离散变量桁架结构拓扑优化设计的混合算法

将相对差商法和混沌优化结合起来,形成求解离散变量桁架结构拓扑优化设计的混合算法。利用相对差商法可以对离散变量快速寻优的特点,及混沌变量的全局遍历性,可以有效地跳出局部最优解,达到拓扑优化全局寻优的目的。通过采用和准最优解的对比及几何稳定性的判断等辅助性技术,降低了重分析次数。同时,高效的重分析方法的结合,提高了求解的效率,也避免了拓扑优化问题中求解的一些困难。算例表明,该算法对于离散变量的拓扑优化设计问题是快速有效的。     

Genetic algorithm for discrete‐sizing optimal design of trusses using the force method

In the process of discrete‐sizing optimal design of truss structures by Genetic Algorithm (GA), analysis should be performed several times. In this article, the force method is employed for the analysis. The advantage of using this method lies in the fact that the matrices corresponding to particular and complementary solutions are formed independently of the mechanical properties of members. These matrices are used several times in the process of the sequential analyses, increasing the speed of optimization. The second feature of the present method is the automatic nature of the prediction of the useful range of sections for a member from a list of profiles with a large number of cross‐sections. The third feature consists of a contraction process developed to increase the efficiency of the GA by which an optimal design for the first sub‐string associated with member cross‐sections is obtained. Improved designs are achieved in subsequent cycles by reducing the length of sub‐strings. Copyright © 2002 John Wiley & Sons, Ltd.     

A Hybrid Method for Structural System Reliability‐Based Design Optimization and its Application to Trusses

Most research studies on structural optimum design have focused on single‐objective optimization of deterministic structures, while little study has been carried out to address multi‐objective optimization of random structures. Statistical parameters and redundancy allocation problems should be considered in structural optimization. In order to address these problems, this paper presents a hybrid method for structural system reliability‐based design optimization (SRBDO) and applies it to trusses. The hybrid method integrates the concepts of the finite element method, radial basis function (RBF) neural networks, and genetic algorithms. The finite element method was used to compute structural responses under random loads. The RBF neural networks were employed to approximate structural responses for the purpose of replacing the structural limit state functions. The system reliabilities were calculated by Monte Carlo simulation method together with the trained RBF neural networks. The optimal parameters were obtained by genetic algorithms, where the system reliabilities were converted into penalty functions in order to address the constrained optimization. The hybrid method applied to trusses was demonstrated by two examples which were a typical 10‐bar truss and a steel truss girder structure. Detailed discussions and parameter analysis for the failure sequences such as web‐bucking failure and beam‐bending failure in the SRBDO were given. This hybrid method provides a new idea for SRBDO of trusses. Copyright © 2015 John Wiley & Sons, Ltd.     

Reanalysis‐based optimal design of trusses

This paper presents a structural reanalysis method and its applications in optimal design of trusses. This reanalysis technique is derived primarily on the basis of a reduced basis formulation, and it has several advantages over previous reduced basis methods. In particular, the reduced system is uncoupled by using a Gram–Schmidt orthonormalization procedure and an error measure is introduced to adaptively monitor whether a good approximate solution is achieved. The latter aspect makes this reanalysis method suitable for use in optimal design problems because the changes in design variables usually vary during a design process. Discussions are presented on the implementation of this reanalysis method using both mathematical programming and optimality criteria‐based optimization schemes. Finally, several example problems of optimal truss design are used to validate the proposed reanalysis‐based design procedure. The presented numerical results indicate that the new reanalysis technique affects very slightly the accuracy of the optimal solutions and it does speed up the design process when the system analysed is large. Copyright © 2000 John Wiley & Sons, Ltd.     

A mixed integer programming for robust truss topology optimization with stress constraints

This paper presents a mixed integer programming (MIP) formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load. A design‐dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization. For a truss with the discrete member cross‐sectional areas, it is shown that the robust topology optimization problem can be reduced to an MIP problem, which is solved globally. Numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty. Copyright © 2010 John Wiley & Sons, Ltd.     

