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Development of an adaptive reliability analysis framework for reinforced concrete frame structures using uncertainty quantification
利用不确定性量化为钢筋混凝土框架结构开发自适应可靠性分析框架

Truong-Thang Nguyen 1 ⋅ 1 ⋅ ^(1)*{ }^{1} \cdot Viet-Hung Dang 1 1 ^(1){ }^{1} (D) Manh-Hung Ha 1 ⋅ 1 ⋅ ^(1)*{ }^{1} \cdot Thanh-Tung Pham 1 ⋅ 1 ⋅ ^(1)*{ }^{1} \cdot Quang-Minh Phan 1 1 ^(1){ }^{1}
Truong-Thang Nguyen 1 ⋅ 1 ⋅ ^(1)*{ }^{1} \cdot Viet-Hung Dang 1 1 ^(1){ }^{1} (D) Manh-Hung Ha 1 ⋅ 1 ⋅ ^(1)*{ }^{1} \cdot Thanh-Tung Pham 1 ⋅ 1 ⋅ ^(1)*{ }^{1} \cdot Quang-Minh Phan 1 1 ^(1){ }^{1}

Accepted: 4 August 2024 / Published online: 22 August 2024
接受：2024 年 8 月 4 日 / 在线发表：2024 年 8 月 22 日
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024
作者独家授权 Springer Science+Business Media, LLC，Springer Nature 2024 的一部分。

Abstract  摘要

Performing reliability analysis for reinforced concrete structures is a tedious and challenging task because it requires conducting a four-nested loop calculation procedure involving millions of data samples to account for the complex behaviours of the structures and multiple random variables. Therefore, the study proposes a novel, practical, and accurate reliability framework that is applicable for multi-component RC frame structures exhibiting different behaviours ranging from linear elastic and non-linear elastic to non-linear plastic. For this purpose, this study first employs a tree-based boosting ensemble model combined with quantile regression, dubbed as QR-LightGBM to calculate the structures’ limit state function and the associated uncertainty estimation at the same time. Next, an active learning process is implemented to improve the computed reliability results progressively. During each active learning step, relevant data samples with potentially high impacts on the model accuracy are determined based on their uncertainty, and then QR-LightGBM is retrained utilizing these samples. By doing so, the prediction performance of the surrogate model is enhanced with a minimized number of actual data samples, thus significantly reducing overall computational resources. The viability and effectiveness of the proposed framework are validated through three case studies involving a simple 1D reinforced concrete beam, a 2D three-story frame, and a 3D fivestory building structure. Furthermore, its performance is quantitatively demonstrated via comparison studies with competing methods such as Monte Carlo simulation, Kriging-based models, and an original LightGBM without active learning.
对钢筋混凝土结构进行可靠性分析是一项繁琐而具有挑战性的任务，因为它需要进行四层循环计算程序，涉及数百万个数据样本，以考虑结构的复杂行为和多个随机变量。因此，本研究提出了一种新颖、实用且精确的可靠性框架，适用于表现出从线性弹性、非线性弹性到非线性塑性等不同行为的多组件 RC 框架结构。为此，本研究首先采用了一种基于树的提升集合模型，并将其与量化回归相结合，称为 QR-LightGBM，以同时计算结构的极限状态函数和相关的不确定性估计。然后，实施主动学习过程，逐步改进计算出的可靠性结果。在每个主动学习步骤中，根据不确定性确定对模型准确性有潜在影响的相关数据样本，然后利用这些样本对 QR-LightGBM 进行再训练。通过这种方法，代用模型的预测性能在实际数据样本数量最少的情况下得到了提升，从而大大减少了总体计算资源。通过涉及简单一维钢筋混凝土梁、二维三层框架和三维四层建筑结构的三个案例研究，验证了所提框架的可行性和有效性。此外，通过与蒙特卡洛模拟、基于克里金的模型和无主动学习的原始 LightGBM 等竞争方法的比较研究，定量证明了该框架的性能。

Keywords Uncertainty quantification • Structural reliability analysis • Active learning • Machine learning • Concrete structure
关键词 不确定性量化 - 结构可靠性分析 - 主动学习 - 机器学习 - 混凝土结构

1 Introduction  1 引言

Reinforced concrete (RC) structures are widely regarded as the most popular load-bearing structures in the real world, ranging from small houses to highrise buildings or largespan bridges. Therefore, ensuring their safety in the face of multiple highly fluctuating factors during their long-term service life is important. An effective way for this purpose is to conduct structural reliability analysis and estimate corresponding reliability indices and failure probabilities. However, structural reliability analysis for RC [ 1 , 2 ] RC [ 1 , 2 ] RC[1,2]\mathrm{RC}[1,2] is highly challenging because it typically involves numerous
钢筋混凝土（RC）结构被广泛认为是现实世界中最常见的承重结构，小到房屋，大到高层建筑或大跨度桥梁。因此，确保其在长期使用过程中面对多种剧烈波动因素时的安全性非常重要。为此，一种有效的方法就是进行结构可靠性分析，并估算相应的可靠性指数和失效概率。然而， RC [ 1 , 2 ] RC [ 1 , 2 ] RC[1,2]\mathrm{RC}[1,2] 的结构可靠性分析具有很高的挑战性，因为它通常涉及众多
random variables such as yield strength of steel, compressive strength of concrete, geometrical uncertainties, stochastic external loads [3], cross-section sizes [4], etc. On the other hand, it is commonly acknowledged that RC is an elastoplastic material, which requires nonlinear analysis to accurately modelling its complex behaviours [5]. The reliability analysis of RC could be further more complicated when accounting for time-varying factors such as corrosion, fire, earthquake, wind loads etc. [6, 7].
随机变量，如钢材屈服强度、混凝土抗压强度、几何不确定性、随机外部荷载 [3]、截面尺寸 [4] 等。另一方面，人们普遍认为 RC 是一种弹塑性材料，需要进行非线性分析才能准确模拟其复杂行为 [5]。如果考虑到腐蚀、火灾、地震、风荷载等时变因素，RC 的可靠性分析可能会更加复杂[6, 7]。[6, 7].
In the authors’ opinion, the methods for addressing structural reliability analysis of RC structure can be categorized into three families: (semi-) analytical methods, simulation techniques, and surrogate models. The analytical approach first constructs a formula approximating the complex original limit state surfaces; after that, searching algorithms are utilized to find the most probable point and calculate the reliability index which is actually the distance between the latter and the origin point [8]. However, it is not always possible
作者认为，解决 RC 结构可靠性分析的方法可分为三类：（半）分析方法、模拟技术和代用模型。分析方法首先构建一个近似复杂原始极限状态曲面的公式，然后利用搜索算法找到最可能的点并计算可靠性指数，可靠性指数实际上就是后者与原点之间的距离[8]。然而，这并不总是可能的
to compute the gradient/Hessian of the limit state function, especially for non-linear problems such as RC structures’ behavior; hence, the applicability of the analytical approach is limited for these scenarios. On the other hand, the simulation technique is a more popular approach that is applicable to almost all structural problems including different structural configurations, various safety criteria, and multiple random variable inputs. Besides the conventional Monte Carlo simulation (MCS), additional sampling techniques such as Importance sampling, Subset simulation, etc. are helpful in reducing the required number of samples when working with problems featuring extremely small failure probabilities [9]. The third family of methods for reliability analysis which is relatively new compared to the other two, is to develop a surrogate model [10-12] that can predict the limit state function with considerably lower computation time compared to direct numerical simulation.
因此，分析方法在这些情况下的适用性有限。另一方面，模拟技术是一种更为流行的方法，几乎适用于所有结构问题，包括不同的结构配置、各种安全标准和多种随机变量输入。除了传统的蒙特卡罗模拟（MCS）外，重要度取样、子集模拟等其他取样技术也有助于在处理失效概率极小的问题时减少所需的取样数量[9]。与其他两种方法相比，可靠性分析的第三种方法相对较新，即开发代用模型 [10-12]，与直接数值模拟相比，它能以更短的计算时间预测极限状态函数。
In the context of structural reliability analysis, different surrogate models have been developed to replace the classical finite element method when assessing the structures’ responses and the associated limit state functions. MendozaLugo [13] proposed a non-parametric Bayesian Network when assessing the failure probability of reinforced concrete bridge pillars. The advantages of the Bayesian Network lie in its non-parametric nature and interpretable causal relationships between random variables and outputs of interests. Some authors incorporate machine/deep learning algorithms into reliability analysis frameworks to boost the calculation process. For example, Pan and Dias [14] adopted the support vector machine model, Hariri-Ardebili and Barak [15] utilized the Extreme gradient boosting method, Hung et al. [16, 17] employed deep learning algorithms such as Long Short term memory, and Transformer architecture, Zhang et al. [18] used Convolutional neural network, and so on. The most popular machine learning widely used in reliability analysis is Gaussian Process, a.k.a, Kriging model thanks to its unique ability to provide uncertainty estimation of prediction results. This capability opens the door for building active learning frameworks which can improve the surrogate model performance with a significantly reduced number of data samples. Some pioneering works leveraging the Kriging model are proposed by Echard et al. in [19-21]. However, the Kriging model needs to construct a large-size matrix for measuring the correlation between data samples. Moreover, the computational resources including the central processing unit (CPU) time and required memory are quadratically proportional to the data size. In fact, the performance of the Kriging model in handling the elastoplastic behavior of RC structure has not been thoroughly investigated in the reviewed works. The Bayesian Neural Network is another practical method able to provide uncertainty quantification, as demonstrated in the work of Li et al. [22] when performing uncertainty quantification in material property
在结构可靠性分析方面，人们开发了不同的替代模型，以取代传统的有限元方法来评估结构的响应和相关的极限状态函数。MendozaLugo [13] 在评估钢筋混凝土桥柱的失效概率时提出了一种非参数贝叶斯网络。贝叶斯网络的优势在于其非参数性质以及随机变量和相关输出之间可解释的因果关系。一些学者在可靠性分析框架中加入了机器/深度学习算法，以改进计算过程。例如，Pan 和 Dias[14]采用了支持向量机模型，Hariri-Ardebili 和 Barak[15]利用了极端梯度提升法，Hung 等人[16, 17]采用了长短期记忆和变压器架构等深度学习算法，Zhang 等人[18]使用了卷积神经网络，等等。在可靠性分析中广泛使用的最流行的机器学习是高斯过程（又称克里金模型），因为它具有对预测结果进行不确定性估计的独特能力。这种能力为构建主动学习框架打开了大门，该框架能在大幅减少数据样本数量的情况下提高代用模型的性能。Echard 等人在 [19-21] 中提出了一些利用克里金模型的开创性工作。然而，克里金模型需要构建一个大型矩阵来测量数据样本之间的相关性。此外，包括中央处理器（CPU）时间和所需内存在内的计算资源与数据大小成二次方。 事实上，Kriging 模型在处理 RC 结构弹塑性行为方面的性能还没有在综述作品中得到深入研究。贝叶斯神经网络是另一种能够提供不确定性量化的实用方法，Li 等人的研究证明了这一点。[22]在对材料特性进行不确定性量化时
prediction, the work of Hung et al. [23] when estimating the time-varying reliability of a bridge subjected to material degradation. Besides, the Dropout neural network is also a promising method for estimating uncertainty in the structures’ nonlinear responses [24]. It is noted that uncertainty quantification is necessary to conduct active learning frameworks, which can improve the surrogate model’s performance with a significantly reduced number of data samples. Various authors have employed active learning in the context of structural reliability analysis. For example, Zhao et al. [25] extended the adaptive Kriging Monte Carlo Simulation (AK-MCS) method, which only considers one new data sample at each iteration, to a parallel method, namely, P-AK-MCS, that can select multiple relevant data samples and append them to the current training dataset to improve the Kriging model. For this purpose, the authors proposed four parallel learning functions to determine various informative candidates without increasing the number of functional evaluations. When studying the safety of the construction equipment aerial building machine for high-rise buildings under extreme wind loads, Wang et al. [26] developed a time-saving active learning approach combining the classical Probability Density Evolution Method with deep learning and Query-by-Dropout-Committee techniques. The proposed method is proven to increase the efficiency by approximately 80 % 80 % 80%80 \% compared to the traditional PDEM while still maintaining reliability error only within 0.5 % 0.5 % 0.5%0.5 \%. Zhang et al. [27] stated that multiple sources of both time-variant and invariant uncertainties, including material performance degradation, external loads, and environmental conditions, should be taken into account when performing structural analysis. The authors utilized a gated recurrent unit neural network to predict the structures’ responses, Latin hypercube sampling to sample random variables, and an active learning process to improve the model accuracy and reduce the required amount of training data (Table 1).
例如，Hung 等人[23]在估算受材料退化影响的桥梁的时变可靠性时所采用的预测方法。此外，Dropout 神经网络也是估计结构非线性响应不确定性的一种有前途的方法[24]。有学者指出，不确定性量化是进行主动学习框架的必要条件，它可以在显著减少数据样本数量的情况下提高代用模型的性能。许多学者在结构可靠性分析中采用了主动学习方法。例如，Zhao 等人[25] 将每次迭代只考虑一个新数据样本的自适应克里金蒙特卡洛模拟（AK-MCS）方法扩展为一种并行方法，即 P-AK-MCS，它可以选择多个相关数据样本并将其附加到当前训练数据集，以改进克里金模型。为此，作者提出了四种并行学习函数，在不增加函数评估次数的情况下确定各种信息候选。在研究极端风载荷下高层建筑施工设备高空作业机的安全性时，Wang 等人[26]开发了一种省时的主动学习方法，该方法将经典的概率密度演化法与深度学习和逐滴查询技术相结合。事实证明，与传统的概率密度演化法相比，该方法的效率提高了约 80 % 80 % 80%80 \% ，同时可靠性误差仍保持在 0.5 % 0.5 % 0.5%0.5 \% 以内。Zhang 等人 [27]指出，在进行结构分析时，应考虑时变和不变不确定性的多种来源，包括材料性能退化、外部载荷和环境条件。作者利用门控递归单元神经网络来预测结构的响应，利用拉丁超立方采样法对随机变量进行采样，并利用主动学习过程来提高模型的准确性并减少所需的训练数据量（表 1）。
This study focuses on the reliability analysis of non-linear reinforced concrete structures with multiple random variables. The authors aim to develop a practical, high-performance and efficient framework for structural reliability analysis of reinforced concrete structures based on a machine learning-based surrogate model, active learning and uncertainty quantification. The framework is abbreviated as RCUQR (Uncertainty Quantification-based Reliability analysis framework for Reinforced Concrete structures). We first harness distributed parallel computing with a scripting approach to conduct extensive non-linear analysis. Next, we combine quantile regression with a high-performance machine learning algorithm for two purposes: predicting the structures’ responses and quantifying uncertainty. Third, an active learning process is leveraged to select the most relevant candidates, effectively improving the prediction model. Additionally, multiple utilities have been implemented, including
本研究重点关注具有多个随机变量的非线性钢筋混凝土结构的可靠性分析。作者旨在开发一种实用、高性能和高效的钢筋混凝土结构可靠性分析框架，该框架以基于机器学习的代用模型、主动学习和不确定性量化为基础。该框架简称为 RCUQR（基于不确定性量化的钢筋混凝土结构可靠性分析框架）。我们首先利用分布式并行计算和脚本方法来进行广泛的非线性分析。接下来，我们将量化回归与高性能机器学习算法相结合，以实现两个目的：预测结构的响应和量化不确定性。第三，利用主动学习过程来选择最相关的候选结构，从而有效改进预测模型。此外，还实现了多种实用功能，包括
Table 1 Existing works utilizing surrogate models and active learning in structural reliability analysis
表 1 在结构可靠性分析中利用代用模型和主动学习的现有研究成果
No  没有 Author  作者 Method  方法 Application  应用 Advantage  优势 Discussion  讨论
Hung et al. [23]
Hung 等人[23]		 Monte Carlo - Bayesian neural network
蒙特卡罗 - 贝叶斯神经网络		 Bridge structure  桥梁结构 Realistic scenario  现实情景 Requirement of considerable amount of data samples for training
需要大量数据样本进行训练
Zhao et al. [25]
赵等人[25］		 Parallel AK-MCS  并行 AK-MCS Benchmark and engineering problems
基准和工程问题		 Parallel computing  并行计算 Based on Kriging model
基于克里金模型
Diaz et al. [5]
迪亚兹等人[5］		 Non-linear FEM and Response Surface method
非线性有限元和响应面法		 Reinforced concrete beam
钢筋混凝土梁		 High accuracy  高精度 No example with high-dimensional problems
无高维问题实例
4 Lieu et al. [12]
Lieu 等人 [12]		 Deep neural network  深度神经网络 Truss structure  桁架结构 Adaptive improvement  适应性改进 RC structures exhibits more complex behaviors than truss structures
与桁架结构相比，RC 结构的行为更为复杂
5 Hong et al. [28]
Hong 等人 [28]		 Surrogate model  代用模型 Benchmark and engineering problems
基准和工程问题		 Beyond Kriging model  超越克里金模型 Emphasize the necessity of alternative surrogate models
强调替代模型的必要性
6 Nguyen et al. [16]
Nguyen 等人 [16].		 Surrogate model, subset simulation
代用模型，子集模拟		 Engineering problems  工程问题 Rapid prediction  快速预测 Effort-intensive data preparation and training process
耗费大量精力的数据准备和培训过程
7 Zhang et al. [29]
张等人[29］		 Physical-informed neural network
物理信息神经网络		 Only benchmark problems  只有基准问题 Suitable for problems described by partial differential equations
适用于偏微分方程描述的问题		 Highlight the importance of focusing to the vicinity of the limit state
强调将重点放在极限状态附近的重要性
8 Wang et al. [30]
Wang 等人[30］		 Convolution neural network
卷积神经网络		 Engineering problems with geotechnical systems
岩土系统的工程问题		 Effectively handle soil properties' uncertainties
有效处理土壤特性的不确定性		 Requirement of converting random variables to equivalent multi-channel "image"
将随机变量转换为等效多通道 "图像 "的要求
9 Moustapha et al. [31]
穆斯塔法等人[31]		 Kriging model + subset simulation
克里金模型 + 子集模拟		 Engineering problem with power transmission
输电工程问题		 Multiple limit state functions
多种极限状态功能		 Requirement of advanced knowledge on deep learning
需要高级深度学习知识
10 Doan and Dinh [32]
Doan 和 Dinh [32]		 Surrogate models + limit state data
代用模型 + 极限状态数据		 Truss structures  桁架结构 Accuracy of surrogate models is improved with limited amount of data
利用有限的数据量提高代用模型的准确性		 Performance of Kriging model on complex structures is still questionable
克里金模型在复杂结构上的性能仍有疑问
11 Das and Tesfamariam [33]
Das 和 Tesfamariam [33]		 Probability density evolution method and stochastic spectral embedding
概率密度演化法和随机频谱嵌入		 Shear building frame subject to earthquake
受地震影响的剪力墙框架		 Computing failure probability with a small number of representative points
利用少量代表性点计算故障概率		 It is argued that limit state data may not easily determined for high-dimensional nonlinear structures
有观点认为，高维非线性结构的极限状态数据可能不易确定
12 Guo et al. [34]
Guo 等人 [34]		 Probability density function-informed method
基于概率密度函数的方法		 Reinforced concrete beams under structural deterioration
结构退化情况下的钢筋混凝土梁		 Account for various environmental actions
对各种环境行动进行核算		 Requirement of defining in advance orthonormal basis functions which may vary for different problems
要求提前定义正交基函数，不同问题的正交基函数可能不同
Guardiani et al. [35]
瓜尔迪亚尼等人[35］		 Support Vector Regression and Extreme Gradient Boosting
支持向量回归和极梯度提升		 Reliability of unsaturated slopes
非饱和斜坡的可靠性		 Assessing the mpact of different factors on the failure probability
评估不同因素对故障概率的影响		 It is desirable to extend the method to more complex structures
最好能将该方法扩展到更复杂的结构中
This study  本研究 LightGBM + active learning + quantile regression
LightGBM + 主动学习 + 量化回归		 Reinforce concrete structures with different complexities
加固不同复杂程度的混凝土结构		 High efficiency and straightforward to apply to various structures
效率高，可直接应用于各种结构		 End-to-end framework with supplement utilities: model selection, visualization, performance and efficiency metrics
带有补充实用程序的端到端框架：模型选择、可视化、性能和效率指标
No Author Method Application Advantage Discussion Hung et al. [23] Monte Carlo - Bayesian neural network Bridge structure Realistic scenario Requirement of considerable amount of data samples for training Zhao et al. [25] Parallel AK-MCS Benchmark and engineering problems Parallel computing Based on Kriging model Diaz et al. [5] Non-linear FEM and Response Surface method Reinforced concrete beam High accuracy No example with high-dimensional problems 4 Lieu et al. [12] Deep neural network Truss structure Adaptive improvement RC structures exhibits more complex behaviors than truss structures 5 Hong et al. [28] Surrogate model Benchmark and engineering problems Beyond Kriging model Emphasize the necessity of alternative surrogate models 6 Nguyen et al. [16] Surrogate model, subset simulation Engineering problems Rapid prediction Effort-intensive data preparation and training process 7 Zhang et al. [29] Physical-informed neural network Only benchmark problems Suitable for problems described by partial differential equations Highlight the importance of focusing to the vicinity of the limit state 8 Wang et al. [30] Convolution neural network Engineering problems with geotechnical systems Effectively handle soil properties' uncertainties Requirement of converting random variables to equivalent multi-channel "image" 9 Moustapha et al. [31] Kriging model + subset simulation Engineering problem with power transmission Multiple limit state functions Requirement of advanced knowledge on deep learning 10 Doan and Dinh [32] Surrogate models + limit state data Truss structures Accuracy of surrogate models is improved with limited amount of data Performance of Kriging model on complex structures is still questionable 11 Das and Tesfamariam [33] Probability density evolution method and stochastic spectral embedding Shear building frame subject to earthquake Computing failure probability with a small number of representative points It is argued that limit state data may not easily determined for high-dimensional nonlinear structures 12 Guo et al. [34] Probability density function-informed method Reinforced concrete beams under structural deterioration Account for various environmental actions Requirement of defining in advance orthonormal basis functions which may vary for different problems Guardiani et al. [35] Support Vector Regression and Extreme Gradient Boosting Reliability of unsaturated slopes Assessing the mpact of different factors on the failure probability It is desirable to extend the method to more complex structures This study LightGBM + active learning + quantile regression Reinforce concrete structures with different complexities High efficiency and straightforward to apply to various structures End-to-end framework with supplement utilities: model selection, visualization, performance and efficiency metrics| No | Author | Method | Application | Advantage | Discussion | | :---: | :---: | :---: | :---: | :---: | :---: | | | Hung et al. [23] | Monte Carlo - Bayesian neural network | Bridge structure | Realistic scenario | Requirement of considerable amount of data samples for training | | | Zhao et al. [25] | Parallel AK-MCS | Benchmark and engineering problems | Parallel computing | Based on Kriging model | | | Diaz et al. [5] | Non-linear FEM and Response Surface method | Reinforced concrete beam | High accuracy | No example with high-dimensional problems | | 4 | Lieu et al. [12] | Deep neural network | Truss structure | Adaptive improvement | RC structures exhibits more complex behaviors than truss structures | | 5 | Hong et al. [28] | Surrogate model | Benchmark and engineering problems | Beyond Kriging model | Emphasize the necessity of alternative surrogate models | | 6 | Nguyen et al. [16] | Surrogate model, subset simulation | Engineering problems | Rapid prediction | Effort-intensive data preparation and training process | | 7 | Zhang et al. [29] | Physical-informed neural network | Only benchmark problems | Suitable for problems described by partial differential equations | Highlight the importance of focusing to the vicinity of the limit state | | 8 | Wang et al. [30] | Convolution neural network | Engineering problems with geotechnical systems | Effectively handle soil properties' uncertainties | Requirement of converting random variables to equivalent multi-channel "image" | | 9 | Moustapha et al. [31] | Kriging model + subset simulation | Engineering problem with power transmission | Multiple limit state functions | Requirement of advanced knowledge on deep learning | | 10 | Doan and Dinh [32] | Surrogate models + limit state data | Truss structures | Accuracy of surrogate models is improved with limited amount of data | Performance of Kriging model on complex structures is still questionable | | 11 | Das and Tesfamariam [33] | Probability density evolution method and stochastic spectral embedding | Shear building frame subject to earthquake | Computing failure probability with a small number of representative points | It is argued that limit state data may not easily determined for high-dimensional nonlinear structures | | 12 | Guo et al. [34] | Probability density function-informed method | Reinforced concrete beams under structural deterioration | Account for various environmental actions | Requirement of defining in advance orthonormal basis functions which may vary for different problems | | | Guardiani et al. [35] | Support Vector Regression and Extreme Gradient Boosting | Reliability of unsaturated slopes | Assessing the mpact of different factors on the failure probability | It is desirable to extend the method to more complex structures | | | This study | LightGBM + active learning + quantile regression | Reinforce concrete structures with different complexities | High efficiency and straightforward to apply to various structures | End-to-end framework with supplement utilities: model selection, visualization, performance and efficiency metrics |
a data generation function, scripts connecting Python with the OpenSees program, model selection, hyperparameter optimization, comparison studies, data visualization for failure cases, and convergence plots. Together, these utilities, along with the three main components mentioned above, form an end-to-end framework for the reliability analysis of non-linear RC structures. Compared with other existing reliability analysis tools such as UQPy [36], this solution offers additional advantages. The existing tools primarily focus on first-order reliability method (FORM) and second-order reliability method (SORM) for relatively simple problems with available analytical limit state functions and on Kriging models for numerical problems. Their current version do not support surrogate models based on high-performance machine learning (ML) or deep learning (DL) algorithms. Additionally, active learning is only available with Kriging models, which require predefined kernel functions and the computation of a correlation matrix. These requirements are not always applicable to other surrogate models, making them unsuitable for integrating active learning with modern high-performance ML/DL-based algorithms. The effectiveness and efficiency of RC-UQR are quantitatively demonstrated via three examples with increasing structural complexity.
数据生成功能、连接 Python 与 OpenSees 程序的脚本、模型选择、超参数优化、对比研究、失效案例数据可视化以及收敛图。这些实用工具与上述三个主要组件共同构成了非线性 RC 结构可靠性分析的端到端框架。与其他现有的可靠性分析工具（如 UQPy [36]）相比，该解决方案具有更多优势。现有的工具主要集中在一阶可靠性方法（FORM）和二阶可靠性方法（SORM）上，用于分析相对简单的问题和可用的极限状态函数，以及用于数值问题的克里金模型。其当前版本不支持基于高性能机器学习（ML）或深度学习（DL）算法的代用模型。此外，主动学习仅适用于 Kriging 模型，这需要预定义的核函数和相关矩阵的计算。这些要求并不总是适用于其他代用模型，因此不适合将主动学习与基于 ML/DL 的现代高性能算法集成。RC-UQR 的有效性和效率通过三个结构复杂度不断增加的示例得到了量化证明。

