Probabilistic Engineering Mechanics最新文献_第7页

Probabilistic volume element model of 2D woven C/SiC composites considering copula dependence between strength and modulus 考虑强度和模量耦合关系的二维编织C/SiC复合材料概率体积元模型

IF 3 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics

Pub Date : 2025-07-01 Epub Date: 2025-06-06 DOI: 10.1016/j.probengmech.2025.103802

Qiang Li , Gang Li , Gang Zhao , Qiang Li

2D woven C/SiC ceramic matrix composites inherently exhibit dispersion in mechanical properties, posing great challenges to their structural design and engineering applications. This dispersion primarily stems from the random distribution of voids and defects within the composite material, as well as the uneven thermochemical damage incurred during manufacturing. To address this issue, this paper proposes a probabilistic volume element model to characterize the dispersion in the mechanical properties of C/SiC composites and investigate the propagation of uncertainties and correlations across scales. To capture the typical nonlinear behavior of C/SiC composites under uniaxial tension, a progressive damage constitutive law was developed within the mesoscale finite element model based on the Linde criterion. Uncertainties in material properties were modeled by implementing Weibull and normal distributions for strength and modulus, respectively. Bivariate copula functions combined with the Bootstrap method were employed to quantify the correlation between strength and modulus, as observed in limited experimental data. Convolutional neural networks were introduced to model the propagation of uncertainty in these correlated parameters. The networks were iteratively updated through transfer learning and optimization algorithms to address the correlation inverse problem, enabling the identification of dependence between mesoscale parameters based on macroscale experimental data, with subsequent quantification using copula functions. Statistical analysis emphasizes the significance of incorporating parameter correlations in multiscale simulations for achieving accurate uncertainty quantification of mechanical properties.

二维编织C/SiC陶瓷基复合材料在力学性能上具有固有的分散性，这对其结构设计和工程应用提出了很大的挑战。这种分散主要源于复合材料中空洞和缺陷的随机分布，以及制造过程中产生的不均匀热化学损伤。为了解决这一问题，本文提出了一个概率体积元模型来表征C/SiC复合材料力学性能的分散性，并研究了不确定性和相关性在尺度上的传播。为了捕捉C/SiC复合材料在单轴拉伸作用下的典型非线性行为，基于Linde准则在中尺度有限元模型中建立了渐进损伤本构律。材料性能的不确定性分别通过实现强度和模量的威布尔分布和正态分布来建模。在有限的实验数据中，采用二元copula函数结合Bootstrap方法来量化强度与模量之间的相关性。引入卷积神经网络对这些相关参数的不确定性传播进行建模。通过迁移学习和优化算法对网络进行迭代更新，以解决相关逆问题，从而基于宏观尺度实验数据识别中尺度参数之间的依赖关系，随后使用copula函数进行量化。统计分析强调了在多尺度模拟中纳入参数相关性的重要性，以实现精确的力学性能不确定性量化。

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引用次数: 0

Randomized prior wavelet neural operator for uncertainty quantification 不确定性量化的随机先验小波神经算子

IF 3.5 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics

Pub Date : 2025-07-01 Epub Date: 2025-08-21 DOI: 10.1016/j.probengmech.2025.103817

Shailesh Garg , Souvik Chakraborty

In this paper, we propose a novel data-driven operator learning framework termed the Randomized Prior Wavelet Neural Operator (RP-WNO). The proposed RP-WNO is an extension of the recently proposed wavelet neural operator, which, while boasts excellent generalizing capabilities, cannot estimate the uncertainty associated with its predictions in its vanilla form. RP-WNO, unlike the vanilla WNO, has an inherent predictive uncertainty quantification module and is expected to be useful for tasks where some form of decision-making is involved. RP-WNO is set in a deterministic framework, which makes it easier to implement than its Bayesian counterpart, especially for large, complex deep-learning architectures. It utilizes randomized prior networks that can account for prior information, and in this paper, we extend the theory of randomized prior networks by using the underlying concept to incorporate seamlessly, a physics-based prior. Three examples, covering datasets originating from two-dimensional partial differential equations, have been shown to test the efficacy of the proposed framework. Two of these examples utilize a randomly initialized prior network, and the remaining example utilizes a physics-based prior along with the randomly initialized prior network. The results produced favorably advocate for the efficacy of the proposed RP-WNO framework.

