This paper develops an efficient algorithm that combines polynomial-chaos kriging (PCK) and adaptive radial-based importance sampling (ARBIS) for reliability analysis. The key idea of ARBIS is to adaptively determine a sphere with the center at the origin and radius equal to the smallest distance of the failure domain to the origin, also known as the optimal $\beta$-sphere, and only those samples outside the optimal $\beta$-sphere have a possibility of failure and thus need to evaluate the limit-state function to judge their states (safe or failure). In the proposed algorithm, both the PCK model and $\beta$-sphere are updated adaptively. In each iteration of determining the optimal $\beta$-sphere, the PCK model is updated sequentially based on an active learning function, which is used to select the most informative sample from the samples between the last and current $\beta$-spheres. Once the stopping criterion is met, the learning process of PCK in this iteration terminates, and the obtained PCK model is then used to determine the next $\beta$-sphere. The updating iteration of the $\beta$-sphere proceeds until the optimal sphere is found. Five representative examples are revisited, in which the results demonstrate the high accuracy and efficiency of the proposed PCK-ARBIS algorithm.

本文开发了一种将多项式混沌克里金法 (PCK) 和自适应径向重要性采样 (ARBIS) 相结合的高效算法，用于可靠性分析。ARBIS 的关键思想是自适应地确定一个以原点为中心，半径等于失效域到原点的最小距离的球体，也称为最优$\beta$-sphere，并且只有那些超出最优值的样本 $\beta$-sphere 有失效的可能性，因此需要评估极限状态函数来判断它们的状态（安全或失效）。在所提出的算法中，PCK 模型和$\beta$-sphere 自适应更新。在每次迭代中确定最优$\beta$-sphere，PCK 模型基于主动学习函数依次更新，用于从最后一个和当前之间的样本中选择信息量最大的样本 $\beta$-领域。一旦满足停止准则，本次迭代中 PCK 的学习过程终止，然后使用获得的 PCK 模型来确定下一个$\beta$-领域。的更新迭代$\beta$-sphere 继续进行，直到找到最佳球体。重新审视了五个有代表性的例子，其中的结果证明了所提出的 PCK-ARBIS 算法的高精度和高效率。