Reliability-based design optimization using a moment method and a kriging metamodel

Reliability-based design optimization (RBDO) has been used for optimizing engineering systems with uncertainties in design variables and system parameters. RBDO involves reliability analysis, which requires a large amount of computational effort, so it is important to select an efficient method for reliability analysis. Of the many methods for reliability analysis, a moment method, which is called the fourth moment method, is known to be less expensive for moderate size problems and requires neither iteration nor the computation of derivatives. Despite these advantages, previous research on RBDO has been mainly based on the first-order reliability method and relatively little attention has been paid to moment-based RBDO. This article considers difficulties in implementing the moment method into RBDO; they are solved using a kriging metamodel with an active constraint strategy. Three numerical examples are tested and the results show that the proposed method is efficient and accurate.     

Reliability-based design optimization with progressive surrogate models

Reliability-based design optimization (RBDO) has traditionally been solved as a nested (bilevel) optimization problem, which is a computationally expensive approach. Unilevel and decoupled approaches for solving the RBDO problem have also been suggested in the past to improve the computational efficiency. However, these approaches also require a large number of response evaluations during optimization. To alleviate the computational burden, surrogate models have been used for reliability evaluation. These approaches involve construction of surrogate models for the reliability computation at each point visited by the optimizer in the design variable space. In this article, a novel approach to solving the RBDO problem is proposed based on a progressive sensitivity surrogate model. The sensitivity surrogate models are built in the design variable space outside the optimization loop using the kriging method or the moving least squares (MLS) method based on sample points generated from low-discrepancy sampling (LDS) to estimate the most probable point of failure (MPP). During the iterative deterministic optimization, the MPP is estimated from the surrogate model for each design point visited by the optimizer. The surrogate sensitivity model is also progressively updated for each new iteration of deterministic optimization by adding new points and their responses. Four example problems are presented showing the relative merits of the kriging and MLS approaches and the overall accuracy and improved efficiency of the proposed approach.     

A multiple-design-point approach for reliability-based design optimization

The application of design-point-based reliability-based design optimization (RBDO) methods is hindered by the challenge of multiple-design-point problems. In this article, to improve the commonality of design-point-based RBDO methods, a novel multiple-design-point (MDP) approach is developed. The MDP approach uses the trace of the design points from consequent reliability analysis iterations to identify whether there are multiple design points, then all of the design points are used to calculate shifting vectors for the sequential optimization and reliability assessment method, and the corresponding probabilistic constraints are moved to the feasible region along these multiple shifting vectors at the same time. With multiple shifted probabilistic constraints, the design feasibility associated with this probabilistic constraint will be satisfied. Two mathematical examples, a speed reducer design and a honeycomb crashworthiness design, are presented to validate the effectiveness of the MDP method. The results show that the MDP approach is effective for handling multiple-design-point problems.     

基于鞍点逼近的机械结构可靠性稳健优化设计

将可靠性优化设计理论与可靠性灵敏度分析方法相结合,讨论了机械零部件稳健优化设计的问题.系统地推导了基于鞍点逼近的可靠性灵敏度公式,并把可靠性灵敏度计算结果融入可靠性稳健优化设计模型之中,将可靠性稳健优化设计归结为满足可靠性要求的多目标优化问题.在基本随机参数概率分布已知的前提下,应用鞍点逼近技术,得到极限状态函数的分布函数与概率密度函数,并且将此结果应用到机械零部件的可靠性灵敏度分析中,进而实现了机械零部件的可靠性稳健优化设计.通过与Monte-Carlo方法计算所得的结果相比可知,应用鞍点逼近技术可以迅速、准确地得到机械零部件可靠性稳健设计信息．     

A new three‐point approximation approach for design optimization problems

In this paper, a newly developed three‐point approximation scheme is proposed. The expression of this scheme consists of a linear combination of the direct and reciprocal linear Taylor expansions as well as of the lumped diagonal terms of the second‐order direct and inverse terms. The unknown parameters of the expression are computed on the basis of the function and gradient values at three points in the design space. Based on this approach, the accuracy of the existing constraint approximation methods can be improved. The effectiveness of the proposed approach is demonstrated on a number of numerical examples. The numerical results are also compared with those of the previously published work. Copyright © 2001 John Wiley & Sons, Ltd.     

