复合材料层合板铺层设计与离散结构选型优化方法研究
|
摘要
层合板的整体性能与单层板的材料性质以及铺层方式(各单层板的厚度和铺设角度和铺层数目)相关,但直接设计单层厚度和铺设角度会由于变量多和多值性等因素引起求解困难。采用能够表征铺层方式的铺层参数为设计变量能够避免上述问题,但需要确定铺层参数的取值范围(可行域),以保证所获得的设计能够实现；根据材料和结构一体化设计思想,合理设计铺层参数和单层板纤维含量在面内的分布可获得更大的优化效益,因而需要研究铺层方式和纤维含量协同设计；可制造性要求最优设计的铺设参数只能从标准的角度系列中选取,因此需要研究离散变量的选型优化问题。
在以上研究需求的驱动下,本文研究并获得了全部铺层参数的合理近似可行域,建立了基于铺层参数的铺层方式和纤维分布协同设计的优化模型和求解方法；研究了考虑工程制造的层合板设计,提出了求解一般离散材料选型优化设计的通用方法—基于广义形函数的参数化方法(Generalized Shape Function based Parameterization);并推广GSFP方法应用到桁架、钢框架结构等的构件选型离散优化设计中,建立了相应的数学模型和求解方法。主要研究内容和成果包括：
(1)基于铺层参数的层合板优化设计模型及铺层参数近似可行域。在单层板材料性能给定的情况下,层合板的刚度矩阵是铺层参数的线性函数,同时铺层参数的可行域被证明为凸域,建立基于铺层参数的层合板优化设计模型,当铺层参数可行域已知时,能够给出问题的良好提法。然而全部铺层参数可行域仍然未知,目前已有的近似可行域也仅是必要而非充分条件。为了获得与真实可行域相近并属于真实可行域的近似域,研究了铺层参数可行域与铺层数目之间的关系,确定并验证了以最小铺层数(6层)对应的厚度和角度来表示的近似可行域是真实可行域的良好近似。基于该近似可行域建立了基于铺层参数的层合板优化设计模型。
(2)基于铺层参数的铺层方式与纤维分布协同优化设计。铺层方式和纤维含量在面内的分布对层合板的性能均有重要影响。根据材料和结构一体化设计思想,以纤维含量和铺层方式随位置的变化为设计参数,建立了基于铺层参数的铺层方式和纤维分布协同设计的优化模型；针对包含拉伸、弯曲、耦合刚度的层合板,利用获得的近似可行域,并考虑工程制造等要求,提出了基于最小铺层数的铺设方式与纤维分布协同设计的两步优化和求解策略。通过数值计算验证了所提方法的合理性与有效性。
(3)离散铺设角度的变刚度层合板优化设计及基于广义形函数的参数化方法(GSFP)。考虑工程制造的铺层方式,通常给定单层板厚度以及供各单层选择的离散角度集合,该问题是离散变量的优化问题。对该问题的求解,提出了基于广义形函数的参数化方法(GSFP),该方法可以作为求解一般离散材料选型优化的通用方法。建立了基于GSFP方法的离散铺设角度的变刚度层合板优化模型。研究表明,采用GSFP万法,离散变量可选择集中元素的排列顺序以及迫使设计变量收敛到离散值的惩罚方式,对优化结果的收敛性有很大影响。针对工程常用离散铺设角度的层合板设计问题给出了可选择集中元素的顺序规则,并提出显式惩罚策略。数值算例验证了GSFP方法的有效性。
(4) GSFP方法在其他类型离散变量选型优化问题中的推广与应用。
(a)基于GSFP方法的桁架结构截面选型离散优化设计。针对实际桁架类结构中构件截面尺寸通常从设计规范规定的标准离散尺寸中选取,建立了基于GSFP方法的桁架结构截面选型离散优化设计模型。给出了可选择集中元素的顺序规则并提出了采用显式惩罚方式来保证优化结果具有较好的收敛性。通过与已有文献中经典算例(benchmark)对比,采用GSFP方法能够获得相当或更好的目标函数值,计算效率相比基于启发式算法提高1-3个数量级,能够处理大规模的离散变量优化设计问题。
(b)基于GSFP方法的钢框架结构截面选型离散优化设计。钢框架结构中影响结构力学性能的截面几何参数之间的关系很难用数学表达式描述；此外,截面尺寸的连续化也无法实现同时对任意类型截面的连续化。为了克服该困难,建立了基于GSFP方法的钢框架截面选型离散优化模型。提出了基于刚度矩阵的Frobenius范数的可选集中元素排序规则,作为提高优化结果收敛性的一种策略。研究表明,采用GSFP方法其计算效率相比遗传算法求解能提高1-2个数量级,并且能够获得相当或者更优的目标值。
(c)基于GSFP方法的自动分组桁架截面选型离散优化设计。工程应用中通常需要对结构组件进行分组,以桁架结构为设计背景,建立了基于GSFP方法和自动分组策略的离散-连续变量混合优化设计模型,并提出了两层级的优化求解策略。研究表明该提法能够获得较好的优化结果,能够应用于实际大规模的优化设计问题。
本论文工作得到国家重点基础研究(973)计划项目(2011CB610304),国家自然科学基金项目(10902019,90816025)的资助,在此表示感谢。
The mechanical properties of laminated plate are attributed to material properties of the ply and the layup configuration (such as fiber orientation angles, ply thickness and number of plies), but it is very difficult to design laminated plates directly by ply thickness and fiber orientation angle due to the existence of large number of design variables and multiple-valued trigonometric functions. In order to avoid the above problem, lamination parameters which represent the layup configuration of laminates can be used as design variables, but the feasible region of lamination parameters needs to be known. According to the idea of material and structure simultaneous optimization method, lamination parameters and in-plane fiber distribution for a given fiber volume fraction can be designed reasonably in order to pursue optimum benefit, so simultaneous optimization scheme of layup configuration and in-plane fiber distribution needs to be studied. Manufacturability requirements of the optimal lamination parameters need select ply angles from the standard angles list, so the optimization problem with discrete variables needs to be studied.
For the demands above, the approximate feasible region of all lamination parameters is obtained, and simultaneous optimization scheme of layup configuration and in-plane fiber distribution for laminated plates is proposed. Manufacturable laminated plate design is investigated, and a generalized shape function based parameterization (GSFP) method is proposed for solving the optimization problem of material selection. The GSFP method is extended to the problem of discrete structures optimization such as truss or steel frame structures, the corresponding mathematical formulations and solving methods are studied. The main content and results are given in the following:
(1) Optimization model of laminated plate based on lamination parameters and approximate feasible region of all lamination parameters. For laminated plate with the same material properties of each ply, the stiffness properties of laminated plate are linear functions for lamination parameters, and the feasible region of lamination parameters is proved to be convex. When the feasible region is known, the optimization model of laminates based on lamination parameters could be proposed and solved well. However, an explicit formulation of the feasible region for all lamination parameters is still unavailable and the known approximate feasible region at present is necessary but not sufficient conditions. In order to obtain reasonable approximate feasible region which is very close and belong to the feasible region, the relationship between the feasible region and the number of plies is studied, it indicates that the feasible region formed by6plies could be large enough to represent the real feasible region. The optimization model of laminated plate based on lamination parameters is proposed by adopting the obtained feasible region.
(2) Simultaneous optimization design of layup configuration and in-plane fiber distribution based on lamination parameters. Layup configuration and in-plane fiber distribution exercise great influence on the properties of laminated plate. According to the idea of material and structure simultaneous optimization method, layup configuration and in-plane fiber distribution for a given fiber volume fraction are designed in every position in the design domain, simultaneous optimization scheme of layup configuration and in-plane fiber distribution based on lamination parameters for maximum stiffness design of laminated plates is proposed. For laminated plate including in-plane, coupling and out-of-plane stiffness, the obtained approximate feasible region is utilized, meanwhile, manufacturability is considered, a two-step simultaneous optimization scheme of layup configuration and in-plane fiber distribution for laminated plates design based on least number of plies is proposed. Numerical examples are presented to validate the proposed optimization scheme.
(3) Optimization design of variable stiffness laminated plate with discrete orientation angle and the method of generalized shape function based parameterization (GSFP). Ply thickness and set of discrete orientation angles are often given for considering manufacturable laminated plate, so the design is a discrete optimization problem. The method of generalized shape function based parameterization (GSFP) is proposed to solve the discrete optimization problem, and the GSFP method can be seen as a common method for solving optimization problem of discrete material selection. The optimization model of variable stiffness laminated plate with discrete orientation angle based on GSFP method is proposed. It is found that sequence of admissible list and methods of punishment have important influence on result of optimization design. Ordering rule for admissible list with engineering orientation angles and quadratic penalty approach are given. Numerical examples are presented to validate the GSFP method.
(4) Extension and application of GSFP method in other problems of discrete structures optimization.
(a) Discrete optimization for cross-section selection of truss structures based on GSFP method. For practical truss structure design, appropriate bar pieces are commonly selected from the list of cross-sections with standard sizes, so discrete optimization for cross-section selection of truss structures based on GSFP method is proposed. In order to make the result of optimization design converge well, ordering rule for admissible list is given, and quadratic penalty approach is utilized. Comparing with benchmark examples, equivalent or better objective value can be obtained. Computing efficiency can be improved by1to3order of magnitude compared to heuristic method, so large-scale optimization design of discrete variables can be processed.
(b) Discrete optimization for cross-section selection of steel frame based on GSFP method. It is difficult to describe the relationship of geometric parameters of section in the frame. Moreover, it is unable to achieve continuous simultaneously for different type sections by continuing size of section. In order to overcome the difficulties above, discrete optimization for cross-section selection of steel frame based on GSFP method is proposed. Ordering rule based on Frobenius norm of stiffness matrix for admissible list is given, which is used to make the result of optimization design converge well. It is found that computing efficiency can be improved by1or2order of magnitude compared to GA method, and equivalent or better objective value can be obtained.
(c) Discrete optimization for cross-section selection of truss structures by automatic grouping and GSFP method. Structural components are usually required to group for engineering application. For background of truss design, discrete optimization for cross-section selection of truss structures by automatic grouping and GSFP method is proposed. The optimization model includes discrete variables and continuous variables, so two-level optimization scheme is given. The good result of optimization design can be obtained, and large-scale optimization design of practical problem can be processed.
