One of the most important goals in civil engineering is to guarantee the safety of the construction. Standards prescribe a required failure probability in the order of $$10^{-4}$$ to $$10^{-6}$$. Generally, it is not possible to compute the failure probability analytically. Therefore, many approximation methods have been developed to estimate the failure probability. Nevertheless, these methods still require a large number of evaluations of the investigated structure, usually finite element (FE) simulations, making full probabilistic design studies not feasible for relevant applications. The aim of this paper is to increase the efficiency of structural reliability analysis by means of reduced order models. The developed method paves the way for using full probabilistic approaches in industrial applications. In the proposed PGD reliability analysis, the solution of the structural computation is directly obtained from evaluating the PGD solution for a specific parameter set without computing a full FE simulation. Additionally, an adaptive importance sampling scheme is used to minimize the total number of required samples. The accuracy of the failure probability depends on the accuracy of the PGD model (mainly influenced on mesh discretization and mode truncation) as well as the number of samples in the sampling algorithm. Therefore, a general iterative PGD reliability procedure is developed to automatically verify the accuracy of the computed failure probability. It is based on a goal-oriented refinement of the PGD model around the adaptively approximated design point. The methodology is applied and evaluated for 1D and 2D examples. The computational savings compared to the method based on a FE model is shown and the influence of the accuracy of the PGD model on the failure probability is studied.

中文翻译：

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在自适应重要性抽样方案中使用PGD模型进行有效的结构可靠性分析

土木工程最重要的目标之一就是确保施工安全。标准规定了所需的失败概率为$$ 10 ^ {-4} $$到$$ 10 ^ {-6} $$。通常，不可能通过分析来计算故障概率。因此，已经开发了许多近似方法来估计故障概率。然而，这些方法仍然需要对所研究的结构进行大量评估，通常是有限元（FE）模拟，这使得对相关应用进行完整的概率设计研究不可行。本文的目的是通过降阶模型来提高结构可靠性分析的效率。所开发的方法为在工业应用中使用完全概率方法铺平了道路。在提出的PGD可靠性分析中，通过计算特定参数集的PGD解决方案可直接获得结构计算的解决方案，而无需计算完整的有限元仿真。另外，自适应重要性采样方案用于最小化所需样本的总数。故障概率的准确性取决于PGD模型的准确性（主要影响网格离散化和模式截断）以及采样算法中的样本数量。因此，开发了通用的迭代PGD可靠性过程来自动验证计算出的故障概率的准确性。它基于围绕自适应逼近设计点的PGD模型的面向目标的改进。该方法适用于一维和二维示例，并对其进行了评估。