在本文中，我们提出了一个新的数据驱动算子学习框架，称为随机先验小波神经算子（RP-WNO）。提出的RP-WNO是最近提出的小波神经算子的扩展，小波神经算子虽然具有出色的泛化能力，但不能以其vanilla形式估计与其预测相关的不确定性。RP-WNO与普通的WNO不同，它有一个固有的预测不确定性量化模块，预计将对涉及某种形式决策的任务有用。RP-WNO设置在确定性框架中，这使得它比贝叶斯算法更容易实现，特别是对于大型、复杂的深度学习架构。它利用可以解释先验信息的随机先验网络，在本文中，我们通过使用底层概念无缝地合并基于物理的先验来扩展随机先验网络的理论。三个例子涵盖了源自二维偏微分方程的数据集，已被证明可以测试所提出框架的有效性。其中两个示例使用随机初始化的先验网络，其余示例使用基于物理的先验以及随机初始化的先验网络。研究结果支持RP-WNO框架的有效性。

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引用次数: 0

Uncertainty quantification of the mechanical response of random porous materials based on manifold space sampling 基于流形空间采样的随机多孔材料力学响应的不确定性量化

IF 3.5 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics

Pub Date : 2025-07-01 Epub Date: 2025-08-05 DOI: 10.1016/j.probengmech.2025.103822

Xianrui Lyu, Xiaodan Ren, Jie Li

The inherent uncertainty of the microstructure in random porous materials propagates to the macroscopic response uncertainty through the underlying physical laws. Accurately characterizing this uncertainty necessitates the use of high-dimensional joint probability density functions, which need to be coupled with nonlinear, cross-scale propagation. This integration poses significant challenges for quantifying macroscopic performance uncertainty. To overcome these challenges, this study proposes an uncertainty analysis framework based on manifold space sampling. Specifically, manifold learning is employed to map the complex, high-dimensional microstructure to a low-dimensional manifold space. Within this manifold space, the uncertainty of the microstructure is comprehensively characterized via the probabilistic distribution of latent variables, enabling effective dimensionality reduction while preserving essential statistical characteristics of the original microstructure. Subsequently, a sampling strategy guided by maximal marginal EF-discrepancy (MF-discrepancy) is used to select representative latent variables, which are then decoded to reconstruct representative microstructure samples. These samples and their corresponding mechanical responses are subsequently input into a physically-based probability density evolution method (PDEM), which transforms the high-dimensional stochastic problem into a set of deterministic partial differential equations. This provides a full probabilistic evolution process of the homogenized stress response, thereby enabling the propagation of microstructural uncertainty to macroscopic performance uncertainty. The accuracy and computational efficiency of the proposed method are validated by comparing its results with the reference values obtained from Monte Carlo simulations (MC) using an sufficiently large sample size. The results demonstrate that the framework offers significant advantages in handling high-dimensional random variables and nonlinear cross-scale propagation, providing an efficient and feasible approach for uncertainty quantification in complex material systems.

随机多孔材料微观结构的固有不确定性通过其内在的物理规律传播为宏观响应的不确定性。准确表征这种不确定性需要使用高维联合概率密度函数，这需要与非线性跨尺度传播相结合。这种整合对量化宏观性能不确定性提出了重大挑战。为了克服这些挑战，本研究提出了一种基于流形空间采样的不确定性分析框架。具体来说，流形学习用于将复杂的高维微观结构映射到低维流形空间。在这个流形空间中，微观结构的不确定性通过潜在变量的概率分布得到全面表征，在保留原始微观结构基本统计特征的同时，实现了有效的降维。然后，采用最大边际ef -差异（mf -差异）指导的采样策略，选择具有代表性的潜在变量，然后对其进行解码，重建具有代表性的微观结构样本。这些样本及其相应的力学响应随后被输入到基于物理的概率密度演化方法（PDEM）中，该方法将高维随机问题转化为一组确定性偏微分方程。这提供了均匀应力响应的完整概率演化过程，从而使微观结构不确定性向宏观性能不确定性传播。在足够大的样本量下，将所得结果与蒙特卡罗模拟（MC）的参考值进行比较，验证了所提方法的精度和计算效率。结果表明，该框架在处理高维随机变量和非线性跨尺度传播方面具有显著优势，为复杂材料系统的不确定性量化提供了一种有效可行的方法。

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引用次数: 0

Research on the non-probabilistic convex modeling method based on the potential connection between probabilistic and convex models 基于概率模型与凸模型潜在联系的非概率凸建模方法研究