Contributions  捐款

In short, the main contributions of this study are summarized as follows:
总之，本研究的主要贡献概述如下：
  • From an application perspective, this paper presents an accessible framework for structural engineers, based on well-known knowledge and open source libraries such as quantile regression and the machine learning LightGBM model. It does not require any specialized software, commercial tools or computers with expensive GPU devices for its development and operation.
    从应用的角度来看，本文基于众所周知的知识和开源库（如量化回归和机器学习 LightGBM 模型），为结构工程师提出了一个易于使用的框架。它的开发和运行不需要任何专业软件、商业工具或配备昂贵 GPU 设备的计算机。
  • Combining three techniques- uncertainty quantification, active learning, and machine learning- to perform reliability analysis is an appealing idea within the structural engineering community. However, according to the authors’ knowledge, this is the first time this approach has been applied to non-linear RC structures. This is likely due to the challenge of preparing data for non-linear RC structures, which requires specialized expertise in RC structure analysis and distributed parallel computing to conduct a large number of simulations on different CPUs simultaneously. In addition, complementary results have been presented to provide insights into different aspects of an end-to-end reliability analysis framework involving model selection, visualizing failure cases, the impact of the number of updated candidates, and providing graphical illustrations of the active learning effect. These com-
    将不确定性量化、主动学习和机器学习这三种技术结合起来进行可靠性分析，在结构工程界是一个很有吸引力的想法。然而，据作者所知，这是首次将这种方法应用于非线性 RC 结构。这可能是由于为非线性 RC 结构准备数据所面临的挑战，这需要 RC 结构分析方面的专业知识和分布式并行计算，以便在不同的 CPU 上同时进行大量模拟。此外，还介绍了一些补充结果，以深入了解端到端可靠性分析框架的不同方面，包括模型选择、失效案例可视化、更新候选模型数量的影响，以及提供主动学习效果的图解。这些成果包括
    prehensive results have not been previously presented or completed in existing works.
    在现有的工作中，还没有提出或完成过全面的成果。
  • From a scientific perspective, the paper emphasizes that data selection is crucial for training a high-performance prediction model in structural reliability analysis. Consequently, directly using a state-of-the-art, well-tuned machine learning model such as LightGBM with randomly selected data samples cannot accurately approximate structures’ responses, leading to unreliable results. To select informative data samples, the active learning process utilizing uncertainty quantification is an effective way which can improve the accuracy of the surrogate model with a small number of additional FEM evaluations. The selected candidates are located either near the limit state surface or in regions where the surrogate model’s predictions are most uncertain.
    本文从科学的角度强调，数据选择对于在结构可靠性分析中训练高性能预测模型至关重要。因此，直接使用最先进的、经过良好调整的机器学习模型（如 LightGBM）来随机选择数据样本，无法准确地近似结构的响应，从而导致不可靠的结果。为了选择有参考价值的数据样本，利用不确定性量化的主动学习过程是一种有效的方法，只需少量额外的有限元评估就能提高代用模型的准确性。所选候选样本要么位于极限状态面附近，要么位于代用模型预测最不确定的区域。
In the rest of the paper, Sect. 2 first presents the nonlinear analysis of reinforced concretes using finite element methods and fiber sections. After that, the overall working flow and main components of the RC-UQR framework are described in detail. In Sect. 2, the RC-UQR is applied to three examples including a 1D RC beam, a 2D three-story RC frame, and a 3D RC building structure. Finally, Sect. 2.1 concludes and suggests perspectives for the next steps of the study.
在本文的其余部分，第 2 节首先介绍了使用有限元方法和纤维截面对钢筋混凝土进行的非线性分析。随后，详细介绍了 RC-UQR 框架的整体工作流程和主要组成部分。在第 2 节中，RC-UQR 被应用于三个实例，包括一维 RC 梁、二维三层 RC 框架和三维 RC 建筑结构。最后，第 2.1 节进行了总结，并对下一步研究提出了展望。

2 Adaptive reliability analysis framework for reinforced concrete frame structures using uncertainty quantification
2 采用不确定性量化的钢筋混凝土框架结构自适应可靠性分析框架

2.1 Numerical simulation of reinforced concrete structures
2.1 钢筋混凝土结构的数值模拟

2.1.1 Fiber section modeling
2.1.1 纤维截面建模

Reinforced concrete structural elements encompass two materials that have completely different properties; thus, accurately simulating their behaviour under various load conditions requires an approach that should be theoretically supported by rigorous mathematical and physical foundations, but not too complicated to numerically implement. Among methods specifically dedicated to RC members, the forced-based distributed plasticity element, a.k.a, fiber elements can effectively handle both the plastic and elastic behaviors. Moreover, unlike the concentrated plasticity model which needs to assume in advance the positions of plastic hinges near the element ends, the distributed plasticity model can account for plasticity at multiple sections along the element length. This method first divides the element into a finite number of cross sections at some control points a priori selected by a numerical integration scheme
钢筋混凝土结构件包含两种性质完全不同的材料；因此，要准确模拟它们在各种荷载条件下的行为，需要一种既有严格的数学和物理基础作为理论支撑，又不至于过于复杂的数值计算方法。在专门针对 RC 构件的方法中，基于强制的分布塑性元素（又称纤维元素）可以有效地处理塑性和弹性行为。此外，集中塑性模型需要预先假定塑性铰链靠近构件两端的位置，而分布塑性模型则不同，它可以考虑构件长度上多个截面的塑性。这种方法首先通过数值积分方案在一些事先选定的控制点上将元素分成有限个截面
such as the Gauss-Lobatto quadrature rule. Subsequently, each of these sections is subdivided into N f i b N f i b N_(fib)N_{f i b} fibers (Fig. 1). The characteristics of a fiber i i ii involve its spatial location ( y , z ) ( y , z ) (y,z)(y, z), and the assigned area A i , f i b A i , f i b A_(i,fib)A_{i, f i b}. Once the stress-strain response of each individual longitudinal fiber is computed, the stress-strain state of the hosted section is calculated using the integration operator. The forced-based distributed plasticity element adopts the hypothesis of small displacements and deformations and the cross-section is assumed to be prismatic before, during and after deformation. This assumption implies the geometric linearity which simplifies the geometric relation between section deformations and fiber strains. It is recalled that the main source of nonlinearity in this study comes from material nonlinearity.
例如高斯-洛巴托正交规则。随后，每个截面被细分为 N f i b N f i b N_(fib)N_{f i b} 纤维（图 1）。纤维 i i ii 的特征涉及其空间位置 ( y , z ) ( y , z ) (y,z)(y, z) 和分配面积 A i , f i b A i , f i b A_(i,fib)A_{i, f i b} 。在计算出每根纵向纤维的应力应变响应后，就可以使用积分算子计算托管截面的应力应变状态。基于强制的分布式塑性元素采用了小位移和小变形假设，并假定横截面在变形前、变形中和变形后都是棱柱形的。这一假设意味着几何线性，简化了截面变形和纤维应变之间的几何关系。在本研究中，非线性的主要来源是材料非线性。

2.1.2 Numerical simulation realization
2.1.2 实现数值模拟

In this study, the numerical simulation of RC structures is realized with the help of the finite element program OpenSees [37] thanks to its widely acknowledged credibility and open-source nature that facilitates its integration into the proposed structural reliability framework. The typical steps of a numerical simulation are presented as below:
在本研究中，有限元程序 OpenSees [37]具有广受认可的可信度和开放源代码的特性，便于将其集成到建议的结构可靠性框架中，因此本研究借助 OpenSees 实现了对 RC 结构的数值模拟。数值模拟的典型步骤如下：
Step 1: Provide the structure’s geometrical configuration and dimensions. For this step, Computer-aid-design software such as AutoCAD is usually utilized to sketch the structures in 2D or 3D dimensional. Some FEM program has a built-in geometrical component for conducting this step, potentially equipping with programming support for automatically conducting parts of the geometric construction.
第 1 步：提供结构的几何构造和尺寸。在此步骤中，通常使用 AutoCAD 等计算机辅助设计软件绘制二维或三维结构草图。某些有限元程序内置了用于执行此步骤的几何组件，并可能配备程序支持，自动执行部分几何结构。
Step 2: Define node coordinates. This step involves specifically determining the Cartesian coordinates of the structures’ essential nodes, which are truss joints and beam-column joints. It is also necessary to number these
步骤 2：确定节点坐标。这一步涉及具体确定结构重要节点的笛卡尔坐标，即桁架连接点和梁柱连接点。还需要对这些节点进行编号
Fig. 1 Distributed plasticity element using fiber section for RC members
图 1 使用纤维截面的分布塑性元件用于 RC 构件
models since it is more practical to work with the nodes’ numbers rather than their coordinates.
模型，因为使用节点编号比使用节点坐标更实用。
Step 3: Define boundary conditions. In this step, nodes with supports such as rollers, pinned, or fixed are restrained in their corresponding degrees of freedom.
步骤 3：定义边界条件。在这一步中，带有滚柱、销轴或固定等支撑的节点会在其相应的自由度上受到约束。
Step 4: Define element connectivity. A structural element is defined by explicitly providing the node numbers of its ends. Furthermore, the structural elements need to be numbered.
步骤 4：定义元素连接。通过明确提供结构元素两端的节点编号来定义结构元素。此外，还需要对结构元素进行编号。
Step 5: Provide the materials’ properties, such as the elastic modulus, the compressive strength of concrete, the yield strength and hardening ratio of steel, the damping ratio, and so on.
步骤 5：提供材料属性，如弹性模量、混凝土抗压强度、钢材屈服强度和硬化比、阻尼比等。
Step 6: Section modelling using the fiber approach. This step is specifically applied to modelling reinforced concrete structural elements, which have been described in detail previously.
步骤 6：使用纤维法进行截面建模。这一步骤专门用于钢筋混凝土结构构件的建模，前面已经详细介绍过。
Step 7: Apply external loads. This step can be quickly done by specializing the load intensities and the number of elements/nodes at which external loads are applied.
步骤 7：施加外部载荷。这一步可以通过调整荷载强度和施加外部荷载的元素/节点数量来快速完成。
Step 8: Define desired outputs. Desired outputs can be nodal displacements, accelerations, and internal forces of elements. It is noted that for complex structures with numerous nodes/elements, it is more convenient to export outputs for a subset of critical nodes/ elements rather than every node/element.
步骤 8：定义期望输出。预期输出可以是节点位移、加速度和元素内力。需要注意的是，对于节点/元素众多的复杂结构，输出关键节点/元素子集的输出比输出每个节点/元素的输出更方便。
Step 9: Conduct analysis and store computed results. After these eight previous steps are in place, the analysis step is carried out by selecting appropriate algorithms, such as Newton Raphson iteration and the Newmark numerical integration scheme.
第 9 步：进行分析并存储计算结果。在上述八个步骤完成后，就可以通过选择合适的算法（如牛顿-拉斐尔森迭代和纽马克数值积分方案）进行分析。
Steps 1, 2, 3, and 4 are typical steps in constructing the geometrical configuration of structures in a finite element program. Meanwhile, the theoretical background of steps 5 and 6 is discussed in the previous subsection. Steps 7 and 8 vary depending on the specific problem. For step 9, the non-linear analysis of RC structures consists of a three-nested loop as below:
步骤 1、2、3 和 4 是在有限元程序中构建结构几何构型的典型步骤。同时，步骤 5 和 6 的理论背景已在上一小节中讨论过。第 7 和第 8 步因具体问题而异。对于第 9 步，RC 结构的非线性分析由以下三层循环组成：
  • The outer “For” loop: the applied load is discretized into a sequence of increment loads F 0 , F 1 = F 0 + Δ F 1 , … , F m = F m − 1 + Δ F m F 0 , F 1 = F 0 + Δ F 1 , … , F m = F m − 1 + Δ F m F_(0),F_(1)=F_(0)+DeltaF_(1),dots,F_(m)=F_(m-1)+DeltaF_(m)F_{0}, F_{1}=F_{0}+\Delta F_{1}, \ldots, F_{m}=F_{m-1}+\Delta F_{m}, where m m mm is the total number of load steps.
    外部 "For "循环：外加载荷被离散化为一系列递增载荷 F 0 , F 1 = F 0 + Δ F 1 , … , F m = F m − 1 + Δ F m F 0 , F 1 = F 0 + Δ F 1 , … , F m = F m − 1 + Δ F m F_(0),F_(1)=F_(0)+DeltaF_(1),dots,F_(m)=F_(m-1)+DeltaF_(m)F_{0}, F_{1}=F_{0}+\Delta F_{1}, \ldots, F_{m}=F_{m-1}+\Delta F_{m} ，其中 m m mm 是载荷的总步数。
  • The intermediate “While” loop: At each load step, the Newton-Raphson iteration is conducted to numerically solve the structure’s equilibrium equation (Eq. 14).
    中间的 "While "循环：在每个载荷步长上，进行牛顿-拉夫逊迭代，数值求解结构的平衡方程（公式 14）。
  • The inner “For” loop: This loop iterates through all structural elements, calculating their fiber stiffnesses, element stiffnesses, and then assembles them into the global structural stiffness matrix for the current load step.
    内部 "For "循环：该循环遍历所有结构元素，计算它们的纤维刚度和元素刚度，然后将它们组合成当前载荷步长的全局结构刚度矩阵。
Moreover, in the context of structural reliability analysis, an additional fourth loop with a large number of samples is required to account for random variables. Collectively, this four-nested loop calculation procedure is highly computationally expensive, or even intractable for complex RC structures in the real world.
此外，在结构可靠性分析中，还需要额外的第四个循环，其中包含大量样本以考虑随机变量。总之，这种四嵌套循环计算程序的计算成本极高，对于现实世界中的复杂 RC 结构而言甚至难以实现。