Enhanced modified reliability index approach for efficient and robust reliability-based design optimization

The reliability index approach (RIA) is one of the effective tools for solving the reliability-based design optimization (RBDO) probabilistic model, which models the uncertainties with probability constraints. However, its wide application in engineering is limited due to low efficiency and convergence problems. The RIA-based modified reliability index approach (MRIA) appears to be very robust and accurate than RIA but yields inefficient for the most probable point (MPP) search with highly nonlinear probabilistic constraints. In this study, an enhanced modified reliability index approach (EMRIA) is developed to improve the efficiency and robustness of searching for MPP and is utilized for RBDO. In the EMRIA, an innovative active set using rigorous inequality is applied to construct the region of exploring for MPP, where the unnecessary probabilistic constraint could be eliminated adaptively during the iterative process. Moreover, the double loop strategy (DLS) is integrated into the EMRIA to strengthen the efficiency and robustness of large-scale RBDO problems. Two numerical examples demonstrated that the EMRIA is an efficient and robust method for MPP search in comparison with current first-order reliability methods. Six RBDO problems quoted also indicate that DLS-based EMRIA has good performance to solve complex RBDO problems.     

A new hybrid reliability-based design optimization method under random and interval uncertainties

This article proposes a new method for hybrid reliability-based design optimization under random and interval uncertainties (HRBDO-RI). In this method, Monte Carlo simulation (MCS) is employed to estimate the upper bound of failure probability, and stochastic sensitivity analysis (SSA) is extended to calculate the sensitivity information of failure probability in HRBDO-RI. Due to a large number of samples involved in MCS and SSA, Kriging metamodels are constructed to substitute true constraints. To avoid unnecessary computational cost on Kriging metamodel construction, a new screening criterion based on the coefficient of variation of failure probability is developed to judge active constraints in HRBDO-RI. Then a projection-outline-based active learning Kriging is achieved by sequentially select update points around the projection outlines on the limit-state surfaces of active constraints. Furthermore, the prediction uncertainty of Kriging metamodel is quantified and considered in the termination of Kriging update. Several examples, including a piezoelectric energy harvester design, are presented to test the accuracy and efficiency of the proposed method for HRBDO-RI.     

基于多目标优化策略的扭杆可靠性稳健优化设计

为提高机械零部件的安全性和稳健性,应用可靠性稳健优化设计理论和多目标决策方法,建立了适合结构可靠性稳健优化设计的多目标优化模型.为能迅速准确地对具有约束条件的多目标优化模型进行求解,提出一种利用模糊理论对约束条件进行处理的方法,然后应用灰色粒子群算法对多目标优化模型进行求解.通过对正态分布参数和任意分布参数的扭杆可靠性稳健优化设计,表明该方法行之有效.     

Reliability-based design and optimization of adaptive marine structures

The objectives of this work are to quantify the influence of material and operational uncertainties on the performance of self-adaptive marine rotors, and to develop a reliability-based design and optimization methodology for adaptive marine structures. Using a previously validated 3D fluid–structure interaction model, performance functions are obtained and used to generate characteristic response surfaces. A first-order reliability method is used to evaluate the influence of uncertainties in material and load parameters and thus optimize the design parameters. The results demonstrate the viability of the proposed reliability-based design and optimization methodology, and demonstrate that a probabilistic approach is more appropriate than a deterministic approach for the design and optimization of adaptive marine structures that rely on fluid–structure interaction for performance improvement.     

Reliability-based robust design optimization of gap size of annular nuclear fuels using kriging and inverse distance weighting methods

In this study, the design optimization of the gap size of annular nuclear fuels used in pressurized water reactors (PWRs) was performed. For this, thermoelastic–plasticity–creep (TEPC) analysis of PWR annular fuels was carried out using an in-house code to investigate the performance of nuclear fuels. Surrogate models based on the kriging and inverse distance weighting models were generated using computational performance data based on optimal Latin hypercube design. Using these surrogate models, the gap size of PWR annular fuel was deterministically optimized using the micro-genetic algorithm to improve the heat transfer efficiency and maintain a lower level of stress. Reliability-based design optimization and reliability-based robust design optimization were conducted to satisfy target reliability and secure the robustness of the PWRs’ performance. The optimal gap size was validated through TEPC analysis and the optimum solutions were compared according to the approximate method and reliability index.     