IF 3 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics

Pub Date : 2025-07-01 Epub Date: 2025-06-07 DOI: 10.1016/j.probengmech.2025.103794

Gang Zhao, Mingdong Wang, Yangyang Liu, Xiaoyu Wang, Wanyue Song

In existing research, probabilistic models and non-probabilistic convex models are typically treated as separate entities, with little attention given to their potential connections. This gap hinders a deeper understanding and rational application of non-probabilistic convex models. To address this issue, this paper proposes a novel non-probabilistic convex modeling method that leverages the potential relationship between probabilistic and convex models to achieve precise uncertainty quantification under limited data conditions. First, the mathematical formulations of the probabilistic and non-probabilistic convex models are presented. Then, a dimension-reduction technique is introduced to provide a feasible way to elucidate the potential connection between these two distinct modeling frameworks, establishing an effective bridge between them. On this basis, a new non-probabilistic convex modeling method is proposed for quantifying uncertainty under limited data. The performance of the proposed convex modeling method is evaluated through numerical examples, and its accuracy and effectiveness are further validated using engineering applications.

在现有的研究中，概率模型和非概率凸模型通常被视为独立的实体，很少关注它们之间的潜在联系。这种差距阻碍了对非概率凸模型的更深入的理解和合理的应用。为了解决这一问题，本文提出了一种新的非概率凸建模方法，利用概率模型和凸模型之间的潜在关系，在有限的数据条件下实现精确的不确定性量化。首先，给出了概率凸模型和非概率凸模型的数学表达式。然后，引入了一种降维技术，提供了一种可行的方法来阐明这两个不同的建模框架之间的潜在联系，在它们之间建立了有效的桥梁。在此基础上，提出了一种新的非概率凸建模方法来量化有限数据下的不确定性。通过数值算例对所提凸建模方法的性能进行了评价，并通过工程应用进一步验证了所提凸建模方法的准确性和有效性。

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引用次数: 0

Joint stationary response prediction of high-dimension strongly nonlinear systems with both uncertain parameters and stochastic excitation by solving FPK equation 求解FPK方程预测具有不确定参数和随机激励的高维强非线性系统联合平稳响应

IF 3 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics

Pub Date : 2025-07-01 Epub Date: 2025-06-21 DOI: 10.1016/j.probengmech.2025.103795

Yangyang Xiao , Lincong Chen , Zhongdong Duan , Jianqiao Sun

Uncertainties in system parameters and dynamic loading are pervasive in engineering and significantly influence the dynamic response of systems. While random response analysis has been studied since the 1960s, predicting responses for high-dimension strongly nonlinear systems under both types of uncertainties remains a significant challenge. This study extends a decoupled Fokker–Planck–Kolmogorov (FPK) equation approach to predict the joint stationary response of high-dimension strongly nonlinear systems with uncertain parameters under additive and/or multiplicative white noise excitations. Leveraging the law of total probability and the subspace method, the decoupled FPK equation governing the unconditional joint probability density function (PDF) of the state variables of interest are derived. These decoupled equations can effectively handle both uncertainties while avoiding the complications of high dimensionality and large numbers of uncertain parameters. Subsequently, the neural network-based methods combined with an efficient hypersphere sampling strategy are used to deal with the decoupled FPK equation, yielding non-Gaussian joint PDFs. Three examples, including the Rayleigh system, the inclined nonlinear cable system, and a high-dimension nonlinear base-isolation frame system with the maximum number of uncertain parameters up to 25, are studied for illustration. Extensive Monte Carlo simulation data validate the accuracy and efficiency of the proposed scheme. The results demonstrate that the proposed approach successfully captures the complex-shaped joint PDF of the strongly nonlinear system, even for the challenging five dimension case. Notably, parameter uncertainties can lead to a reduction of up to 20% in the peak PDF of the responses and an increase in the tail PDF by several orders of magnitude compared to deterministic systems.