2.2 Adaptive reliability analysis framework based on uncertainty quantification
2.2 基于不确定性量化的自适应可靠性分析框架

2.2.1 Quantile regression LightGBM model
2.2.1 定量回归 LightGBM 模型

Among a vast number of machine learning/deep learning models for data analysis, tree-based models are currently one of the most favourite ones thanks to their high performance and especially their expressivity, which is built from explicit rules based on the values of features. Another advantage of tree-based methods is that they have fewer hyperparameters to tune from problem to problem compared to the neural network-based counterparts. Furthermore, the performance of tree-based models can be improved by combining different tree models into an ensemble model which is more accurate, robust and applicable to problems featuring complex relationships between inputs and outputs. This is because according to the ensemble learning theory [38], ensemble models benefit from diversity, complementariness, and generalization. In other words, they do not solely depend on a single model configuration, or a specific set of
在大量用于数据分析的机器学习/深度学习模型中，基于树的模型是目前最受欢迎的模型之一，这要归功于它们的高性能，尤其是基于特征值的显式规则所带来的表现力。基于树的方法的另一个优势是，与基于神经网络的方法相比，它们在不同问题上需要调整的超参数较少。此外，还可以通过将不同的树模型组合成一个集合模型来提高树模型的性能，这种集合模型更准确、更稳健，而且适用于输入和输出之间关系复杂的问题。这是因为根据集合学习理论[38]，集合模型得益于多样性、互补性和泛化。换句话说，集合模型并不完全依赖于单一的模型配置，或一组特定的参数。
hyperparameters, and are able to leverage different strategies to discover the impacts of the features and their interrelationships on the outputs.
超参数，并能利用不同的策略来发现特征及其相互关系对输出的影响。
Individual trees in the ensemble model can grow in two fashions: leaf-wise or level-wise. Specifically, in level-wise fashion, at each calculation iteration, a new branch will be added to every leaf of the current level. Meanwhile, in leafwise fashion, only the leaf having the most impact on the loss function is deepened, i.e., prioritizing tree depth over width. This usually result in a lower loss function value with fewer leaves, leading to faster computational time. The popular leaf-wise ensemble model is LightGBM [39] whose mechanism is highlighted in Fig. 2 and mathematically described as follows.
集合模型中的单棵树可以以两种方式生长：按树叶生长或按层次生长。具体来说，在逐级增长时，每次计算迭代都会在当前级别的每一片叶子上添加一个新的分支。与此同时，在按树叶顺序计算时，只有对损失函数影响最大的树叶才会被加深，也就是说，树的深度优先于宽度。这通常会以较少的树叶获得较低的损失函数值，从而加快计算时间。常用的叶式集合模型是 LightGBM [39]，其机制如图 2 所示，数学描述如下。
Assuming that at learning iteration t t tt, the ensemble model consists of t t tt tree models:
假设在学习迭代 t t tt 时，集合模型由 t t tt 个树模型组成：
f ^ ( χ ) = ∑ i = 0 t − 1 f ^ i ( χ ) f ^ ( χ ) = ∑ i = 0 t − 1   f ^ i ( χ ) widehat(f)(chi)=sum_(i=0)^(t-1) widehat(f)_(i)(chi)\widehat{f}(\chi)=\sum_{i=0}^{t-1} \widehat{f}_{i}(\chi),
Next, the residual error r ( y i , f ( x i ) ) r y i , f x i r(y_(i),f(x_(i)))r\left(y_{i}, f\left(x_{i}\right)\right) between the results predicted by the ensemble model and actual values is calculated. Subsequently, a new individual model f ^ t ( χ ) = h ( χ , θ ) f ^ t ( χ ) = h ( χ , θ ) widehat(f)_(t)(chi)=h(chi,theta)\widehat{f}_{t}(\chi)=h(\chi, \theta) developed from f ^ t − 1 ( χ ) f ^ t − 1 ( χ ) hat(f)_(t-1)(chi)\hat{f}_{t-1}(\chi) to estimate the residual is designed, whose parameters θ θ theta\theta are determined by:
接着，计算集合模型预测结果与实际值之间的残差 r ( y i , f ( x i ) ) r y i , f x i r(y_(i),f(x_(i)))r\left(y_{i}, f\left(x_{i}\right)\right) 。随后，设计一个由 f ^ t − 1 ( χ ) f ^ t − 1 ( χ ) hat(f)_(t-1)(chi)\hat{f}_{t-1}(\chi) 发展而来的新的单独模型 f ^ t ( χ ) = h ( χ , θ ) f ^ t ( χ ) = h ( χ , θ ) widehat(f)_(t)(chi)=h(chi,theta)\widehat{f}_{t}(\chi)=h(\chi, \theta) 来估计残差，其参数 θ θ theta\theta 由以下公式确定：
θ t = argmin ( ∑ i = 1 N sample ( r i , t − h ( χ i , θ ) ) 2 ) θ t = argmin ∑ i = 1 N sample    r i , t − h χ i , θ 2 theta_(t)=argmin(sum_(i=1)^(N_("sample "))(r_(i,t)-h(chi_(i),theta))^(2))\theta_{t}=\operatorname{argmin}\left(\sum_{i=1}^{N_{\text {sample }}}\left(r_{i, t}-h\left(\chi_{i}, \theta\right)\right)^{2}\right)
The ensemble model at learning iteration t t tt is then updated as follows:
然后，学习迭代 t t tt 中的集合模型更新如下：
Fig. 2 Illustration of the Light Gradient Boosting Machine (LightGBM) model
图 2 光梯度提升机（LightGBM）模型图示
Fig. 3 Schematic representation of the proposed RC-UQR framework for reliability analysis of non-linear reinforced concrete structures
图 3 用于非线性钢筋混凝土结构可靠性分析的拟议 RC-UQR 框架示意图
f ^ ( χ ) = ∑ i = 0 t − 1 f ^ i ( χ ) + h ( χ , θ t ) f ^ ( χ ) = ∑ i = 0 t − 1   f ^ i ( χ ) + h χ , θ t widehat(f)(chi)=sum_(i=0)^(t-1) widehat(f)_(i)(chi)+h(chi,theta_(t))\widehat{f}(\chi)=\sum_{i=0}^{t-1} \widehat{f}_{i}(\chi)+h\left(\chi, \theta_{t}\right)
In addition to the conventional LightGBM model which provides a single predicted value for a given input data, a variant version of LightGBM, namely quantile regressionLightGBM (QR-LightGBM) is derived to yield additional uncertainty estimation of prediction. Recall that a quantile value Q ( τ ) Q ( τ ) Q(tau)Q(\tau) is determined by:
除了为给定输入数据提供单一预测值的传统 LightGBM 模型外，我们还推导出了 LightGBM 的变体版本，即量化回归 LightGBM (QR-LightGBM)，以获得额外的预测不确定性估计。回想一下，量值 Q ( τ ) Q ( τ ) Q(tau)Q(\tau) 是通过以下方式确定的：
Q ( τ ) = { y ∈ R : F ( y ) ≥ τ } Q ( τ ) = { y ∈ R : F ( y ) ≥ τ } Q(tau)={y inR:F(y) >= tau}Q(\tau)=\{y \in \mathbb{R}: F(y) \geq \tau\} with τ ∈ ( 0 , 1 ) τ ∈ ( 0 , 1 ) tau in(0,1)\tau \in(0,1)   Q ( τ ) = { y ∈ R : F ( y ) ≥ τ } Q ( τ ) = { y ∈ R : F ( y ) ≥ τ } Q(tau)={y inR:F(y) >= tau}Q(\tau)=\{y \in \mathbb{R}: F(y) \geq \tau\} 与 τ ∈ ( 0 , 1 ) τ ∈ ( 0 , 1 ) tau in(0,1)\tau \in(0,1) 的关系
To yield the quantile value Q ( τ ) Q ( τ ) Q(tau)Q(\tau), the QR -LightGBM model will be trained by using the quantile regression loss function L τ ( y , y ^ ) L τ ( y , y ^ ) L_(tau)(y, hat(y))L_{\tau}(y, \hat{y}) instead of a normal regression loss function directly measuring the discrepancy between predicted and actual results such as MAE = | y − y ^ | = | y − y ^ | =|y- hat(y)|=|y-\hat{y}|. The quantile regression loss function is mathematically expressed as follows [40]:
为了得到量化值 Q ( τ ) Q ( τ ) Q(tau)Q(\tau) ，将使用量化回归损失函数 L τ ( y , y ^ ) L τ ( y , y ^ ) L_(tau)(y, hat(y))L_{\tau}(y, \hat{y}) 来训练 QR -LightGBM 模型，而不是使用直接测量预测结果与实际结果之间差异（如 MAE = | y − y ^ | = | y − y ^ | =|y- hat(y)|=|y-\hat{y}| ）的普通回归损失函数。量化回归损失函数的数学表达式如下 [40]：
L τ ( y , y τ ^ ) = { τ ( y − y τ ^ ) if y − y ^ > 0 ( 1 − τ ) ( y − y τ ^ ) otherwise L τ y , y τ ^ = τ y − y τ ^  if  y − y ^ > 0 ( 1 − τ ) y − y τ ^  otherwise  L_(tau)(y,( widehat(y_(tau))))={[tau(y-( widehat(y_(tau))))" if "y- hat(y) > 0],[(1-tau)(y-( widehat(y_(tau))))" otherwise "]:}L_{\tau}\left(y, \widehat{y_{\tau}}\right)=\left\{\begin{array}{c}\tau\left(y-\widehat{y_{\tau}}\right) \text { if } y-\hat{y}>0 \\ (1-\tau)\left(y-\widehat{y_{\tau}}\right) \text { otherwise }\end{array}\right.
Then, by calculating the upper and lower quantiles Q ( τ high ) , Q ( τ low ) Q τ high  , Q τ low  Q(tau_("high ")),Q(tau_("low "))Q\left(\tau_{\text {high }}\right), Q\left(\tau_{\text {low }}\right), we can estimate the prediction interval Γ = [ Q ( τ low ) , Q ( τ high ) ] Γ = Q τ low  , Q τ high  Gamma=[Q(tau_("low ")),Q(tau_("high "))]\Gamma=\left[Q\left(\tau_{\text {low }}\right), Q\left(\tau_{\text {high }}\right)\right]. For example, a prediction interval with a coverage of 90 % 90 % 90%90 \% is defined by Γ = [ Q ( 0.05 ) , Q ( 0.95 ) ] Γ = [ Q ( 0.05 ) , Q ( 0.95 ) ] Gamma=[Q(0.05),Q(0.95)]\Gamma=[Q(0.05), Q(0.95)]. Based on Γ Γ Gamma\Gamma, we can estimate the standard deviation of the predicted values, denoted by σ g ^ ( χ ) σ g ^ ( χ ) sigma_( widehat(g))(chi)\sigma_{\widehat{g}}(\chi).
然后，通过计算上量化值 Q ( τ high ) , Q ( τ low ) Q τ high  , Q τ low  Q(tau_("high ")),Q(tau_("low "))Q\left(\tau_{\text {high }}\right), Q\left(\tau_{\text {low }}\right) 和下量化值 Q ( τ high ) , Q ( τ low ) Q τ high  , Q τ low  Q(tau_("high ")),Q(tau_("low "))Q\left(\tau_{\text {high }}\right), Q\left(\tau_{\text {low }}\right) ，我们可以估算出预测区间 Γ = [ Q ( τ low ) , Q ( τ high ) ] Γ = Q τ low  , Q τ high  Gamma=[Q(tau_("low ")),Q(tau_("high "))]\Gamma=\left[Q\left(\tau_{\text {low }}\right), Q\left(\tau_{\text {high }}\right)\right] 。例如，覆盖范围为 90 % 90 % 90%90 \% 的预测区间由 Γ = [ Q ( 0.05 ) , Q ( 0.95 ) ] Γ = [ Q ( 0.05 ) , Q ( 0.95 ) ] Gamma=[Q(0.05),Q(0.95)]\Gamma=[Q(0.05), Q(0.95)] 定义。根据 Γ Γ Gamma\Gamma ，我们可以估算出预测值的标准偏差，用 σ g ^ ( χ ) σ g ^ ( χ ) sigma_( widehat(g))(chi)\sigma_{\widehat{g}}(\chi) 表示。

2.2.2 Adaptive reliability analysis framework for reinforced concrete structures using quantile regression LightGBM
2.2.2 使用量化回归 LightGBM 的钢筋混凝土结构自适应可靠性分析框架