横向力作用下的悬臂桁架结构拓扑优化

研究了应力约束下最小重量悬臂梁桁架结构的拓扑优化设计。根据Michell理论,首先用解析方法和有限元方法建立满应力类桁架连续体结构。然后选择其中部分杆件形成离散桁架作为近最优结构,并建立桁架的拓扑优化解析表达式。采用解析方法证明最优拓扑结构的腹杆中间结点在节长的四分之一位置。最后采用解析和数值方法对自由端受集中力和侧边受均布力作用的桁架进一步拓扑优化,确定了桁架的节数和每节的长度,最后得到拓扑优化桁架结构。得到的拓扑优化桁架比工程上普遍采用的45°腹杆桁架的体积少20%以上。     

A GA‐based technique for layout optimization of truss with stress and displacement constraints

Various developments of increasing complexity involved in layout optimization are discussed. The use of conventional GA in layout optimization is briefly mentioned with emphasis on its limitations and conditions imposed in finding the optimal design. The proposed new technique is applied to the benchmark example of Michell's truss for verification. The approach has also been applied to new examples of bridge truss and crane truss problems in order to demonstrate the generality and robustness for topology optimization. The approach is extended to include dual stress‐displacements constraints since many practical problems involve these two constraints simultaneously. Two‐bar and 10‐bar trusses are solved as examples for layout optimization with both stress and displacement constraints with satisfactory results. The effect of mutation on the final topology is also discussed. The major drawbacks of the ground structure approach are overcome in this proposed new method. The optimal designs obtained demonstrate the ability, robustness and generality of using the proposed new technique in layout optimization problems. Copyright © 2001 John Wiley & Sons, Ltd.     

An optimization technique using the characteristics of genetic algorithm

Optimization problems could happen often in discrete or discontinuous search space. Therefore, the traditional gradient‐based methods are not able to apply to this kind of problems. The discrete design variables are considered reasonably and the heuristic techniques are generally adopted to solve this problem, and the genetic algorithm based on stochastic search technique is one of these. The genetic algorithm method with discrete variables can be applied to structural optimization problems, such as composite laminated structures or trusses. However, the discrete optimization adopted in genetic algorithm gives rise to a troublesome task that is a mapping between each strings and discrete variables. And also, its solution quality could be restricted in some cases. In this study, a technique using the genetic algorithm characteristics is developed to utilize continuous design variables instead of discrete design variables in discontinuous solution spaces. Additionally, the proposed algorithm, which is manipulating a fitness function artificially, is applied to example problems and its results are compared with the general discrete genetic algorithm. The example problems are to optimize support positions of an unstable structure with discontinuous solution spaces.     

桁架结构智能布局优化设计

结构的布局优化由于涉及尺寸、形状和拓扑三个层次的综合设计而成为优化问题中的难点,结合桁架结构提出了一个基于多个初始基结构的布局优化方法。以智能生成的、型式多样合理的基结构代替传统模型中的单一基结构,然后从不同基结构下的拓扑优化结果中找出最优设计。在克服传统基结构法有可能限制求解空间而丢失最优解这一局限性的同时,将形状和拓扑优化设计有效分离,降低了求解的难度,并且结合拓扑变化法,实现了桁架结构从选型生成、分析计算到优化设计的一体化智能设计过程。算例表明:利用该文提出的方法进行桁架结构的最优布局设计是可靠有效的。     

An enhanced genetic algorithm for structural topology optimization

Genetic algorithms (GAs) have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures. In this paper, a versatile, robust and enhanced GA is proposed for structural topology optimization by using problem‐specific knowledge. The original discrete black‐and‐white (0–1) problem is directly solved by using a bit‐array representation method. To address the related pronounced connectivity issue effectively, the four‐neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns. A simpler version of the perimeter control approach is developed to obtain a well‐posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge‐free design. To handle the problem of representation degeneracy effectively, a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner. An efficient FEM‐based function evaluation method is developed to reduce the computational cost. A dynamic penalty method is presented for the GA to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy. With all these enhancements and appropriate choice of the GA operators, the present GA can achieve significant improvements in evolving into near‐optimum solutions and viable topologies with checkerboard free, mesh independent and hinge‐free characteristics. Numerical results show that the present GA can be more efficient and robust than the conventional GAs in solving the structural topology optimization problems of minimum compliance design, minimum weight design and optimal compliant mechanisms design. It is suggested that the present enhanced GA using problem‐specific knowledge can be a powerful global search tool for structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

