Multi-objective robust optimization design of a front-end underframe structure for a high-speed train

A multi-objective robust design optimization of a front-end underframe structure for application in high-speed trains is proposed and the structural parameter uncertainty is considered. A finite element model of the structure is developed and verified by dynamic impact experiments. The sensitivity analysis demonstrates that the thicknesses of the centre sill have significant influences on structural crushing behaviours. The specific energy absorption and the initial peak crushing force (F_p) are taken as optimization objectives. Compared with the baseline structure, the 6-sigma robust design shows that the F_p and the structural mass are reduced by 54.86% and 13.06%, respectively, and the robust optimum is more reliable. The 6-sigma robust optimal solution has an efficient energy-absorbing capacity while satisfying the design constraint. Thus, 6-sigma robust optimization can be applied for high-speed trains.     

飞机总体设计中的一种多目标鲁棒优化方法

 提出一种基于灵敏度的多目标鲁棒优化方法。针对各维设计变量存在扰动的情况,在原约束多目标优化模型上,附加偏差目标函数,并采用最差估计法对约束条件进行鲁棒可行性调整。采用全局敏度方程方法来计算目标函数和约束函数对设计变量的敏度,进而采用Pareto遗传算法搜索约束多目标优化问题的非劣解集,设计者可以根据不同的设计准则从中选择合适的设计点。将上述方法用于飞机总体参数优化设计,并与采用常规优化方法所得的优化结果进行了分析和比较。     

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.     

Simulated variance optimization for robust design

A method to aid robust design in the presence of design parameter uncertainty is described. For a given relationship between a performance measure (or output parameter) and the uncertain design parameters a probabilistic simulation is used to obtain the variance of the performance measure as a function of the nominal design parameter values. The optimum values of the latter are then obtained as those corresponding to a minimum of the computed variance, determined by means of a particular non-linear optimization algorithm in the presence of constraints. The latter are in the form of limits on the nominal values of the design parameters and a specified value for the performance measure at the nominal design point, i.e. the deterministic design target. Some problems inherent in this type of procedure are discussed and methods of solution are described. A specific example is studied and the results from the present method are compared with those previously obtained by use of another procedure. © 1998 John Wiley & Sons, Ltd.     

Tolerancing and optimization for model-based robust engineering design

There are various methods for performing tolerancing and robust design within a computer-aided design (CAD) framework. Recent work on fitting statistical emulators to CAD systems can be used to facilitate fast optimization geared towards robustness against input variation. After discussing available methods for tolerancing within a common framework, a comprehensive strategy for robust design is developed which involves a combination of circuit simulation, emulation and global optimization.     

A surrogate assisted parallel multiobjective evolutionary algorithm for robust engineering design

A number of multi-objective evolutionary algorithms have been proposed in recent years and many of them have been used to solve engineering design optimization problems. However, designs need to be robust for real-life implementation, i.e. performance should not degrade substantially under expected variations in the variable values or operating conditions. Solutions of constrained robust design optimization problems should not be too close to the constraint boundaries so that they remain feasible under expected variations. A robust design optimization problem is far more computationally expensive than a design optimization problem as neighbourhood assessments of every solution are required to compute the performance variance and to ensure neighbourhood feasibility. A framework for robust design optimization using a surrogate model for neighbourhood assessments is introduced in this article. The robust design optimization problem is modelled as a multi-objective optimization problem with the aim of simultaneously maximizing performance and minimizing performance variance. A modified constraint-handling scheme is implemented to deal with neighbourhood feasibility. A radial basis function (RBF) network is used as a surrogate model and the accuracy of this model is maintained via periodic retraining. In addition to using surrogates to reduce computational time, the algorithm has been implemented on multiple processors using a master–slave topology. The preliminary results of two constrained robust design optimization problems indicate that substantial savings in the actual number of function evaluations are possible while maintaining an acceptable level of solution quality.     

Multi-objective design optimization of reconfigurable machine tools: a modified fuzzy-Chebyshev programming approach

A reconfigurable manufacturing system (RMS) is designed for rapid adjustment of functionalities in response to market changes. A RMS consists of a number of reconfigurable machine tools (RMTs) for processing different jobs using different processing modules. The potential benefits of a RMS may not be materialized if not properly designed. This paper focuses on RMT design optimization considering three important yet conflicting factors: configurability, cost and process accuracy. The problem is formulated as a multi-objective model. A mechanism is developed to generate and evaluate alternative designs. A modified fuzzy-Chebyshev programming (MFCP) method is proposed to achieve a preferred compromise of the design objectives. Unlike the original fuzzy-Chebyshev programming (FCP) method which imposes an identical satisfaction level for all objectives regardless of their relative importance, the MFCP respects their priority order. This method also features an adaptive satisfaction-level-dependent process to dynamically adjust objective weights in the search process. A particle swarm optimization algorithm (PSOA) is developed to provide quick solutions. The application of the proposed approach is demonstrated using a reconfigurable boring machine. Our computational results have shown that the combined MFCP and PSOA algorithm is efficient and robust. The advantages of the MFCP over the original FCP are also illustrated based on the results.     