Given a predefined safety threshold y s y s y_(s)y_{s}, the limit state function is calculated as:
给定一个预定义的安全临界值 y s y s y_(s)y_{s} ，极限状态函数的计算公式为
g ( X ) = y − y s g ( X ) = y − y s g(X)=y-y_(s)g(X)=y-y_{s}
where X X XX is a set of random variables. If g ( X ) ≥ 0 g ( X ) ≥ 0 g(X) >= 0g(X) \geq 0, the structure is in a safe condition, otherwise if g ( X ) < 0 g ( X ) < 0 g(X) < 0g(X)<0, a failure
其中 X X XX 是一组随机变量。如果 g ( X ) ≥ 0 g ( X ) ≥ 0 g(X) >= 0g(X) \geq 0 ，则结构处于安全状态，否则，如果 g ( X ) < 0 g ( X ) < 0 g(X) < 0g(X)<0 ，则出现故障
occurs. With a database of N sample N sample  N_("sample ")N_{\text {sample }} samples, the failure probability is calculated by:
发生。在 N sample N sample  N_("sample ")N_{\text {sample }} 样本数据库中，故障概率的计算方法是：
P f = 1 N sample × ∑ i = 1 N sample I ( g ( X i ) ) P f = 1 N sample  × ∑ i = 1 N sample    I g X i P_(f)=(1)/(N_("sample "))xxsum_(i=1)^(N_("sample "))I(g(X_(i)))P_{f}=\frac{1}{N_{\text {sample }}} \times \sum_{i=1}^{N_{\text {sample }}} I\left(g\left(X_{i}\right)\right) with I ( g ( X i ) ) = { 0 if g ( X i ) ≥ 0 1 if g ( X i ) < 0 I g X i = 0  if  g X i ≥ 0 1  if  g X i < 0 I(g(X_(i)))={[0" if "g(X_(i)) >= 0],[1" if "g(X_(i)) < 0]:}I\left(g\left(X_{i}\right)\right)=\left\{\begin{array}{l}0 \text { if } g\left(X_{i}\right) \geq 0 \\ 1 \text { if } g\left(X_{i}\right)<0\end{array}\right.   P f = 1 N sample × ∑ i = 1 N sample I ( g ( X i ) ) P f = 1 N sample  × ∑ i = 1 N sample    I g X i P_(f)=(1)/(N_("sample "))xxsum_(i=1)^(N_("sample "))I(g(X_(i)))P_{f}=\frac{1}{N_{\text {sample }}} \times \sum_{i=1}^{N_{\text {sample }}} I\left(g\left(X_{i}\right)\right) 与 I ( g ( X i ) ) = { 0 if g ( X i ) ≥ 0 1 if g ( X i ) < 0 I g X i = 0  if  g X i ≥ 0 1  if  g X i < 0 I(g(X_(i)))={[0" if "g(X_(i)) >= 0],[1" if "g(X_(i)) < 0]:}I\left(g\left(X_{i}\right)\right)=\left\{\begin{array}{l}0 \text { if } g\left(X_{i}\right) \geq 0 \\ 1 \text { if } g\left(X_{i}\right)<0\end{array}\right. 的关系
The estimated value of P f P f P_(f)P_{f} and the required value of N sample N sample  N_("sample ")N_{\text {sample }} can be related via the following equation:
P f P f P_(f)P_{f} 的估计值和 N sample N sample  N_("sample ")N_{\text {sample }} 的要求值可以通过下式联系起来：
N sample = 1 − P f P f × CoV 2 N sample  = 1 − P f P f × CoV 2 N_("sample ")=(1-P_(f))/(P_(f)xxCoV^(2))N_{\text {sample }}=\frac{1-P_{f}}{P_{f} \times \operatorname{CoV}^{2}}
where CoV is an user-defined coefficient of variance of P f P f P_(f)P_{f}. Usually, N sample N sample  N_("sample ")N_{\text {sample }} is substantial, reaching millions or more. On the other hand, using finite element method (FEM) to calculate outputs of nonlinear RC structures requires a time-consuming four-nested loop calculation procedure as mentioned above. Therefore, in this study, the LightGBM model is utilized as a surrogate model to estimate outputs with reduced time complexity. Building an accurate surrogate model for non-linear RC structures requires not only a sufficient quantity of training data samples but also relevant samples from different structures’ behaviours ranging from linear elastic, non-linear elastic to non-linear plastic. In other words, using a large number of redundant samples in the elastic range may not help the surrogate model accurately predict the plastic responses. That is why in this study, we develop an active learning framework that can identify relevant data samples to progressively improve the performance of LightGBM.
其中 CoV 是用户定义的 P f P f P_(f)P_{f} 方差系数。通常情况下， N sample N sample  N_("sample ")N_{\text {sample }} 的值很大，可达数百万或更多。另一方面，使用有限元法（FEM）计算非线性 RC 结构的输出需要上述耗时的四嵌套循环计算程序。因此，在本研究中，利用 LightGBM 模型作为代用模型来估算输出，同时降低时间复杂性。为非线性 RC 结构建立精确的代用模型不仅需要足够数量的训练数据样本，还需要从线性弹性、非线性弹性到非线性塑性等不同结构行为的相关样本。换句话说，在弹性范围内使用大量冗余样本可能无助于代用模型准确预测塑性响应。因此，在本研究中，我们开发了一种主动学习框架，可以识别相关数据样本，从而逐步提高 LightGBM 的性能。
For this purpose, on the basis of the intuition that relevant samples are either those near the limit state surface, i.e., g ( X ) ≈ 0 g ( X ) ≈ 0 g(X)~~0g(X) \approx 0, or those with high uncertainty estimation, i.e., large σ g ^ ( X ) σ g ^ ( X ) sigma_( widehat(g))(X)\sigma_{\widehat{g}}(X), a U-function is introduced to rank the potential impact of data samples on the performance of the LightGBM model. This function is the ratio between the mean value μ g ^ ( χ ) μ g ^ ( χ ) mu_( hat(g))(chi)\mu_{\hat{g}}(\chi) and standard deviation g ^ ( X ) g ^ ( X ) widehat(g)(X)\widehat{g}(X) as follows [20]:
为此，根据相关样本要么是接近极限状态面（即 g ( X ) ≈ 0 g ( X ) ≈ 0 g(X)~~0g(X) \approx 0 ）的样本，要么是不确定性估计较高（即 σ g ^ ( X ) σ g ^ ( X ) sigma_( widehat(g))(X)\sigma_{\widehat{g}}(X) 较大）的样本这一直觉，我们引入了一个 U 函数来排列数据样本对 LightGBM 模型性能的潜在影响。该函数是平均值 μ g ^ ( χ ) μ g ^ ( χ ) mu_( hat(g))(chi)\mu_{\hat{g}}(\chi) 与标准偏差 g ^ ( X ) g ^ ( X ) widehat(g)(X)\widehat{g}(X) 之间的比率，如下所示 [20]：
Fig. 4 Python functions to create tcl scripts (a) and run OpenSees programs (b)
图 4 用于创建 tcl 脚本（a）和运行 OpenSees 程序（b）的 Python 函数
# Python function to update TCL script for OpenSees
def gen_tcl(X, fname = 'base.tcl', fout='id_'):
    # X : is a vector containing values of random variables
    # base.tcl is the reference tcl script
    list_key = ['LCol', 'HCol', 'BCol', 'coverCol', 'barAreaTop', 'fc', 'Ec', 'Fy', 'Es']
    with open(fname, 'r') as f:
        file = f.readlines()
        for i, line in enumerate(file):
            for j, k in enumerate(list_key):
                    if f'set {k} ' in line:
                        file[i] = f'set {k} {X[j]}\n'
            if 'recorder Node' in line:
                file[i] = f'recorder Node -file {fout}disp_{int(X[-1])}.txt -time -node 2 -dof 1 disp\n'
    # Indexing the tcl file by the sample ID
    with open(fout + f'{int(X[-1])}.tcl', 'w') as fw:
        fw.writelines(file)
        fw.close()
    return None
a) Creating tcl scripts with given random variable values
a) 使用给定的随机变量值创建 tcl 脚本
# Run OpenSees program with tcl scripts and the dataframe of random variables
in ='ex_3.tcl'
fout='ex_3/id_
for i in range(Nsample):
    X = df.iloc[i].values
    gen_tcl(x, fname = fin, fout=fout)
    subprocess.run(f"Opensees {fout}{i}.tcl")
b) Running OpenSees program via Python subprocess module
b) 通过 Python 子进程模块运行 OpenSees 程序
def objective(trial):
    # Define hyperparameters
    learning_rate = trial.suggest_float("learning_rate", 1.0e-3, 1.0)
    max_depth= trial.suggest_int("max_depth", 2,10)
    num_leaves = trial.suggest_int('num_leaves', 2, 100)
    n_estimators=trial.suggest_int("n_estimators", 2, 1000)
    min_child_weight=trial.suggest_int("min_child_weight", 2,100)
    # Create and train the model
    model = LGBMRegressor(learning_rate=learning_rate,
                    max_depth=max_depth,
                    num_leaves = num_leaves,
            n_estimators=n_estimators,
            min_child_weight=min_child_weight)
    model.fit(X_train, y_train)
    y_pred = model.predict(X_valid)
    e_xgb = 1-r2_score(y_valid, y_pred)
    return e_xgb
study = = == optuna.create_study(direction=‘minimize’)
研究 = = == optuna.create_study(direction='minimize')
study.optimize(objective, n_trials=300)
Fig. 5 Hyperparameter Optimization step with the aid of the Optuna library
图 5 借助 Optuna 库的超参数优化步骤
U ( χ ) = | μ g ^ ( χ ) | σ g ^ ( χ ) U ( χ ) = μ g ^ ( χ ) σ g ^ ( χ ) U(chi)=(|mu_( hat(g))(chi)|)/(sigma_( widehat(g))(chi))U(\chi)=\frac{\left|\mu_{\hat{g}}(\chi)\right|}{\sigma_{\widehat{g}}(\chi)}
It can be seen that a small value of U ( X ) U ( X ) U(X)U(X) implies that either μ g ^ ( χ ) μ g ^ ( χ ) mu_( hat(g))(chi)\mu_{\hat{g}}(\chi) is near zero or σ g ^ ( χ ) σ g ^ ( χ ) sigma_( hat(g))(chi)\sigma_{\hat{g}}(\chi) is significantly high, or both scenarios. Consequently, we select k k kk samples with the smallest values of U ( X ) U ( X ) U(X)U(X) to include in the current Design of Experiment (DoE) to retrain and improve the LightGBM model.
可以看出， U ( X ) U ( X ) U(X)U(X) 的小值意味着 μ g ^ ( χ ) μ g ^ ( χ ) mu_( hat(g))(chi)\mu_{\hat{g}}(\chi) 接近零或 σ g ^ ( χ ) σ g ^ ( χ ) sigma_( hat(g))(chi)\sigma_{\hat{g}}(\chi) 明显偏高，或者两种情况都有。因此，我们选择 k k kk 中 U ( X ) U ( X ) U(X)U(X) 值最小的样本纳入当前的实验设计 (DoE)，以重新训练和改进 LightGBM 模型。
The working flow of the proposed framework is summarized in Algorithm 1 and schematically presented in Fig. 3, which consists of three blocks: Data preparation & Model initialization, Active learning process and Reliability results. In the first block, the key random variables having significant effects on the structure’s reliability are specified, including their probability characteristics such as mean, standard deviation values and type of probability distributions. Next, a Monte Carlo population with a large number of data samples N M C N M C N_(MC)N_{M C} is generated, N M C N M C N_(MC)N_{M C} can be roughly estimated by Eq. (8) using an user-defined coefficient of variance (CoV) value and a rough estimation of P f P f P_(f)P_{f}. Besides, an initial DoE dataset is generated with N D o E ≪ N M C N D o E ≪ N M C N_(DoE)≪N_(MC)N_{D o E} \ll N_{M C}. After that, FEM evaluations on DoE are carried out, the outputs of interest and limit state function values are derived, and appended to DoE, forming a training dataset for the LightGBM model.
拟议框架的工作流程概述于算法 1，示意图见图 3，由三个部分组成：数据准备与模型初始化、主动学习过程和可靠性结果。在第一个模块中，指定了对结构可靠性有重要影响的关键随机变量，包括它们的概率特征，如平均值、标准偏差值和概率分布类型。接下来，生成一个具有大量数据样本 N M C N M C N_(MC)N_{M C} 的蒙特卡罗群体， N M C N M C N_(MC)N_{M C} 可以通过公式 (8) 使用用户定义的方差系数 (CoV) 值和 P f P f P_(f)P_{f} 的粗略估计值进行粗略估计。此外，还生成了一个初始 DoE 数据集 N D o E ≪ N M C N D o E ≪ N M C N_(DoE)≪N_(MC)N_{D o E} \ll N_{M C} 。然后，对 DoE 进行有限元评估，得出相关输出和极限状态函数值，并将其添加到 DoE 中，形成 LightGBM 模型的训练数据集。
Specifically, a reference FEM model of the structures is first created. This model is then exported to a script written in the Tcl programming language. After that, a Python function is developed to automatically update the script with random variable values from the pandas data frame generated earlier. The Python subprocess module is employed to execute the OpenSees program with the updated Tcl scripts. For each data sample, an input Tcl script is prepared, and the computed results are stored in a separate ASCII file (Fig. 4). Notice that the input Tcl scripts and output ASCII files are indexed with the same ID of the data sample. Thanks to
具体来说，首先创建结构的参考有限元模型。然后将该模型导出到用 Tcl 编程语言编写的脚本中。然后，开发一个 Python 函数，利用之前生成的 pandas 数据帧中的随机变量值自动更新脚本。Python 子进程模块用于使用更新后的 Tcl 脚本执行 OpenSees 程序。为每个数据样本准备一个输入 Tcl 脚本，并将计算结果存储在一个单独的 ASCII 文件中（图 4）。请注意，输入 Tcl 脚本和输出 ASCII 文件的索引与数据样本的 ID 相同。感谢
the open-source nature of OpenSees, Python, and Tcl, it is feasible to install these programs on various computation servers and perform the numerical simulations in parallel to reduce waiting times. The FEM models, tcl scripts, and Python functions of the case studies in this study are available in the GitHub repository at https://github.com/dvhun g2sky/RC-UQR.
由于 OpenSees、Python 和 Tcl 的开源特性，可以将这些程序安装在不同的计算服务器上，并行执行数值模拟，以减少等待时间。本研究中案例研究的有限元模型、tcl 脚本和 Python 函数可在 GitHub 存储库中获取：https://github.com/dvhun g2sky/RC-UQR。
Next, a Hyperparameter Optimization (HO) step is conducted using DoE to select the optimal set of hyperparameters of LightGBM. The HO in this study is realized with the aid of the Bayesian Optimization algorithm and the Optuna library [41]. The key hyperparameters of LightGBM include the tree depth, the maximum number of estimators, the learning rate, and the minimum number of samples in a leaf (Fig. 5).
接下来，使用 DoE 进行超参数优化（HO）步骤，以选择 LightGBM 的最佳超参数集。本研究中的超参数优化是借助贝叶斯优化算法和 Optuna 库[41]实现的。LightGBM 的关键超参数包括树的深度、估计器的最大数量、学习率和叶中样本的最小数量（图 5）。
The second block is the active learning process which begins by training/retraining LightGBM-QR with the current DoE dataset. Subsequently, the limit state functions, failure probability P f P f P_(f)P_{f} and U-function are calculated by applying the LightGBM-QR model to the Monte Carlo dataset. It is noted that the inference time of LightGBM is very fast, taking only a few milliseconds for each data sample. Hence, this step is relatively fast even with a very large number of samples. After that, the samples are ranked based on their U-function values, and the top- k samples with the lowest U-function values are selected to update the DoE dataset. Afterwards, the stopping condition is verified to determine if the reliability results converge. If the condition is not satisfied, FEM evaluations are then conducted on the newly added k k kk samples, and another active learning iteration is repeated. On the other hand, if the calculated P f P f P_(f)P_{f} has converged but the associated CoV is larger than the desired value it may be necessary to increase N M C N M C N_(MC)N_{M C} and repeat the process. Once the stopping condition is met, the final value of P f P f P_(f)P_{f} and the total computation time are reported.
第二部分是主动学习过程，首先使用当前的 DoE 数据集对 LightGBM-QR 进行训练/再训练。随后，通过将 LightGBM-QR 模型应用于蒙特卡罗数据集，计算出极限状态函数、故障概率 P f P f P_(f)P_{f} 和 U 函数。值得注意的是，LightGBM 的推理时间非常快，每个数据样本只需几毫秒。因此，即使样本数量非常多，这一步骤也相对较快。之后，根据 U 函数值对样本进行排序，并选择 U 函数值最低的前 k 个样本更新 DoE 数据集。然后，验证停止条件，以确定可靠性结果是否收敛。如果不满足条件，则对新添加的 k k kk 样本进行有限元评估，并重复另一次主动学习迭代。另一方面，如果计算出的 P f P f P_(f)P_{f} 已经收敛，但相关的 CoV 大于期望值，则可能需要增加 N M C N M C N_(MC)N_{M C} 并重复该过程。一旦满足停止条件，就会报告 P f P f P_(f)P_{f} 的最终值和总计算时间。
Algorithm 1: Pseudocode of the proposed RC-UQR framework
算法 1：拟议 RC-UQR 框架的伪代码
Input: Structural configuration and random variables with predefined probability distributions
输入结构配置和具有预定概率分布的随机变量
Output: Failure probability P f P f P_(f)P_{f}, reliability index R i R i R_(i)R_{i}, CPU time, insights into failure cases
输出：故障概率 P f P f P_(f)P_{f} 、可靠性指数 R i R i R_(i)R_{i} 、 CPU 时间、对故障案例的深入了解
/* Data preparation and ML model building */
/* 数据准备和 ML 模型构建 */
  1. Generate a Monte Carlo (MC) population with N M C N M C N_(MC)N_{M C} samples
    用 N M C N M C N_(MC)N_{M C} 样本生成蒙特卡罗 (MC) 种群
  2. Generate an initial DoE population with N D o E ≪ N M C N D o E ≪ N M C N_(DoE)≪N_(MC)N_{D o E} \ll N_{M C} samples
    用 N D o E ≪ N M C N D o E ≪ N M C N_(DoE)≪N_(MC)N_{D o E} \ll N_{M C} 样本生成初始 DoE 群体
  3. Perform FEM with DoE, and calculate the limit state function g ( X ) g ( X ) g(X)g(X)
    使用 DoE 进行有限元分析，并计算极限状态函数 g ( X ) g ( X ) g(X)g(X)
  4. Perform Hyperparameter Optimization for LightGBM using the DoE dataset.
    使用 DoE 数据集对 LightGBM 进行超参数优化。
    /* Conduct the active learning loop */
    /* 进行主动学习循环 */
  5. While stopping condition=FALSE do
  6. Train/retrain the QR-LightGBM with the current DoE dataset
    使用当前的 DoE 数据集训练/重新训练 QR-LightGBM
  7. quad\quad For i ← 1 i ← 1 i larr1i \leftarrow 1 to N M C N M C N_(MC)N_{M C} do
    quad\quad 对于 i ← 1 i ← 1 i larr1i \leftarrow 1 到 N M C N M C N_(MC)N_{M C} 做
Compute the limit state function g ( X i ) g X i g(X_(i))g\left(X_{i}\right) and uncertainty quantification σ g ^ ( χ ) σ g ^ ( χ ) sigma_( hat(g))(chi)\sigma_{\hat{g}}(\chi)
计算极限状态函数 g ( X i ) g X i g(X_(i))g\left(X_{i}\right) 和不确定性量化 σ g ^ ( χ ) σ g ^ ( χ ) sigma_( hat(g))(chi)\sigma_{\hat{g}}(\chi)
9.
Compute the U-function by Eq. (16)
根据公式 (16) 计算 U 函数
10. End for  10.结束
11. Estimate P f , R i P f , R i P_(f),R_(i)P_{f}, R_{i} and CoV .
11.估计 P f , R i P f , R i P_(f),R_(i)P_{f}, R_{i} 和 CoV。
12. Sort samples in the MC population by their U-function values
12.按 U 函数值对 MC 群体中的样本进行排序
13. Select k k kk samples with the lowest U-function values
13.选择 U 函数值最低的 k k kk 样品
14. Perform FEM with k k kk newly added samples and update the DoE population
14.使用 k k kk 新添加的样本进行有限元计算，并更新 DoE 群体
15. End while  15.结束同时

3 Numerical applications
3 数值应用

3.1 Example 1: 1D reinforced concrete column
3.1 示例 1：一维钢筋混凝土柱

In the first case-study, we consider a RC column subjected to a horizontal load applied at its top end as shown in Fig. 6. This simple example facilitates a sanity check of the implementation of the proposed method before addressing more complex structures. For this example there are 10 input random variables including the column length, width, height, concrete and reinforcing steel properties, and the load intensity. The statistical properties of these random variables are listed in Table 1, inspired by works in the literature [42, 43]. Since the external load is usually the most uncertain factor; its CoV is set larger than those of others. The non-linear
在第一个案例研究中，我们考虑了一根在上端承受水平荷载的 RC 柱，如图 6 所示。这个简单的例子有助于在处理更复杂的结构之前，对所建议方法的实施进行合理性检查。在这个例子中，有 10 个输入随机变量，包括柱子的长度、宽度、高度、混凝土和钢筋属性以及荷载强度。受文献 [42, 43] 的启发，表 1 列出了这些随机变量的统计属性。由于外部荷载通常是最不确定的因素，因此其 CoV 值比其他因素要大。非线性
analysis of the RC column is realized with the help of the open-source finite element program OpenSees, using fiber sections, and non-linear constitutive laws of concrete and steel as discussed in the previous section. The limit state function for this example is expressed as follows:
在开源有限元程序 OpenSees 的帮助下，利用纤维截面以及上一节讨论的混凝土和钢的非线性结构定律，实现了对 RC 柱的分析。本例的极限状态函数如下所示：
g ( X ) = y − y limit = y − 0.04 g ( X ) = y − y limit  = y − 0.04 g(X)=y-y_("limit ")=y-0.04g(X)=y-y_{\text {limit }}=y-0.04
where y y yy and y limit y limit  y_("limit ")y_{\text {limit }} are the top horizontal displacement and safety threshold ( y limit = 0.04 y limit  = 0.04 y_("limit ")=0.04y_{\text {limit }}=0.04 ), respectively.
其中， y y yy 和 y limit y limit  y_("limit ")y_{\text {limit }} 分别为顶部水平位移和安全阈值（ y limit = 0.04 y limit  = 0.04 y_("limit ")=0.04y_{\text {limit }}=0.04 ）。
Next, the working flow depicted in Fig. 3 is carried out to perform structural reliability analysis of the RC column. Specifically, an initial DoE dataset with a relatively small size of 500 samples is generated with random variables sampled according to Table 2 . Subsequently, 500 simulations with these samples are conducted, and the resulting
接下来，按照图 3 所示的工作流程对 RC 柱进行结构可靠性分析。具体来说，首先生成一个初始 DoE 数据集，该数据集规模较小，只有 500 个样本，随机变量的取样如表 2 所示。随后，利用这些样本进行 500 次模拟，并得出
Fig. 6 Schematic representation of the reinforced concrete column
图 6 钢筋混凝土柱示意图
Table 2 Random variables for the 1D RC column
表 2 1D RC 柱的随机变量
Variable  可变
L col L col  L_("col ")\boldsymbol{L}_{\text {col }}
( m ) ( m ) (m)(\mathrm{m})
L_("col ") (m)| $\boldsymbol{L}_{\text {col }}$ | | :--- | | $(\mathrm{m})$ |
H col H col  H_("col ")\boldsymbol{H}_{\text {col }}
( m ) ( m ) (m)(\mathrm{m})
H_("col ") (m)| $\boldsymbol{H}_{\text {col }}$ | | :--- | | $(\mathrm{m})$ |
B col B col  B_("col ")\boldsymbol{B}_{\text {col }}
( m ) ( m ) (m)(\mathrm{m})
B_("col ") (m)| $\boldsymbol{B}_{\text {col }}$ | | :--- | | $(\mathrm{m})$ |
  封面 ( cm ) ( cm ) (cm)(\mathrm{cm})
cover
( cm ) ( cm ) (cm)(\mathrm{cm})
cover (cm)| cover | | :--- | | $(\mathrm{cm})$ |
A rebar A rebar  A^("rebar ")\boldsymbol{A}^{\text {rebar }}
( cm 2 ) cm 2 (cm^(2))\left(\mathrm{cm}^{2}\right)
A^("rebar ") (cm^(2))| $\boldsymbol{A}^{\text {rebar }}$ | | :--- | | $\left(\mathrm{cm}^{2}\right)$ |
f c f c f_(c)\boldsymbol{f}_{\boldsymbol{c}}
( MPa ) ( MPa ) (MPa)(\mathrm{MPa})
f_(c) (MPa)| $\boldsymbol{f}_{\boldsymbol{c}}$ | | :--- | | $(\mathrm{MPa})$ |
E c E c E_(c)\boldsymbol{E}_{\boldsymbol{c}}
( GPa ) ( GPa ) (GPa)(\mathrm{GPa})
E_(c) (GPa)| $\boldsymbol{E}_{\boldsymbol{c}}$ | | :--- | | $(\mathrm{GPa})$ |
f y f y f_(y)\boldsymbol{f}_{\boldsymbol{y}}
( MPa ) ( MPa ) (MPa)(\mathrm{MPa})
f_(y) (MPa)| $\boldsymbol{f}_{\boldsymbol{y}}$ | | :--- | | $(\mathrm{MPa})$ |
E s E s E_(s)\boldsymbol{E}_{\boldsymbol{s}}
( GPa ) ( GPa ) (GPa)(\mathrm{GPa})
E_(s) (GPa)| $\boldsymbol{E}_{\boldsymbol{s}}$ | | :--- | | $(\mathrm{GPa})$ |
F F F\boldsymbol{F}
( kN ) ( kN ) (kN)(\mathrm{kN})
F (kN)| $\boldsymbol{F}$ | | :--- | | $(\mathrm{kN})$ |
Mean  平均值 8.0 1.5 1.2 6 75.36 27.0 25.0 460 200 750
CoV 20 20 20 20 20 20 20 20 20 20
Dist. Type  地区类型 log log log\log log log log\log log log log\log log log log\log log log log\log normal  正常 normal  正常 log log log\log log log log\log normal  正常
Variable "L_("col ") (m)" "H_("col ") (m)" "B_("col ") (m)" "cover (cm)" "A^("rebar ") (cm^(2))" "f_(c) (MPa)" "E_(c) (GPa)" "f_(y) (MPa)" "E_(s) (GPa)" "F (kN)" Mean 8.0 1.5 1.2 6 75.36 27.0 25.0 460 200 750 CoV 20 20 20 20 20 20 20 20 20 20 Dist. Type log log log log log normal normal log log normal| Variable | $\boldsymbol{L}_{\text {col }}$ <br> $(\mathrm{m})$ | $\boldsymbol{H}_{\text {col }}$ <br> $(\mathrm{m})$ | $\boldsymbol{B}_{\text {col }}$ <br> $(\mathrm{m})$ | cover <br> $(\mathrm{cm})$ | $\boldsymbol{A}^{\text {rebar }}$ <br> $\left(\mathrm{cm}^{2}\right)$ | $\boldsymbol{f}_{\boldsymbol{c}}$ <br> $(\mathrm{MPa})$ | $\boldsymbol{E}_{\boldsymbol{c}}$ <br> $(\mathrm{GPa})$ | $\boldsymbol{f}_{\boldsymbol{y}}$ <br> $(\mathrm{MPa})$ | $\boldsymbol{E}_{\boldsymbol{s}}$ <br> $(\mathrm{GPa})$ | $\boldsymbol{F}$ <br> $(\mathrm{kN})$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Mean | 8.0 | 1.5 | 1.2 | 6 | 75.36 | 27.0 | 25.0 | 460 | 200 | 750 | | CoV | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | | Dist. Type | $\log$ | $\log$ | $\log$ | $\log$ | $\log$ | normal | normal | $\log$ | $\log$ | normal |
Model  模型 MAE RMSE R2 MAPE CPU time (s)  CPU 时间（秒）
Light Gradient Boosting Machine (LightGBM)
光梯度增强机（LightGBM）		 0.0016 0.0025 0.7302 0.0771 0.23
Extreme Gradient Boosting (XGBoost)
极梯度提升（XGBoost）		 0.0016 0.0026 0.7218 0.0782 1.88
AdaBoost Regressor  AdaBoost 调节器 0.0019 0.0029 0.6543 0.0890 0.07
Random Forest Regressor  随机森林回归器 0.0019 0.0029 0.6568 0.0903 0.23
Decision Tree Regressor  决策树回归器 0.0030 0.0043 0.5382 0.1321 0.02
Support Vector Machine  支持向量机 0.0048 0.0058 0.4068 0.910 0.03
Model MAE RMSE R2 MAPE CPU time (s) Light Gradient Boosting Machine (LightGBM) 0.0016 0.0025 0.7302 0.0771 0.23 Extreme Gradient Boosting (XGBoost) 0.0016 0.0026 0.7218 0.0782 1.88 AdaBoost Regressor 0.0019 0.0029 0.6543 0.0890 0.07 Random Forest Regressor 0.0019 0.0029 0.6568 0.0903 0.23 Decision Tree Regressor 0.0030 0.0043 0.5382 0.1321 0.02 Support Vector Machine 0.0048 0.0058 0.4068 0.910 0.03| Model | MAE | RMSE | R2 | MAPE | CPU time (s) | | :--- | :--- | :--- | :--- | :--- | :--- | | Light Gradient Boosting Machine (LightGBM) | 0.0016 | 0.0025 | 0.7302 | 0.0771 | 0.23 | | Extreme Gradient Boosting (XGBoost) | 0.0016 | 0.0026 | 0.7218 | 0.0782 | 1.88 | | AdaBoost Regressor | 0.0019 | 0.0029 | 0.6543 | 0.0890 | 0.07 | | Random Forest Regressor | 0.0019 | 0.0029 | 0.6568 | 0.0903 | 0.23 | | Decision Tree Regressor | 0.0030 | 0.0043 | 0.5382 | 0.1321 | 0.02 | | Support Vector Machine | 0.0048 | 0.0058 | 0.4068 | 0.910 | 0.03 |
Table 3 Model selection results for the RC column example
表 3 RC 柱示例的模型选择结果
Appendix. In order to justify the usage of LightGBM, a comparison study was conducted to highlight its performance in comparison with competing methods which were also finetuned via Hyperparameter Optimization. The comparison results are presented in Table 3, clearly demonstrating that LightGBM provides the best prediction results in terms of various metrics such as mean absolute error (MAE), root mean squared error (RMSE), R2 score, and mean absolute percentage error.
附录。为了证明使用 LightGBM 的合理性，我们进行了一项比较研究，以突出其与竞争方法的性能比较，这些竞争方法也通过超参数优化进行了微调。比较结果如表 3 所示，清楚地表明 LightGBM 在各种指标（如平均绝对误差 (MAE)、均方根误差 (RMSE)、R2 分数和平均绝对百分比误差）方面提供了最佳预测结果。
After fine-tuning, LightGBM is integrated into the active learning process to calculate the failure probability and reliability index of the RC column. The computed reliability results are presented in Table 4. Notice that the calculations with these surrogate models are repeated ten times, and the mean and standard deviation values are reported in the table. In addition, we analyze the RC column’s reliability by using other methods such as the conventional Monte Carlo simulation (MCS) method with FEM, MCS using the LightGBM without active learning, and the Kriging-based MCS. For conventional FEM-MCS, the estimated value of P f P f P_(f)P_{f} is 0.00224 ; hence, if we select a coefficient of variance of 5 % 5 % 5%5 \%, the required total number of samples is about 100,000 according to
经过微调后，LightGBM 被集成到主动学习过程中，用于计算 RC 柱的失效概率和可靠性指数。可靠性计算结果见表 4。请注意，这些代用模型的计算重复了十次，表中报告了平均值和标准偏差值。此外，我们还使用其他方法分析了 RC 柱的可靠性，如使用有限元的传统蒙特卡洛模拟 (MCS)方法、使用 LightGBM（无主动学习）的 MCS 方法以及基于克里金法的 MCS 方法。对于传统的 FEM-MCS 方法， P f P f P_(f)P_{f} 的估计值为 0.00224；因此，如果我们选择方差系数 5 % 5 % 5%5 \% ，则所需的样本总数约为 100,000 个。
Eq. (8). On the other hand, the proposed RC-UQR method achieves a result P f = 0.00219 P f = 0.00219 P_(f)=0.00219P_{f}=0.00219, which is close to that of FEM-MCS but with significantly fewer FEM runs (about 2000), thus significantly reducing the overall CPU time compared to FEM-MCS. It is also observed that directly utilizing LightGBM-FEM leads to more pronounced errors; for example, the failure probability of LighGBMMCS is only 0.00182 , which is less accurate than that of RC-UQR. Meanwhile, the Kriging-based MCS method is also a highly efficient method; however, its reliability results are less accurate compared to RC-UQR when dealing with the RC columns’ non-linear behaviors. It can be seen that among 3 three alternatives to FEM-MCS, the RC-UQR method achieves the highest accuracy. Furthermore, thanks to the ability to select the relevant candidates, RC-UQR and Kriging-based method has lower variances, for example, their standard deviation of P f P f P_(f)P_{f} is only about 5.0 E − 5 5.0 E − 5 5.0E-55.0 \mathrm{E}-5 significantly less than that of LightGBMMCS (6.7E-4). Although, its total CPU time is higher than the two others, but still nearly 10 × 10 × 10 xx10 \times faster than FEMMCS. Therefore, it can state that the proposed method has the most balance between accuracy and efficiency.
式 (8)。另一方面，拟议的 RC-UQR 方法得到的结果 P f = 0.00219 P f = 0.00219 P_(f)=0.00219P_{f}=0.00219 与 FEM-MCS 接近，但 FEM 运行次数明显减少（约 2000 次），因此与 FEM-MCS 相比，大大减少了 CPU 的总体时间。我们还观察到，直接使用 LightGBM-FEM 会导致更明显的误差；例如，LightGBMMCS 的失败概率仅为 0.00182，不如 RC-UQR 精确。同时，基于 Kriging 的 MCS 方法也是一种高效方法，但在处理 RC 柱的非线性行为时，其可靠性结果的准确性不如 RC-UQR 方法。可以看出，在 FEM-MCS 的三种替代方法中，RC-UQR 方法的精度最高。此外，由于 RC-UQR 和基于 Kriging 的方法能够选择相关候选对象，因此它们的方差较小，例如，它们的标准偏差 P f P f P_(f)P_{f} 仅约 5.0 E − 5 5.0 E − 5 5.0E-55.0 \mathrm{E}-5 明显小于 LightGBMMCS 的标准偏差（6.7E-4）。虽然它的总 CPU 时间高于其他两个，但仍比 FEMMCS 快近 10 × 10 × 10 xx10 \times 。因此，可以说所提出的方法在精度和效率之间取得了最大的平衡。
Fig. 7 Parallel coordinate graph highlighting failure cases (in red) for the RC column
图 7 显示 RC 柱故障情况（红色）的平行坐标图