基于RBF网络的步行式底盘内力计算优化方法

为了确定步行式底盘局部结构在作业时的最大受力状态,提出了一种基于RBF神经网络的两级优化模型求解方法,第一级优化模型用逐步二次规划法找到局部结构在给定位置参数下的最大受力状态,通过正交试验设计,利用RBF网络构造出局部结构界面最大受力状态与位置参数之间的非线性映射关系;第二级优化模型用GA求解RBF网络的最大值,并通过二分法不断缩小位置参数的搜索空间,提高RBF网络的逼近水平。研究表明,计算结果可为步行式底盘设计提供理论依据,该方法是解决复杂结构系统中非线性、多变量优化问题的有效手段。     

Prediction and optimization of friction welding parameters for super duplex stainless steel (UNS S32760) joints

Friction welding finds widespread industrial use as a mass production process for joining materials. Friction welding process allows welding of several materials that are extremely difficult to fusion weld. Friction welding process parameters play a significant role in making good quality joints. To produce a good quality joint it is important to set up proper welding process parameters. This can be done by employing optimization techniques. This paper presents a multi objective optimization method for optimizing the process parameters during friction welding process. The proposed method combines the response surface methodology (RSM) with an intelligent optimization algorithm, i.e. genetic algorithm (GA). Corrosion resistance and impact strength of friction welded super duplex stainless steel (SDSS) (UNS S32760) joints were investigated considering three process parameters: friction force (F), upset force (U) and burn off length (B). Mathematical models were developed and the responses were adequately predicted. Direct and interaction effects of process parameters on responses were studied by plotting graphs. Burn off length has high significance on corrosion current followed by upset force and friction force. In the case of impact strength, friction force has high significance followed by upset force and burn off length. Multi objective optimization for maximizing the impact strength and minimizing the corrosion current (maximizing corrosion resistance) was carried out using GA with the RSM model. The optimization procedure resulted in the creation of nondominated optimal points which can aid the process operator to fix the input control variables. The selection of a point from the Pareto front will always be a trade-off between the corrosion resistance and impact strength of the weld depending on the application.     

改进蚁群算法设计拉式膜片弹簧

 通过对拉式膜片弹簧载荷-变形特性的综合分析,考虑各种约束条件,提出了一种新的多目标优化设计数学模型．该模型以在摩擦片磨损极限范围内,弹簧压紧力变化的平均值最小及驾驶员作用在分离轴承装置上的分离操纵力的平均值最小为共同优化目标,使离合器后备系数稳定,离合器分离力的平均作用力较小．蚁群算法是一种新型的元启发式优化算法,该算法具有较强的发现较好解的能力,但同时也存在一些缺点,如容易出现停滞现象、收敛速度慢等．将遗传算法和蚁群算法结合起来,在蚁群算法的每一次迭代中,首先根据信息量选择解分量的初值,然后使用变异操作来确定解的值．最后,通过实例与其他优化方法的结果进行比较．结果表明,该算法有较好的收敛速度及稳定性．     

A surrogate assisted evolutionary optimization method with application to the transonic airfoil design

A multi-layer perceptron neural network (NN) method is used for efficient estimation of the expensive objective functions in the evolutionary optimization with the genetic algorithm (GA). The estimation capability of the NN is improved by dynamic retraining using the data from successive generations. In addition, the normal distribution of the training data variables is used to determine well-trained parts of the design space for the NN approximation. The efficiency of the method is demonstrated by two transonic airfoil design problems considering inviscid and viscous flow solvers. Results are compared with those of the simple GA and an alternative surrogate method. The total number of flow solver calls is reduced by about 40% using this fitness approximation technique, which in turn reduces the total computational time without influencing the convergence rate of the optimization algorithm. The accuracy of the NN estimation is considerably improved using the normal distribution approach compared with the alternative method.     

基于遗传算法的大型风机复合材料叶片根部强度优化设计

为了提高复合材料叶片承担载荷的能力, 尤其是承受最大弯矩的叶片根部的承载能力, 研究了遗传算法的优化原理并将遗传算法应用到复合材料叶片根部铺层的优化设计中。针对复合材料层压结构遗传算法优化设计中, 层压结构参数具有离散型的特点, 提出了适合复合材料层压结构遗传算法优化设计的整数编码策略, 以整数来表征层压结构参数。在分析层压结构强度的基础上, 针对结构强度优化的目标构造了可用于遗传算法的适应度函数。同时参考了一定的铺层规则, 在铺层角度限制为工程中常用的四种角度的前提下, 应用遗传算法对叶片根部进行了铺层优化设计。结果表明, 由于遗传算法特有的处理离散型问题的优势, 在叶片根部的铺层优化设计中应用遗传算法是可行和可信的。     

