An efficient algorithm for estimating profust failure probability function under the assumption of probable input and fuzzy state

Under the assumption of probable input and fuzzy state (profust), profust (also named as generalized) failure probability function (G-FPF), which varies with random input distribution parameters (DP) in the interested region, can reflect the effect of DP on structure safety and decouples the generalized reliability-based design optimization. The direct double-loop analysis of G-FPF, which repeatedly estimates the G-FPF values at different DP realizations, is timeconsuming. Thus, this paper proposes a single-loop importance sampling (IS) method to estimate G-FPF by combining a variance reduction technique with a sample information-sharing strategy. The proposed method has two innovations. The first is constructing an optimal unified IS density (ISD), which is independent of the DP and envelops the interested DP region. By sharing the sample of the unified ISD, the double-loop analysis for G-FPF can be avoided, and by fusing the IS variance reduction technique, the efficiency of estimating G-FPF can be improved further. The second is designing an adaptive strategy to update the Kriging model of performance function, so that the computational cost, which is measured by the number of performance function evaluations while ensuring the acceptable precision of G-FPF estimation, can be reduced in approaching and sampling the optimal unified ISD as well as predicting the performance function at the sample of the optimal unified ISD. Moreover, the proposed method has wide applicability, and it has no restriction on the nonlinearity of the performance function and the size of the interested DP region, which is sufficiently verified by the presented examples.