Fig. 8 Schematic representation of the 2D three-story frame
图 8 二维三层框架示意图
Table 5 Random variables for the 2D three-story RC frame
表 5 二维三层 RC 框架的随机变量
Variable  可变 L col ( m ) L col  ( m ) {:[L_("col ")],[(m)]:}\begin{aligned} & \boldsymbol{L}_{\text {col }} \\ & (\mathrm{m}) \end{aligned} H col ( m ) H col  ( m ) {:[H_("col ")],[(m)]:}\begin{aligned} & \boldsymbol{H}_{\text {col }} \\ & (\mathrm{m}) \end{aligned} B col (m) B col   (m)  {:[B_("col ")],[" (m) "]:}\begin{aligned} & \boldsymbol{B}_{\text {col }} \\ & \text { (m) } \end{aligned} L beam ( m ) L beam  ( m ) {:[L_("beam ")],[(m)]:}\begin{aligned} & \boldsymbol{L}_{\text {beam }} \\ & (\mathrm{m}) \end{aligned} H beam (m) H beam   (m)  {:[H_("beam ")],[" (m) "]:}\begin{aligned} & \boldsymbol{H}_{\text {beam }} \\ & \text { (m) } \end{aligned} B beam (m) B beam   (m)  {:[B_("beam ")],[" (m) "]:}\begin{aligned} & \boldsymbol{B}_{\text {beam }} \\ & \text { (m) } \end{aligned}
  封面（厘米）
Cover
(cm)
Cover (cm)| Cover | | :--- | | (cm) |
 A cobar rebl ( cm 2 ) A cobar  rebl  cm 2 {:[A_("cobar ")^("rebl ")],[(cm^(2))]:}\begin{aligned} & \boldsymbol{A}_{\text {cobar }}^{\text {rebl }} \\ & \left(\mathrm{cm}^{2}\right) \end{aligned}
Mean  平均值 3.5 0.6 0.6 8.0 0.8 0.6 6 50.24
CoV 20 20 20 20 20 20 20 20
Dist. Type  地区类型 log log log\log log log log\log log log log\log log log log\log log log log\log log log log\log log log log\log log log log\log
Variable  可变 A bearn rebar ( cm 2 ) A bearn  rebar  cm 2 {:[A_("bearn ")^("rebar ")],[(cm^(2))]:}\begin{aligned} & \boldsymbol{A}_{\text {bearn }}^{\text {rebar }} \\ & \left(\mathrm{cm}^{2}\right) \end{aligned}
   f c f c f_(c)f_{c} （兆帕）
f c f c f_(c)f_{c}
(MPa)
f_(c) (MPa)| $f_{c}$ | | :--- | | (MPa) |
 E c (GPa) E c  (GPa)  {:[E_(c)],[" (GPa) "]:}\begin{aligned} & \boldsymbol{E}_{c} \\ & \text { (GPa) } \end{aligned} f y ( MPa ) f y ( MPa ) {:[f_(y)],[(MPa)]:}\begin{aligned} & f_{y} \\ & (\mathrm{MPa}) \end{aligned} E s ( GPa ) E s ( GPa ) {:[E_(s)],[(GPa)]:}\begin{aligned} & \boldsymbol{E}_{s} \\ & (\mathrm{GPa}) \end{aligned} F 2 ( kN ) F 2 ( kN ) {:[F_(2)],[(kN)]:}\begin{aligned} & \boldsymbol{F}_{2} \\ & (\mathrm{kN}) \end{aligned} F 1 ( kN ) F 1 ( kN ) {:[F_(1)],[(kN)]:}\begin{aligned} & \boldsymbol{F}_{1} \\ & (\mathrm{kN}) \end{aligned}
Mean  平均值 15.70 27.0 25.0 460 200 375 750
CoV 20 20 20 20 20 20 20
Dist. Type  地区类型 log log log\log normal  正常 normal  正常 log log log\log log log log\log normal  正常 normal  正常
Variable "L_(col ) (m)" "H_(col ) (m)" "B_(col ) (m) " "L_(beam ) (m)" "H_(beam ) (m) " "B_(beam ) (m) " "Cover (cm)" "A_(cobar )^(rebl ) (cm^(2))" Mean 3.5 0.6 0.6 8.0 0.8 0.6 6 50.24 CoV 20 20 20 20 20 20 20 20 Dist. Type log log log log log log log log Variable "A_(bearn )^(rebar ) (cm^(2))" "f_(c) (MPa)" "E_(c) (GPa) " "f_(y) (MPa)" "E_(s) (GPa)" "F_(2) (kN)" "F_(1) (kN)" Mean 15.70 27.0 25.0 460 200 375 750 CoV 20 20 20 20 20 20 20 Dist. Type log normal normal log log normal normal | Variable | $\begin{aligned} & \boldsymbol{L}_{\text {col }} \\ & (\mathrm{m}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{H}_{\text {col }} \\ & (\mathrm{m}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{B}_{\text {col }} \\ & \text { (m) } \end{aligned}$ | $\begin{aligned} & \boldsymbol{L}_{\text {beam }} \\ & (\mathrm{m}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{H}_{\text {beam }} \\ & \text { (m) } \end{aligned}$ | $\begin{aligned} & \boldsymbol{B}_{\text {beam }} \\ & \text { (m) } \end{aligned}$ | Cover <br> (cm) | $\begin{aligned} & \boldsymbol{A}_{\text {cobar }}^{\text {rebl }} \\ & \left(\mathrm{cm}^{2}\right) \end{aligned}$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Mean | 3.5 | 0.6 | 0.6 | 8.0 | 0.8 | 0.6 | 6 | 50.24 | | CoV | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | | Dist. Type | $\log$ | $\log$ | $\log$ | $\log$ | $\log$ | $\log$ | $\log$ | $\log$ | | Variable | $\begin{aligned} & \boldsymbol{A}_{\text {bearn }}^{\text {rebar }} \\ & \left(\mathrm{cm}^{2}\right) \end{aligned}$ | $f_{c}$ <br> (MPa) | $\begin{aligned} & \boldsymbol{E}_{c} \\ & \text { (GPa) } \end{aligned}$ | $\begin{aligned} & f_{y} \\ & (\mathrm{MPa}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{E}_{s} \\ & (\mathrm{GPa}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{F}_{2} \\ & (\mathrm{kN}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{F}_{1} \\ & (\mathrm{kN}) \end{aligned}$ | | | Mean | 15.70 | 27.0 | 25.0 | 460 | 200 | 375 | 750 | | | CoV | 20 | 20 | 20 | 20 | 20 | 20 | 20 | | | Dist. Type | $\log$ | normal | normal | $\log$ | $\log$ | normal | normal | |
Table 6 Comparison of reliability results between the RC-UQR method and the competing method for the 2D 3 -story frame
表 6 二维三层框架的 RC-UQR 方法与竞争方法的可靠性结果比较
Results  成果 FEM-MCS RC-UQR LightGBM-MCS Kriging-based MCS  基于克里金法的监控监
Failure probability P f P f P_(f)P_{f}
故障概率 P f P f P_(f)P_{f} 		0.00197 0.00189 ± 4.53 E − 5 0.00189 ± 4.53 E − 5 0.00189+-4.53E-50.00189 \pm 4.53 \mathrm{E}-5 0.00072 ± 2.1 E − 4 0.00072 ± 2.1 E − 4 0.00072+-2.1E-40.00072 \pm 2.1 \mathrm{E}-4 0.00101 ± 4.54 E − 5 0.00101 ± 4.54 E − 5 0.00101+-4.54E-50.00101 \pm 4.54 \mathrm{E}-5
Reliability index R i R i R_(i)R_{i}
可靠性指数 R i R i R_(i)R_{i} 		2.883 2.896 ± 0.007 2.896 ± 0.007 2.896+-0.0072.896 \pm 0.007 3.197 ± 0.089 3.197 ± 0.089 3.197+-0.0893.197 \pm 0.089 3.092 ± 0.013 3.092 ± 0.013 3.092+-0.0133.092 \pm 0.013
Initial DoE size  初始 DoE 大小 0 500 500 500
Total FEM runs  FEM 运行总数 100,000 5000 500 2000
CPU time for FEM (s)
有限元计算 CPU 时间（秒）		 90,000 4500 450 1800
CPU time for HO (s)
HO 的 CPU 时间（秒）		 0 462 462 102
CPU time for active learning (s)
用于主动学习的 CPU 时间（秒）		 0 233 0 676
Total CPU time (s)
CPU 总时间（秒）		 90,000 5195 912 2578
Results FEM-MCS RC-UQR LightGBM-MCS Kriging-based MCS Failure probability P_(f) 0.00197 0.00189+-4.53E-5 0.00072+-2.1E-4 0.00101+-4.54E-5 Reliability index R_(i) 2.883 2.896+-0.007 3.197+-0.089 3.092+-0.013 Initial DoE size 0 500 500 500 Total FEM runs 100,000 5000 500 2000 CPU time for FEM (s) 90,000 4500 450 1800 CPU time for HO (s) 0 462 462 102 CPU time for active learning (s) 0 233 0 676 Total CPU time (s) 90,000 5195 912 2578| Results | FEM-MCS | RC-UQR | LightGBM-MCS | Kriging-based MCS | | :--- | :--- | :--- | :--- | :--- | | Failure probability $P_{f}$ | 0.00197 | $0.00189 \pm 4.53 \mathrm{E}-5$ | $0.00072 \pm 2.1 \mathrm{E}-4$ | $0.00101 \pm 4.54 \mathrm{E}-5$ | | Reliability index $R_{i}$ | 2.883 | $2.896 \pm 0.007$ | $3.197 \pm 0.089$ | $3.092 \pm 0.013$ | | Initial DoE size | 0 | 500 | 500 | 500 | | Total FEM runs | 100,000 | 5000 | 500 | 2000 | | CPU time for FEM (s) | 90,000 | 4500 | 450 | 1800 | | CPU time for HO (s) | 0 | 462 | 462 | 102 | | CPU time for active learning (s) | 0 | 233 | 0 | 676 | | Total CPU time (s) | 90,000 | 5195 | 912 | 2578 |
To gain more insights into the reliability results, a parallel coordinate graph [44] is plotted to graphically present all samples and their corresponding limit state function values in Fig. 7. In the figure, vertical coordinates denote random variables, and the right-most vertical coordinate represents g ( X ) g ( X ) g(X)g(X). A data sample including its random variable input and computed output, is represented by a polyline. If a data sample yields a negative value of g ( X ) g ( X ) g(X)g(X), it is highlighted in red color. Apparently, there is a high likelihood that a long column under substantial applied loads will result in failure cases.
为了更深入地了解可靠性结果，我们绘制了平行坐标图 [44]，在图 7 中以图形方式展示了所有样本及其相应的极限状态函数值。图中，纵坐标表示随机变量，最右侧的纵坐标表示 g ( X ) g ( X ) g(X)g(X) 。数据样本包括随机变量输入和计算输出，用折线表示。如果一个数据样本产生了 g ( X ) g ( X ) g(X)g(X) 的负值，则用红色突出显示。显然，长柱在承受巨大荷载的情况下很有可能出现故障。

3.2 Example 2: 2D reinforced concrete three-story frame
3.2 示例 2：二维钢筋混凝土三层框架

The second example involves a 2D three-story RC frame structure subjected to horizontal loads as depicted in Fig. 8. The width and total height of the frame are 8 m and 10.5 m , respectively. The nominal cross-sections of the columns and beams are 600 × 600 600 × 600 600 xx600600 \times 600 and 600 × 800 mm 2 600 × 800 mm 2 600 xx800mm^(2)600 \times 800 \mathrm{~mm}^{2}, respectively. There are in total 15 random variables, their statistical characteristics are enumerated in Table 5. The output of interest in this example is the top floor horizontal displacement. On the other hand, the limit state function is defined as follows:
第二个例子涉及一个二维三层 RC 框架结构，该结构承受水平荷载，如图 8 所示。框架的宽度和总高度分别为 8 米和 10.5 米。柱和梁的标称截面分别为 600 × 600 600 × 600 600 xx600600 \times 600 和 600 × 800 mm 2 600 × 800 mm 2 600 xx800mm^(2)600 \times 800 \mathrm{~mm}^{2} 。总共有 15 个随机变量，其统计特征列于表 5。本例中关注的输出是顶层水平位移。另一方面，极限状态函数定义如下：
g ( X ) = y top − y limit = y top − 0.154 g ( X ) = y top  − y limit  = y top  − 0.154 g(X)=y_("top ")-y_("limit ")=y_("top ")-0.154g(X)=y_{\text {top }}-y_{\text {limit }}=y_{\text {top }}-0.154
Following the working flow in Fig. 3 and Algorithm 1, the FEM model, DoE dataset, LightGBM, and active learning process are carried out similarly as done in the first example. The reliability results and required computational resources for the second example are presented in detail in Table 6. For RC-UQR, starting with a 500 -size DoE dataset, after 150 iterations, we achieve convergent results of P f = 0.00187 P f = 0.00187 P_(f)=0.00187P_{f}=0.00187 and R i = 2.899 R i = 2.899 R_(i)=2.899R_{i}=2.899. The final size of DoE is about 5000 , i.e., at each iteration, 30 new candidates are appended to DoE. The CPU time for conducting 5000 FEM evaluation is 4500 s . Furthermore, when accounting for CPU times for the HO step, the active learning process, the total CPU times is 5195 s . Meanwhile, the FEM-MCS method provides P f P f P_(f)P_{f} and R i R i R_(i)R_{i} of 0.00197 and 2.883 , respectively, at the expense of 1E5 FEM evaluations and a CPU time of around 90 , 000 s 90 , 000 s 90,000s90,000 \mathrm{~s}. Apparently, RC-UQR can produce reliability results comparable to those of FEM-MCS. The table also points out again that without an active learning process, the LightGBM-MCS method misses a considerable number of fail cases, leading to a considerably smaller P f P f P_(f)P_{f}, i.e., higher R i R i R_(i)R_{i} compared to reference results. On the other hand, the Kriging-based MCS, though it delivers better results compared to LightGBMMCS, still falls short compared to FEM-MCS and RC-UQR. These results reconfirm that RC-UQR emerges as the best alternative method to the time-intensive FEM-MCS in terms of accuracy. Although it requires additional FEM runs due to the active learning process; hence its total CPU time is higher than others but still drastically lower compared to
按照图 3 和算法 1 中的工作流程，有限元模型、DoE 数据集、LightGBM 和主动学习过程与第一个示例类似。第二个实例的可靠性结果和所需计算资源详见表 6。对于 RC-UQR，从 500 大小的 DoE 数据集开始，经过 150 次迭代后，我们获得了 P f = 0.00187 P f = 0.00187 P_(f)=0.00187P_{f}=0.00187 和 R i = 2.899 R i = 2.899 R_(i)=2.899R_{i}=2.899 的收敛结果。DoE 的最终大小约为 5000，即每次迭代都会有 30 个新候选添加到 DoE 中。进行 5000 次 FEM 评估所需的 CPU 时间为 4500 s。此外，如果将主动学习过程 HO 步骤所需的 CPU 时间计算在内，总的 CPU 时间为 5195 s。同时，FEM-MCS 方法提供的 P f P f P_(f)P_{f} 和 R i R i R_(i)R_{i} 分别为 0.00197 和 2.883，但需要 1E5 次 FEM 评估和大约 90 , 000 s 90 , 000 s 90,000s90,000 \mathrm{~s} 的 CPU 时间。显然，RC-UQR 可以产生与 FEM-MCS 相媲美的可靠性结果。该表还再次指出，在没有主动学习过程的情况下，LightGBM-MCS 方法会遗漏大量故障案例，导致 P f P f P_(f)P_{f} 大大降低，即 R i R i R_(i)R_{i} 高于参考结果。另一方面，与 LightGBMMCS 相比，基于克里金法的 MCS 虽然能提供更好的结果，但与 FEM-MCS 和 RC-UQR 相比仍有不足。这些结果再次证明，就精确度而言，RC-UQR 是耗时的 FEM-MCS 的最佳替代方法。虽然由于主动学习过程，RC-UQR 需要额外的有限元运行，因此其 CPU 总耗时高于其他方法，但与 FEM-MCS 和 RC-UQR 相比，RC-UQR 的 CPU 总耗时仍大大降低。
Fig. 9 Evolution curves of computed reliability index against the number of active learning iterations with different numbers of added candidates at each iteration
图 9 在每次迭代增加不同数量候选者的情况下，计算出的可靠性指数随主动学习迭代次数的变化曲线

