Compromise design of stochastic dynamical systems: A reliability-based approach

This paper presents a procedure for obtaining compromise designs of structural systems under stochastic excitation. In particular, an effective strategy for determining specific Pareto optimal solutions is implemented. The design goals are defined in terms of deterministic performance functions and/or performance functions involving reliability measures. The associated reliability problems are characterized by means of a large number of uncertain parameters (hundreds or thousands). The designs are obtained by formulating a compromise programming problem which is solved by a first-order interior point algorithm. The sensitivity information required by the proposed solution strategy is estimated by an approach that combines an advanced simulation technique with local approximations of some of the quantities associated with structural performance. An efficient Pareto sensitivity analysis with respect to the design variables becomes possible with the proposed formulation. Such information is used for decision making and tradeoff analysis. Numerical validations show that only a moderate number of stochastic analyses (reliability estimations) has to be performed in order to find compromise designs. Two example problems are presented to illustrate the effectiveness of the proposed approach.     

平流层飞艇的多目标优化与决策

该文建立了以平流层飞艇阻力最小、自重最轻、极限承载力最大及刚度最大为优化目标的多目标优化模型;采用强度Pareto进化算法(SPEA)进行了多目标优化设计;基于优化所得的Pareto解集,采用基于信噪比的决策方法选择满足实际需要的最终方案。结果表明:采用的SPEA算法是合理有效的,可以得到非劣解分布较均匀的Pareto曲面;通过基于信噪比的决策方法,可从非劣解集中获得满足实际要求的最稳健设计方案。     

Multi-objective genetic algorithm for solving N-version program design problem

N-version programming (NVP) is a programming approach for constructing fault tolerant software systems. Generally, an optimization model utilized in NVP selects the optimal set of versions for each module to maximize the system reliability and to constrain the total cost to remain within a given budget. In such a model, while the number of versions included in the obtained solution is generally reduced, the budget restriction may be so rigid that it may fail to find the optimal solution. In order to ameliorate this problem, this paper proposes a novel bi-objective optimization model that maximizes the system reliability and minimizes the system total cost for designing N-version software systems. When solving multi-objective optimization problem, it is crucial to find Pareto solutions. It is, however, not easy to obtain them. In this paper, we propose a novel bi-objective optimization model that obtains many Pareto solutions efficiently.We formulate the optimal design problem of NVP as a bi-objective 0–1 nonlinear integer programming problem. In order to overcome this problem, we propose a Multi-objective genetic algorithm (MOGA), which is a powerful, though time-consuming, method to solve multi-objective optimization problems. When implementing genetic algorithm (GA), the use of an appropriate genetic representation scheme is one of the most important issues to obtain good performance. We employ random-key representation in our MOGA to find many Pareto solutions spaced as evenly as possible along the Pareto frontier. To pursue improve further performance, we introduce elitism, the Pareto-insertion and the Pareto-deletion operations based on distance between Pareto solutions in the selection process.The proposed MOGA obtains many Pareto solutions along the Pareto frontier evenly. The user of the MOGA can select the best compromise solution among the candidates by controlling the balance between the system reliability and the total cost.     

Formulation and application of weight-function-based physical programming

Physical programming is effective in multi-objective optimization since it assists the designer to find the most preferred solution. Preference-function-based physical programming (PFPP) abandons the weighted-sum approach and its performance in generating Pareto solutions is susceptible to the transformation of pseudo-preferences. With the aim of integrating a weighted-sum approach into physical programming and generating well-distributed Pareto solutions, a weight-function-based physical programming (WFPP) method has been proposed. The approach forms a weight function for each normalized criterion and uses the variable weighted sum of all criteria as the aggregate objective function. Implementation for numerical and engineering design problems indicates that WFPP works as well as PFPP. The design process of generating Pareto solutions by WFPP is further presented, where the pseudo-preferences are allowed to transform in different ranges. Examples and results demonstrate that solutions generated by WFPP have better diversity performance than those of PFPP, especially when the pseudo-preferences are far from the true Pareto front.     

结构主动控制的一体化多目标优化研究

基于Pareto多目标遗传算法提出了结构主动控制系统的一体化多目标优化设计方法,对作动器位置与主动控制器进行同步优化设计.外界激励采用平稳过滤白噪声来模拟,在状态空间下通过求解Lyapunov方程,得到结构响应和主动控制力的均方值.主动控制器采用LQG控制算法来进行设计.以结构位移和加速度均方值最大值与相应无控响应均方值的最大值之比,以及所需控制力均方值之和作为多目标同步优化的目标函数.优化过程还考虑了结构与激励参数对优化结果的影响.最后以某6层平面框架有限元模型为例进行了计算机仿真分析,结果表明所提出的主动控制系统多目标一体化优化方法简单,高效,实用,具有较好的普适性.     