This work is supported by the National Key Basic Research Program of China (Nos.2011CB610304), the National Natural Science Foundation of China through grant Nos.(10902019,90816025). The financial contributions are greatly acknowledged.
在以上研究需求的驱动下,本文研究并获得了全部铺层参数的合理近似可行域,建立了基于铺层参数的铺层方式和纤维分布协同设计的优化模型和求解方法；研究了考虑工程制造的层合板设计,提出了求解一般离散材料选型优化设计的通用方法—基于广义形函数的参数化方法(Generalized Shape Function based Parameterization);并推广GSFP方法应用到桁架、钢框架结构等的构件选型离散优化设计中,建立了相应的数学模型和求解方法。主要研究内容和成果包括：
(1)基于铺层参数的层合板优化设计模型及铺层参数近似可行域。在单层板材料性能给定的情况下,层合板的刚度矩阵是铺层参数的线性函数,同时铺层参数的可行域被证明为凸域,建立基于铺层参数的层合板优化设计模型,当铺层参数可行域已知时,能够给出问题的良好提法。然而全部铺层参数可行域仍然未知,目前已有的近似可行域也仅是必要而非充分条件。为了获得与真实可行域相近并属于真实可行域的近似域,研究了铺层参数可行域与铺层数目之间的关系,确定并验证了以最小铺层数(6层)对应的厚度和角度来表示的近似可行域是真实可行域的良好近似。基于该近似可行域建立了基于铺层参数的层合板优化设计模型。
(2)基于铺层参数的铺层方式与纤维分布协同优化设计。铺层方式和纤维含量在面内的分布对层合板的性能均有重要影响。根据材料和结构一体化设计思想,以纤维含量和铺层方式随位置的变化为设计参数,建立了基于铺层参数的铺层方式和纤维分布协同设计的优化模型；针对包含拉伸、弯曲、耦合刚度的层合板,利用获得的近似可行域,并考虑工程制造等要求,提出了基于最小铺层数的铺设方式与纤维分布协同设计的两步优化和求解策略。通过数值计算验证了所提方法的合理性与有效性。
(3)离散铺设角度的变刚度层合板优化设计及基于广义形函数的参数化方法(GSFP)。考虑工程制造的铺层方式,通常给定单层板厚度以及供各单层选择的离散角度集合,该问题是离散变量的优化问题。对该问题的求解,提出了基于广义形函数的参数化方法(GSFP),该方法可以作为求解一般离散材料选型优化的通用方法。建立了基于GSFP方法的离散铺设角度的变刚度层合板优化模型。研究表明,采用GSFP万法,离散变量可选择集中元素的排列顺序以及迫使设计变量收敛到离散值的惩罚方式,对优化结果的收敛性有很大影响。针对工程常用离散铺设角度的层合板设计问题给出了可选择集中元素的顺序规则,并提出显式惩罚策略。数值算例验证了GSFP方法的有效性。
(4) GSFP方法在其他类型离散变量选型优化问题中的推广与应用。
(a)基于GSFP方法的桁架结构截面选型离散优化设计。针对实际桁架类结构中构件截面尺寸通常从设计规范规定的标准离散尺寸中选取,建立了基于GSFP方法的桁架结构截面选型离散优化设计模型。给出了可选择集中元素的顺序规则并提出了采用显式惩罚方式来保证优化结果具有较好的收敛性。通过与已有文献中经典算例(benchmark)对比,采用GSFP方法能够获得相当或更好的目标函数值,计算效率相比基于启发式算法提高1-3个数量级,能够处理大规模的离散变量优化设计问题。
(b)基于GSFP方法的钢框架结构截面选型离散优化设计。钢框架结构中影响结构力学性能的截面几何参数之间的关系很难用数学表达式描述；此外,截面尺寸的连续化也无法实现同时对任意类型截面的连续化。为了克服该困难,建立了基于GSFP方法的钢框架截面选型离散优化模型。提出了基于刚度矩阵的Frobenius范数的可选集中元素排序规则,作为提高优化结果收敛性的一种策略。研究表明,采用GSFP方法其计算效率相比遗传算法求解能提高1-2个数量级,并且能够获得相当或者更优的目标值。
(c)基于GSFP方法的自动分组桁架截面选型离散优化设计。工程应用中通常需要对结构组件进行分组,以桁架结构为设计背景,建立了基于GSFP方法和自动分组策略的离散-连续变量混合优化设计模型,并提出了两层级的优化求解策略。研究表明该提法能够获得较好的优化结果,能够应用于实际大规模的优化设计问题。
本论文工作得到国家重点基础研究(973)计划项目(2011CB610304),国家自然科学基金项目(10902019,90816025)的资助,在此表示感谢。
The mechanical properties of laminated plate are attributed to material properties of the ply and the layup configuration (such as fiber orientation angles, ply thickness and number of plies), but it is very difficult to design laminated plates directly by ply thickness and fiber orientation angle due to the existence of large number of design variables and multiple-valued trigonometric functions. In order to avoid the above problem, lamination parameters which represent the layup configuration of laminates can be used as design variables, but the feasible region of lamination parameters needs to be known. According to the idea of material and structure simultaneous optimization method, lamination parameters and in-plane fiber distribution for a given fiber volume fraction can be designed reasonably in order to pursue optimum benefit, so simultaneous optimization scheme of layup configuration and in-plane fiber distribution needs to be studied. Manufacturability requirements of the optimal lamination parameters need select ply angles from the standard angles list, so the optimization problem with discrete variables needs to be studied.
For the demands above, the approximate feasible region of all lamination parameters is obtained, and simultaneous optimization scheme of layup configuration and in-plane fiber distribution for laminated plates is proposed. Manufacturable laminated plate design is investigated, and a generalized shape function based parameterization (GSFP) method is proposed for solving the optimization problem of material selection. The GSFP method is extended to the problem of discrete structures optimization such as truss or steel frame structures, the corresponding mathematical formulations and solving methods are studied. The main content and results are given in the following:
(1) Optimization model of laminated plate based on lamination parameters and approximate feasible region of all lamination parameters. For laminated plate with the same material properties of each ply, the stiffness properties of laminated plate are linear functions for lamination parameters, and the feasible region of lamination parameters is proved to be convex. When the feasible region is known, the optimization model of laminates based on lamination parameters could be proposed and solved well. However, an explicit formulation of the feasible region for all lamination parameters is still unavailable and the known approximate feasible region at present is necessary but not sufficient conditions. In order to obtain reasonable approximate feasible region which is very close and belong to the feasible region, the relationship between the feasible region and the number of plies is studied, it indicates that the feasible region formed by6plies could be large enough to represent the real feasible region. The optimization model of laminated plate based on lamination parameters is proposed by adopting the obtained feasible region.
(2) Simultaneous optimization design of layup configuration and in-plane fiber distribution based on lamination parameters. Layup configuration and in-plane fiber distribution exercise great influence on the properties of laminated plate. According to the idea of material and structure simultaneous optimization method, layup configuration and in-plane fiber distribution for a given fiber volume fraction are designed in every position in the design domain, simultaneous optimization scheme of layup configuration and in-plane fiber distribution based on lamination parameters for maximum stiffness design of laminated plates is proposed. For laminated plate including in-plane, coupling and out-of-plane stiffness, the obtained approximate feasible region is utilized, meanwhile, manufacturability is considered, a two-step simultaneous optimization scheme of layup configuration and in-plane fiber distribution for laminated plates design based on least number of plies is proposed. Numerical examples are presented to validate the proposed optimization scheme.
(3) Optimization design of variable stiffness laminated plate with discrete orientation angle and the method of generalized shape function based parameterization (GSFP). Ply thickness and set of discrete orientation angles are often given for considering manufacturable laminated plate, so the design is a discrete optimization problem. The method of generalized shape function based parameterization (GSFP) is proposed to solve the discrete optimization problem, and the GSFP method can be seen as a common method for solving optimization problem of discrete material selection. The optimization model of variable stiffness laminated plate with discrete orientation angle based on GSFP method is proposed. It is found that sequence of admissible list and methods of punishment have important influence on result of optimization design. Ordering rule for admissible list with engineering orientation angles and quadratic penalty approach are given. Numerical examples are presented to validate the GSFP method.
(4) Extension and application of GSFP method in other problems of discrete structures optimization.
(a) Discrete optimization for cross-section selection of truss structures based on GSFP method. For practical truss structure design, appropriate bar pieces are commonly selected from the list of cross-sections with standard sizes, so discrete optimization for cross-section selection of truss structures based on GSFP method is proposed. In order to make the result of optimization design converge well, ordering rule for admissible list is given, and quadratic penalty approach is utilized. Comparing with benchmark examples, equivalent or better objective value can be obtained. Computing efficiency can be improved by1to3order of magnitude compared to heuristic method, so large-scale optimization design of discrete variables can be processed.
(b) Discrete optimization for cross-section selection of steel frame based on GSFP method. It is difficult to describe the relationship of geometric parameters of section in the frame. Moreover, it is unable to achieve continuous simultaneously for different type sections by continuing size of section. In order to overcome the difficulties above, discrete optimization for cross-section selection of steel frame based on GSFP method is proposed. Ordering rule based on Frobenius norm of stiffness matrix for admissible list is given, which is used to make the result of optimization design converge well. It is found that computing efficiency can be improved by1or2order of magnitude compared to GA method, and equivalent or better objective value can be obtained.
(c) Discrete optimization for cross-section selection of truss structures by automatic grouping and GSFP method. Structural components are usually required to group for engineering application. For background of truss design, discrete optimization for cross-section selection of truss structures by automatic grouping and GSFP method is proposed. The optimization model includes discrete variables and continuous variables, so two-level optimization scheme is given. The good result of optimization design can be obtained, and large-scale optimization design of practical problem can be processed.
This work is supported by the National Key Basic Research Program of China (Nos.2011CB610304), the National Natural Science Foundation of China through grant Nos.(10902019,90816025). The financial contributions are greatly acknowledged.