Fig. 10 Parallel coordinate graph highlighting failure cases (in red) for the 2D three-story frame
图 10 平行坐标图，突出显示二维三层框架的故障情况（红色部分
FEM-MCS ( 20 × 20 × 20 xx20 \times faster). This reaffirms the excellent balance of RC-UQR between performance and computational resources.
FEM-MCS（ 20 × 20 × 20 xx20 \times 更快）。这再次证明了 RC-UQR 在性能和计算资源之间的出色平衡。
One important hyperparameter of the active learning process is the number of new candidates appended to the current DoE in each active learning iteration. Intuitively, using a small number of candidates will ensure the relevance of candidates but may slow down the active learning process. Meanwhile, using a larger number of candidates may accelerate the active learning process, but could introduce redundant samples, thus increasing the number of FEM evaluations and the total CPU times. Therefore, a parameter study is executed to select the optimal value k k kk for the number of new candidates. We vary k k kk in the range of [ 5 , 10 , 15 , 20 , 30 , 40 , 50 ] [ 5 , 10 , 15 , 20 , 30 , 40 , 50 ] [5,10,15,20,30,40,50][5,10,15,20,30,40,50]. For each value of k k kk, the active learning process is conducted, and the evolution of the reliability index is recorded. The results of the parametric studies are illustrated in Fig. 9. It can be seen that k = 50 k = 50 k=50k=50 and 40 require about 80 and 120 iterations to achieve convergent
主动学习过程中的一个重要超参数是每次主动学习迭代中添加到当前 DoE 中的新候选者数量。直观地说，使用少量候选者可以确保候选者的相关性，但可能会减慢主动学习过程。同时，使用较多的候选样本可能会加速主动学习过程，但可能会引入冗余样本，从而增加有限元求值次数和 CPU 总占用时间。因此，我们对参数进行了研究，以选择新候选样本数量的最佳值 k k kk 。我们在 [ 5 , 10 , 15 , 20 , 30 , 40 , 50 ] [ 5 , 10 , 15 , 20 , 30 , 40 , 50 ] [5,10,15,20,30,40,50][5,10,15,20,30,40,50] 的范围内改变 k k kk 。对于 k k kk 的每个值，我们都进行了主动学习，并记录了可靠性指数的变化情况。参数研究结果如图 9 所示。从图中可以看出， k = 50 k = 50 k=50k=50 和 40 分别需要迭代 80 次和 120 次才能达到收敛。
results, respectively. Meanwhile using k = 5 k = 5 k=5k=5, the process has not converged even after 200 iterations. Based on this figure, we observe that using k = 40 k = 40 k=40k=40 balances the speed of convergence and the model performance.
结果。同时，使用 k = 5 k = 5 k=5k=5 时，即使迭代 200 次，过程也没有收敛。从图中我们可以看出，使用 k = 40 k = 40 k=40k=40 可以平衡收敛速度和模型性能。
Furthermore, the reliability results are visualized via the parallel coordinate graph as shown in Fig. 10. It can be seen that the column length (frame height), external load applied at the top story have the most influence on the limit state function. In addition, samples with negative g ( X ) g ( X ) g(X)g(X) also exhibit below-average values for beam height and column rebar area.
此外，如图 10 所示，可靠性结果可通过平行坐标图直观显示。可以看出，柱长（框架高度）、施加在顶层的外部荷载对极限状态函数的影响最大。此外，负 g ( X ) g ( X ) g(X)g(X) 样本的梁高和柱钢筋面积也低于平均值。

3.3 Example 3: 3D reinforced concrete five-story building
3.3 示例 3：三维钢筋混凝土五层楼建筑

For the third example, a 3D 5-story RC building structure is investigated. The structure configuration, including the dimensions, external loads is demonstrated in Fig. 11. It is noteworthy that, for 3D simulation, each column-beam
在第三个例子中，研究了三维 5 层 RC 建筑结构。图 11 展示了该结构的构造，包括尺寸和外部荷载。值得注意的是，在三维模拟中，每根柱梁
Fig. 11 Schematic representation of the 3D five-story building structures
图 11 三维五层建筑结构示意图
Table 7 Random variables for the 3D RC building structure
表 7 三维 RC 建筑结构的随机变量
Table 8 Comparison of reliability results between the RC-UQR method and the competing method for the 3D five-story building structure
表 8 在三维五层建筑结构中，RC-UQR 方法与竞争方法的可靠性结果比较
Variable  可变 L col ( m ) L col  ( m ) {:[L_("col ")],[(m)]:}\begin{aligned} & \boldsymbol{L}_{\text {col }} \\ & (\mathrm{m}) \end{aligned} H col ( m ) H col  ( m ) H_("col ")_((m))\underset{(\mathrm{m})}{\boldsymbol{H}_{\text {col }}} B col ( m ) B col  ( m ) {:[B_("col ")],[(m)]:}\begin{aligned} & \boldsymbol{B}_{\text {col }} \\ & (\mathrm{m}) \end{aligned} L beam ( m ) L beam  ( m ) {:[L_("beam ")],[(m)]:}\begin{aligned} & \boldsymbol{L}_{\text {beam }} \\ & (\mathrm{m}) \end{aligned} H beam ( m ) H beam  ( m ) H_("beam ")_((m))\underset{(\mathrm{m})}{\boldsymbol{H}_{\text {beam }}} B beam ( m ) B beam  ( m ) {:[B_("beam ")],[(m)]:}\begin{aligned} & \boldsymbol{B}_{\text {beam }} \\ & (\mathrm{m}) \end{aligned} L girder ( m ) L girder  ( m ) {:[L_("girder ")],[(m)]:}\begin{aligned} & \boldsymbol{L}_{\text {girder }} \\ & (\mathrm{m}) \end{aligned}
Mean  平均值 3.5 0.6 0.6 9.0 0.8 0.6 7.0
CoV 20 20 20 20 20 20 20
Dist. Type  地区类型 log log log\log log log log\log log log log\log log log log\log log log log\log log log log\log Log  日志
Variable  可变 H girder ( m ) H girder  ( m ) {:[H_("girder ")],[(m)]:}\begin{aligned} & \boldsymbol{H}_{\text {girder }} \\ & (\mathrm{m}) \end{aligned} B girder (m) B girder   (m)  {:[B_("girder ")],[" (m) "]:}\begin{aligned} & \boldsymbol{B}_{\text {girder }} \\ & \text { (m) } \end{aligned} Cover (cm)  封面（厘米） A rebar ( cm col ) A rebar  cm col  {:[A^("rebar ")],[(cm^("col "))]:}\begin{aligned} & A^{\text {rebar }} \\ & \left(\mathrm{cm}^{\text {col }}\right) \end{aligned} A bearn rebar ( cm 2 ) A bearn  rebar  cm 2 {:[A_("bearn ")^("rebar ")],[(cm^(2))]:}\begin{aligned} & \boldsymbol{A}_{\text {bearn }}^{\text {rebar }} \\ & \left(\mathrm{cm}^{2}\right) \end{aligned} A girger rebr ( cm 2 ) A girger  rebr  cm 2 {:[A_("girger ")^("rebr ")],[(cm^(2))]:}\begin{aligned} & \boldsymbol{A}_{\text {girger }}^{\text {rebr }} \\ & \left(\mathrm{cm}^{2}\right) \end{aligned} f c f c f_(c)f_{c} (MPa)   f c f c f_(c)f_{c} （兆帕）
Mean  平均值 0.6 0.6 5 50.24 15.70 15.70 27.0
CoV 20 20 20 20 20 20 20
Dist. Type  地区类型 log log log\log normal  正常 normal  正常 log log log\log log log log\log log log log\log normal  正常
Variable  可变 E c (GPa) E c  (GPa)  {:[E_(c)],[" (GPa) "]:}\begin{aligned} & \boldsymbol{E}_{\boldsymbol{c}} \\ & \text { (GPa) } \end{aligned} f y ( MPa ) f y ( MPa ) {:[f_(y)],[(MPa)]:}\begin{aligned} & f_{y} \\ & (\mathrm{MPa}) \end{aligned} E s ( GPa ) E s ( GPa ) {:[E_(s)],[(GPa)]:}\begin{aligned} & \boldsymbol{E}_{s} \\ & (\mathrm{GPa}) \end{aligned} F 2 ( kN ) F 2 ( kN ) {:[F_(2)],[(kN)]:}\begin{aligned} & \boldsymbol{F}_{2} \\ & (\mathrm{kN}) \end{aligned} F 3 ( kN ) F 3 ( kN ) {:[F_(3)],[(kN)]:}\begin{aligned} & \boldsymbol{F}_{3} \\ & (\mathrm{kN}) \end{aligned} F 4 ( kN ) F 4 ( kN ) {:[F_(4)],[(kN)]:}\begin{aligned} & \boldsymbol{F}_{4} \\ & (\mathrm{kN}) \end{aligned} F 5 ( kN ) F 5 ( kN ) {:[F_(5)],[(kN)]:}\begin{aligned} & \boldsymbol{F}_{5} \\ & (\mathrm{kN}) \end{aligned}
Mean  平均值 25.0 460 200 500 600 750 900
CoV 20 20 20 20 20 20 10
Dist. Type  地区 类型 normal  正常 log log log\log log log log\log log log log\log log log log\log log log log\log normal  正常
Variable "L_(col ) (m)" H_("col ")_((m)) "B_(col ) (m)" "L_(beam ) (m)" H_("beam ")_((m)) "B_(beam ) (m)" "L_(girder ) (m)" Mean 3.5 0.6 0.6 9.0 0.8 0.6 7.0 CoV 20 20 20 20 20 20 20 Dist. Type log log log log log log Log Variable "H_(girder ) (m)" "B_(girder ) (m) " Cover (cm) "A^(rebar ) (cm^(col ))" "A_(bearn )^(rebar ) (cm^(2))" "A_(girger )^(rebr ) (cm^(2))" f_(c) (MPa) Mean 0.6 0.6 5 50.24 15.70 15.70 27.0 CoV 20 20 20 20 20 20 20 Dist. Type log normal normal log log log normal Variable "E_(c) (GPa) " "f_(y) (MPa)" "E_(s) (GPa)" "F_(2) (kN)" "F_(3) (kN)" "F_(4) (kN)" "F_(5) (kN)" Mean 25.0 460 200 500 600 750 900 CoV 20 20 20 20 20 20 10 Dist. Type normal log log log log log normal| Variable | $\begin{aligned} & \boldsymbol{L}_{\text {col }} \\ & (\mathrm{m}) \end{aligned}$ | $\underset{(\mathrm{m})}{\boldsymbol{H}_{\text {col }}}$ | $\begin{aligned} & \boldsymbol{B}_{\text {col }} \\ & (\mathrm{m}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{L}_{\text {beam }} \\ & (\mathrm{m}) \end{aligned}$ | $\underset{(\mathrm{m})}{\boldsymbol{H}_{\text {beam }}}$ | $\begin{aligned} & \boldsymbol{B}_{\text {beam }} \\ & (\mathrm{m}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{L}_{\text {girder }} \\ & (\mathrm{m}) \end{aligned}$ | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Mean | 3.5 | 0.6 | 0.6 | 9.0 | 0.8 | 0.6 | 7.0 | | CoV | 20 | 20 | 20 | 20 | 20 | 20 | 20 | | Dist. Type | $\log$ | $\log$ | $\log$ | $\log$ | $\log$ | $\log$ | Log | | Variable | $\begin{aligned} & \boldsymbol{H}_{\text {girder }} \\ & (\mathrm{m}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{B}_{\text {girder }} \\ & \text { (m) } \end{aligned}$ | Cover (cm) | $\begin{aligned} & A^{\text {rebar }} \\ & \left(\mathrm{cm}^{\text {col }}\right) \end{aligned}$ | $\begin{aligned} & \boldsymbol{A}_{\text {bearn }}^{\text {rebar }} \\ & \left(\mathrm{cm}^{2}\right) \end{aligned}$ | $\begin{aligned} & \boldsymbol{A}_{\text {girger }}^{\text {rebr }} \\ & \left(\mathrm{cm}^{2}\right) \end{aligned}$ | $f_{c}$ (MPa) | | Mean | 0.6 | 0.6 | 5 | 50.24 | 15.70 | 15.70 | 27.0 | | CoV | 20 | 20 | 20 | 20 | 20 | 20 | 20 | | Dist. Type | $\log$ | normal | normal | $\log$ | $\log$ | $\log$ | normal | | Variable | $\begin{aligned} & \boldsymbol{E}_{\boldsymbol{c}} \\ & \text { (GPa) } \end{aligned}$ | $\begin{aligned} & f_{y} \\ & (\mathrm{MPa}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{E}_{s} \\ & (\mathrm{GPa}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{F}_{2} \\ & (\mathrm{kN}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{F}_{3} \\ & (\mathrm{kN}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{F}_{4} \\ & (\mathrm{kN}) \end{aligned}$ | $\begin{aligned} & \boldsymbol{F}_{5} \\ & (\mathrm{kN}) \end{aligned}$ | | Mean | 25.0 | 460 | 200 | 500 | 600 | 750 | 900 | | CoV | 20 | 20 | 20 | 20 | 20 | 20 | 10 | | Dist. Type | normal | $\log$ | $\log$ | $\log$ | $\log$ | $\log$ | normal |
Results  成果 FEM-MCS RC-UQR LightGBM-MCS Kriging-based MCS  基于克里金法的监控监
Failure probability P f P f P_(f)P_{f}
故障概率 P f P f P_(f)P_{f} 		0.00162 0.00153 ± 4.1 E − 5 0.00153 ± 4.1 E − 5 0.00153+-4.1E-50.00153 \pm 4.1 \mathrm{E}-5 0.00098 ± 3.3 E − 4 0.00098 ± 3.3 E − 4 0.00098+-3.3E-40.00098 \pm 3.3 \mathrm{E}-4 0.00118 ± 5.0 E − 5 0.00118 ± 5.0 E − 5 0.00118+-5.0E-50.00118 \pm 5.0 \mathrm{E}-5
Reliability index R i R i R_(i)R_{i}
可靠性指数 R i R i R_(i)R_{i} 		2.944 2.962 ± 0.008 2.962 ± 0.008 2.962+-0.0082.962 \pm 0.008 3.112 ± 0.104 3.112 ± 0.104 3.112+-0.1043.112 \pm 0.104 3.041 ± 0.014 3.041 ± 0.014 3.041+-0.0143.041 \pm 0.014
Initial DoE size  初始 DoE 大小 0 500 500 500
Total FEM runs  FEM 运行总数 1 e 5 7500 500 1000
CPU time for FEM (s)
有限元计算 CPU 时间（秒）		 300,000 24,000 1500 3000
CPU time for HO (s)
HO 的 CPU 时间（秒）		 0 283 283 124
CPU time for active learning (s)
用于主动学习的 CPU 时间（秒）		 0 374 0 514
Total CPU time (s)
CPU 总时间（秒）		 300,000 24,657 1783 3638
Results FEM-MCS RC-UQR LightGBM-MCS Kriging-based MCS Failure probability P_(f) 0.00162 0.00153+-4.1E-5 0.00098+-3.3E-4 0.00118+-5.0E-5 Reliability index R_(i) 2.944 2.962+-0.008 3.112+-0.104 3.041+-0.014 Initial DoE size 0 500 500 500 Total FEM runs 1 e 5 7500 500 1000 CPU time for FEM (s) 300,000 24,000 1500 3000 CPU time for HO (s) 0 283 283 124 CPU time for active learning (s) 0 374 0 514 Total CPU time (s) 300,000 24,657 1783 3638| Results | FEM-MCS | RC-UQR | LightGBM-MCS | Kriging-based MCS | | :--- | :--- | :--- | :--- | :--- | | Failure probability $P_{f}$ | 0.00162 | $0.00153 \pm 4.1 \mathrm{E}-5$ | $0.00098 \pm 3.3 \mathrm{E}-4$ | $0.00118 \pm 5.0 \mathrm{E}-5$ | | Reliability index $R_{i}$ | 2.944 | $2.962 \pm 0.008$ | $3.112 \pm 0.104$ | $3.041 \pm 0.014$ | | Initial DoE size | 0 | 500 | 500 | 500 | | Total FEM runs | 1 e 5 | 7500 | 500 | 1000 | | CPU time for FEM (s) | 300,000 | 24,000 | 1500 | 3000 | | CPU time for HO (s) | 0 | 283 | 283 | 124 | | CPU time for active learning (s) | 0 | 374 | 0 | 514 | | Total CPU time (s) | 300,000 | 24,657 | 1783 | 3638 |
connection has six degrees of freedom instead of three, as in 2D simulation, making FEM more computationally demanding. In this example, there are a total of 21 random variables describing the structure’s geometry, such as the beam length, the girder length, the column height, and the material properties such as the concrete compressive strength, steel yield strength, Young modulus, rebar area, and external loads. The mean, standard deviation values and types of distribution of these random variables are described in Table 7. The limit state function of the third example is written as follows:
连接具有六个自由度，而不是二维模拟中的三个自由度，因此有限元计算要求更高。在本例中，共有 21 个随机变量描述了结构的几何形状，如梁长、梁长、柱高，以及材料属性，如混凝土抗压强度、钢材屈服强度、杨氏模量、钢筋面积和外部荷载。这些随机变量的平均值、标准偏差值和分布类型见表 7。第三个示例的极限状态函数写法如下
g ( X ) = y top − y limit = y − H building 50 = y − 0.35 g ( X ) = y top  − y limit  = y − H building  50 = y − 0.35 g(X)=y_("top ")-y_("limit ")=y-(H_("building "))/(50)=y-0.35g(X)=y_{\text {top }}-y_{\text {limit }}=y-\frac{H_{\text {building }}}{50}=y-0.35
In principle, the data generation for DoE , finite element models, hyperparameter optimization and active learning process are performed in a manner similar to that of the two previous examples. Table 8 presents the reliability results for the third example. It takes significantly higher computation times compared to these two previous 2D examples. Specifically, it takes more than 80 h of CPU times for FEMMCS to complete 1E5 simulations, providing results of P f P f P_(f)P_{f} = 0.00162 , R i = 2.944 = 0.00162 , R i = 2.944 =0.00162,R_(i)=2.944=0.00162, R_{i}=2.944 with a CoV = 0.08 CoV = 0.08 CoV=0.08\mathrm{CoV}=0.08. Such a high CPU time underlies the necessity of faster alternative methods for computing the reliability of 3D RC structures in realworld scenarios. LightGBM-MCS and Kriging-based MCS can perform reliability analysis for this 3D structure with CPU times of around 1782 s and 3683 s, respectively. However, their corresponding failure probability results are not satisfied, i.e., being noticeably smaller than the reference values ( 0.00092 and 0.00108 vs. 0.00162 ). Meanwhile, the proposed RC-UQR delivers results that are approximate to the reference one, i.e., P f = 0.00151 P f = 0.00151 P_(f)=0.00151P_{f}=0.00151 vs. P f = 0.00162 P f = 0.00162 P_(f)=0.00162P_{f}=0.00162 with a
原则上，DoE 的数据生成、有限元模型、超参数优化和主动学习过程的执行方式与前两个示例类似。表 8 列出了第三个示例的可靠性结果。与前两个二维示例相比，第三个示例的计算时间明显更长。具体来说，FEMMCS 完成 1E5 仿真需要超过 80 小时的 CPU 时间，提供 P f P f P_(f)P_{f} = 0.00162 , R i = 2.944 = 0.00162 , R i = 2.944 =0.00162,R_(i)=2.944=0.00162, R_{i}=2.944 和 CoV = 0.08 CoV = 0.08 CoV=0.08\mathrm{CoV}=0.08 的结果。如此高的 CPU 时间表明，有必要采用更快的替代方法来计算真实世界中三维 RC 结构的可靠性。LightGBM-MCS 和基于克里金法的 MCS 可对该三维结构进行可靠性分析，CPU 运算时间分别约为 1782 秒和 3683 秒。然而，它们相应的失效概率结果并不令人满意，即明显小于参考值（0.00092 和 0.00108 对 0.00162）。与此同时，拟议的 RC-UQR 提供了与参考值近似的结果，即 P f = 0.00151 P f = 0.00151 P_(f)=0.00151P_{f}=0.00151 与 P f = 0.00162 P f = 0.00162 P_(f)=0.00162P_{f}=0.00162 的 a
Fig. 13 Graphical illustration of the LightGBM performance before and after active learning
图 13 主动学习前后 LightGBM 性能图示
CPU time approximately 10 × 10 × 10 xx10 \times faster than FEM-MCS. These results consistently justify that RC-UQR possesses a remarkable balancing properties, providing useful reliability results with amenable computation time. Furthermore, it is a highly versatile framework applicable to different 2D and 3D RC structures.
CPU 计算时间约 10 × 10 × 10 xx10 \times 快于 FEM-MCS。这些结果一致证明，RC-UQR 具有显著的平衡特性，可提供有用的可靠性结果，且计算时间适中。此外，它还是一个适用于不同二维和三维 RC 结构的多功能框架。
In terms of result expressivity, the parallel coordinate graph in Fig. 12 demonstrates that the column length, column section height, beam section height, applied load at the top story, steel yield strength, and beam rebar area are among the parameters having the most impact on the limit state function.
在结果表达方面，图 12 中的平行坐标图表明，柱长、柱截面高度、梁截面高度、顶层施加荷载、钢屈服强度和梁钢筋面积等参数对极限状态函数的影响最大。
Fig. 12 Parallel coordinate graph highlighting failure cases (in red) for the 3 D five-story building structure
图 12 平行坐标图，突出显示了 3 D 五层楼房结构的故障情况（红色）。
Fig. 14 Failure probability and reliability index curves in functions of different safety thresholds
图 14 不同安全临界值下的故障概率和可靠性指数曲线
In order to further clarify the effect brought by the active learning process, we plot the prediction results obtained by LightGBM before and after active learning in Fig. 13. In the figure, the Y - and X -axis denote the actual and predicted values of the limit state function. Obviously, before active learning, the LightBGM misses a considerable number of failure cases, as it assigns positive values to g ( X ) g ( X ) g(X)g(X) for failure cases. However, after active learning process, LightGBM can provide accurate results for samples near limit state surfaces, i.e., g ( X ) ∼ 0 g ( X ) ∼ 0 g(X)∼0g(X) \sim 0, as these samples lie close to the ideal 45-degree line. This ideal line denotes that prediction results perfectly match with actual ones. For sample far from limit surface, i.e. | g ( X ) | | g ( X ) | |g(X)||g(X)| is significantly larger than 0 , LightGBM may not provide exact values of g ( X ) g ( X ) g(X)g(X), it still identify correctly the signs of g ( X ) g ( X ) g(X)g(X); thus, providing a relatively good estimation of P f P f P_(f)P_{f}.
为了进一步阐明主动学习过程带来的效果，我们在图 13 中绘制了 LightGBM 在主动学习前后的预测结果。图中，Y 轴和 X 轴分别表示极限状态函数的实际值和预测值。显然，在主动学习之前，LightBGM 错过了相当多的故障案例，因为它为故障案例的 g ( X ) g ( X ) g(X)g(X) 赋值为正值。然而，在主动学习过程之后，LightGBM 可以为极限状态表面附近的样本（即 g ( X ) ∼ 0 g ( X ) ∼ 0 g(X)∼0g(X) \sim 0 ）提供准确的结果，因为这些样本接近理想的 45 度线。这条理想线表示预测结果与实际结果完全吻合。对于远离极限表面的样本，即 | g ( X ) | | g ( X ) | |g(X)||g(X)| 明显大于 0 的样本，LightGBM 可能无法提供 g ( X ) g ( X ) g(X)g(X) 的精确值，但它仍能正确识别 g ( X ) g ( X ) g(X)g(X) 的符号；因此，它能提供相对较好的 P f P f P_(f)P_{f} 估计值。
In addition to a safety threshold y limit = 0.35 m y limit  = 0.35 m y_("limit ")=0.35my_{\text {limit }}=0.35 \mathrm{~m} as in Eq. (18), we applied RC-UQR for different safety threshold values in the range [ 0.3 , 0.45 ] m [ 0.3 , 0.45 ] m [0.3,0.45]m[0.3,0.45] \mathrm{m}. We then plotted the computed failure probability and reliability index curves as demonstrated in Fig. 14. It is noted that for each value of y limit y limit  y_("limit ")y_{\text {limit }}, the active learning process was reconducted as described in the previous sections. It can be seen that, with low safety threshold values, i.e., low-reliability indices, the results produced by RC-UQR closely match the reference results from FEM-MCS. However, for extreme cases with high safety threshold values, i.e., very small P f P f P_(f)P_{f}, the deviations between the two methods become noticeable. This can be explained by the fact that for extremely small P f P f P_(f)P_{f}, the number of data samples near the failure limit surface is very scarce, making it more challenging to train the surrogate models to match FEM accurately.
除了公式 (18) 中的安全阈值 y limit = 0.35 m y limit  = 0.35 m y_("limit ")=0.35my_{\text {limit }}=0.35 \mathrm{~m} 外，我们还对 [ 0.3 , 0.45 ] m [ 0.3 , 0.45 ] m [0.3,0.45]m[0.3,0.45] \mathrm{m} 范围内的不同安全阈值应用了 RC-UQR 。然后，我们绘制了计算出的故障概率和可靠性指数曲线，如图 14 所示。需要注意的是，对于 y limit y limit  y_("limit ")y_{\text {limit }} 的每个值，都要按照前面章节所述重新进行主动学习过程。可以看出，在安全阈值较低，即可靠性指数较低的情况下，RC-UQR 得出的结果与 FEM-MCS 得出的参考结果非常接近。然而，在安全阈值较高的极端情况下，即 P f P f P_(f)P_{f} 非常小的情况下，两种方法之间的偏差就变得非常明显。这是因为在 P f P f P_(f)P_{f} 非常小的情况下，失效极限面附近的数据样本数量非常少，这使得训练代用模型使其与 FEM 精确匹配更具挑战性。
In summary, we have demonstrated the applicability and versatility of the proposed RC-UQR method via three examples, including a single RC column with 10 random variables, a 2D frame structure featuring 3 columns and 3
总之，我们通过三个实例展示了所提出的 RC-UQR 方法的适用性和多功能性，包括带有 10 个随机变量的单根 RC 柱、带有 3 根柱和 3 个随机变量的 2D 框架结构、带有 3 个随机变量的单根 RC 柱和带有 3 个随机变量的 2D 框架结构。
beams with 15 random variables, and a 3D spatial frame structure comprised of 20 columns and 20 beams with 21 random variables. It is relatively straightforward to apply RC-UQR to different structures without major changes in the framework configurations, irrespective of the structures’ complexity. The results presented in Tables 3,5 , and 7 show that for more complex structures with numerous elements, FEM quickly become time-consuming. For example, the CPU time for a single simulation increases from around 0.09 s for a RC column ( 177 s for 2000 FEMs) to 3 s for a 3D structure ( 30,000 s for 10,000 FEM), but the CPU time for the HO step and active learning does not vary significantly across the examples. On the other hand, for more complex structures that potentially exhibit highly non-linear behaviours, the accuracy of surrogate models tends to decrease. However, RC-UQR is still able to provide results with satisfied accuracy, i.e., the relative errors of RC-UQR on the failure probability for the three examples are 2.2 % 2.2 % 2.2%2.2 \%, 5.0 % 5.0 % 5.0%5.0 \%, and 6.8 % 6.8 % 6.8%6.8 \%, respectively. Whereas the performance of the conventional Kriging-based MCS and ML-based models without active learning decreases considerably. For instance, the relative errors of these two methods for the 1D column are 21 % 21 % 21%21 \% and 13 % 13 % 13%13 \%, but increase to 43 % 43 % 43%43 \% and 33 % 33 % 33%33 \% for the 3D frame structure.
一个由 15 个随机变量的梁组成的三维空间框架结构，以及一个由 21 个随机变量的 20 个柱和 20 个梁组成的三维空间框架结构。将 RC-UQR 应用于不同的结构相对比较简单，不需要对框架结构进行较大的改动，也不考虑结构的复杂程度。表 3、表 5 和表 7 中的结果表明，对于具有大量元素的复杂结构，有限元计算很快就会变得耗时。例如，单次模拟的 CPU 时间从 RC 柱的 0.09 秒左右（2000 个有限元 177 秒）增加到三维结构的 3 秒（10,000 个有限元 30,000 秒），但 HO 步骤和主动学习的 CPU 时间在不同示例中没有显著差异。另一方面，对于可能表现出高度非线性行为的更复杂结构，代用模型的精度往往会降低。然而，RC-UQR 仍能提供令人满意的精度，即 RC-UQR 对三个示例的失效概率的相对误差分别为 2.2 % 2.2 % 2.2%2.2 \% 、 5.0 % 5.0 % 5.0%5.0 \% 和 6.8 % 6.8 % 6.8%6.8 \% 。而传统的基于克里金法的 MCS 模型和基于 ML 的模型在没有主动学习的情况下，性能会大大降低。例如，这两种方法对一维柱的相对误差分别为 21 % 21 % 21%21 \% 和 13 % 13 % 13%13 \% ，但对三维框架结构的相对误差则增加到 43 % 43 % 43%43 \% 和 33 % 33 % 33%33 \% 。

