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reliability-based design optimization of systems in Voronoi tessellation-based partitioned design space

STRUCTURES 2025年 71卷

作者： Khalid, Mohd Aman Bansal, Sahil Indian Inst Technol Delhi Dept Civil Engn New Delhi 110016 India

reliability-based design optimization (RBDO) is a well-known design methodology involving reliability constraints to achieve an optimal design with a failure probability lower than the predefined threshold. Evaluating reliability constraints, or equivalently the Failure Probability Functions (FPFs), has proven challenging due to its higher computational cost within the optimization loop, lower accuracy, and limited applicability resulting from their approximations. To address these issues, this paper introduces a novel stochastic simulation-based decoupling approach for performing RBDO. The key novelty of the proposed approach is the transformation of FPFs approximating or evaluation into a density estimation problem using Voronoi tessellation while decoupling reliability constraints evaluation from the optimization loop. The proposed approach initiates with the establishment of an augmented reliability formulation. Subsequently, akin to Stochastic Subset optimization (SSO), the original design space undergoes iterative reduction to obtain a feasible design space that satisfies reliability constraints. At each iteration, Voronoi tessellation is implemented to estimate the probability density value of design variables conditioned on the failure events. Ultimately, a deterministic optimization problem is solved within the feasible design space to minimize the cost function. The efficacy and computational efficiency of the proposed approach are demonstrated through its successful application to various RBDO benchmark examples.

关键词： reliability-based design optimization RBDO optimization under uncertainty probabilistic constraints stochastic simulation decoupling Voronoi Tessellation

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reliability-based design optimization incorporating extended Optimal Quantification

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PROBABILISTIC ENGINEERING MECHANICS 2025年 80卷

作者： Miska, Niklas Balzani, Daniel Ruhr Univ Bochum Chair Continuum Mech Univ Str 150 D-44801 Bochum Germany

reliability-based design optimization (RBDO) approaches aim to identify the best design of an engineering problem, whilst the probability of failure (PoF) remains below an acceptable value. Thus, the incorporation of the sharpest bounds on the PoF under given constraints on the uncertain input quantities strongly strengthens the significance of RBDO results, since unjustified assumptions on the input quantities are avoided. In this contribution, the extended Optimal Uncertainty Quantification framework is embedded within an RBDO context in terms of a double loop approach. By that, the mathematically sharpest bounds on the PoF as well as on the cost function can be computed for all design candidates and compared with acceptable values. The extended OUQ allows the incorporation of aleatory as well as epistemic uncertainties, where the definition of probability density functions is not necessarily required and just given data on the input can be included. Specifically, not only bounds on the values themselves, but also bounds on moment constraints can be taken into account. Thus, inadmissible assumptions on the data can be avoided, while the optimal design of a problem can be identified. The capability of the resulting framework is firstly shown by means of a benchmark problem under the influence of polymorphic uncertainties. Afterwards, a realistic engineering problem is analyzed, where the positioning of laser-hardened lines within a steel sheet for a car crash structure are optimized.

关键词： reliability-based design optimization Optimal Uncertainty Quantification Epistemic and aleatory uncertainties Laser-hardened car components

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A decoupled method for time-dependent reliability-based design optimization

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Science China(Technological Sciences) 2025年第1期68卷 308-320页

作者： Dequan ZHANG Meide YANG Xu HAN State Key Laboratory of Reliability and Intelligence of Electrical Equipment School of Mechanical EngineeringHebei University of TechnologyTianjin 300401China State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body College of Mechanical and Vehicle EngineeringHunan UniversityChangsha 410082China

Time-dependent reliability-based design optimization(TRBDO)has received extensive attention because of its ability to achieve optimal solutions that help meet the requirement for whole lifecycle reliability by quantitatively considering dynamic ***,directly solving TRBDO problems is computationally expensive,if not prohibitive,owing to the need to repeatedly evaluate time-dependent probabilistic *** address this challenge,an efficient decoupled method called sequential optimization and time-dependent reliability assessment(SOTRA)is proposed in this *** method transforms the original TRBDO problem,initially formulated probabilistically,into a problem using percentile formulation after discretizing time-dependent performance *** adopting the equivalent minimum performance target point(EMPTP)concept,the TRBDO problem is further converted into an equivalent deterministic optimization problem,which is subsequently solved through a sequential iteration process involving deterministic optimization and time-dependent reliability *** efficiently and robustly search an EMPTP for reliability analysis,a time-dependent self-adaptive finite-step length method is *** verify the proposed SOTRA method against existing TRBDO methods,a numerical example,a benchmark structural design case of a simply supported beam,and an engineering application for flexible wheel design are exemplified in this *** results demonstrate that the proposed SOTRA method exhibits high efficiency and robustness in solving TRBDO problems.