引文
[1]Kaw A K. Mechanics of Composite Materials [M].2rd ed. Boca Raton:CRC press,2006.
[2]Gurdal Z, Haftka R T, Hajela P. Design and optimization of laminated composite materials [M]. New York:Wiley-Interscience,1999.
[3]Khani A, Ijsselmuiden S T, Abdalla M M, et al. Design of variable stiffness panels for maximum strength using lamination parameters [J]. Composites Part B:Engineering,2010,42:546-552.
[4]Setoodeh S, Abdalla M M, Gurdal Z. Design of variable-stiffness laminates using lamination parameters [J].Composites Part B:Engineering,2006,37:301-309.
[5]Waldhart C. Analysis of tow-placed, variable-stiffness laminates [D]. Blacksburg:Faculty of the Virginia Polytechnic Institute,1996.
[6]Blom A W, Setoodeh S, Hol J, et al. Design of variable-stiffness conical shells for maximum fundamental eigenfrequency [J]. Computers & Structures,2008,86(9):870-878.
[7]Ghiasi H, Fayazbakhsh K, Pasini D, et al.Optimum stacking sequence design of composite materials Part Ⅱ:Variable stiffness design.Composite Structures [J].2010,93(1):1-13.
[8]程耿东.工程结构优化设计基础[M].北京：水利电力出版社,1984.
[9]Assie A E, Kabeel A M, Mahmoud F F. Effect of loading and lamination parameters on the optimum design of laminated plates [J]. Journal of Mechanical Science and Technology,2011, 25(5):1149-1158.
[10]Hammer V B, Bendsoe M P, Lipton R, et al. Parametrization in laminate design for optimal compliance [J]. International Journal of Solids and Structures,1997,34(4):415-434.
[11]Fukunaga H, Sekine H. Stiffness design method of symmetric laminates using lamination parameters [J]. AIAA Journal,1992,30:2791-2793.
[12]Weise T. Global Optimization Algorithms:theory and Application [M/OL]. URLs: http://www.lania.mx/-ccoello/EMOO/,2007.
[13]Fox R L. Optimization methods for engineering design [M]. Addison-Wesley Publishing Compony, 1971.
[14]Haftka R T, Giirdal Z. Elements of structural optimization [M]. Netherlands:Kluwer Academic Publishers,1992.
[15]Bonnans J F. Numerical optimization:theoretical and practical aspects [M].2rd ed. New York: Springer-Verlag Inc,2006.
[16]Bendsoe M P. Optimization of structural topology, shape and material [M]. Berlin Heidelberg: Springer-Verlag Inc,1995.
[17]Coelho P G, Fernandes P R, Guedes J M, et al. A hierarchical model for concurrent material and topology optimisation of three-dimensional structures [J]. Structural and Multidisciplinary Optimization,2008,35(2):107-115.
[18]Rodrigues H, Guedes J M, Bendsoe M P. Hierarchical optimization of material and structure [J]. Structural and Multidisciplinary Optimization,2002,24(1):1-10.
[19]Zowe J, Ko Vara M, Bends E M P. Free material optimization via mathematical programming [J]. Mathematical Programming,1997,79(1):445-466.
[20]刘岭,阎军,程耿东.考虑均一微结构的结构/材料两级协同优化[J].计算力学学报,2008,25：29-34.
[21]Kocvara M, Stingl M, Zowe J. Free material optimization:recent progress [J]. Optimization:A Journal of Mathematical Programming and Operations Research,2008,57(1):79-100.
[22]Bendsoe M P, Diaz A R, Lipton R, et al. Optimal design of material properties and material distribution for multiple loading conditions [J]. International Journal for Numerical Methods in Engineering,1995,38(7):1149-1170.
[23]Zhang W H, Sun S P. Scale-related topology optimization of cellular materials and structures [J]. International Journal for Numerical Methods in Engineering,2006,68(9):993-1011.
[24]张卫红,孙士平.多孔材料/结构尺度关联的一体化拓扑优化技术[J].力学学报,2006,38：522-529.
[25]阎军,刘岭,刘晓峰,等.考虑尺寸效应的模块化结构两层级优化设计[J].力学学报,2010,2：268-274.
[26]Liu L, Yan J, Cheng G. Optimum structure with homogeneous optimum truss-like material [J]. Computers & Structures,2008,86(13):1417-1425.
[27]Bendsoe M P. Optimal shape design as a material distribution problem [J]. Structural and Multidisciplinary Optimization,1989,1(4):193-202.
[28]Bendsoe M P, Sigmund O. Material interpolation schemes in topology optimization [J]. Archive of Applied Mechanics,1999,69(9):635-654.
[29]Sigmund O. Systematic design of electrothermomechanical microactuators using topology optimization [J]. Modelling and Simulation of Microsystems, Semiconductors, Sensors and Actuators,1998,350-355.
[30]Sigmund O. Tailoring materials for specific needs [J]. Journal of Intelligent Material Systems and Structures,1994,5(6):736-742.
[31]Sigmund O. Tailoring materials with prescribed elastic properties [J]. Mechanics of Materials,1995, 20(4):351-368.
[32]Sigmund O, Torquato S. Composites with extremal thermal expansion coefficients [J]. Applied Physics Letters,1996,69(21):3203-3205.
[33]Jensen J S, Sigmund O. Systematic design of photonic crystal structures using topology optimization:Low-loss waveguide bends [J]. Applied physics letters,2004,84(12):2022-2024.
[34]Jonsmann J, Sigmund O, Bouwstra S. Compliant electro-thermal microactuators [C]. IEEE International MEMS 99 Conference, Twelfth IEEE International Conference on Micro Electro Mechanical Systems,1999.
[35]Gersborg-Hansen A, Sigmund O, Haber R B. Topology optimization of channel flow problems [J]. Structural and Multidisciplinary Optimization,2005,30(3):181-192.
[36]Madsen J I. Design optimization of internal flow devices [D]. Denmark:Alborg University,1998.
[37]Sigmund O. Design of multiphysics actuators using topology optimization-Part Ⅱ:Two-material structures [J]. Computer Methods in Applied Mechanics and Engineering,2001,190(49): 6605-6627.
[38]Luo Q, Tong L. An adhesively laminated plate element for PZT smart plates [J]. Computational Mechanics,2004,34(3):224-236.
[39]Wang S Y, Tai K, Quek S T. Topology optimization of piezoelectric sensors/actuators for torsional vibration control of composite plates [J]. Smart Materials and Structures,2006,15:253-269.
[40]Christensen S T, Sorokin S V, Olhoff N. On analysis and optimization in structural acoustics-Part I: Problem formulation and solution techniques [J]. Structural and Multidisciplinary Optimization, 1998,16(2):83-95.
[41]Christensen S T, Sorokin S V, Olhoff N. On analysis and optimization in structural acoustics-Part II: Exemplifications for axisymmetric structures [J]. Structural and Multidisciplinary Optimization, 1998,16(2):96-107.
[42]Diihring M B, Jensen J S, Sigmund O. Acoustic design by topology optimization [J]. Journal of Sound and Vibration,2008,317(3):557-575.
[43]Jones R M. Mechanics of Composite Materials [M].2rd ed. Philadelphia:Taylor & Francis Group, 1999.
[44]Vasiliev V V, Morozov E V. Advanced mechanics of composite materials [M]. Oxford:Elsevier Science,2007.
[45]Bruyneel M, Fleury C. Composite structures optimization using sequential convex programming [J]. Advances in Engineering Software,2002,33(7):697-711.
[46]Bruyneel M, Duysinx P, Fleury C. Composite structures design for strength and stiffness with respect to ply thickness and/or fibers orientation [C]. Dalian:Proceedings of the 4th World Congress of Structural and Multidisciplinary Optimization WCSMO4,2001.
[47]Bruyneel M, Colson B, Jetteur P, et al. Recent progress in the optimal design of composite structures:industrial solution procedures on case studies [J]. International Journal for Simulation and Multidisciplinary Design Optimization,2008,2(4):283-288.
[48]Bruyneel M, Duysinx P, Fleury C. A family of MMA approximations for structural optimization [J]. Structural and Multidisciplinary Optimization,2002,24(4):263-276.
[49]Bruyneel M, Colson B, Remouchamps A. Discussion on some convergence problems in buckling optimization [J]. Structural and Multidisciplinary Optimization,2008,35(2):181-186.
[50]Svanberg K. The method of moving asymptotes-a new method for structural optimization [J]. International Journal for Numerical Methods in Engineering,1987,24:359-373.
[51]Fukunaga H, Vanderplaats G N. Strength optimization of laminated composites with respect to layer thickness and/or layer orientation angle [J]. Computers & Structures,1991,40(6):1429-1439.
[52]Miki M. Material design of composite laminates with required in-plane elastic properties [C]. Tokyo:Progress in Science and Engineering of Composites, ICCM-IV,1981, pp:1725-1731.
[53]Diaconu C G, Sato M, Sekine H. Feasible region in general design space of lamination parameters for laminated composites [J].AIAA Journal,2002,40(3):559-565.
[54]Grenestedt J L, Gudmundson P. Layup optimization of composite material structures [C].IUTAM Symposium on Optimal Design with Advanced Materials,1993,311-336.
[55]Fukunaga H, Hirano Y. Stability optimization of laminated composite plates under in-plane loads [C]. Tokyo:Proceedings of the Fourth International Conference on Composite Materials,1982, 565-572.
[56]Ringertz U T. On finding the optimal distribution of material properties [J]. Structural and Multidisciplinary Optimization,1993,5(4):265-267.
[57]Autio M. Coupled thermal-structural problems in the optimization of laminated plates [J]. Structural and Multidisciplinary Optimization,1998,15(1):49-56.
[58]Grediac M. A procedure for designing laminated plates with required stiffness properties, Application to thin quasi-isotropic quasi-homogeneous uncoupled laminates [J]. Journal of Composite Materials,1999,33(20):1939-1956.
[59]Autio M. Determining the real lay-up of a laminate corresponding to optimal lamination parameters by genetic search [J]. Structural and Multidisciplinary Optimization,2000,20(4):301-310.
[60]Autio M. Optimization of coupled thermal-structural problems of laminated plates with lamination parameters [J]. Structural and Multidisciplinary Optimization,2001,21(1):40-51.
