A novel single-loop Kriging importance sampling method for estimating failure probability upper bound under random-interval mixed uncertainties

Abstract

For reliability analysis under wide existing random and interval mixed uncertainties (RIMU), the failure probability upper bound (FPUB), a conservative index to quantify the reliability level of the structure, is the basis for structural safety assessment under RIMU. To accurately and efficiently estimate FPUB, this paper proposes a novel importance sampling method based single-loop Kriging model (IS-SLK), in which the IS variance reduction is combined with the SLK of the performance function, and a new most reduction learning function under RIMU (MRLF-RI) is established. By introducing the IS probability density function (PDF), the proposed IS-SLK firstly expresses the FPUB as the product of the approximate FPUB estimated by current SLK and the correction term, then it designs a strategy of two-step adaptive updating SLK to calibrate FPUB. The innovation of the IS-SLK includes two aspects. Firstly, it constructs a quasi optimal IS PDF to approach the optimal one based on adaptively updating SLK, and the simple reject sampling method is employed to extract the sample of IS PDF, on which the required size of candidate sample pool for obtaining the convergent FPUB can be reduced and the robustness of the algorithm is improved. Secondly, based on the comprehensive consideration of multiple factors, which include the misclassification probability of structural state, joint PDF and Kriging prediction at the IS candidate sample, as well as the logical relationship between the states of random input candidate samples and those at different realizations of interval input, the new learning function named as MRLF-RI is established to adaptively select the training sample contributing most to improve the estimation accuracy of FPUB. Moreover, by deriving the upper bound of the expected relative error, a convergence criterion of MRLF-RI is constructed to further improve the efficiency. Compared with the up-to-date methods, the proposed IS-SLK is more efficient, and its superiority is verified by four examples.