系统参数和动载荷的不确定性在工程中普遍存在，对系统的动态响应有重要影响。虽然随机响应分析自20世纪60年代以来一直在研究，但预测两种不确定性下高维强非线性系统的响应仍然是一个重大挑战。本文扩展了一种解耦Fokker-Planck-Kolmogorov （FPK）方程方法，用于预测具有不确定参数的高维强非线性系统在加性和/或乘性白噪声激励下的联合平稳响应。利用总概率定律和子空间方法，导出了控制感兴趣状态变量无条件联合概率密度函数（PDF）的解耦FPK方程。这种解耦方程既能有效地处理不确定性，又能避免高维数和大量不确定参数的复杂性。然后，利用神经网络方法结合高效的超球采样策略对解耦FPK方程进行处理，得到非高斯联合pdf。通过对瑞利系统、倾斜非线性索系统和最大不确定参数达25个的高维非线性基础隔震框架系统三个实例的研究进行了说明。大量的蒙特卡罗仿真数据验证了该方案的准确性和有效性。结果表明，该方法成功地捕获了强非线性系统的复杂形状关节PDF，即使在具有挑战性的五维情况下也是如此。值得注意的是，与确定性系统相比，参数不确定性可导致响应的峰值PDF减少高达20%，而尾部PDF增加几个数量级。

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引用次数: 0

Eigenvalue analysis of stochastic structural systems: A quantum computing approach 随机结构系统的特征值分析：量子计算方法

IF 3.5 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics

Pub Date : 2025-07-01 Epub Date: 2025-08-05 DOI: 10.1016/j.probengmech.2025.103814

Leonidas Taliadouros, Ilias G. Mavromatis, Ioannis A. Kougioumtzoglou

A quantum computing approach is developed for eigenvalue analysis of stochastic structural systems. Specifically, within a Monte Carlo simulation (MCS) solution framework, both the subspace-search variational quantum eigensolver (SSVQE) and the variational quantum deflation (VQD) algorithms are employed and appropriately adapted for treating the random eigenvalue problem in structural dynamics. Compared to alternative quantum-based solution efforts in the literature, the herein-developed approach yields statistics for the complete set of the system eigenvalues. Further, certain advantageous properties of the system parameter matrices, such as symmetry and sparsity, are also exploited for enhancing the efficiency of the approach. Furthermore, an efficient strategy is proposed for addressing the challenging problem of initialization of the SSVQE and VQD algorithms, and calibration of the associated hyperparameters. Two representative examples exhibiting random parameter matrices are considered. They relate to a chain-like system that is widely used in structural dynamics for modeling, for instance, shear-type building structures, and to a system with cyclic symmetry that is of relevance to the dynamics of rotating machines such as turbine blades. The accuracy degree of the approach is demonstrated by comparing eigenvalue statistics obtained based on classical computing (Python) with estimates obtained by employing a quantum computer simulator and, for a specific case, an actual quantum computer (IBM Sherbrooke). The latter achievement has its own merit since, for the first time, an MCS approach is employed on a real quantum computer for conducting eigenvalue analysis of a stochastic structural system. This serves as a proof-of-concept that quantum computing can, potentially, treat challenging stochastic dynamics problems in the near future.

提出了一种用于随机结构系统特征值分析的量子计算方法。具体而言，在蒙特卡罗模拟（MCS）求解框架内，采用了子空间搜索变分量子特征求解器（SSVQE）和变分量子紧缩（VQD）算法，并适当地适用于处理结构动力学中的随机特征值问题。与文献中其他基于量子的解决方案相比，本文开发的方法产生了系统特征值完整集合的统计信息。此外，还利用了系统参数矩阵的对称性和稀疏性等优点，提高了方法的效率。此外，提出了一种有效的策略来解决SSVQE和VQD算法的初始化问题，以及相关超参数的校准问题。考虑了两个具有代表性的随机参数矩阵的例子。它们涉及到广泛用于结构动力学建模的链状系统，例如剪切型建筑结构，以及与旋转机器（如涡轮叶片）动力学相关的循环对称系统。通过比较基于经典计算（Python）获得的特征值统计与使用量子计算机模拟器获得的估计，以及在特定情况下，使用实际量子计算机（IBM Sherbrooke）获得的估计，证明了该方法的准确性。后一项成就有其自身的优点，因为这是第一次在真实的量子计算机上使用MCS方法对随机结构系统进行特征值分析。这证明了量子计算在不久的将来可以潜在地处理具有挑战性的随机动力学问题。

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引用次数: 0

Reliable uncertainty quantification for fiber orientation in composite molding processes using multilevel polynomial surrogates 复合材料成型过程中纤维取向的多级多项式替代可靠不确定度量化

IF 3 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics

Pub Date : 2025-07-01 Epub Date: 2025-07-16 DOI: 10.1016/j.probengmech.2025.103806