Optimization of injection moulding conditions with user-definable objective functions based on a genetic algorithm

This paper applies a Genetic Algorithm (GA) method to optimize injection moulding conditions, such as melt temperature, mould temperature and injection time. A GA is very suitable for moulding conditions optimization where complex patterns of local minima are possible. Existing work in the literature has limited versatility because the optimization algorithm is hard-wired with specific objective function. However, for most of the practical applications, the appropriateness of optimization objective functions depends on each specific moulding problem. The paper develops a multi-objective GA optimization strategy, where the objective functions may be defined by the designers, including using different criteria and/or weights. For parts with general quality requirements, an objective function is also recommended with some quality measuring criteria, which are either more accurately represented or cover more moulding defects than those from existing simulation-based optimization approaches. The paper also elaborates on the effective GA attributes suited to moulding conditions optimization, such as population size, crossover rate and mutation rate. A case study demonstrates the effectiveness of the proposed approach and algorithm. The optimization results are compared with those from an exhaustive search method to determine the algorithm's accuracy in finding global optimum. It is found to be favourable.     

Game formulations for structural optimization under uncertainty

We consider structural optimization (SO) under uncertainty formulated as a mathematical game between two players –– a “designer” and “nature”. The first player wants to design a structure that performs optimally, whereas the second player tries to find the worst possible conditions to impose on the structure. Several solution concepts exist for such games, including Stackelberg and Nash equilibria and Pareto optima. Pareto optimality is shown not to be a useful solution concept. Stackelberg and Nash games are, however, both of potential interest, but these concepts are hardly ever discussed in the literature on SO under uncertainty. Based on concrete examples of topology optimization of trusses and finite element-discretized continua under worst-case load uncertainty, we therefore analyze and compare the two solution concepts. In all examples, Stackelberg equilibria exist and can be found numerically, but for some cases we demonstrate nonexistence of Nash equilibria. This motivates a view of the Stackelberg solution concept as the correct one. However, we also demonstrate that existing Nash equilibria can be found using a simple so-called decomposition algorithm, which could be of interest for other instances of SO under uncertainty, where it is difficult to find a numerically efficient Stackelberg formulation.     

Optimization of geometrically non‐linear symmetric systems with coincident critical points

A new formulation is presented for optimum design of an elastic symmetric structure for specified non‐linear buckling load factor. It is shown that the method of sensitivity analysis of bifurcation load factor developed in Ohsaki and Vetani (Int. J. Numer. Methods Engng. 1996; 39 : 1707–1720) can also be applied for the case where the structure reaches coincident critical points including a limit point. Based on the method of sensitivity analysis, an algorithm is presented for finding optimum designs for specified coincident critical points. The well‐known danger of designing a structure that exhibits coincident buckling is discussed in detail. It is shown in the examples of trusses that the structural volume may be successfully reduced as a result of optimization even if the reduction of the maximum load factor due to possible asymmetric initial imperfection is considered. Copyright © 2000 John Wiley & Sons, Ltd.     

Discrete size optimization of steel trusses using a refined big bang–big crunch algorithm

This article presents a methodology that provides a method for design optimization of steel truss structures based on a refined big bang–big crunch (BB-BC) algorithm. It is shown that a standard formulation of the BB-BC algorithm occasionally falls short of producing acceptable solutions to problems from discrete size optimum design of steel trusses. A reformulation of the algorithm is proposed and implemented for design optimization of various discrete truss structures according to American Institute of Steel Construction Allowable Stress Design (AISC-ASD) specifications. Furthermore, the performance of the proposed BB-BC algorithm is compared to its standard version as well as other well-known metaheuristic techniques. The numerical results confirm the efficiency of the proposed algorithm in practical design optimization of truss structures.     

A regional genetic algorithm for the discrete optimal design of truss structures

A regional genetic algorithm (R‐GA) is used for the discrete optimal design of truss structures. The chromosomes are selected from a sub‐region centred on the continuous optimum. This approach replaces genetic rebirth as previously proposed by other authors, thereby significantly reducing computational costs. As a pure discrete method, the R‐GA method does not require heuristic arguments or approximations. This makes the algorithm highly effective when buckling and slenderness constraints with scatter in the data are introduced. A large set of numerical test examples is used to illustrate the capabilities of the method. The algorithm is shown to be effective and robust, making it suitable for the optimal design of very large truss structures. Copyright © 1999 John Wiley & Sons, Ltd.