In engineering practices, it is a time-consuming procedure to estimate the small failure probability of highly nonlinear and dimensional limit state functions and Kriging-based methods are more effective representatives. However, it is an important challenge to construct the candidate importance sample pool for Kriging-based small failure probability analysis methods with multiple input random variables when the Metropolis-Hastings (M-H) algorithm with acceptance-rejection sampling principle is employed. To address the challenge and estimate the reliability of structures in a more efficient and accurate way, an active learning Kriging model based on the Gibbs importance sampling algorithm (AK-Gibbs) is proposed, especially for the small failure probabilities with nonlinear and high-dimensional limit state functions. A new active learning function that can be directly linked to the global error is first constructed. Weighting coefficients of the joint probability density function in the new active learning function are then determined to select the most probable points (MPPs) and update samples efficiently and accurately. The Gibbs importance sampling algorithm is derived based on the Gibbs algorithm to effectively establish the candidate importance sample pool. An improved global error-based stopping criterion is finally constructed to avoid pre-mature or late-mature for the estimation of small failure probabilities with complicated failure domains. Two numerical and four engineering examples are respectively employed to elaborate and validate the effectiveness of the proposed method.

中文翻译：

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AK-Gibbs：基于吉布斯重要性采样算法的主动学习克里金模型，适用于小故障概率

在工程实践中，估计高度非线性和维数极限状态函数的小失效概率是一个耗时的过程，基于克里格的方法是更有效的代表。然而，当采用接受-拒绝抽样原理的Metropolis-Hastings（MH）算法时，为基于克里金法的多输入随机变量小故障概率分析方法构建候选重要性样本池是一个重要的挑战。为了应对这一挑战并以更有效、更准确的方式估计结构的可靠性，提出了一种基于吉布斯重要性采样算法（AK-Gibbs）的主动学习克里金模型，特别是对于非线性和高维的小失效概率极限状态函数。首先构建了一个可以直接链接到全局误差的新的主动学习函数。然后确定新主动学习函数中联合概率密度函数的加权系数，以选择最可能点（MPP）并高效准确地更新样本。在Gibbs算法的基础上衍生出Gibbs重要性采样算法，有效建立候选重要性样本池。最终构建了一种改进的基于误差的全局停止准则，以避免对复杂失效域的小失效概率的估计过早或晚熟。分别采用两个数值实例和四个工程实例来阐述和验证所提方法的有效性。