Stjepan Salatovic , Sebastian Krumscheid , Florian Wittemann , Luise Kärger

Fiber orientation is decisive for the mechanical performance of composite materials. During manufacturing, variations in material and process parameters can influence fiber orientation. We employ multilevel polynomial surrogates to model the propagation of uncertain material properties in the injection molding process. To ensure reliable uncertainty quantification, a key focus is deriving novel error bounds for statistical measures of a quantity of interest. Numerical experiments employ the Cross-WLF viscosity model and Hagen–Poiseuille flow to investigate the impact of uncertainties in fiber length and matrix temperature on the fractional anisotropy of fiber orientation. The Folgar–Tucker equation and the improved anisotropic rotary diffusion model, incorporating analytical solutions, are used for verification. Results show that the method improves significantly upon standard Monte Carlo estimation, while also providing error guarantees. These findings offer the first step towards a reliable and practical tool for optimizing fiber-reinforced polymer manufacturing processes in the future.

纤维取向对复合材料的力学性能起决定性作用。在制造过程中，材料和工艺参数的变化会影响纤维的取向。我们采用多水平多项式替代模型来模拟不确定材料性能在注射成型过程中的传播。为了确保可靠的不确定度量化，一个关键的焦点是为感兴趣的数量的统计度量推导新的误差界限。数值实验采用Cross-WLF黏度模型和hahagen - poiseuille流动研究了纤维长度和基体温度的不确定性对纤维取向分数各向异性的影响。采用Folgar-Tucker方程和改进的各向异性旋转扩散模型，并结合解析解进行验证。结果表明，该方法在提供误差保证的同时，在标准蒙特卡罗估计的基础上有了显著的改进。这些发现为未来优化纤维增强聚合物制造工艺的可靠实用工具迈出了第一步。

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引用次数: 0

Reliability-based design optimization of key components in a gantry machining center 基于可靠性的龙门加工中心关键部件设计优化

IF 3 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics

Pub Date : 2025-07-01 Epub Date: 2025-06-06 DOI: 10.1016/j.probengmech.2025.103786

Yumo Chen, Xianzhen Huang, Mingfei Ma, Jiaxin Luo, Boyang Ding

The gantry machining center is a critical piece of equipment in modern manufacturing, with its structural design playing a crucial role in determining machining precision, production efficiency, and product quality. To achieve both high performance and a lightweight design, this paper presents an optimization method for the key components of the gantry machining center based on reliability. The goal is to minimize the system's mass while ensuring that the static deformation does not increase, first-order modal frequency does not decrease, and the reliability meets the predetermined confidence level. Sensitivity analysis is performed to identify the parameters with significant impacts on the gantry machining center's performance. To overcome the time-consuming nature of the finite element method (FEM), an adaptive Kriging surrogate model is employed. An efficient metaheuristic algorithm is then used to solve for the optimal design. The effectiveness of the proposed optimization method is verified through comparative analysis. The results show that the reliability optimization method can effectively balance the mass and reliability of the gantry machining center, significantly improving the system's performance stability under random uncertainties, thus providing a theoretical foundation for the structural optimization of gantry machining centers.

龙门加工中心是现代制造业中的关键设备，其结构设计对加工精度、生产效率和产品质量起着至关重要的作用。为实现龙门加工中心的高性能和轻量化设计，提出了一种基于可靠性的关键部件优化方法。目标是在保证静变形不增加、一阶模态频率不降低、可靠性满足预定置信水平的前提下，使系统质量最小。通过灵敏度分析，识别出对龙门加工中心性能影响较大的参数。为了克服有限元法耗时的缺点，采用了自适应Kriging代理模型。然后采用一种高效的元启发式算法求解最优设计。通过对比分析验证了所提优化方法的有效性。结果表明，该可靠性优化方法能有效地平衡龙门加工中心的质量和可靠性，显著提高了系统在随机不确定性下的性能稳定性，为龙门加工中心的结构优化提供了理论基础。

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引用次数: 0

A comparative analysis of intrusive and non-intrusive PCE methods for random mode computation 随机模态计算中侵入式与非侵入式PCE方法的比较分析