Multi-objective design optimization of antenna structures using sequential domain patching with automated patch size determination

In this article, a simple yet efficient and reliable technique for fully automated multi-objective design optimization of antenna structures using sequential domain patching (SDP) is discussed. The optimization procedure according to SDP is a two-step process: (i) obtaining the initial set of Pareto-optimal designs representing the best possible trade-offs between considered conflicting objectives, and (ii) Pareto set refinement for yielding the optimal designs at the high-fidelity electromagnetic (EM) simulation model level. For the sake of computational efficiency, the first step is realized at the level of a low-fidelity (coarse-discretization) EM model by sequential construction and relocation of small design space segments (patches) in order to create a path connecting the extreme Pareto front designs obtained beforehand. The second stage involves response correction techniques and local response surface approximation models constructed by reusing EM simulation data acquired in the first step. A major contribution of this work is an automated procedure for determining the patch dimensions. It allows for appropriate selection of the number of patches for each geometry variable so as to ensure reliability of the optimization process while maintaining its low cost. The importance of this procedure is demonstrated by comparing it with uniform patch dimensions.     

Optimization of aluminium sheet hot stamping process using a multi-objective stochastic approach

This article aims to investigate the means to obtain optimal hot stamping process parameters and the influence of the stochastic variability of these parameters on forming quality. A multi-objective stochastic approach, integrating response surface methodology (RSM), multi-objective genetic algorithm optimization non-dominated sorting genetic algorithm II (NSGA-II) and the Monte Carlo simulation (MCS) method is proposed in this article to achieve this goal. RSM was used to establish the relationship between the process parameters and forming quality indices. NSGA-II was utilized to obtain a Pareto frontier, which consists of a series of optimal process parameters. The MCS method was employed to study and reduce the influence of a stochastic property of these process parameters on forming quality. The results confirmed the efficiency of the proposed multi-objective stochastic approach during optimization of the hot stamping process. Robust optimal process parameters guaranteeing good forming quality were also obtained using this approach.     

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.     

Finite-element analysis and structural optimization design study for cradle seat of CNC machine tool

In order to improve the machining precision of the five-axis CNC machine tool, the cradle seat must have good structure and mechanical properties. Using the CAD/CAE-integrated design platform, static finite-element analysis and sensitivity analysis of the original cradle seat were conducted. Based on the above analysis, the weak part of the original cradle seat was discovered. A multi-objective optimization design process for the cradle seat’s weak part was carried out, and six groups of non-inferior solutions were obtained. From the six groups of optimization design schemes and the original design scheme, the optimal design scheme was selected using fuzzy matter-element method and the entropy-weight method. The cradle seat’s mass was reduced by 2.76%, and the maximum deformation was reduced by 27.38%. The impact test analysis results showed that the dynamic performance of the cradle seat, after optimization design, was greatly improved, which proves that the proposed structural optimization design method based on sensitivity analysis is reasonable and feasible.     

基于声振耦合的装载机驾驶室多目标形貌优化设计

对某装载机驾驶室及室内声腔进行建模得到声振耦合模型,通过SIMO法模态试验验证所建模型的准确性,测取悬置点激励力并进行频响分析及室内噪声预测。结合耦合模态频率和噪声曲线峰值频率确定关键优化频率,在驾驶室的最大扭矩工况下进行静力学分析,采用折衷规划法和平均频率法将驾驶室静态整体刚度和多阶关键频率归一为Euclidean距离的多目标函数,对驾驶室进行多目标形貌优化。结果表明:此优化方法在驾驶室结构优化上的应用综合提高了结构整体刚度和多阶关键固有频率,避免了单频优化时频率震荡现象,得到了优化目标的整体Pareto最优解,室内噪声总声压级降低了3.03 d B。     