Multiobjective robust optimization method for drawbead design in sheet metal forming

It is recognized that fracture and wrinkling in sheet metal forming can be eliminated via an appropriate drawbead design. Although deterministic multiobjective optimization algorithms and finite element analysis (FEA) have been applied in this respect to improve formability and shorten design cycle, the design could become less meaningful or even unacceptable when considering practical variation in design variables and noises of system parameters. To tackle this problem, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on drawbead design, where the six sigma principle is adopted to measure the variations, a dual response surface method is used to construct surrogate model and a multiobjective particle swarm optimization is developed to generate robust Pareto solutions. In this paper, the procedure of drawbead design is divided into two stages: firstly, equivalent drawbead restraining forces (DBRF) are obtained by developing a multiobjective robust particle swarm optimization, and secondly the DBRF model is integrated into a single-objective particle swarm optimization (PSO) to optimize geometric parameters of drawbead. The optimal design showed a good agreement with the physical drawbead geometry and remarkably improve the formability and robust. Thus, the presented method provides an effective solution to geometric design of drawbead for improving product quality.     

Distributionally robust optimization for engineering design under uncertainty

This paper addresses the challenge of design optimization under uncertainty when the designer only has limited data to characterize uncertain variables. We demonstrate that the error incurred when estimating a probability distribution from limited data affects the out-of-sample performance (ie, performance under the true distribution) of optimized designs. We demonstrate how this can be mitigated by reformulating the engineering design problem as a distributionally robust optimization (DRO) problem. We present computationally efficient algorithms for solving the resulting DRO problem. The performance of the DRO approach is explored in a practical setting by applying it to an acoustic horn design problem. The DRO approach is compared against traditional approaches to optimization under uncertainty, namely, sample-average approximation and multiobjective optimization incorporating a risk reduction objective. In contrast with the multiobjective approach, the proposed DRO approach does not use an explicit risk reduction objective but rather specifies a so-called ambiguity set of possible distributions and optimizes against the worst-case distribution in this set. Our results show that the DRO designs, in some cases, significantly outperform those designs found using the sample-average or the multiobjective approach.     

Gearbox design for uncertain load requirements using active robust optimization

Design and optimization of gear transmissions have been intensively studied, but surprisingly the robustness of the resulting optimal design to uncertain loads has never been considered. Active Robust (AR) optimization is a methodology to design products that attain robustness to uncertain or changing environmental conditions through adaptation. In this study the AR methodology is utilized to optimize the number of transmissions, as well as their gearing ratios, for an uncertain load demand. The problem is formulated as a bi-objective optimization problem where the objectives are to satisfy the load demand in the most energy efficient manner and to minimize production cost. The results show that this approach can find a set of robust designs, revealing a trade-off between energy efficiency and production cost. This can serve as a useful decision-making tool for the gearbox design process, as well as for other applications.     

A robust optimization methodology for preliminary aircraft design

This article focuses on a robust optimization of an aircraft preliminary design under operational constraints. According to engineers' know-how, the aircraft preliminary design problem can be modelled as an uncertain optimization problem whose objective (the cost or the fuel consumption) is almost affine, and whose constraints are convex. It is shown that this uncertain optimization problem can be approximated in a conservative manner by an uncertain linear optimization program, which enables the use of the techniques of robust linear programming of Ben-Tal, El Ghaoui, and Nemirovski [Robust Optimization, Princeton University Press, 2009]. This methodology is then applied to two real cases of aircraft design and numerical results are presented.     

Development of a sequential optimization procedure for robust design and tolerance design within a bi-objective paradigm

Many practitioners and researchers have implemented robust design and tolerance design as quality improvement and process optimization tools for more than two decades. Robust design is an enhanced process/product design methodology for determining the best settings of control factors while minimizing process bias and variability. Tolerance design is aimed at determining the best tolerance limits for minimizing the total cost incurred by both the customer and manufacturer by balancing quality loss due to variations in product performance and the cost of controlling these variations. Although robust design and tolerance design have received much attention from researchers and practitioners, there is ample room for improvement. First, most researchers consider robust design and tolerance design as separate research fields. Second, most research work is based on a single quality characteristic. The primary goal of this paper is to integrate a sequential robust design–tolerance design optimization procedure within a bi-objective paradigm, which, the authors believe, is the first attempt in the robust design and tolerance design literature. Models are proposed and numerical examples along with sensitivity analysis are performed for verification purposes.     

Multi-objective feasibility enhanced particle swarm optimization

This article introduces a new method entitled multi-objective feasibility enhanced partical swarm optimization (MOFEPSO), to handle highly-constrained multi-objective optimization problems. MOFEPSO, which is based on the particle swarm optimization technique, employs repositories of non-dominated and feasible positions (or solutions) to guide feasible particle flight. Unlike its counterparts, MOFEPSO does not require any feasible solutions in the initialized swarm. Additionally, objective functions are not assessed for infeasible particles. Such particles can only fly along sensitive directions, and particles are not allowed to move to a position where any previously satisfied constraints become violated. These unique features help MOFEPSO gradually increase the overall feasibility of the swarm and to finally attain the optimal solution. In this study, multi-objective versions of a classical gear-train optimization problem are also described. For the given problems, the article comparatively evaluates the performance of MOFEPSO against several popular optimization algorithms found in the literature.     