Reliability analysis and reliability-based design optimization of roadway horizontal curves using a first-order reliability method

This article presents reliability analysis and reliability-based optimization of roadway minimum radius design based on vehicle dynamics, mainly focusing on exit ramps and interchanges. The performance functions are formulated as failure modes of vehicle rollover and sideslip. To accurately describe the failure modes, analytical models for rollover and sideslip are derived considering nonlinear characteristics of vehicle behaviour using the commercial software TruckSim. The probability of an accident is evaluated using the first-order reliability method and numerical studies are conducted using a single-unit truck model. To propose a practical application for the study, the reliability analysis for the minimum radius recommended by American Association of State Highway and Transportation Officials is conducted. The results show that, even if there are deviations from assumed design conditions of the current design guideline, the proposed design method can guarantee given target margins of safety against rollover and sideslip. Based on the reliability analysis, reliability-based design optimization is carried out and the results indicate new recommendations for minimum radii satisfying given target reliability levels.     

基于变频率区间约束的结构材料优化设计

针对频率约束的结构材料优化问题,基于结构拓扑优化思想,提出变频率区间约束的结构材料优化方法。借鉴均匀化及ICM(独立、连续、映射)方法,以微观单元拓扑变量倒数为设计变量,导出宏观单元等效质量矩阵及导数,进而获得频率一阶近似展开式。结合变频率区间约束思想,获得以结构质量为目标函数、频率为约束条件的连续体微结构拓扑优化近似模型;采用对偶方法求解。通过算例验证该方法的有效性及可行性,表明考虑质量矩阵变化影响所得优化结果更合理。     

考虑可靠度可信区间的结构可靠性优化设计

 在贝叶斯统计理论和结构可靠性优化设计方法的基础上,研究了结构在小样本情况下考虑可靠度可信区间的结构可靠性优化设计问题.将结构可靠度作为随机变量,根据先验信息和样本信息,采用贝叶斯推断技术获得结构可靠度的概率分布,给出了可靠度的点估计及区间估计.建立了考虑可靠度可信区间的结构可靠性优化设计模型,提出了考虑可靠度可信区间的结构可靠性优化设计方法.所提出的方法为解决小样本情况下的结构可靠性优化设计问题提供了新的解决方案.数值算例验证了所提出的结构可靠性优化设计方法的有效性和正确性.     

A reliability-based robust optimization design for the drum brake using adaptive Kriging surrogate model

To decrease random parameters’ influence on the drum brake reliability, the reliability-based robust optimization design (RBROD) of the electric vehicle brake is proposed. Based on the assumption that the maximum temperature of the brake cannot exceed the allowable temperature, a performance function model of thermal–mechanical coupling reliability of drum brakes is established by the adaptive Kriging method, and the analysis of reliability sensitivity and RBROD are conducted. The accuracy of the proposed model is verified by temperature measurement experiment under emergency braking condition. The robust optimization design improves the drum brake reliability to 0.99998 and reduce the influence of the design parameters on the reliability, with the absolute values of the reliability sensitivity and the weight of the drum brake are significantly smaller. Therefore, the objectives of reliability design, robustness design, and optimization design are simultaneously achieved by the proposed methods. Besides, the relative error of the proposed method is 0.373%, the number of function evaluations is 39, and the comparison with four meta-model methods show that the proposed method holds high-accuracy and high-efficiency. This study provides a high-precision theoretical explanation for the robust optimization design of drum brake.     

Adaptive improved response surface method for reliability-based design optimization

Reliability-based design optimization (RBDO) requires the evaluation of probabilistic constraints (or reliability), which can be very time consuming. Therefore, a practical solution for efficient reliability analysis is needed. The response surface method (RSM) and dimension reduction (DR) are two well-known approximation methods that construct the probabilistic limit state functions for reliability analysis. This article proposes a new RSM-based approximation approach, named the adaptive improved response surface method (AIRSM), which uses the moving least-squares method in conjunction with a new weight function. AIRSM is tested with two simplified designs of experiments: saturated design and central composite design. Its performance on reliability analysis is compared with DR in terms of efficiency and accuracy in multiple RBDO test problems.     