硅片机器人大臂结构多目标拓扑优化设计

目的为了改善硅片机器人结构的静、动力学性能,实现结构的轻量化设计。方法引入多目标优化理论,并结合层次分析法,实现硅片机器人大臂的结构优化设计。依据设计方案分析大臂的受力情况,采用固体各向同性材料惩罚模型分别构建多目标优化数值模型,并运用优化准则法进行优化求解。引入数理统计中的层次分析法确定各子目标函数的权重比,依据折衷规划组合方法构造关于静刚度和一阶固有频率的总目标函数。结果优化结果可知,大臂的柔度从26.890 mm/N降到13.221 mm/N,一阶固有频率从556.86 Hz增加到629.90 Hz,结构质量从1.48 kg减少到0.583 kg。结论多目标优化结果表明,基于多目标优化理论对硅片机器人大臂结构的改进设计,不仅有效地提高了其静刚度特性,一阶固有频率的提高说明大臂结构能抑制振动能力的提升,还实现了大臂结构轻量化设计。层次分析法的引入为多目标优化问题中各子目标函数的重要性提供了客观的理论依据。     

Multi-objective design optimization of variable stiffness composite cylinders

The bending-induced buckling improvement in a variable stiffness (VS) composite cylinder (made by fiber steering) is studied. For such a cylinder, the effect of the variation of the direction of the load on its buckling performance of the cylinder is also examined. Compromise programming, as a multi-objective optimization method, is used to design for buckling of the VS cylinder subjected to bending load in either of the two opposite directions. Different combinations of weight factors for the structural performance in the two opposite directions were also applied to obtain the Pareto frontier as the main decision making tool for the designers in a multi-objective design problem.     

Shape optimization of preform tools in forging of aerofoil using a metamodel-assisted multi-island genetic algorithm

Preform design plays an important role in improving the material flow, mechanical properties and reducing defects for forgings with complex shapes. In this paper, a study on shape optimization of preform tools in forging of an airfoil is carried out based on a multi-island genetic algorithm combined with a metamodel technique. An optimal Latin hypercube sampling technique is employed for sampling with the expected coverage of parameter space. Finite element (FE) simulations of multistep forging processes are implemented to obtain the objective function values for evaluating the forging qualities. For facilitating the optimization process, a radial basis function surrogate model is established to predict the responses of the hot forging process to the variation of the preform tool shape. In consideration of the compromise between different optimal objectives, a set of Pareto-optimal solutions are identified by the suggested genetic algorithm to provide more selections. Finally, according to the proposed fitness function, the best solution of multi-objective optimization on the Pareto front is confirmed and the corresponding preform tool shape proves optimal performances with substantially improved forging qualities via FE validation.     

UUV耐压结构多目标优化设计

UUV(unmanned underwater vehicle,无人水下航行器)在海洋民用与军事领域具有广阔的应用前景。UUV耐压结构作为影响UUV负载能力及保障UUV航行任务安全高效执行的重要部件,其优化设计有重要意义。为了最大程度地实现减重目标,有效平衡耐压结构质量、结构强度和稳定性之间的矛盾,进而提升UUV综合性能,提出一种基于组合加权响应面法的多目标优化方法。通过试验设计得到初始采样点,利用有限元工具计算响应值并构建代理模型;然后,以折衷规划法对子目标进行归一化处理,采用组合加权法设定子目标权重系数,以进行耐压结构的多目标优化设计。以某型UUV为例,利用所提方法对其梯形肋骨耐压结构进行多目标优化设计,优化后耐压结构质量减轻了6.6%,肋骨应力下降了6.7%,同时满足稳定性要求。在此基础上,分别以质量为优化目标和以质量、结构强度和稳定性为综合优化目标,对不同肋骨形式耐压结构进行优化设计。结果表明:梯形肋骨耐压结构的综合优化效果最佳。该研究方法适用于UUV耐压结构的多目标优化,研究结果可为UUV耐压结构优化设计提供理论指导,具有实际工程意义。     

Multi-objective optimization of a triple layer piezoelectric bender with a flexible extension using genetic algorithm

This article presents an electromechanical analysis for a piezoelectric bimorph actuator with a flexible extension, which is used to increase the tip deflection. The performance measuring attributes of such an actuator are derived, and a genetic algorithm is used for multi-objective optimization. The analysis reveals that for a thick flexible extension, the length of the extension provides Pareto optimal solutions for multi-objective optimization. The analysis also shows that as the thickness of the flexible extension decreases, the Pareto optimal solutions converge to a single solution for multi-objective optimization. We have considered nonlinear deflection behavior of piezoelectric materials at high electric fields, and series and parallel electrical connections in the analysis.     

