Meta model-based importance sampling combined with adaptive Kriging method for estimating failure probability function

The failure probability function (FPF) can provide the entire effect of the distribution parameter in the concerned region on failure probability (FP), which is important for decoupling the optimal parameter search and constraint analysis in the FP -constrained design optimization. To improve the efficiency of solving the FPF, this paper combines meta model -based importance sampling with adaptive Kriging (Meta -IS -AK). First, based on the theoretical optimal importance sampling density (ISD) for estimating the FP (ISD-FP) at the given distribution parameter, the theoretical optimal ISD for solving FPF (ISD-FPF) is constructed using an integral operator. Then, a Kriging model of the performance function is adaptively trained to establish the quasi -optimal ISD-FPF for approximating the optimal performance function, and a simple rejection sampling (SRS) method is designed to extract the samples of the quasi -optimal ISD-FPF. For recognizing the state of the ISD-FPF sample and completing the IS estimation of FPF, the Kriging model is secondly updated to establish ISD-FPF in the sample pool comprising these ISD-FPF samples. Using the proposed method, the whole FPF corresponding to arbitrary distribution parameters can be obtained using one IS reliability analysis based on the constructed ISD-FPF. Four examples and an application illustrate the advantages of the proposed method.