An inverse reliability analysis method for reliability-based design optimization with random and dependent interval variables constrained within ellipsoids

In practical design problems, interval variables exist. Many existing methods can handle only independent interval variables. Some interval variables, however, are dependent. In this work, dependent interval variables constrained within a multi-ellipsoid convex set are considered and incorporated into reliability-based design optimization (RBDO). An efficient RBDO method is proposed by employing the sequential single-loop procedure, which separates the coupled reliability analysis procedure from the deterministic optimization procedure. In the reliability analysis procedure, a single-loop optimization for the inverse reliability analysis is performed, and an efficient inverse reliability analysis method for searching for the worst-case most probable point (WMPP) is developed. The search method contains two stages. The first stage deals the situation where the WMPP is on the boundary of the feasible region, while the second stage accommodates the situation where the WMPP is inside the feasible region by interpolation. Three examples are used for a demonstration.     

Design space optimization using a numerical design continuation method

A generalized optimization problem in which design space is also a design to be found is defined and a numerical implementation method is proposed. In conventional optimization, only a portion of structural parameters is designated as design variables while the remaining set of other parameters related to the design space are often taken for granted. A design space is specified by the number of design variables, and the layout or configuration. To solve this type of design space problems, a simple initial design space is selected and gradually improved while the usual design variables are being optimized. To make the design space evolve into a better one, one may increase the number of design variables, but, in this transition, there are discontinuities in the objective and constraint functions. Accordingly, the sensitivity analysis methods based on continuity will not apply to this discontinuous stage. To overcome the difficulties, a numerical continuation scheme is proposed based on a new concept of a pivot phase design space. Two new categories of structural optimization problems are formulated and concrete examples shown. Copyright © 2001 John Wiley & Sons, Ltd.     

有频率禁区的结构优化设计

该文以一座矮塔斜拉桥——大连长山大桥为工程背景,对地震作用下结构的动力优化问题进行研究,通过优化改善桥梁结构的静动力特性,提高结构抗震性能,确保它们在动力环境下安全可靠的工作。在ANSYS中建立长山大桥空间有限元模型,然后采用响应面法以桥梁振型频率为优化目标,进行结构动力特性的优化设计;采用试验设计的优化方法以地震作用下结构的动力响应为优化目标,进行结构动力响应的优化设计;采用神经网络建立网络模型,快速有效的预测结构地震响应,为桥梁结构优化设计提供参考作用。研究表明：进行结构动力优化设计可以在保证桥梁结构整体性不变的前提下,充分发挥结构自身潜能,减小桥梁地震动力响应。     

Reliability-based design optimization of a car body using dimension- reduced Chebyshev polynomial

A reliability-based design optimization (RBDO) method of a car body is presented on basis of dimension-reduced Chebyshev polynomial method (DCM). To improve calculation efficiency and save computational time, complex models are often approximated by metamodels in reliability analysis. Traditional metamodels require a large number of sample points, which is time-consuming. To improve the efficiency, DCM is proposed to approximate the performance function of the car body. First, the performance function is decomposed by the dimension-reduction method into a sum of univariate functions, which are then fitted through Chebyshev polynomials. The reliability of the car body is predicted by the Taylor expansion method and the fourth-moment method. Finally, the result of RBDO is obtained using an improved adaptive genetic algorithm. The proposed method saves on the calculation time with high precision. Besides, the improved adaptive genetic algorithm reduces the number of iterations in the car body optimization and improves the efficiency.     

Multi-objective system reliability-based optimization method for design of a fully parametric concept car body

This article investigates multi-objective optimization under reliability constraints with applications in vehicle structural design. To improve computational efficiency, an improved multi-objective system reliability-based design optimization (MOSRBDO) method is developed, and used to explore the lightweight and high-performance design of a concept car body under uncertainty. A parametric model knowledge base is established, followed by the construction of a fully parametric concept car body of a multi-purpose vehicle (FPCCB-MPV) based on the knowledge base. The structural shape, gauge and topology optimization are then designed on the basis of FPCCB-MPV. The numerical implementation of MOSRBDO employs the double-loop method with design optimization in the outer loop and system reliability analysis in the inner loop. Multi-objective particle swarm optimization is used as the outer loop optimization solver. An improved multi-modal radial-based importance sampling (MRBIS) method is utilized as the system reliability solver for multi-constraint analysis in the inner loop. The accuracy and efficiency of the MRBIS method are demonstrated on three widely used test problems. In conclusion, MOSRBDO has been successfully applied for the design of a full parametric concept car body. The results show that the improved MOSRBDO method is more effective and efficient than the traditional MOSRBDO while achieving the same accuracy, and that the optimized body-in-white structure signifies a noticeable improvement from the baseline model.