关键词： reliability-based design optimization dynamic uncertainties time-dependent reliability analysis decoupled method sequential optimization and reliability analysis

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A hybrid single-loop approach combining the target beta-hypersphere sampling and active learning Kriging for reliability-based design optimization

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AEROSPACE SCIENCE AND TECHNOLOGY 2025年 161卷

作者： Hu, Huanhuan Wang, Pan Chang, Haoqi Yang, Rong Yang, Weizhu Li, Lei Northwestern Polytech Univ Sch Mech Civil Engn & Architecture Xian 710072 Shaanxi Peoples R China

In engineering design, system-level requirements typically provide each subsystem with specific target reliability indexes. This makes reliability-based design optimization (RBDO) under the prescribed target reliability index particularly relevant for practical applications. However, solving complex nonlinear RBDO problems often involves nested double-loop optimization, leading to prohibitive computational costs and potential convergence issues. To address these challenges, this study proposes a minimum performance measure-based hybrid singleloop approach (TSPM-AK-HSLA) that integrates target beta-hypersphere sampling and active learning Kriging. First, a novel sampling strategy combining target beta-hypersphere and local enhancement is introduced to accurately identify the minimum performance target point (MPTP) without requiring gradient calculations or iterative search direction adjustments. Second, an identification criterion for the active constraint is incorporated to determine whether the Kriging model needs updating within the local region around the approximate MPTP, thereby focusing sampling efforts for improved efficiency. Finally, an adaptive strategy is employed to implement the hybrid single-loop approach, accelerating convergence while maintaining robustness for nonlinear problems. Comparative analyses with existing methods, along with two numerical MPTP search examples and two nonlinear RBDO examples demonstrate the superior efficiency and accuracy of the proposed approach. The RBDO application for an engineering clamping mechanism of the aircraft engine guides the design.

关键词： reliability-based design optimization Kriging model Combining sampling Hybrid single-loop approach Minimum performance target point (MPTP)

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A new hybrid inverse reliability method for searching MPTP and its application in reliability-based design optimization

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reliability ENGINEERING & SYSTEM SAFETY 2025年 253卷

作者： Lai, Xiongming Yang, Tao Zhang, Yong Wang, Cheng Liao, Shuirong Zeng, Xianbiao Zhang, Xiaodong Huaqiao Univ Coll Mech Engn & Automat Xiamen 361021 Peoples R China Huaqiao Univ Coll Comp Sci & Technol Xiamen 361021 Peoples R China

reliability-based design optimization (RBDO) is a nested optimization approach. This paper emphasizes the importance of improving computational efficiency by reducing the number of evaluations of the performance function, which is the bottleneck for practical engineering applications. Aim at this issue, the paper proposes to reduce the function evaluation number caused by gradient calculations as the important direction for improving RBDO's efficiency, as it takes up the majority of the total evaluation number of RBDO. Within the above important direction, the paper introduces a new hybrid inverse reliability method for searching the minimum performance target point (MPTP). It integrates the traditional Advanced Mean Value (AMV) with an innovative MPTP-searching approach, where the new format of gradient computation is dependent on one variable other than muliti-ones. This can effectively eliminate a drawback of conventional gradient computation, whose evaluation number is influenced by the dimensionality of random variables. Furthermore, its operational simplicity is highlighted by the absence of chaos control factors, making it more convenient for real engineering applications compared to methods like Chaos Control (CC) or Modified Chaos Control (MCC). Lastly, the numerical examples are investigated to validate the accuracy and efficiency of the proposed method.