[61]Topal U, Uzman U. Thermal buckling load optimization of laminated composite plates [J]. Thin-Walled Structures,2008,46(6):667-675.
[62]Topal U, Uzman U. Thermal buckling load optimization of laminated skew plates [J]. Materials & Design,2009 30(7):2569-2575.
[63]Abdalla M M, Setoodeh S, Gurdal Z. Design of variable stiffness composite panels for maximum fundamental frequency using lamination parameters [J]. Composite Structures,2007,81(2): 283-291.
[64]Thuwis G, De Breuker R, Abdalla M, et al. Aeroelastic tailoring using lamination parameters [J]. Structural and Multidisciplinary Optimization,2010,41(4):637-646.
[65]Diaconu C G, Sato M, Sekine H. Buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads using lamination parameters [J]. Composite Structures,2002,58(4):423-433.
[66]Setoodeh S, Abdalla M M, Gurdal Z. Approximate feasible regions for lamination parameters [C]. Portsmouth:11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference,2006,
[67]Bradford B C, David D P, Hannu H. The quickhull algorithm for convex hulls [J]. ACM Transactions on Mathematical Software,1996,22(4):469-483.
[68]Liu B, Haftka R T, Trompette P. Maximization of buckling loads of composite panels using flexural lamination parameters [J]. Structural and Multidisciplinary Optimization,2004,26(1): 28-36.
[69]Diaconu C G, Sekine H. Layup optimization for buckling of laminated composite shells with restricted layer angles [J]. AlAA Journal,2004,42(10):2153-2166.
[70]Foldager J, Hansen J S, Olhoff N. A general approach forcing convexity of ply angle optimization in composite laminates [J]. Structural and Multidisciplinary Optimization,1998,16(2):201-211.
[71]Dawe D J, Wang S. Spline finite strip analysis of the buckling and vibration of rectangular composite laminated plates [J]. International Journal of Mechanical Sciences,1995,37(6):645-667.
[72]Mota Soares C M, Mota Soares C A, Mateus H C. A model for the optimum design of thin laminated plate-shell structures for static, dynamic and buckling behavior [J]. Composite Structures, 1995,32(1-4):69-79.
[73]Singha M K, Ramachandra L S, Bandyopadhyay J N. Optimum design of laminated composite plates for maximum thermal buckling loads [J]. Journal of Composite Materials,2000,34(23): 1982-1997.
[74]Lanzi L, Giavotto V. Post-buckling optimization of composite stiffened panels:computations and experiments [J]. Composite Structures,2006,73(2):208-220.
[75]Farshi B, Rabiei R. Optimum design of composite laminates for frequency constraints [J]. Composite Structures,2007,81(4):587-597.
[76]Topal U, Uzman U. Frequency optimization of laminated folded composite plates [J]. Materials & Design,2009,30(3):494-501.
[77]Lopatin A V, Morozov E V. Fundamental frequency of an orthotropic rectangular plate with an internal centre point support [J]. Composite Structures,2011,93(10):2487-2495.
[78]Qatu M S, Sullivan R W, Wang W. Recent research advances on the dynamic analysis of composite shells:2000-2009 [J]. Composite Structures,2010,93(1):14-31.
[79]Setoodeh S, Abdalla M M, Usselmuiden S T, et al. Design of variable-stiffness composite panels for maximum buckling load [J]. Composite Structures,2009,87(1):109-117.
[80]Lopes C S, Gurdal Z, Camanho P P. Variable-stiffness composite panels:buckling and first-ply failure improvements over straight-fibre laminates [J]. Computers & Structures,2008,86(9): 897-907.
[81]Correia V M F, Soares C M M, Soares C A M. Design sensitivity analysis and optimal design of composite structures using higher order discrete models [J]. Engineering Optimization,1997, 29(1-4):85-111.
[82]Stegmann J, Lund E. Discrete material optimization of general composite shell structures [J]. International Journal for Numerical Methods in Engineering,2005,62(14):2009-2027.
[83]Lund E, Stegmann J. On structural optimization of composite shell structures using a discrete constitutive parametrization [J]. Wind Energy,2005,8(1):109-124.
[84]唐文艳,顾元宪,赵国忠.复合材料层合板铺层顺序优化遗传算法[J].大连理工大学学报,2004,44(002)：186-189.
[85]Upadhyay A, Kalyanaraman V. Optimum design of fibre composite stiffened panels using Genetic Algorithms [J]. Engineering Optimization,2000,33(2):201-220.
[86]穆朋刚,赵美英,陈鹏飞,等.基于蚁群算法的复合材料层合板的铺层顺序优化[J].玻璃钢／复合材料,2007,11(006)：14-17.
[87]Sigmund O. On the usefulness of non-gradient approaches in topology optimization [J]. Structural and Multidisciplinary Optimization,2011,43(5):589-596.
[88]Hirano Y. Optimum design of laminated plates under shear [J]. Journal of Composite Materials, 1979,13(4):329-334.
[89]Tauchert T R, Adibhatla S. Design of laminated plates for maximum stiffness [J]. Journal of Composite Materials,1984,18(1):58-69.
[90]Cheng K. Sensitivity analysis and a mixed approach to the optimization of symmetric layered composite plates [J]. Engineering Optimization,1986,9(4):233-247.
[91]Mahadevan S, Liu X. Probabilistic optimum design of composite laminates [J]. Journal of Composite Materials,1998,32(1):68-82.
[92]Fleury C, Schmit L A. Dual methods and approximation concepts in structural synthesis [R].USA: NASA Contractor Report,1980.
[93]Fleury C. First and second order convex approximation strategies in structural optimization [J]. Structural and Multidisciplinary Optimization,1989,1(1):3-10.
[94]Fleury C. CONLIN:an efficient dual optimizer based on convex approximation concepts [J]. Structural and Multidisciplinary Optimization,1989,1(2):81-89.
[95]Svanberg K.The method of moving asymptotes-a new method for structural optimization [J]. International Journal for Numerical Methods in Engineering,1987,24(2):359-373.
[96]Zillober C.A globally convergent version of the method of moving asymptotes [J]. Structural and Multidisciplinary Optimization,1993,6(3):166-174.
[97]Chandra R, Singh S P, Gupta K. Damping studies in fiber-reinforced composites-a review [J]. Composite Structures,1999,46(1):41-51.
[98]Sobieszczanski-Sobieski J, Haftka R T. Multidisciplinary aerospace design optimization:survey of recent developments [J]. Structural and Multidisciplinary Optimization,1997,14(1):1-23.
[99]Todoroki A, Haftka R T. Stacking sequence optimization by a genetic algorithm with a new recessive gene like repair strategy [J]. Composites Part B:Engineering,1998,29(3):277-285.
[100]唐文艳.结构优化中的遗传算法研究和应用[D].大连：大连理工大学,2002.
[101]晏飞,李为吉.基于自适应遗传算法的复合材料层合板铺层顺序优化设计[J].西北工业大学学报,2001,19(1)：156-159.
[102]任茶仙,张铎.复合材料层合结构铺层顺序优化设计的免疫遗传算法[J].强度与环境,2007,34(2)：44-50.
[103]Aymerich F, Serra M. Optimization of laminate stacking sequence for maximum buckling load using the ant colony optimization (ACO) metaheuristic [J]. Composites Part A:Applied Science and Manufacturing,2008,39(2):262-272.
[104]Sigmund O, Torquato S. Design of materials with extreme thermal expansion using a three-phase topology optimization method [J]. Journal of the Mechanics and Physics of Solids,1997,45(6): 1037-1067.
[105]Duysinx P, Gao T, Zhang W, et al. New developments for an efficient solution of the discrete material topology optimization of composite structures [C]. Rioskilde:32nd RISO International Symposium on Material Science,2011.
[106]Hvejsel C F, Lund E. Material interpolation schemes for unified topology and multi-material optimization [J]. Structural and Multidisciplinary Optimization,2011,43(6):811-825.
[107]Bruyneel M. SFP——a new parameterization based on shape functions for optimal material selection: application to conventional composite plies [J]. Structural and Multidisciplinary Optimization, 2011,43(1):17-27.
[108]Gao T, Zhang W, Duysinx P. A bi-value coding parameterization scheme for the discrete optimal orientation design of the composite laminate [J]. International Journal for Numerical Methods in Engineering,2012, DOI:10.1002/nme.4270.
[109]Arora J S, Huang M W. Hsieh C C. Methods for optimization of nonlinear problems with discrete variables:a review [J]. Structural and Multidisciplinary Optimization,1994,8(2):69-85.
[110]Arora J S, Huang M W. Discrete structural optimization with commercially available sections [J]. Struct. Mech. Earthquake Eng,1996,13(2):93-110.
[111]孙焕纯,柴山,王跃方.离散变量结构优化设计[M].大连：大连理工大学出版社,]995.
[112]孙焕纯,柴山,王跃方.离散变量结构优化设计的发展,现状及展望[J].力学与实践,1997,19(4)：7-11.
[113]Arora J S.Introduction to optimum design.New York:Academic Press,2004.
[114]Arora J S.Optimization of structural and mechanical systems [M]. London:World Scientific Pub Co Inc,2007.
[115]Land A H, Doig A G. An automatic method of solving discrete programming problems [J]. Econometrica:Journal of the Econometric Society,1960,28(3):497-520.
[116]Dakin R J. A tree-search algorithm for mixed integer programming problems [J]. The Computer Journal,1965,8(3):250-255.
[117]Ringertz U. On methods for discrete structural optimization [J]. Engineering Optimization,1988, 13(1):47-64.
[118]Ohsaki M. Optimization of finite dimensional structures [M]. Taylor & Francis Group:CRC Press, 2010.
[119]Stender J. Parallel genetic algorithms:theory and applications [M]. Netherlands:Ios Press Inc, 1993.
[120]Stolpe M, Canh N N, Stainko R. The Branch and Cut Method in the PLATO-N Project [C]. Rio de Janeiro:EngOpt 2008:International Conference on Engineering Optimization,2008.
[121]Shin D, Gurdal Z, Griffin O H. A penalty approach for nonlinear optimization with discrete design variables [J]. Engineering Optimization,1989,16(1):29-42.