Multi-objective optimization of engineering systems using game theory and particle swarm optimization

This study proposes particle swarm optimization (PSO) based algorithms to solve multi-objective engineering optimization problems involving continuous, discrete and/or mixed design variables. The original PSO algorithm is modified to include dynamic maximum velocity function and bounce method to enhance the computational efficiency and solution accuracy. The algorithm uses a closest discrete approach (CDA) to solve optimization problems with discrete design variables. A modified game theory (MGT) approach, coupled with the modified PSO, is used to solve multi-objective optimization problems. A dynamic penalty function is used to handle constraints in the optimization problem. The methodologies proposed are illustrated by several engineering applications and the results obtained are compared with those reported in the literature.     

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．     

Reliability-based robust multi-objective optimization applied to engineering system design

ABSTRACT

Probabilistic and non-probabilistic methods have been proposed to deal with design problems under uncertainties. Reliability-based design and robust design are probabilistic strategies traditionally used for this purpose. In the present contribution, reliability-based robust design optimization (RBRDO) is formulated as a multi-objective problem considering the interaction of both approaches. The proposed methodology is based on the differential evolution algorithm associated with two strategies to deal with reliability and robustness, respectively, namely inverse reliability analysis and the effective mean concept. This multi-objective optimization problem considers the maximization of reliability and robustness coefficients as additional objective functions. The effectiveness of the methodology is illustrated by two classical test cases and a rotor-dynamics application. The results demonstrate that the proposed methodology is an alternative method to solve RBRDO problems.     

基于疲劳寿命的旋翼气动弹性多目标优化研究

基于Hamilton原理基础上推导了旋翼桨叶有限元动力学模型和疲劳寿命计算模型。以动力学特性的固有频率,自转惯量为约束,以剖面特性参数的挥、摆、扭刚度及桨叶线性密度为设计变量,进行最小质量及最大疲劳寿命的多目标优化。采用满足溢出分析的优化算法（Satisficing Trade-off Analysis）。结果在满足各约束条件下,实现旋翼桨叶质量减少7.27％,疲劳寿命循环次数由3.98 108次到4.73 108次,寿命提高了18.7%,优化效果明显。     

Multi-objective optimization design using amended particle swarm optimization and Taguchi method for a six-phase copper rotor induction motor

A multi-objective optimization design technique for a six-phase copper rotor induction motor is proposed. The amended particle swarm optimization (PSO) and Taguchi methods combined with finite element analysis are used in this design technique. The objectives in the first-stage optimization are the minimization of manufacturing cost and starting current. In the second-stage optimization, the objectives are the maximization of efficiency, power factor and output torque. The Taguchi method can optimize the machine parameters of performance characteristics in electrical discharge machining. The experimental results are further transformed into the signal-to-noise ratios and amended PSO coefficients based on amended PSO analysis with regard to multiple performance characteristics index values. The results of the optimizations showed significant reduction in terms of the use of magnets as well as improvement in the machine performance. Finally, the experimental results confirm the validity of the proposed optimization design approach.     

Dynamic reliability-based robust design optimization with time-variant probabilistic constraints

With the increasing complexity of engineering systems, ensuring high system reliability and system performance robustness throughout a product life cycle is of vital importance in practical engineering design. Dynamic reliability analysis, which is generally encountered due to time-variant system random inputs, becomes a primary challenge in reliability-based robust design optimization (RBRDO). This article presents a new approach to efficiently carry out dynamic reliability analysis for RBRDO. The key idea of the proposed approach is to convert time-variant probabilistic constraints to time-invariant ones by efficiently constructing a nested extreme response surface (NERS) and then carry out dynamic reliability analysis using NERS in an iterative RBRDO process. The NERS employs an efficient global optimization technique to identify the extreme time responses that correspond to the worst case scenario of system time-variant limit state functions. With these extreme time samples, a kriging-based time prediction model is built and used to estimate extreme responses for any given arbitrary design in the design space. An adaptive response prediction and model maturation mechanism is developed to guarantee the accuracy and efficiency of the proposed NERS approach. The NERS is integrated with RBRDO with time-variant probabilistic constraints to achieve optimum designs of engineered systems with desired reliability and performance robustness. Two case studies are used to demonstrate the efficacy of the proposed approach.     

考虑随机因素的汽车悬架参数多目标稳健优化

为了稳健地提高汽车平顺性,减少轮胎对路面破坏。以某载货汽车的四自由度悬架模型为研究对象,同时考虑制造精度的影响,以车身垂直加速度、前后轮胎动载荷的均方根值的统计均值和方差为目标函数,采用蒙特卡罗抽样法和自适应多目标遗传算法对悬架参数进行稳健优化。结果表明：优化后的悬架弹簧刚度减少而阻尼系数增大,性能有大幅度改善,与传统的优化设计相比更符合实际情况,且具有更强的抗干扰能力。