Sensitivity analysis and shape optimization of axisymmetric structures

An efficient method is developed for sensitivity analysis in shape optimization of axisymmetric structures. The technique of isoparametric mapping is used to generate the finite element mesh from a small set of master elements and master nodes. Co-ordinates of selected master nodes are used as design variables. Shape function values of master elements at derived finite element nodes obtained during the isoparametric mapping process are utilized to calculate the gradients of weight and response of the structures with respect to the design variables. Analytic formulations of the gradients are developed for sensitivity analysis of axisymmetric structures. An optimization procedure using a sequential linear programming method is applied to effectively utilize the calculated gradients. Numerical examples of optimum design of disks subject to thermo-mechanical loadings are presented.     

Application of the Kalai-Smorodinsky approach in multi-objective optimization of metal forming processes

The problem of multi-objective optimization (MOP) is approached from the theoretical background of the Game Theory, which consists in finding a compromise between two rational players of a bargaining problem. In particular, the Kalai and Smorodinsky (K-S) model offers a balanced and attractive solution resulting from cooperative players. This approach allows avoiding the computationally expensive and uncertain reconstruction of the full Pareto Frontier usually required by MOPs. The search for the K-S solution can be implemented into methodologies with useful applications in engineering MOPs where two or more functions must be minimized. This paper presents an optimization algorithm aimed at rapidly finding the K-S solution where the MOP is transformed into a succession of single objective problems (SOP). Each SOP is solved by meta-model assisted evolution strategies used in interaction with an FEM simulation software for metal forming applications. The proposed method is first tested and demonstrated with known mathematical multi-objective problems, showing its ability to find a solution lying on the Pareto Frontier, even with a largely incomplete knowledge of it. The algorithm is then applied to the FEM optimization problem of wire drawing process with one and two passes, in order to simultaneously minimize the pulling force and the material damage. The K-S solutions are compared to results previously suggested in literature using more conventional methodologies and engineering expertise. The paper shows that K-S solutions are very promising for finding quite satisfactory engineering compromises, in a very efficient manner, in metal forming applications.     

A two-stage approach for multi-objective decision making with applications to system reliability optimization

This paper proposes a two-stage approach for solving multi-objective system reliability optimization problems. In this approach, a Pareto optimal solution set is initially identified at the first stage by applying a multiple objective evolutionary algorithm (MOEA). Quite often there are a large number of Pareto optimal solutions, and it is difficult, if not impossible, to effectively choose the representative solutions for the overall problem. To overcome this challenge, an integrated multiple objective selection optimization (MOSO) method is utilized at the second stage. Specifically, a self-organizing map (SOM), with the capability of preserving the topology of the data, is applied first to classify those Pareto optimal solutions into several clusters with similar properties. Then, within each cluster, the data envelopment analysis (DEA) is performed, by comparing the relative efficiency of those solutions, to determine the final representative solutions for the overall problem. Through this sequential solution identification and pruning process, the final recommended solutions to the multi-objective system reliability optimization problem can be easily determined in a more systematic and meaningful way.     

Shape, sizing optimization and material selection based on mixed variables and genetic algorithm

In this work, we explore simultaneous designs of materials selection and structural optimization. As the material selection turns out to be a discrete process that finds the optimal distribution of materials over the design domain, it cannot be performed with common gradient-based optimization methods. In this paper, material selection is considered together with the shape and sizing optimization in a framework of multiobjective optimization of tracking the Pareto curve. The idea of mixed variables is often introduced in the case of mono-objective optimization. However, in the case of multi-objective optimization, we still face some hard key points related to the convexity and the continuity of the Pareto domain, which underline the originality of this work. In addition to the above aspect, there is a lack in the literature concerning the industrial applications that consider the mixed parameters. Continuous variables refer to structural parameters such as thickness, diameter and spring elastic constants while material ID is defined as binary design variable for each material. Both mechanical and thermal loads are considered in this work with the aim of minimizing the maximum stress and structural weight simultaneously. The efficiency of the design procedure is demonstrated through various numerical examples.     

Springback control of sheet metal forming based on the response-surface method and multi-objective genetic algorithm

An automated optimization method by the integration of the response-surface method and an optimization algorithm is presented to control springback of stamping parts. In order to minimize both objective functions of springback and thickness deformation simultaneously, a multi-objective genetic algorithm is applied to find all the optimal solutions at one run instead of transforming multi-objective functions into a single objective function. The response-surface model is employed as a fast analysis tool to surrogate the time-consuming finite-element procedure in the iterations of the multi-objective genetic algorithm. An example is studied to illustrate the application of the approach proposed, and it is concluded that the proposed method is more efficient than the traditional manual finite-element procedure and the ‘trial and error’ approach for springback controlling.