[122]Yu X, Zhang S, Johnson E. A discrete post-processing method for structural optimization [J]. Engineering with Computers,2003,19(2):213-220.
[123]Li Y, Tan T, Li X. A gradient-based approach for discrete optimum design [J]. Structural and Multidisciplinary Optimization,2010,41(6):881-892.
[124]Walls R, Elvin A. An algorithm for grouping members in a structure [J]. Engineering Structures, 2010,32(6):1760-1768.
[125]Barbosa H J C, Lemonge A C C. A genetic algorithm encoding for a class of cardinality constraints New York:Proceedings of the 2005 Conference on Genetic and Evolutionary Computation,2005.
[126]Kaveh A, Zolghadr A. A multi-set charged system search for truss optimization with variables of different natures element grouping [J]. Civil Engineering,2011,55(2):87-98.
[127]Biedermann J D, Grierson D E. A generic model for building design [J]. Engineering with Computers,1995,11(3):173-184.
[128]Barbosa H J C, Lemonge A C C, Borges C C H. A genetic algorithm encoding for cardinality constraints and automatic variable linking in structural optimization [J]. Engineering Structures, 2008,30(12):3708-3723.
[129]Togan V, Daloglu A T. Optimization of 3d trusses with adaptive approach in genetic algorithms [J]. Engineering Structures,2006,28(7):1019-1027.
[130]Togan V, Daloglu A T. An improved genetic algorithm with initial population strategy and self-adaptive member grouping [J]. Computers & Structures,2008,86(11-12):1204-1218.
[131]Lemonge A C C, Barbosa H J C, Coutinho A L G A, et al. Multiple cardinality constraints and automatic member grouping in the optimal design of steel framed structures [J]. Engineering Structures,2011,33(2):433-444.
[132]Lemonge A C C, Silva M M, Barbosa H J C. Design optimization of geometrically nonlinear truss structures considering cardinality constraints [C]. Juiz de Fora:2011 IEEE Congress on Evolutionary Computation (CEC),2011.
[133]Shea K, Cagan J, Fenves S J. A shape annealing approach to optimal truss design with dynamic grouping of members [J]. Journal of Mechanical Design,1997,119(3):388-394.
[134]刘晓峰,阎军,程耿东.采用自动分组遗传算法的频率约束下桁架拓扑优化[J].计算力学学报,2011,28(1)：1-7.
[135]刘晓峰.自动分组遗传算法的改进及在结构工程中的应用[D].大连：大连理工大学,2011.
[136]刘书田,林哲祺.考虑作动器联接方式的结构形状控制优化[J].工程力学,2009,26(2)：227-241.
[137]Liu S, Lin Z. Integrated design optimization of voltage channel distribution and control voltages for tracking the dynamic shapes of smart plates [J]. Smart Materials and Structures,2010,19(12): 1250-1263.
[138]Durand L P. Composite materials research progress [M]. New York:Nova Science Pub Inc,2008.
[139]Miki M. Material design of composite laminates with required in-plane elastic properties [C]. Tokyo:Progress in Science and Engineering of Composites, ICCM-IV,1981, pp:1725-1731.
[140]Miki M, Sugiyama Y. Optimum design of laminated composite plates using lamination parameters [J]. AIAA Journal,1993,31(5):921-922.
[141]Bloomfield M W, Diaconu C G, Weaver P M. On feasible regions of lamination parameters for lay-up optimization of laminated composites [J]. Proceedings of the Royal Society A:Mathematical, Physical and Engineering Science,2008,456(2104):1123-1143.
[142]Storn R, Price K. Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces [J]. Journal of Global Optimization,1997,11(4):341-359.
[143]Brest J, Greiner S, Boskovic B, et al. Self-adapting control parameters in differential evolution:A comparative study on numerical benchmark problems [J]. IEEE Transactions on Evolutionary Computation,2006,10(6):646-657.
[144]Brest J, Bo Kovi B, Greiner S, et al. Performance comparison of self-adaptive and adaptive differential evolution algorithms [J]. Soft Computing-A Fusion of Foundations, Methodologies and Applications,2007,11(7):617-629.
[145]方开泰.均匀设计与均匀设计表[M].北京：科学出版社,1994.
[146]Fang K. T, Lin D K J, Winker P, et al. Uniform design:theory and application [J].Technometrics, 2000,42(3):237-248.
[147]Sigmund O. Material interpolation schemes in topology optimization [J]. Archive of Applied Mechanics,1999,69(9/10):635-654.
[148]Schmit L A, Fleury C. Discrete-continuous variable structural synthesis using dual methods [J]. AIAA Journal,1980,18:1515-1524.
[149]Christensen P W, Klarbring A. An introduction to structural optimization [M].Springer Verlag, 2008.
[150]晏飞,戴德海,朱智春.基于遗传算法的纤维增强复合材料层合板刚度设计方法[J].强度与环境,2004,31(002)：7-12.
[151]Gibiansky L V, Sigmund O. Multiphase composites with extremal bulk modulus [J]. Journal of the Mechanics and Physics of Solids,2000,48(3):461-498.
[152]Svanberg K. A class of globally convergent optimization methods based on conservative convex separable approximations [J]. SIAM Journal on Optimization,2002,12(2):555-573.
[153]Hvejsel C F, Lund E, Stolpe M. Optimization strategies for discrete multi-material stiffness optimization [J]. Structural and Multidisciplinary Optimization,2011,44:149-163.
[154]Thanedar P B. Survey of discrete variable optimization for structural design [J]. Journal of Structural Engineering,1995,121(2):301-306.
[155]Rajeev S, Krishnamoorthy C S. Discrete optimization of structures using genetic algorithms [J]. Journal of Structural Engineering,1992,118(5):1233-1250.
[156]Lee K S, Geem Z W. A new structural optimization method based on the harmony search algorithm [J]. Computers & Structures,2004,82(9):781-798.
[157]Camp C V, Bichon B J. Design of space trusses using ant colony optimization [J]. Journal of Structural Engineering,2004,130(5):741-751.
[158]Kripka M. Discrete optimization of trusses by simulated annealing [J]. Journal of the Brazilian Society of Mechanical Sciences and Engineering,2004,26(2):170-173.
[159]Kaveh A, Talatahari S. A hybrid particle swarm and ant colony optimization for design of truss structures [J]. Asian Journal of Civil Engineering (building and housing),2008,9(4):329-348.
[160]Sonmez M. Discrete optimum design of truss structures using artificial bee colony algorithm [J]. Structural and Multidisciplinary Optimization,2011,43:85-97.
[161]Wu S J, Chow P T. Steady-state genetic algorithms for discrete optimization of trusses [J]. Computers & Structures,1995,56(6):979-991.
[162]Erbatur F, Hasancebi O, Tutuncu I, et al. Optimal design of planar and space structures with genetic algorithms [J]. Computers & Structures,2000,75(2):209-224.
[163]He S, Wu Q H, Wen J Y, et al. A particle swarm optimizer with passive congregation [J]. Biosystems,2004,78(1-3):135-147.
[164]Togan V, Daloglu A T. Optimization of 3d trusses with adaptive approach in genetic algorithms [J], Engineering Structures,2006,28(7):1019-1027.
[165]Isaacs A, Ray T, Smith W. An Efficient Hybrid Algorithm for Optimization of Discrete Structures [J]. Simulated Evolution and Learning,2008,5361:625-634.
[166]Togan V, Daloglu A T. An improved genetic algorithm with initial population strategy and self-adaptive member grouping [J]. Computers & Structures,2008,86(11-12):1204-1218.
[167]Sandgren E. Nonlinear integer and discrete programming in mechanical design optimization [J]. Journal of Mechanical Design,1990,11(2):223-231.
[168]Gandomi A, Yang X S. Benchmark Problems in Structural Optimization [J]. Computational Optimization, Methods and Algorithms,2011,356:259-281.
[169]Cai J, THIERAUF G. Discrete optimization of structures using an improved penalty function method [J]. Engineering Optimization,1993,21(4):293-306.
[170]Coello C A C. Discrete optimization of trusses using genetic algorithms [C]. Spanish:The Sixth International Conference on Computation,1994.
[171]Camp C, Pezeshk S, Cao G. Optimized design of two-dimensional structures using a genetic algorithm [J]. Journal of Structural Engineering,1998,124(5):551-559.
[172]Tong W H, Liu G R. An optimization procedure for truss structures with discrete design variables and dynamic constraints [J]. Computers & Structures,2001,79(2):155-162.
[173]Nanakorn P, Meesomklin K. An adaptive penalty function in genetic algorithms for structural design optimization [J]. Computers & Structures,2001,79(29):2527-2539.
[174]Turkkan N. Discrete optimization of structures using a floating-point genetic algorithm [C]. Annual Conference of the Canadian Society for Civil Engineering,2003.
[175]De Sousa F L, Takahashi W K. Discrete optimal design of trusses by generalized extremal optimization [C]. Rio de Janeiro:6th World Congresses of Structural and Multidisciplinary Optimization,2005.
[176]Ming-Zhu D. An improved Templeman's algorithm for the optimum design of trusses with discrete member sizes [J]. Engineering Optimization,1986,9(4):303-312.
[177]Lee K S, Geem Z W, Lee S, et al. The harmony search heuristic algorithm for discrete structural optimization [J]. Engineering Optimization,2005,37(7):663-684.
[178]Li L J, Huang Z B, Liu F. A heuristic particle swarm optimization method for truss structures with discrete variables [J].Computers & Structures,2009,87(7-8):435-443.
[179]Dede T, Bekiroglu S, Ayvaz Y. Weight minimization of trusses with genetic algorithm [J]. Applied Soft Computing,2011,11(2):2565-2575.
[180]Ghasemi M R, Hinton E, Wood R D. Optimization of trusses using genetic algorithms for discrete and continuous variables [J]. Engineering Computations,1999,16(3):272-303.
[181]Cai J, Thierauf G. Discrete structural optimization using evolution strategies [J]. In:Neural Networks and Combinatorial Optimization in Civil and Structural Engineering, Edinburg. Civil-Comp,95-100.