IF 3 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics

Pub Date : 2025-07-01 DOI: 10.1016/j.probengmech.2025.103792

Eric Jacquelin , Sondipon Adhikari , Denis Brizard

Random eigenmodes present a significant challenge in the analysis of uncertain dynamical systems, particularly when traditional Monte Carlo methods become computationally prohibitive for high-dimensional problems. While Polynomial Chaos Expansion (PCE) offers a promising alternative, the choice between intrusive (physics-based) and non-intrusive (data-driven) implementations remains a critical yet understudied decision. This paper presents the first comprehensive comparison of these PCE approaches for random eigenmode computation, examining their theoretical foundations, implementation complexities, and computational efficiency. Through systematic analysis of a three-degree-of-freedom system with varying uncertainty parameters, we demonstrate that intrusive PCE achieves superior accuracy for low-dimensional problems, while non-intrusive PCE shows better scalability for higher-dimensional systems. Our findings reveal a previously undocumented trade-off between implementation complexity and computational efficiency, establishing clear criteria for approach selection based on problem dimensionality and accuracy requirements. These insights extend beyond modal analysis to the broader field of uncertainty quantification in computational mechanics, providing practical guidelines for selecting optimal PCE strategies in various engineering applications. The methodological framework presented here opens new possibilities for efficient uncertainty analysis in large-scale dynamical systems.

随机特征模态在不确定动力系统的分析中提出了一个重大的挑战，特别是当传统的蒙特卡罗方法对高维问题的计算变得令人望而却步时。虽然多项式混沌展开（PCE）提供了一个很有前途的替代方案，但在侵入式（基于物理的）和非侵入式（数据驱动的）实现之间的选择仍然是一个关键但尚未得到充分研究的决定。本文首次对这些随机特征模计算的PCE方法进行了全面比较，考察了它们的理论基础、实现复杂性和计算效率。通过对具有不同不确定参数的三自由度系统的系统分析，我们证明了侵入式PCE在低维问题上具有更好的精度，而非侵入式PCE在高维问题上具有更好的可扩展性。我们的研究结果揭示了以前没有记录的实现复杂性和计算效率之间的权衡，建立了基于问题维度和精度要求的方法选择的明确标准。这些见解从模态分析扩展到计算力学中更广泛的不确定性量化领域，为在各种工程应用中选择最佳PCE策略提供了实用指南。本文提出的方法框架为大规模动力系统的高效不确定性分析开辟了新的可能性。

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引用次数: 0

Variations in the reliability performance of free-form single-layer grid structures during deterministic optimization, and optimization method improvement 确定性优化过程中自由形式单层网格结构可靠性性能的变化及优化方法的改进

IF 3 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics

Pub Date : 2025-07-01 Epub Date: 2025-07-18 DOI: 10.1016/j.probengmech.2025.103813

Dong Li, Baoshi Jiang, Bowen Hou

Structural optimization can be categorized into deterministic optimization (DO) and reliability‐based design optimization (RBDO). Although RBDO yields designs with better reliability than DO, it incurs substantially greater computational costs. Existing studies typically compare the final outcomes of RBDO and DO, without examining how structural reliability evolves during DO. In this study, we track how the reliability of a free‐form single‐layer grid structure evolves during DO. The strain energy is adopted as the objective, and the probability density evolution method is used to compute the structural-response probability density function. Based on the observed trends, we propose an enhanced optimization strategy that balances the simultaneous improvements in the vertical and horizontal mechanical performance and reliability performance. The improved method is then applied to a cantilever‐type free‐form surface structure to assess its generality. The results indicate that the mechanical and reliability performances vary in the same manner along the load direction. Moreover, by improving the objective function, the proposed method effectively enhances both the mechanical and reliability performances under horizontal seismic and vertical loads. It achieves concurrent improvements in both directions and performs well across different structural types.

结构优化可分为确定性优化（DO）和基于可靠性的设计优化（RBDO）。虽然RBDO产生的设计比DO具有更好的可靠性，但它的计算成本要高得多。现有的研究通常比较RBDO和DO的最终结果，而没有研究DO过程中结构可靠性的演变。在这项研究中，我们跟踪了自由形式单层网格结构在DO过程中的可靠性演变。以应变能为目标，采用概率密度演化法计算结构-响应概率密度函数。基于观察到的趋势，我们提出了一种增强的优化策略，以平衡垂直和水平力学性能和可靠性性能的同时改进。然后将改进的方法应用于悬臂型自由曲面结构，以评估其普遍性。结果表明，沿荷载方向，结构的力学性能和可靠性均呈相同的变化规律。通过对目标函数的改进，有效地提高了结构在水平地震和竖向荷载作用下的力学性能和可靠性。它实现了两个方向的同步改进，并且在不同的结构类型中表现良好。

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引用次数: 0