关键词： reliability-based design optimization Single loop method Convergence Performance function reliability index

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Efficient global adaptive Kriging approximation method in terms of moment for reliability-based design optimization

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2025年 437卷

作者： Yang, Meide Zhang, Hongfei Zhang, Dequan Wang, Fang Han, Xu Hebei Univ Technol Sch Mech Engn State Key Lab Reliabil & Intelligence Elect Equipm Tianjin 300401 Peoples R China

reliability-based design optimization (RBDO) methods based on the most probable point (MPP) have been extensively studied and applied to practical engineering problems. Nevertheless, these methods are not viable when MPP is not straightforward to be searched or multiple MPPs may exhibit. Fortunately, moment method can circumvent the computation of partial derivatives for performance function and iteration to search for MPPs, which is considered as an effective way to solve such problem. However, direct application of moment method to RBDO often incurs high computational cost, which greatly hinders its practicability. To enhance the computational efficiency of the moment-based RBDO methods, an efficient global adaptive Kriging approximation method for RBDO is proposed in this study. The strategy is that a new initial design of experiment scheme according to Gaussian-Hermite integration nodes is innovatively proposed. On this basis, a feasibility check criterion for probabilistic constraints and a selection strategy for candidate samples are respectively proposed to efficiently establish Kriging models of performance functions in the probabilistic constraints. In addition, an enhanced univariate dimension-reduction method with high robustness is presented to calculate the first four-order statistical moments of the above constructed Kriging models. Consequently, the failure probability of each probabilistic constraint can be calculated by Edgeworth series. Finally, a deterministic optimization algorithm is executed to derive the optimal solution. Three numerical examples and two structural examples are exemplified to demonstrate the effectiveness of the proposed moment-based method compared to prevailing MPP-based and Kriging-based RBDO methods.

关键词： reliability-based design optimization Univariate dimension-reduction method Edgeworth series Kriging model Active learning

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Polynomial chaos kriging-assisted reliability-based design optimization for a deep mixing wall

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COMPUTERS AND GEOTECHNICS 2025年 179卷

作者： Zhang, Tingting Abdallah, Adel Cuisinier, Olivier Masrouri, Farimah Univ Lorraine Lab LEMTA UMR 7563 CNRS 2 Ave Foret Haye F-54500 Vandoeuvre Les Nancy France

This paper presents a reliability-based design optimization (RBDO) for Deep Mixing Walls (DMW), taking into account parameter uncertainties, required safety levels and construction costs. A procedure that integrates Polynomial Chaos Kriging (PCK) and constrained (1 + 1) Covariance Matrix Adaptation Evolution Strategy (constrained (1 + 1)-CMA-ES) is proposed, referred to as PCK-RBDO-ES, for the optimization of DMW design. Specifically, PCK is employed to reduce required numerical simulations and improve efficiency of reliability assessments, while the constrained (1 + 1)-CMA-ES method is used for global optimization of design parameters. Unconfined Compressive Strength (UCS) is considered as the design variable to be optimized in this study and an existing DMW was selected as a reference case. The PCK-RBDO-ES results are first presented and compared to validate the effectiveness and efficiency. Then, the influence of the DMW uncertainties and adjacent soil uncertainties on the UCS evaluations is discussed, and the effect of the target reliability requirement is further investigated. The results indicate that the proposed PCK-RBDO-ES procedure is effective and provides useful insights for DMW design and construction. Uncertainties on UCS and stiffness of DMW, as well as on the friction angle of soft to medium clay layer, are shown to have significant impacts on the design optimization. This study can help designers to manage project risks, and facilitate risk-aware and cost-effective decision-making.

关键词： Deep mixing wall reliability-based design optimization Polynomial chaos kriging Constrained (1+1)-CMA-ES Unconfined compressive strength

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An integral method for reliability-based design optimization

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2025年 441卷

作者： Chen, Zhenzhong Wang, Wenhao Pan, Qianghua Guo, Guangming Li, Xiaoke Chen, Ge Gan, Xuehui Donghua Univ Coll Mech Engn Shanghai 201620 Peoples R China China Special Equipment Inspect & Res Inst 2 Hepingjie West Garden Beijing 100029 Peoples R China Beijing Acad Sci & Technol Intelligent Equipment Res Inst Beijing 100061 Peoples R China Zhengzhou Univ Light Ind Sch Mech & Elect Engn Henan Key Lab Mech Equipment Intelligent Mfg Zhengzhou 450002 Peoples R China