[182]Saka M P. Optimum design of steel frames with tapered members [J]. Computers & Structures, 1997,63(4):797-811.
[183]Huang M W, Arora J S. Optimal design of steel structures using standard sections [J]. Structural and Multidisciplinary Optimization,1997,14(1):24-35.
[184]Torregosa R F, Kanok-Nukulchai W. Weight optimization of steel frames using genetic algorithm [J]. Advances in Structural Engineering,2002,5(2):99-111.
[185]GB50017-2003,刚结构设计规范[S].北京：中国计划出版社,2003.
[186]钱令希.工程结构优化设计[M].水利电力出版社,1983.
[187]Beckers M. Topology optimization using a dual method with discrete variables [J]. Structural and Multidisciplinary Optimization,1999,17(1):14-24.
[188]Olsen G R, Vanderplaats G N. Method for nonlinear optimization with discrete design variables [J]. AIAA Journal,1989,27(11):1584-1589.
[189]Teller E, Metropolis N, Rosenbluth A. Equation of state calculations by fast computing machines [J]. Journal of Chemical Physics,1953,21(13):1087-1092.
[2]Gurdal Z, Haftka R T, Hajela P. Design and optimization of laminated composite materials [M]. New York:Wiley-Interscience,1999.
[3]Khani A, Ijsselmuiden S T, Abdalla M M, et al. Design of variable stiffness panels for maximum strength using lamination parameters [J]. Composites Part B:Engineering,2010,42:546-552.
[4]Setoodeh S, Abdalla M M, Gurdal Z. Design of variable-stiffness laminates using lamination parameters [J].Composites Part B:Engineering,2006,37:301-309.
[5]Waldhart C. Analysis of tow-placed, variable-stiffness laminates [D]. Blacksburg:Faculty of the Virginia Polytechnic Institute,1996.
[6]Blom A W, Setoodeh S, Hol J, et al. Design of variable-stiffness conical shells for maximum fundamental eigenfrequency [J]. Computers & Structures,2008,86(9):870-878.
[7]Ghiasi H, Fayazbakhsh K, Pasini D, et al.Optimum stacking sequence design of composite materials Part Ⅱ:Variable stiffness design.Composite Structures [J].2010,93(1):1-13.
[8]程耿东.工程结构优化设计基础[M].北京：水利电力出版社,1984.
[9]Assie A E, Kabeel A M, Mahmoud F F. Effect of loading and lamination parameters on the optimum design of laminated plates [J]. Journal of Mechanical Science and Technology,2011, 25(5):1149-1158.
[10]Hammer V B, Bendsoe M P, Lipton R, et al. Parametrization in laminate design for optimal compliance [J]. International Journal of Solids and Structures,1997,34(4):415-434.
[11]Fukunaga H, Sekine H. Stiffness design method of symmetric laminates using lamination parameters [J]. AIAA Journal,1992,30:2791-2793.
[12]Weise T. Global Optimization Algorithms:theory and Application [M/OL]. URLs: http://www.lania.mx/-ccoello/EMOO/,2007.
[13]Fox R L. Optimization methods for engineering design [M]. Addison-Wesley Publishing Compony, 1971.
[14]Haftka R T, Giirdal Z. Elements of structural optimization [M]. Netherlands:Kluwer Academic Publishers,1992.
[15]Bonnans J F. Numerical optimization:theoretical and practical aspects [M].2rd ed. New York: Springer-Verlag Inc,2006.
[16]Bendsoe M P. Optimization of structural topology, shape and material [M]. Berlin Heidelberg: Springer-Verlag Inc,1995.
[17]Coelho P G, Fernandes P R, Guedes J M, et al. A hierarchical model for concurrent material and topology optimisation of three-dimensional structures [J]. Structural and Multidisciplinary Optimization,2008,35(2):107-115.
[18]Rodrigues H, Guedes J M, Bendsoe M P. Hierarchical optimization of material and structure [J]. Structural and Multidisciplinary Optimization,2002,24(1):1-10.
[19]Zowe J, Ko Vara M, Bends E M P. Free material optimization via mathematical programming [J]. Mathematical Programming,1997,79(1):445-466.
[20]刘岭,阎军,程耿东.考虑均一微结构的结构/材料两级协同优化[J].计算力学学报,2008,25：29-34.
[21]Kocvara M, Stingl M, Zowe J. Free material optimization:recent progress [J]. Optimization:A Journal of Mathematical Programming and Operations Research,2008,57(1):79-100.
[22]Bendsoe M P, Diaz A R, Lipton R, et al. Optimal design of material properties and material distribution for multiple loading conditions [J]. International Journal for Numerical Methods in Engineering,1995,38(7):1149-1170.
[23]Zhang W H, Sun S P. Scale-related topology optimization of cellular materials and structures [J]. International Journal for Numerical Methods in Engineering,2006,68(9):993-1011.
[24]张卫红,孙士平.多孔材料/结构尺度关联的一体化拓扑优化技术[J].力学学报,2006,38：522-529.
[25]阎军,刘岭,刘晓峰,等.考虑尺寸效应的模块化结构两层级优化设计[J].力学学报,2010,2：268-274.
[26]Liu L, Yan J, Cheng G. Optimum structure with homogeneous optimum truss-like material [J]. Computers & Structures,2008,86(13):1417-1425.
[27]Bendsoe M P. Optimal shape design as a material distribution problem [J]. Structural and Multidisciplinary Optimization,1989,1(4):193-202.
[28]Bendsoe M P, Sigmund O. Material interpolation schemes in topology optimization [J]. Archive of Applied Mechanics,1999,69(9):635-654.
[29]Sigmund O. Systematic design of electrothermomechanical microactuators using topology optimization [J]. Modelling and Simulation of Microsystems, Semiconductors, Sensors and Actuators,1998,350-355.
[30]Sigmund O. Tailoring materials for specific needs [J]. Journal of Intelligent Material Systems and Structures,1994,5(6):736-742.
[31]Sigmund O. Tailoring materials with prescribed elastic properties [J]. Mechanics of Materials,1995, 20(4):351-368.
[32]Sigmund O, Torquato S. Composites with extremal thermal expansion coefficients [J]. Applied Physics Letters,1996,69(21):3203-3205.
[33]Jensen J S, Sigmund O. Systematic design of photonic crystal structures using topology optimization:Low-loss waveguide bends [J]. Applied physics letters,2004,84(12):2022-2024.
[34]Jonsmann J, Sigmund O, Bouwstra S. Compliant electro-thermal microactuators [C]. IEEE International MEMS 99 Conference, Twelfth IEEE International Conference on Micro Electro Mechanical Systems,1999.
[35]Gersborg-Hansen A, Sigmund O, Haber R B. Topology optimization of channel flow problems [J]. Structural and Multidisciplinary Optimization,2005,30(3):181-192.
[36]Madsen J I. Design optimization of internal flow devices [D]. Denmark:Alborg University,1998.
[37]Sigmund O. Design of multiphysics actuators using topology optimization-Part Ⅱ:Two-material structures [J]. Computer Methods in Applied Mechanics and Engineering,2001,190(49): 6605-6627.
[38]Luo Q, Tong L. An adhesively laminated plate element for PZT smart plates [J]. Computational Mechanics,2004,34(3):224-236.
[39]Wang S Y, Tai K, Quek S T. Topology optimization of piezoelectric sensors/actuators for torsional vibration control of composite plates [J]. Smart Materials and Structures,2006,15:253-269.
[40]Christensen S T, Sorokin S V, Olhoff N. On analysis and optimization in structural acoustics-Part I: Problem formulation and solution techniques [J]. Structural and Multidisciplinary Optimization, 1998,16(2):83-95.
[41]Christensen S T, Sorokin S V, Olhoff N. On analysis and optimization in structural acoustics-Part II: Exemplifications for axisymmetric structures [J]. Structural and Multidisciplinary Optimization, 1998,16(2):96-107.
[42]Diihring M B, Jensen J S, Sigmund O. Acoustic design by topology optimization [J]. Journal of Sound and Vibration,2008,317(3):557-575.
[43]Jones R M. Mechanics of Composite Materials [M].2rd ed. Philadelphia:Taylor & Francis Group, 1999.
[44]Vasiliev V V, Morozov E V. Advanced mechanics of composite materials [M]. Oxford:Elsevier Science,2007.
[45]Bruyneel M, Fleury C. Composite structures optimization using sequential convex programming [J]. Advances in Engineering Software,2002,33(7):697-711.
[46]Bruyneel M, Duysinx P, Fleury C. Composite structures design for strength and stiffness with respect to ply thickness and/or fibers orientation [C]. Dalian:Proceedings of the 4th World Congress of Structural and Multidisciplinary Optimization WCSMO4,2001.
[47]Bruyneel M, Colson B, Jetteur P, et al. Recent progress in the optimal design of composite structures:industrial solution procedures on case studies [J]. International Journal for Simulation and Multidisciplinary Design Optimization,2008,2(4):283-288.
[48]Bruyneel M, Duysinx P, Fleury C. A family of MMA approximations for structural optimization [J]. Structural and Multidisciplinary Optimization,2002,24(4):263-276.
[49]Bruyneel M, Colson B, Remouchamps A. Discussion on some convergence problems in buckling optimization [J]. Structural and Multidisciplinary Optimization,2008,35(2):181-186.
[50]Svanberg K. The method of moving asymptotes-a new method for structural optimization [J]. International Journal for Numerical Methods in Engineering,1987,24:359-373.
[51]Fukunaga H, Vanderplaats G N. Strength optimization of laminated composites with respect to layer thickness and/or layer orientation angle [J]. Computers & Structures,1991,40(6):1429-1439.
[52]Miki M. Material design of composite laminates with required in-plane elastic properties [C]. Tokyo:Progress in Science and Engineering of Composites, ICCM-IV,1981, pp:1725-1731.
[53]Diaconu C G, Sato M, Sekine H. Feasible region in general design space of lamination parameters for laminated composites [J].AIAA Journal,2002,40(3):559-565.
[54]Grenestedt J L, Gudmundson P. Layup optimization of composite material structures [C].IUTAM Symposium on Optimal Design with Advanced Materials,1993,311-336.
