampling (AIS) method based on the von Mises-Fisher-Nakagami mixture (vMFNM) model for estimating prior and posterior AFP. In the proposed method, the parameterized vMFNM is used to approach the optimal IS density for estimating AFP, and by minimizing the difference of two densities, the parameters of the vMFNM can be determined and then the AFP can be efficiently estimated. To further improve the efficiency of the proposed algorithm, the strategies of layered constructing vMFNM model and adaptively training surrogate model of performance function are incorporated into the proposed method, which further reduces the performance function evaluations for estimating small AFP. The example results show that the computational accuracy and efficiency of the proposed method are significantly better than those of existing methods, especially for the problem with high-dimensional inputs and small magnitude of the AFP.