In reliability-based design optimization (RBDO), the aim is to develop an optimal design characterized by high reliability through fulfilling design requirements at the targeted probability threshold. The goal of reliability optimization is to obtain excellent algorithms by focusing on evaluation and optimization. In RBDO, due to the selection of evaluation methods and the problem of updating reliable point methods, it is often impossible to obtain accurate failure probability results with less computation, and the results are inaccurate, or the computation amount is increased. The integral-based reliability analysis method: Hyperspherical cap area integral method (HCAIM) can achieve an accuracy close to that of the MCS method while having a very low computational load, thus enabling the efficient and precise calculation of failure probabilities. Therefore, by introducing a reliability evaluation method based on Integral Method for reliability optimization (IMRO), as a decoupling method, high accuracy failure probability calculation results can be obtained with relatively small calculation amount, and then optimization results can be obtained by IMRO and sequence optimization methods. First, a reliability analysis example is utilized to demonstrate the accuracy of the reliability analysis part of IMRO. Then, a set of nonlinear challenges and diverse engineering case studies were utilized to assess the algorithm's performance. The calculation results after comparison prove that IMRO is more accurate in dealing with nonlinear problems and engineering examples, and can better meet the requirements.

关键词： reliability-based design optimization Hyperspherical cap area integral method Second order reliability method Sequential optimization and reliability Assessment

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reliability-based design optimization of offshore wind turbine support structures using RBF surrogate model

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Frontiers of Structural and Civil Engineering 2023年第7期17卷 1086-1099页

作者： Changhai YU Xiaolong LV Dan HUANG Dongju JIANG Department of Engineering Mechanics Hohai UniversityNanjing 211100China

An efficient reliability-based design optimization method for the support structures of monopile offshore wind turbines is proposed ***,parametric finite element analysis(FEA)models of the support structure are established by considering stochastic ***,a surrogate model is constructed using a radial basis function(RBF)neural network to replace the time-consuming *** uncertainties of loads,material properties,key sizes of structural components,and soil properties are *** uncertainty of soil properties is characterized by the variabilities of the unit weight,friction angle,and elastic modulus of *** reliability is determined via Monte Carlo simulation,and five limit states are considered,i.e.,structural stresses,tower top displacements,mudline rotation,buckling,and natural *** on the RBF surrogate model and particle swarm optimization algorithm,an optimal design is established to minimize the *** show that the proposed method can yield an optimal design that satisfies the target reliability and that the constructed RBF surrogate model significantly improves the optimization ***,the uncertainty of soil parameters significantly affects the optimization results,and increasing the monopile diameter is a cost-effective approach to cope with the uncertainty of soil parameters.

关键词： reliability-based design optimization offshore wind turbine parametric finite element analysis RBF surrogate model uncertain soil parameter

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reliability-based design optimization for flexible mechanism with particle swarm optimization and advanced extremum response surface method

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Journal of Central South University 2016年第8期23卷 2001-2007页

作者：张春宜宋鲁凯费成巍郝广平刘令君 School of Mechanical and Power Engineering Harbin University of Science and Technology School of Computer Science and Engineering Beihang University Department of Mechanical Engineering The Hong Kong Polytechnic University

To improve the computational efficiency of the reliability-based design optimization(RBDO) of flexible mechanism, particle swarm optimization-advanced extremum response surface method(PSO-AERSM) was proposed by integrating particle swarm optimization(PSO) algorithm and advanced extremum response surface method(AERSM). Firstly, the AERSM was developed and its mathematical model was established based on artificial neural network, and the PSO algorithm was investigated. And then the RBDO model of flexible mechanism was presented based on AERSM and PSO. Finally, regarding cross-sectional area as design variable, the reliability optimization of flexible mechanism was implemented subject to reliability degree and uncertainties based on the proposed approach. The optimization results show that the cross-section sizes obviously reduce by 22.96 mm^2 while keeping reliability degree. Through the comparison of methods, it is demonstrated that the AERSM holds high computational efficiency while keeping computational precision for the RBDO of flexible mechanism, and PSO algorithm minimizes the response of the objective function. The efforts of this work provide a useful sight for the reliability optimization of flexible mechanism, and enrich and develop the reliability theory as well.

关键词： reliability-based design optimization flexible robot manipulator artificial neural network particle swarm optimization advanced extremum response surface method

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