[55]Fukunaga H, Hirano Y. Stability optimization of laminated composite plates under in-plane loads [C]. Tokyo:Proceedings of the Fourth International Conference on Composite Materials,1982, 565-572.
[56]Ringertz U T. On finding the optimal distribution of material properties [J]. Structural and Multidisciplinary Optimization,1993,5(4):265-267.
[57]Autio M. Coupled thermal-structural problems in the optimization of laminated plates [J]. Structural and Multidisciplinary Optimization,1998,15(1):49-56.
[58]Grediac M. A procedure for designing laminated plates with required stiffness properties, Application to thin quasi-isotropic quasi-homogeneous uncoupled laminates [J]. Journal of Composite Materials,1999,33(20):1939-1956.
[59]Autio M. Determining the real lay-up of a laminate corresponding to optimal lamination parameters by genetic search [J]. Structural and Multidisciplinary Optimization,2000,20(4):301-310.
[60]Autio M. Optimization of coupled thermal-structural problems of laminated plates with lamination parameters [J]. Structural and Multidisciplinary Optimization,2001,21(1):40-51.
[61]Topal U, Uzman U. Thermal buckling load optimization of laminated composite plates [J]. Thin-Walled Structures,2008,46(6):667-675.
[62]Topal U, Uzman U. Thermal buckling load optimization of laminated skew plates [J]. Materials & Design,2009 30(7):2569-2575.
[63]Abdalla M M, Setoodeh S, Gurdal Z. Design of variable stiffness composite panels for maximum fundamental frequency using lamination parameters [J]. Composite Structures,2007,81(2): 283-291.
[64]Thuwis G, De Breuker R, Abdalla M, et al. Aeroelastic tailoring using lamination parameters [J]. Structural and Multidisciplinary Optimization,2010,41(4):637-646.
[65]Diaconu C G, Sato M, Sekine H. Buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads using lamination parameters [J]. Composite Structures,2002,58(4):423-433.
[66]Setoodeh S, Abdalla M M, Gurdal Z. Approximate feasible regions for lamination parameters [C]. Portsmouth:11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference,2006,
[67]Bradford B C, David D P, Hannu H. The quickhull algorithm for convex hulls [J]. ACM Transactions on Mathematical Software,1996,22(4):469-483.
[68]Liu B, Haftka R T, Trompette P. Maximization of buckling loads of composite panels using flexural lamination parameters [J]. Structural and Multidisciplinary Optimization,2004,26(1): 28-36.
[69]Diaconu C G, Sekine H. Layup optimization for buckling of laminated composite shells with restricted layer angles [J]. AlAA Journal,2004,42(10):2153-2166.
[70]Foldager J, Hansen J S, Olhoff N. A general approach forcing convexity of ply angle optimization in composite laminates [J]. Structural and Multidisciplinary Optimization,1998,16(2):201-211.
[71]Dawe D J, Wang S. Spline finite strip analysis of the buckling and vibration of rectangular composite laminated plates [J]. International Journal of Mechanical Sciences,1995,37(6):645-667.
[72]Mota Soares C M, Mota Soares C A, Mateus H C. A model for the optimum design of thin laminated plate-shell structures for static, dynamic and buckling behavior [J]. Composite Structures, 1995,32(1-4):69-79.
[73]Singha M K, Ramachandra L S, Bandyopadhyay J N. Optimum design of laminated composite plates for maximum thermal buckling loads [J]. Journal of Composite Materials,2000,34(23): 1982-1997.
[74]Lanzi L, Giavotto V. Post-buckling optimization of composite stiffened panels:computations and experiments [J]. Composite Structures,2006,73(2):208-220.
[75]Farshi B, Rabiei R. Optimum design of composite laminates for frequency constraints [J]. Composite Structures,2007,81(4):587-597.
[76]Topal U, Uzman U. Frequency optimization of laminated folded composite plates [J]. Materials & Design,2009,30(3):494-501.
[77]Lopatin A V, Morozov E V. Fundamental frequency of an orthotropic rectangular plate with an internal centre point support [J]. Composite Structures,2011,93(10):2487-2495.
[78]Qatu M S, Sullivan R W, Wang W. Recent research advances on the dynamic analysis of composite shells:2000-2009 [J]. Composite Structures,2010,93(1):14-31.
[79]Setoodeh S, Abdalla M M, Usselmuiden S T, et al. Design of variable-stiffness composite panels for maximum buckling load [J]. Composite Structures,2009,87(1):109-117.
[80]Lopes C S, Gurdal Z, Camanho P P. Variable-stiffness composite panels:buckling and first-ply failure improvements over straight-fibre laminates [J]. Computers & Structures,2008,86(9): 897-907.
[81]Correia V M F, Soares C M M, Soares C A M. Design sensitivity analysis and optimal design of composite structures using higher order discrete models [J]. Engineering Optimization,1997, 29(1-4):85-111.
[82]Stegmann J, Lund E. Discrete material optimization of general composite shell structures [J]. International Journal for Numerical Methods in Engineering,2005,62(14):2009-2027.
[83]Lund E, Stegmann J. On structural optimization of composite shell structures using a discrete constitutive parametrization [J]. Wind Energy,2005,8(1):109-124.
[84]唐文艳,顾元宪,赵国忠.复合材料层合板铺层顺序优化遗传算法[J].大连理工大学学报,2004,44(002)：186-189.
[85]Upadhyay A, Kalyanaraman V. Optimum design of fibre composite stiffened panels using Genetic Algorithms [J]. Engineering Optimization,2000,33(2):201-220.
[86]穆朋刚,赵美英,陈鹏飞,等.基于蚁群算法的复合材料层合板的铺层顺序优化[J].玻璃钢／复合材料,2007,11(006)：14-17.
[87]Sigmund O. On the usefulness of non-gradient approaches in topology optimization [J]. Structural and Multidisciplinary Optimization,2011,43(5):589-596.
[88]Hirano Y. Optimum design of laminated plates under shear [J]. Journal of Composite Materials, 1979,13(4):329-334.
[89]Tauchert T R, Adibhatla S. Design of laminated plates for maximum stiffness [J]. Journal of Composite Materials,1984,18(1):58-69.
[90]Cheng K. Sensitivity analysis and a mixed approach to the optimization of symmetric layered composite plates [J]. Engineering Optimization,1986,9(4):233-247.
[91]Mahadevan S, Liu X. Probabilistic optimum design of composite laminates [J]. Journal of Composite Materials,1998,32(1):68-82.
[92]Fleury C, Schmit L A. Dual methods and approximation concepts in structural synthesis [R].USA: NASA Contractor Report,1980.
[93]Fleury C. First and second order convex approximation strategies in structural optimization [J]. Structural and Multidisciplinary Optimization,1989,1(1):3-10.
[94]Fleury C. CONLIN:an efficient dual optimizer based on convex approximation concepts [J]. Structural and Multidisciplinary Optimization,1989,1(2):81-89.
[95]Svanberg K.The method of moving asymptotes-a new method for structural optimization [J]. International Journal for Numerical Methods in Engineering,1987,24(2):359-373.
[96]Zillober C.A globally convergent version of the method of moving asymptotes [J]. Structural and Multidisciplinary Optimization,1993,6(3):166-174.
[97]Chandra R, Singh S P, Gupta K. Damping studies in fiber-reinforced composites-a review [J]. Composite Structures,1999,46(1):41-51.
[98]Sobieszczanski-Sobieski J, Haftka R T. Multidisciplinary aerospace design optimization:survey of recent developments [J]. Structural and Multidisciplinary Optimization,1997,14(1):1-23.
[99]Todoroki A, Haftka R T. Stacking sequence optimization by a genetic algorithm with a new recessive gene like repair strategy [J]. Composites Part B:Engineering,1998,29(3):277-285.
[100]唐文艳.结构优化中的遗传算法研究和应用[D].大连：大连理工大学,2002.
[101]晏飞,李为吉.基于自适应遗传算法的复合材料层合板铺层顺序优化设计[J].西北工业大学学报,2001,19(1)：156-159.
[102]任茶仙,张铎.复合材料层合结构铺层顺序优化设计的免疫遗传算法[J].强度与环境,2007,34(2)：44-50.
[103]Aymerich F, Serra M. Optimization of laminate stacking sequence for maximum buckling load using the ant colony optimization (ACO) metaheuristic [J]. Composites Part A:Applied Science and Manufacturing,2008,39(2):262-272.
[104]Sigmund O, Torquato S. Design of materials with extreme thermal expansion using a three-phase topology optimization method [J]. Journal of the Mechanics and Physics of Solids,1997,45(6): 1037-1067.
[105]Duysinx P, Gao T, Zhang W, et al. New developments for an efficient solution of the discrete material topology optimization of composite structures [C]. Rioskilde:32nd RISO International Symposium on Material Science,2011.
[106]Hvejsel C F, Lund E. Material interpolation schemes for unified topology and multi-material optimization [J]. Structural and Multidisciplinary Optimization,2011,43(6):811-825.
[107]Bruyneel M. SFP——a new parameterization based on shape functions for optimal material selection: application to conventional composite plies [J]. Structural and Multidisciplinary Optimization, 2011,43(1):17-27.
[108]Gao T, Zhang W, Duysinx P. A bi-value coding parameterization scheme for the discrete optimal orientation design of the composite laminate [J]. International Journal for Numerical Methods in Engineering,2012, DOI:10.1002/nme.4270.
[109]Arora J S, Huang M W. Hsieh C C. Methods for optimization of nonlinear problems with discrete variables:a review [J]. Structural and Multidisciplinary Optimization,1994,8(2):69-85.
[110]Arora J S, Huang M W. Discrete structural optimization with commercially available sections [J]. Struct. Mech. Earthquake Eng,1996,13(2):93-110.
[111]孙焕纯,柴山,王跃方.离散变量结构优化设计[M].大连：大连理工大学出版社,]995.
[112]孙焕纯,柴山,王跃方.离散变量结构优化设计的发展,现状及展望[J].力学与实践,1997,19(4)：7-11.
[113]Arora J S.Introduction to optimum design.New York:Academic Press,2004.
[114]Arora J S.Optimization of structural and mechanical systems [M]. London:World Scientific Pub Co Inc,2007.
