An efficient method combining active learning Kriging and Monte Carlo simulation for profust failure probability

For more and more complicated engineering structures, it is a challenge to efficiently estimate the profust failure probability based on the probability inputs and fuzzy state assumption. By combining active learning Kriging with Monte Carlo simulation (AK-MCS), an efficient method is proposed to estimate the profust failure probability. Firstly, the profust failure probability is transformed into an integral of the classical failure probability by introducing a variable related to the fuzzy state assumption. This integral is further reorganized as a weighted sum of a series of classical failure probabilities by Gaussian quadrature, and the series of the classical failure probabilities have the similar limit state function constructions constrained by different thresholds. Secondly, MCS is used according to the probability input distribution to generate the sample pool, in which the active learning Kriging is used to establish the surrogates of the series of similar limit state functions with different thresholds. An improved learning function is proposed by minimizing the U-learning function minima corresponding to all limit state functions, so that the candidate with the largest effect on the surrogating quality of all limit states can be selected as a training point to update the Kriging model. Once the updating process of the Kriging model converges, all limit state functions can be identified by the Kriging model, and the profust failure probability can be estimated by using the Kriging model without any extra model evaluation. Several examples are used to demonstrate the feasibility of the proposed strategy for estimating the profust failure probability.