4 Conclusions  4 结论

Throughout the paper, different aspects of the proposed reliability analysis framework for reinforced concrete frame structures have been presented, including the theoretical background, the overall working flow, the pseudo-code, the implementation details, the application examples and the computation results. In short, the main findings of the study are summarized as follows:
本文通篇介绍了所提出的钢筋混凝土框架结构可靠性分析框架的各个方面，包括理论背景、总体工作流程、伪代码、实现细节、应用实例和计算结果。总之，研究的主要发现总结如下：
  • Accurately predicting the reinforced concrete’s nonlinear responses is challenging, especially in the context of limited data samples that are expensive to generate/ obtain. That is why directly using a state-of-the-art and well-tuned machine learning model such as LightGBM with randomly selected data samples cannot approximate structures’ responses, leading to unusable reliability results.
    准确预测钢筋混凝土的非线性响应具有挑战性，尤其是在数据样本有限且生成/获取成本高昂的情况下。这就是为什么直接使用最先进的、经过良好调整的机器学习模型（如 LightGBM）和随机选择的数据样本无法逼近结构的响应，从而导致无法使用的可靠性结果。
  • The active learning process utilizing uncertainty quantification to identify informative candidates is an effective way to improve the accuracy of the surrogate model with a small number of additional FEM evaluations. These candidates are located either near the limit state surface or in regions where the surrogate model is the most uncertain about its prediction values.
    利用不确定性量化来识别信息候选的主动学习过程，是一种只需少量额外有限元评估就能提高代用模型精度的有效方法。这些候选点要么位于极限状态面附近，要么位于代用模型预测值最不确定的区域。
  • Calculation results obtained in three case studies with different structural complexities reaffirm the applicability of the proposed method. More specifically, RC-UQR consistently provides reliability results close to the reference results obtained by the standard MCS methods but with significantly lower total CPU times, i.e., up to about ten times faster. RC-UQR can also provide more accurate results compared to the popular Kriging-based models when working with RC concretes’ non-linear behaviours.
    在三个具有不同结构复杂性的案例研究中获得的计算结果再次证明了所提方法的适用性。更具体地说，RC-UQR 所提供的可靠性结果始终接近标准 MCS 方法所获得的参考结果，但 CPU 总运行时间却大大减少，即最多可快 10 倍。在处理 RC 混凝土的非线性行为时，RC-UQR 还能提供比基于克里金法的流行模型更精确的结果。
  • Quantile regression is a simple yet effective and practical technique to provide uncertainty quantification, which, in principle, is applicable to any data-driven models, but does not require significant changes in the main architecture of the models.
    定量回归是一种简单而有效的不确定性量化实用技术，原则上适用于任何数据驱动的模型，而且不需要对模型的主要结构进行重大改动。
  • The proposed framework can serve as a practical and valuable tool for engineers and researchers to accurately estimate the reliability of 2D and 3D nonlinear reinforced
    所提出的框架可作为工程师和研究人员准确估算二维和三维非线性加固材料可靠性的实用而有价值的工具。
    concrete structures with multiple random variables. Furthermore, thanks to the high efficiency and accuracy of RC-UQR, one can perform a large number of calculations and visualize failure cases via parallel coordinates, as presented above. This allows identifying the critical range of feature values likely to result in failures, thus informing owners and engineers on areas for improvement to enhance the structures’ safety.
    多随机变量的混凝土结构。此外，得益于 RC-UQR 的高效率和高精确度，我们可以进行大量计算，并通过平行坐标可视化失效情况，如上所述。这样就可以确定可能导致失效的特征值临界范围，从而告知业主和工程师需要改进的地方，以提高结构的安全性。
A noticeable limitation of the proposed RC-UQR method is that when working with a new RC structure, even a minor alteration in structural layouts requires modifying and retraining the RC-UQR model accordingly. Therefore, it is desirable to build a versatile model that needs to be trained once with data from similar structures, and then it can be employed to perform reliability analysis on these structures under different loading scenarios. A second aspect for enhancing the proposed framework is that the current version only addresses a single safety criterion; hence, it is desirable to expand RC-UQR to encompass multiple safety criteria by designing a compound U-function to select representative samples near the limit state hyperplanes of various failure regions. Additionally, utilizing a multi-output surrogate model is also helpful in mitigating computational effort. Another exciting yet challenging direction is to improve the performance of RC-UQR in extreme scenarios, such as those with extremely small failure probability. This can be achieved by using advanced deep learning architectures or by devising a more sophisticated workflow.
所提出的 RC-UQR 方法有一个明显的局限性，即在处理新的 RC 结构时，即使结构布局稍有改动，也需要相应地修改和重新训练 RC-UQR 模型。因此，最好能建立一个通用模型，只需使用类似结构的数据进行一次训练，然后就能在不同荷载情况下对这些结构进行可靠性分析。加强拟议框架的第二个方面是，当前版本只针对单一安全标准；因此，最好通过设计复合 U 函数，在各种失效区域的极限状态超平面附近选择具有代表性的样本，从而将 RC-UQR 扩展到多个安全标准。此外，利用多输出代理模型也有助于减少计算工作量。另一个令人兴奋而又充满挑战的方向是提高 RC-UQR 在极端情况下的性能，如故障概率极小的情况。这可以通过使用先进的深度学习架构或设计更复杂的工作流程来实现。

Concrete  混凝土

Reinforcing steel  钢筋
Fig. 15 Non-linear constitutive laws of concrete (left) and reinforcing steel (right)
图 15 混凝土（左）和钢筋（右）的非线性构造规律

Appendix  附录

Material constitutive laws
材料构成定律

In this study, non-linearity mainly arises from the non-linear constitutive law of materials. For concrete, the Scott, Park, and Priestley’s model proposed in [45] is adopted thanks to its clear intuition, computationally efficient and the ability to capture the concrete’s essential characteristics. The compression part of the model consists of three branches, as shown in Fig. 15, in which the ascending branch goes from the origin ( 0 , 0 ) ( 0 , 0 ) (0,0)(0,0) to the peak of stress ( ε 0 , σ c c ) ε 0 , σ c c (epsi_(0),sigma_(c)_(c))\left(\varepsilon_{0}, \sigma_{c}{ }_{c}\right). At the beginning, the stress is linearly proportional to the strain with an initial slope E 0 E 0 E_(0)E_{0}. Afterwards, the descending branch from the peak to a residual point ( ε c u , σ c u ) ε c u , σ c u (epsi_(cu),sigma_(cu))\left(\varepsilon_{c u}, \sigma_{c u}\right) denotes a linearly inverse proportional relationship between the stress and the strain. From the residual point onwards is the residual branch characterized by a constant stress σ c u σ c u sigma_(cu)\sigma_{c u} and continuously increasing strains. On the other hand, the tension part of the model is composed of two short linear branches characterized by the point with the peak tensile stress σ t = 0.31 σ c ( M P a ) σ t = 0.31 σ c ( M P a ) sigma_(t)=0.31sqrt(sigma_(c))(MPa)\sigma_{t}=0.31 \sqrt{\sigma_{c}}(M P a) and a second point with the ultimate tensile strains ε t u ε t u epsi_(tu)\varepsilon_{t u}. For tensile strains beyond ε t u ε t u epsi_(tu)\varepsilon_{t u}, the corresponding stress is assumed to be zero. Some authors adopted a slope E t ∼ 0.05 E 0 E t ∼ 0.05 E 0 E_(t)∼0.05E_(0)E_{t} \sim 0.05 E_{0} [46] for the descending branch, leading to a rough estimation of ε t u ε t u epsi_(tu)\varepsilon_{t u} as follows:
在本研究中，非线性主要源于材料的非线性构成规律。对于混凝土，我们采用 Scott、Park 和 Priestley 在文献[45]中提出的模型，因为该模型直观明了、计算效率高，而且能够捕捉到混凝土的基本特征。如图 15 所示，模型的压缩部分由三个分支组成，其中上升分支从原点 ( 0 , 0 ) ( 0 , 0 ) (0,0)(0,0) 到应力峰值 ( ε 0 , σ c c ) ε 0 , σ c c (epsi_(0),sigma_(c)_(c))\left(\varepsilon_{0}, \sigma_{c}{ }_{c}\right) 。开始时，应力与应变成线性比例，初始斜率为 E 0 E 0 E_(0)E_{0} 。之后，从峰值到残差点 ( ε c u , σ c u ) ε c u , σ c u (epsi_(cu),sigma_(cu))\left(\varepsilon_{c u}, \sigma_{c u}\right) 的下降分支表示应力和应变之间的线性反比关系。从残余点开始是残余分支，其特点是应力 σ c u σ c u sigma_(cu)\sigma_{c u} 恒定，应变持续增加。另一方面，模型的拉伸部分由两个短的线性分支组成，其特征是拉伸应力峰值点 σ t = 0.31 σ c ( M P a ) σ t = 0.31 σ c ( M P a ) sigma_(t)=0.31sqrt(sigma_(c))(MPa)\sigma_{t}=0.31 \sqrt{\sigma_{c}}(M P a) 和极限拉伸应变 ε t u ε t u epsi_(tu)\varepsilon_{t u} 的第二个点。当拉伸应变超过 ε t u ε t u epsi_(tu)\varepsilon_{t u} 时，相应的应力假定为零。一些作者采用斜率 E t ∼ 0.05 E 0 E t ∼ 0.05 E 0 E_(t)∼0.05E_(0)E_{t} \sim 0.05 E_{0} [46]来计算下降分支，从而粗略估算出 ε t u ε t u epsi_(tu)\varepsilon_{t u} 如下：
ε t u ≈ f t E 0 + f t 0.05 E 0 ≈ 21 f t E 0 ε t u ≈ f t E 0 + f t 0.05 E 0 ≈ 21 f t E 0 epsi_(tu)~~(f_(t))/(E_(0))+(f_(t))/(0.05E_(0))~~21(f_(t))/(E_(0))\varepsilon_{t u} \approx \frac{f_{t}}{E_{0}}+\frac{f_{t}}{0.05 E_{0}} \approx 21 \frac{f_{t}}{E_{0}}
For reinforcing steel, the nonlinear institutive law is simulated by using the Giuffré-Menegotto-Pinto isotropic hardening model [47] which consists of two linear branches plus a transition between these two as depicted in Fig. 15 (right). The elastic branch is characterized by a slope that equals to the elastic modulus E s E s E_(s)E_{s}. When the stress goes beyond an yield stress value σ y σ y sigma_(y)\sigma_{y}, reinforcing steel would exhibit plastic behaviour which is described via a smaller slope of E 1 = k h × E s E 1 = k h × E s E_(1)=k_(h)xxE_(s)E_{1}=k_{h} \times E_{s} where k h k h k_(h)k_{h} is the hardening ratio. The transition between the elastic and plastic branches is defined by a curve with a curvature of R 0 R 0 R_(0)R_{0}.
对于钢筋，采用 Giuffré-Menegotto-Pinto 各向同性硬化模型 [47]模拟非线性结构定律，该模型由两个线性分支和两个分支之间的过渡组成，如图 15（右图）所示。弹性分支的斜率等于弹性模量 E s E s E_(s)E_{s} 。当应力超过屈服应力值 σ y σ y sigma_(y)\sigma_{y} 时，钢筋将表现出塑性行为，其斜率较小，为 E 1 = k h × E s E 1 = k h × E s E_(1)=k_(h)xxE_(s)E_{1}=k_{h} \times E_{s} ，其中 k h k h k_(h)k_{h} 为硬化比。弹性分支和塑性分支之间的过渡由曲率为 R 0 R 0 R_(0)R_{0} 的曲线定义。
Concretely, some key parameters of concrete and steel materials such as the concrete compressive strength, reinforcing steel yield stress, these materials’ elastic moduli, etc. would be considered as input random variables in structural reliability analysis.
具体而言，混凝土和钢材的一些关键参数，如混凝土抗压强度、钢筋屈服应力、这些材料的弹性模量等，将被视为结构可靠性分析中的输入随机变量。

