This paper presents a multi-fidelity integration rule for statistical moments and failure probability evaluations. The contribution-degree analysis is first conducted for dividing the random inputs as relatively important and unimportant ones, where the multi-dimensional Gaussian-weighted integral respect to moments estimation can be separated into two lower-dimensional integrals in an additive form. A flexible spherical-radial cubature rule is derived to evaluate the integral consisting of important random variables, where the free parameter is optimally determined via a moment-matching strategy. A low-degree spherical-radial cubature rule, whose algebraic degree of accuracy is between 3 and 5, is then applied to estimate the integral related to unimportant variables. In this regard, a multi-fidelity integration rule, where different numerical schemes are employed, is established accordingly for estimating the statistical moments of the limit state function, which can ensure the balance of precision and efficiency. The maximum entropy method is then applied to obtain the entire probability distribution of the limit state function based on the statistical moments, where the failure probability can be straightforwardly assessed. The efficiency and accuracy of the proposed method are demonstrated through five numerical examples for both the statistical moments and failure probability evaluations.

本文提出了一种用于统计矩和故障概率评估的多保真积分规则。首先进行贡献度分析，将随机输入分为相对重要和不重要的两个，其中关于矩估计的多维高斯加权积分可以以加法形式分成两个较低维的积分。导出了一个灵活的球-径向体积规则来评估由重要随机变量组成的积分，其中自由参数通过力矩匹配策略最优确定。然后应用代数精度在 3 到 5 之间的低次球面-径向体积规则来估计与不重要变量相关的积分。在这方面，多保真集成规则，在采用不同数值方案的情况下，相应地建立估计极限状态函数的统计矩，可以保证精度和效率的平衡。然后应用最大熵方法基于统计矩获得极限状态函数的整个概率分布，其中可以直接评估失效概率。通过统计矩和失效概率评估的五个数值例子证明了所提出方法的效率和准确性。然后应用最大熵方法基于统计矩获得极限状态函数的整个概率分布，其中可以直接评估失效概率。通过统计矩和失效概率评估的五个数值例子证明了所提出方法的效率和准确性。然后应用最大熵方法基于统计矩获得极限状态函数的整个概率分布，其中可以直接评估失效概率。通过统计矩和失效概率评估的五个数值例子证明了所提出方法的效率和准确性。