# An efficient reliability analysis method combining adaptive Kriging and modified importance sampling for small failure probability

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## Abstract

A vital challenge problem of structural reliability analysis is how to estimate the small failure probability with a minimum number of model evaluations. The Adaptive Kriging combined with Importance Sampling method (AK-IS) which is developed from the adaptive Kriging combined with Monte Carlo simulation (AK-MCS) is a viable method to deal with this challenge. The aim of this paper is to reduce the number of model evaluations of the existing AK-IS algorithm. Firstly, we use a contributive weight function to divide the candidate samples of model input variables generated in AK-IS. The candidate samples are used to select the best next sample to update the Kriging model in AK-IS. Secondly, select the best next sample only in the important area obtained according to the contributive weight value to failure probability to update the Kriging model until the stopping condition is satisfied. Thirdly, use the Kriging model constructed in the important area to predict the other area and update the important area by adding the point with the maximum contributive weight value in the area except the important area ceaselessly until the probability of the accurate identification on the limit state function’s signs (positive limit state value or negative limit state value) of all the importance sampling points satisfies a criterion. Finally, the updated Kriging model is used to estimate the failure probability especially for the small failure probability. The proposed method uses a thought from local to global in order to reduce the computational cost of AK-IS and simultaneously guarantees the accuracy of estimation. A non-linear oscillator system, a roof truss structure and a planar ten-bar structure are analyzed by the proposed method. The results demonstrate the efficiency and accuracy of the proposed method in structural reliability analysis especially for small failure probability.

## Keywords

Reliability analysis Contributive weight function Kriging Importance sampling Small failure probability## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant 51,475,370, 51,775,439), the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant CX201708), and the Natural Science Basic Research Plan in Shaanxi Province of China (2017JQ1007) behind (Grant CX201708).

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