Fiber element formulation
纤维元件配方

Formally, the calculation process of the fiber element can be written as follows. First, the section deformation vector at a point x x xx on the element length is denoted by
从形式上看，纤维元件的计算过程可以写成如下。首先，元素长度上 x x xx 点的截面变形矢量记为
d ( x ) = [ χ z ( x ) , χ y ( x ) , ε ( x ) ] T d ( x ) = χ z ( x ) , χ y ( x ) , ε ( x ) T d(x)=[chi_(z)(x),chi_(y)(x),epsi(x)]^(T)d(x)=\left[\chi_{z}(x), \chi_{y}(x), \varepsilon(x)\right]^{T}, where ε ( x ) ε ( x ) epsi(x)\varepsilon(x) is the axial strain, χ z ( x ) χ z ( x ) chi_(z)(x)\chi_{z}(x) and χ y ( x ) χ y ( x ) chi_(y)(x)\chi_{y}(x) are the curvatures about the z - and y -axis, respectively.
d ( x ) = [ χ z ( x ) , χ y ( x ) , ε ( x ) ] T d ( x ) = χ z ( x ) , χ y ( x ) , ε ( x ) T d(x)=[chi_(z)(x),chi_(y)(x),epsi(x)]^(T)d(x)=\left[\chi_{z}(x), \chi_{y}(x), \varepsilon(x)\right]^{T} ，其中 ε ( x ) ε ( x ) epsi(x)\varepsilon(x) 是轴向应变， χ z ( x ) χ z ( x ) chi_(z)(x)\chi_{z}(x) 和 χ y ( x ) χ y ( x ) chi_(y)(x)\chi_{y}(x) 分别是绕 z 轴和 y 轴的曲率。
At the current calculation iteration step i i ii, it is assumed that all variables at the previous step i − 1 i − 1 i-1i-1 are calculated, including the structural stiffness matrix K s i − 1 K s i − 1 K_(s)^(i-1)K_{s}^{i-1}. The next step involves updating the stiffness matrix K s i K s i K_(s)^(i)K_{s}^{i} and calculating the structure displacements, member and section deformations, etc. given load increment Δ P i Δ P i DeltaP^(i)\Delta P^{i} at step i i ii. Firstly, using K s i − 1 K s i − 1 K_(s)^(i-1)K_{s}^{i-1} and Δ P i Δ P i DeltaP^(i)\Delta P^{i} we calculate displacement increment Δ p i Δ p i Deltap^(i)\Delta p^{i} by solving the following system of linear equation:
在当前计算迭代步骤 i i ii 中，假定上一步骤 i − 1 i − 1 i-1i-1 中的所有变量都已计算完毕，包括结构刚度矩阵 K s i − 1 K s i − 1 K_(s)^(i-1)K_{s}^{i-1} 。下一步包括更新刚度矩阵 K s i K s i K_(s)^(i)K_{s}^{i} ，并在步骤 i i ii 中计算给定荷载增量 Δ P i Δ P i DeltaP^(i)\Delta P^{i} 的结构位移、构件和截面变形等。首先，利用 K s i − 1 K s i − 1 K_(s)^(i-1)K_{s}^{i-1} 和 Δ P i Δ P i DeltaP^(i)\Delta P^{i} 计算位移增量 Δ p i Δ p i Deltap^(i)\Delta p^{i} ，解出以下线性方程组：
K s i − 1 × Δ p i = Δ P i K s i − 1 × Δ p i = Δ P i K_(s)^(i-1)xx Deltap^(i)=DeltaP^(i)\boldsymbol{K}_{s}^{i-1} \times \Delta \boldsymbol{p}^{i}=\Delta \boldsymbol{P}^{i}
Based on the calculated displacement increment Δ p i Δ p i Deltap^(i)\Delta \boldsymbol{p}^{i}, we would derive the element deformation increment, and then the section deformation increment Δ d i ( x ) Δ d i ( x ) Deltad^(i)(x)\Delta d^{i}(x).
根据计算出的位移增量 Δ p i Δ p i Deltap^(i)\Delta \boldsymbol{p}^{i} ，我们将得出元素变形增量，然后得出截面变形增量 Δ d i ( x ) Δ d i ( x ) Deltad^(i)(x)\Delta d^{i}(x) 。
On the other hand, the fiber deformation is represented by the vector e ( x ) = [ ε 1 f i b ( x , y 1 , z 1 ) , … , ε n f i b ( x , y 1 , z 1 ) ] T e ( x ) = ε 1 f i b x , y 1 , z 1 , … , ε n f i b x , y 1 , z 1 T e(x)=[epsi_(1)^(fib)(x,y_(1),z_(1)),dots,epsi_(n)^(fib)(x,y_(1),z_(1))]^(T)\boldsymbol{e}(x)=\left[\varepsilon_{1}^{f i b}\left(x, y_{1}, z_{1}\right), \ldots, \varepsilon_{n}^{f i b}\left(x, y_{1}, z_{1}\right)\right]^{T}, is related to section deformation d ( x ) d ( x ) d(x)\boldsymbol{d}(x) via a linear geometric matrix I ( x ) I ( x ) I(x)\boldsymbol{I}(x), as follows:
另一方面，纤维变形由矢量 e ( x ) = [ ε 1 f i b ( x , y 1 , z 1 ) , … , ε n f i b ( x , y 1 , z 1 ) ] T e ( x ) = ε 1 f i b x , y 1 , z 1 , … , ε n f i b x , y 1 , z 1 T e(x)=[epsi_(1)^(fib)(x,y_(1),z_(1)),dots,epsi_(n)^(fib)(x,y_(1),z_(1))]^(T)\boldsymbol{e}(x)=\left[\varepsilon_{1}^{f i b}\left(x, y_{1}, z_{1}\right), \ldots, \varepsilon_{n}^{f i b}\left(x, y_{1}, z_{1}\right)\right]^{T} 表示，并通过线性几何矩阵 I ( x ) I ( x ) I(x)\boldsymbol{I}(x) 与截面变形 d ( x ) d ( x ) d(x)\boldsymbol{d}(x) 相关，如下所示：
e ( x ) = I ( x ) × d ( x ) → Δ e i , f i b ( x ) = I ( x ) × Δ d i ( x ) e ( x ) = I ( x ) × d ( x ) → Δ e i , f i b ( x ) = I ( x ) × Δ d i ( x ) e(x)=I(x)xx d(x)rarr Deltae^(i,fib)(x)=I(x)xx Deltad^(i)(x)\boldsymbol{e}(x)=\boldsymbol{I}(x) \times \boldsymbol{d}(x) \rightarrow \Delta \boldsymbol{e}^{i, f i b}(x)=\boldsymbol{I}(x) \times \Delta \boldsymbol{d}^{i}(x)
where  其中
I ( x ) = [ − y 1 f i b z 1 f i b 1 ⋮ ⋮ ⋮ − y n f i b z n f i b 1 ] I ( x ) = − y 1 f i b z 1 f i b 1 ⋮ ⋮ ⋮ − y n f i b z n f i b 1 I(x)=[[-y_(1)^(fib),z_(1)^(fib),1],[vdots,vdots,vdots],[-y_(n)^(fib),z_(n)^(fib),1]]\boldsymbol{I}(x)=\left[\begin{array}{ccc}-y_{1}^{f i b} & z_{1}^{f i b} & 1 \\ \vdots & \vdots & \vdots \\ -y_{n}^{f i b} & z_{n}^{f i b} & 1\end{array}\right]
Hence, the fiber deformation at current step i i ii is updated by:
因此，当前步骤 i i ii 中的纤维变形将通过以下方式更新：
e i , f i b ( x ) = e i − 1 , f i b ( x ) + Δ e i , f i b ( x ) e i , f i b ( x ) = e i − 1 , f i b ( x ) + Δ e i , f i b ( x ) e^(i,fib)(x)=e^(i-1,fib)(x)+Deltae^(i,fib)(x)\boldsymbol{e}^{i, f i b}(x)=\boldsymbol{e}^{i-1, f i b}(x)+\Delta \boldsymbol{e}^{i, f i b}(x)
Next, based on the material stress-strain relations presented in the previous subsection, the stress σ i , f i b ( x ) σ i , f i b ( x ) sigma^(i,fib)(x)\sigma^{i, f i b}(x) and the corresponding tangent modulus E f i b i E f i b i E_(fib)^(i)E_{f i b}^{i}, which is the slope of the stress-strain curve of fibers, are determined. Then the section stiffness matrix at the current step i i ii is derived from all fibers as follows:
接下来，根据上一小节介绍的材料应力-应变关系，确定应力 σ i , f i b ( x ) σ i , f i b ( x ) sigma^(i,fib)(x)\sigma^{i, f i b}(x) 和相应的切线模量 E f i b i E f i b i E_(fib)^(i)E_{f i b}^{i} （即纤维应力-应变曲线的斜率）。然后，根据所有纤维得出当前步长下的截面刚度矩阵 i i ii 如下：
k i ( x ) = [ ∑ i f b = 1 n E i f b i A i f i y i f b 2 ∑ i f b = 1 n E i f i b i A i f i y i f i b z i f i b − ∑ i f i b = 1 n E i f i b i A i f i y i f i b ∑ i f i b = 1 n E i f i i A i f i b y i f i z i f i b ∑ i f b = 1 n E i f i b i A i f i b z i f i b 2 ∑ i f i b = 1 n E i f i b i A i f i b z i f i b − ∑ i f i b = 1 n E i f i b i A i f i b y i f i b ∑ i f i b = 1 n E i f i b i A i f i b z i f i b ∑ i f i b = 1 n E i f i b i A i f i b ] k i ( x ) = ∑ i f b = 1 n   E i f b i A i f i y i f b 2 ∑ i f b = 1 n   E i f i b i A i f i y i f i b z i f i b − ∑ i f i b = 1 n   E i f i b i A i f i y i f i b ∑ i f i b = 1 n   E i f i i A i f i b y i f i z i f i b ∑ i f b = 1 n   E i f i b i A i f i b z i f i b 2 ∑ i f i b = 1 n   E i f i b i A i f i b z i f i b − ∑ i f i b = 1 n   E i f i b i A i f i b y i f i b ∑ i f i b = 1 n   E i f i b i A i f i b z i f i b ∑ i f i b = 1 n   E i f i b i A i f i b k^(i)(x)=[[sum_(ifb=1)^(n)E_(ifb)^(i)A_(ifi)y_(ifb)^(2),sum_(ifb=1)^(n)E_(ifib)^(i)A_(ifi)y_(ifib)z_(ifib),-sum_(ifib=1)^(n)E_(ifib)^(i)A_(ifi)y_(ifib)],[sum_(ifib=1)^(n)E_(ifi)^(i)A_(ifib)y_(ifi)z_(ifib),sum_(ifb=1)^(n)E_(ifib)^(i)A_(ifib)z_(ifib)^(2),sum_(ifib=1)^(n)E_(ifib)^(i)A_(ifib)z_(ifib)],[-sum_(ifib=1)^(n)E_(ifib)^(i)A_(ifib)y_(ifib),sum_(ifib=1)^(n)E_(ifib)^(i)A_(ifib)z_(ifib),sum_(ifib=1)^(n)E_(ifib)^(i)A_(ifib)]]\boldsymbol{k}^{i}(\boldsymbol{x})=\left[\begin{array}{ccc}\sum_{i f b=1}^{n} E_{i f b}^{i} A_{i f i} y_{i f b}^{2} & \sum_{i f b=1}^{n} E_{i f i b}^{i} A_{i f i} y_{i f i b} z_{i f i b} & -\sum_{i f i b=1}^{n} E_{i f i b}^{i} A_{i f i} y_{i f i b} \\ \sum_{i f i b=1}^{n} E_{i f i}^{i} A_{i f i b} y_{i f i} z_{i f i b} & \sum_{i f b=1}^{n} E_{i f i b}^{i} A_{i f i b} z_{i f i b}^{2} & \sum_{i f i b=1}^{n} E_{i f i b}^{i} A_{i f i b} z_{i f i b} \\ -\sum_{i f i b=1}^{n} E_{i f i b}^{i} A_{i f i b} y_{i f i b} & \sum_{i f i b=1}^{n} E_{i f i b}^{i} A_{i f i b} z_{i f i b} & \sum_{i f i b=1}^{n} E_{i f i b}^{i} A_{i f i b}\end{array}\right]
By performing the numerical integration scheme, the element stiffness matrix K i K i K^(i)K^{i} is obtained from section stiffness matrices k i ( x ) k i ( x ) k^(i)(x)k^{i}(x). Subsequently, by assembling the element stiffness matrix K e i K e i K_(e)^(i)\boldsymbol{K}_{e}^{i}, we would obtain the updated structural stiffness matrix K s i K s i K_(s)^(i)\boldsymbol{K}_{s}^{i}. After that, the whole process presented above is repeated until the convergence criterion is reached.
通过执行数值积分方案，可从截面刚度矩阵 k i ( x ) k i ( x ) k^(i)(x)k^{i}(x) 中获得元素刚度矩阵 K i K i K^(i)K^{i} 。随后，通过组合元素刚度矩阵 K e i K e i K_(e)^(i)\boldsymbol{K}_{e}^{i} ，我们将得到更新的结构刚度矩阵 K s i K s i K_(s)^(i)\boldsymbol{K}_{s}^{i} 。之后，重复上述整个过程，直到达到收敛标准。

Simulation results obtained by finite element method
有限元法获得的模拟结果

Fig. 16 Displacement-load curves obtained by FEM on the initial DoE dataset for three investigated case studies
图 16 对三个调查案例研究的初始 DoE 数据集进行有限元计算得出的位移-荷载曲线


Hyperparameter optimization results
超参数优化结果

Table 9 Hyperparameter optimization results for three examples under investigation
表 9 三个研究实例的超参数优化结果
No  没有 Example  示例 Hyperparameter optimization results
超参数优化结果
1 1D RC column  1D RC 柱

{'learning_rate'： 0.20，'num_estimators'： 480，'max_depth'： 2 2
{'learning_rate': 0.20,'num_estimators': 480, 'max_depth': 2
2
{'learning_rate': 0.20,'num_estimators': 480, 'max_depth': 2 2| {'learning_rate': 0.20,'num_estimators': 480, 'max_depth': 2 | | :--- | | | | 2 |
2D three-story frame  二维三层框架

叶片数"：75，"最小子代重量"：23}。{'learning_rate': 0.25,'num_estimators': 885, 'max_depth'：3, 3
'num_leaves': 75, 'min_child_weight': 23}
{'learning_rate': 0.25,'num_estimators': 885, 'max_depth': 3,
3
'num_leaves': 75, 'min_child_weight': 23} {'learning_rate': 0.25,'num_estimators': 885, 'max_depth': 3, 3| 'num_leaves': 75, 'min_child_weight': 23} | | :--- | | {'learning_rate': 0.25,'num_estimators': 885, 'max_depth': 3, | | 3 |
'num_leaves': 4, 'min_child_weight': 7}
num_leaves'：4, 'min_child_weight': 7}
'num_leaves': 4, 'min_child_weight': 7}| 'num_leaves': 4, 'min_child_weight': 7} | | :--- |
3D RC five-story building
3D RC 五层楼建筑

{'learning_rate'：0.30, 'num_estimators': 985, 'max_depth': 2, 'num_leaves': 82, 'min_child_weight': 21}
{'learning_rate': 0.30, 'num_estimators': 985, 'max_depth': 2,
'num_leaves': 82, 'min_child_weight': 21}
{'learning_rate': 0.30, 'num_estimators': 985, 'max_depth': 2, 'num_leaves': 82, 'min_child_weight': 21}| {'learning_rate': 0.30, 'num_estimators': 985, 'max_depth': 2, | | :--- | | 'num_leaves': 82, 'min_child_weight': 21} |
No Example Hyperparameter optimization results 1 1D RC column "{'learning_rate': 0.20,'num_estimators': 480, 'max_depth': 2 2" 2D three-story frame "'num_leaves': 75, 'min_child_weight': 23} {'learning_rate': 0.25,'num_estimators': 885, 'max_depth': 3, 3" "'num_leaves': 4, 'min_child_weight': 7}" 3D RC five-story building "{'learning_rate': 0.30, 'num_estimators': 985, 'max_depth': 2, 'num_leaves': 82, 'min_child_weight': 21}"| No | Example | Hyperparameter optimization results | | :--- | :--- | :--- | | 1 | 1D RC column | {'learning_rate': 0.20,'num_estimators': 480, 'max_depth': 2 <br> <br> 2 | | 2D three-story frame | 'num_leaves': 75, 'min_child_weight': 23} <br> {'learning_rate': 0.25,'num_estimators': 885, 'max_depth': 3, <br> 3 | 'num_leaves': 4, 'min_child_weight': 7} | | | 3D RC five-story building | {'learning_rate': 0.30, 'num_estimators': 985, 'max_depth': 2, <br> 'num_leaves': 82, 'min_child_weight': 21} |
Table 9  表 9
Fig. 17 Convergence plots presenting the evolution of the failure probabilities against the number of FEM runs obtained by investigated methods (RC-UQR, FEM-MCS, LGBM-MCS, and Kriging-MCS)
图 17 收敛图，显示故障概率随研究方法（RC-UQR、FEM-MCS、LGBM-MCS 和 Kriging-MCS）获得的有限元运行次数的变化情况
  1. Viet-Hung Dang
    hungdv@huce.edu.vn
    1 Faculty of Building and Industrial Construction, Hanoi University of Civil Engineering, Hanoi, Vietnam
    1 河内土木工程大学建筑与工业建设学院，越南河内