AK-ARBIS: An improved AK-MCS based on the adaptive radial-based importance sampling for small failure probability

The pivotal problem in reliability analysis is how to use a smaller number of model evaluations to get more accurate failure probabilities. To achieve this aim, an iterative method based on the Monte Carlo simulation and the adaptive Kriging (AK) model (abbreviated as AK-MCS) has been proposed in 2011 by Echard et al. But for small failure probability, the number of the candidate points is extremely large for convergent solution. These points need to be evaluated by the current Kriging model to select the best next sample for updating the Kriging model in AK-MCS method, and the large candidate points will make the adaptive updating process of Kriging model much more time-consuming. Therefore, to improve the applicability of the AK-MCS method for small failure probability, the adaptive radial-based importance sampling (ARBIS) is employed to reduce the number of candidate points in the AK-MCS method, and an ARBIS combined with AK model method (abbreviated as AK-ARBIS) is proposed. The idea of the ARBIS is adaptively to find the optimal -sphere, i.e., the largest sphere of the safe domain, and then samples inside the optimal -sphere is directly recognized as safety and do not need to call the true limit state function to judge their states (safe or failed). During the adaptive process of finding the optimal -sphere, the Kriging model is ceaselessly updated layer after layer based on the U learning scheme in each sampling pool which only contains the samples between the current spherical rings. The updating process of Kriging model stops until the optimal -sphere is adaptively found and the convergent condition is satisfied. By finding the optimal -sphere, the total number of candidate samples is reduced which only includes the samples outside the optimal -sphere. Besides, the whole candidate sampling pool is partitioned into several sub-candidate sampling pool sequentially. The proposed method not only inherits the advantage of the AK-MCS but also reduces the reliability analysis time of the AK-MCS from two aspects. One is the size reduction of the candidate sampling pool, the other is the reduction of the actual limit state function evaluations because the sampling points locating inside the adaptively searched optimal -sphere do not need to participate in the training process. By analyzing a highly nonlinear numerical case, a non-linear oscillator system, a simplified wing box structural model, an aero-engine turbine disk and a planar ten-bar structure, the effectiveness and the accuracy of the proposed AK-ARBIS method for estimating the small failure probability are verified.