A combined Importance Sampling and active learning Kriging reliability method for small failure probability with random and correlated interval variables

Abstract

The existing hybrid reliability analysis (HRA) method (Yang et al., 2015; Zhang et al., 2015; Yang et al., 2015) is found not suitable for estimating small failure probabilities. Meanwhile, the previous ALK-HRA algorithm (ALK-HRA: an active learning HRA method combining Kriging and Monte Carlo simulation) reduces its numerical efficiency when number of uncertain variables increases. Furthermore, the ALK-HRA approach with both random and interval/ellipsoid variables cannot deal with complex “multi-source uncertainty” problems. In order to overcome these issues, therefore, the following strategies is proposed: 1) First, a more general HRA (MGHRA) method with both random and parallelepiped convex variables is developed. Within the MGHRA method, the parallelepiped convex model is employed to describe independent and correlated interval variables in a unified framework. 2) Sequentially, we propose an original and implementable approach called ALK-MGHRA-IS for active learning MGHRA method and Kriging-based Importance Sampling. The MGHRA method, which is capable of handling the complicated “multi-source uncertainty” problems, associates the Kriging metamodel, and its advantageous stochastic property with Importance Sampling to accurately evaluate bounds of small failure probabilities with respect to interval variables. Actually, the calculated failure probability is still a random variable when the approximations of the proposed method are employed. The proposed method enables the correction of the FORM-UUA approximation with only a few function computations. To further improve the efficiency of the proposed ALK-MGHRA-IS, an optimization method based on Karush–Kuhn–Tucker conditions (KKT) is introduce to relieve the burden of searching the extreme values. Four numerical examples are investigated to demonstrate the efficiency and accuracy of the proposed method.