[115]Land A H, Doig A G. An automatic method of solving discrete programming problems [J]. Econometrica:Journal of the Econometric Society,1960,28(3):497-520.
[116]Dakin R J. A tree-search algorithm for mixed integer programming problems [J]. The Computer Journal,1965,8(3):250-255.
[117]Ringertz U. On methods for discrete structural optimization [J]. Engineering Optimization,1988, 13(1):47-64.
[118]Ohsaki M. Optimization of finite dimensional structures [M]. Taylor & Francis Group:CRC Press, 2010.
[119]Stender J. Parallel genetic algorithms:theory and applications [M]. Netherlands:Ios Press Inc, 1993.
[120]Stolpe M, Canh N N, Stainko R. The Branch and Cut Method in the PLATO-N Project [C]. Rio de Janeiro:EngOpt 2008:International Conference on Engineering Optimization,2008.
[121]Shin D, Gurdal Z, Griffin O H. A penalty approach for nonlinear optimization with discrete design variables [J]. Engineering Optimization,1989,16(1):29-42.
[122]Yu X, Zhang S, Johnson E. A discrete post-processing method for structural optimization [J]. Engineering with Computers,2003,19(2):213-220.
[123]Li Y, Tan T, Li X. A gradient-based approach for discrete optimum design [J]. Structural and Multidisciplinary Optimization,2010,41(6):881-892.
[124]Walls R, Elvin A. An algorithm for grouping members in a structure [J]. Engineering Structures, 2010,32(6):1760-1768.
[125]Barbosa H J C, Lemonge A C C. A genetic algorithm encoding for a class of cardinality constraints New York:Proceedings of the 2005 Conference on Genetic and Evolutionary Computation,2005.
[126]Kaveh A, Zolghadr A. A multi-set charged system search for truss optimization with variables of different natures element grouping [J]. Civil Engineering,2011,55(2):87-98.
[127]Biedermann J D, Grierson D E. A generic model for building design [J]. Engineering with Computers,1995,11(3):173-184.
[128]Barbosa H J C, Lemonge A C C, Borges C C H. A genetic algorithm encoding for cardinality constraints and automatic variable linking in structural optimization [J]. Engineering Structures, 2008,30(12):3708-3723.
[129]Togan V, Daloglu A T. Optimization of 3d trusses with adaptive approach in genetic algorithms [J]. Engineering Structures,2006,28(7):1019-1027.
[130]Togan V, Daloglu A T. An improved genetic algorithm with initial population strategy and self-adaptive member grouping [J]. Computers & Structures,2008,86(11-12):1204-1218.
[131]Lemonge A C C, Barbosa H J C, Coutinho A L G A, et al. Multiple cardinality constraints and automatic member grouping in the optimal design of steel framed structures [J]. Engineering Structures,2011,33(2):433-444.
[132]Lemonge A C C, Silva M M, Barbosa H J C. Design optimization of geometrically nonlinear truss structures considering cardinality constraints [C]. Juiz de Fora:2011 IEEE Congress on Evolutionary Computation (CEC),2011.
[133]Shea K, Cagan J, Fenves S J. A shape annealing approach to optimal truss design with dynamic grouping of members [J]. Journal of Mechanical Design,1997,119(3):388-394.
[134]刘晓峰,阎军,程耿东.采用自动分组遗传算法的频率约束下桁架拓扑优化[J].计算力学学报,2011,28(1)：1-7.
[135]刘晓峰.自动分组遗传算法的改进及在结构工程中的应用[D].大连：大连理工大学,2011.
[136]刘书田,林哲祺.考虑作动器联接方式的结构形状控制优化[J].工程力学,2009,26(2)：227-241.
[137]Liu S, Lin Z. Integrated design optimization of voltage channel distribution and control voltages for tracking the dynamic shapes of smart plates [J]. Smart Materials and Structures,2010,19(12): 1250-1263.
[138]Durand L P. Composite materials research progress [M]. New York:Nova Science Pub Inc,2008.
[139]Miki M. Material design of composite laminates with required in-plane elastic properties [C]. Tokyo:Progress in Science and Engineering of Composites, ICCM-IV,1981, pp:1725-1731.
[140]Miki M, Sugiyama Y. Optimum design of laminated composite plates using lamination parameters [J]. AIAA Journal,1993,31(5):921-922.
[141]Bloomfield M W, Diaconu C G, Weaver P M. On feasible regions of lamination parameters for lay-up optimization of laminated composites [J]. Proceedings of the Royal Society A:Mathematical, Physical and Engineering Science,2008,456(2104):1123-1143.
[142]Storn R, Price K. Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces [J]. Journal of Global Optimization,1997,11(4):341-359.
[143]Brest J, Greiner S, Boskovic B, et al. Self-adapting control parameters in differential evolution:A comparative study on numerical benchmark problems [J]. IEEE Transactions on Evolutionary Computation,2006,10(6):646-657.
[144]Brest J, Bo Kovi B, Greiner S, et al. Performance comparison of self-adaptive and adaptive differential evolution algorithms [J]. Soft Computing-A Fusion of Foundations, Methodologies and Applications,2007,11(7):617-629.
[145]方开泰.均匀设计与均匀设计表[M].北京：科学出版社,1994.
[146]Fang K. T, Lin D K J, Winker P, et al. Uniform design:theory and application [J].Technometrics, 2000,42(3):237-248.
[147]Sigmund O. Material interpolation schemes in topology optimization [J]. Archive of Applied Mechanics,1999,69(9/10):635-654.
[148]Schmit L A, Fleury C. Discrete-continuous variable structural synthesis using dual methods [J]. AIAA Journal,1980,18:1515-1524.
[149]Christensen P W, Klarbring A. An introduction to structural optimization [M].Springer Verlag, 2008.
[150]晏飞,戴德海,朱智春.基于遗传算法的纤维增强复合材料层合板刚度设计方法[J].强度与环境,2004,31(002)：7-12.
[151]Gibiansky L V, Sigmund O. Multiphase composites with extremal bulk modulus [J]. Journal of the Mechanics and Physics of Solids,2000,48(3):461-498.
[152]Svanberg K. A class of globally convergent optimization methods based on conservative convex separable approximations [J]. SIAM Journal on Optimization,2002,12(2):555-573.
[153]Hvejsel C F, Lund E, Stolpe M. Optimization strategies for discrete multi-material stiffness optimization [J]. Structural and Multidisciplinary Optimization,2011,44:149-163.
[154]Thanedar P B. Survey of discrete variable optimization for structural design [J]. Journal of Structural Engineering,1995,121(2):301-306.
[155]Rajeev S, Krishnamoorthy C S. Discrete optimization of structures using genetic algorithms [J]. Journal of Structural Engineering,1992,118(5):1233-1250.
[156]Lee K S, Geem Z W. A new structural optimization method based on the harmony search algorithm [J]. Computers & Structures,2004,82(9):781-798.
[157]Camp C V, Bichon B J. Design of space trusses using ant colony optimization [J]. Journal of Structural Engineering,2004,130(5):741-751.
[158]Kripka M. Discrete optimization of trusses by simulated annealing [J]. Journal of the Brazilian Society of Mechanical Sciences and Engineering,2004,26(2):170-173.
[159]Kaveh A, Talatahari S. A hybrid particle swarm and ant colony optimization for design of truss structures [J]. Asian Journal of Civil Engineering (building and housing),2008,9(4):329-348.
[160]Sonmez M. Discrete optimum design of truss structures using artificial bee colony algorithm [J]. Structural and Multidisciplinary Optimization,2011,43:85-97.
[161]Wu S J, Chow P T. Steady-state genetic algorithms for discrete optimization of trusses [J]. Computers & Structures,1995,56(6):979-991.
[162]Erbatur F, Hasancebi O, Tutuncu I, et al. Optimal design of planar and space structures with genetic algorithms [J]. Computers & Structures,2000,75(2):209-224.
[163]He S, Wu Q H, Wen J Y, et al. A particle swarm optimizer with passive congregation [J]. Biosystems,2004,78(1-3):135-147.
[164]Togan V, Daloglu A T. Optimization of 3d trusses with adaptive approach in genetic algorithms [J], Engineering Structures,2006,28(7):1019-1027.
[165]Isaacs A, Ray T, Smith W. An Efficient Hybrid Algorithm for Optimization of Discrete Structures [J]. Simulated Evolution and Learning,2008,5361:625-634.
[166]Togan V, Daloglu A T. An improved genetic algorithm with initial population strategy and self-adaptive member grouping [J]. Computers & Structures,2008,86(11-12):1204-1218.
[167]Sandgren E. Nonlinear integer and discrete programming in mechanical design optimization [J]. Journal of Mechanical Design,1990,11(2):223-231.
[168]Gandomi A, Yang X S. Benchmark Problems in Structural Optimization [J]. Computational Optimization, Methods and Algorithms,2011,356:259-281.
[169]Cai J, THIERAUF G. Discrete optimization of structures using an improved penalty function method [J]. Engineering Optimization,1993,21(4):293-306.
[170]Coello C A C. Discrete optimization of trusses using genetic algorithms [C]. Spanish:The Sixth International Conference on Computation,1994.
[171]Camp C, Pezeshk S, Cao G. Optimized design of two-dimensional structures using a genetic algorithm [J]. Journal of Structural Engineering,1998,124(5):551-559.
[172]Tong W H, Liu G R. An optimization procedure for truss structures with discrete design variables and dynamic constraints [J]. Computers & Structures,2001,79(2):155-162.
[173]Nanakorn P, Meesomklin K. An adaptive penalty function in genetic algorithms for structural design optimization [J]. Computers & Structures,2001,79(29):2527-2539.
[174]Turkkan N. Discrete optimization of structures using a floating-point genetic algorithm [C]. Annual Conference of the Canadian Society for Civil Engineering,2003.
[175]De Sousa F L, Takahashi W K. Discrete optimal design of trusses by generalized extremal optimization [C]. Rio de Janeiro:6th World Congresses of Structural and Multidisciplinary Optimization,2005.
[176]Ming-Zhu D. An improved Templeman's algorithm for the optimum design of trusses with discrete member sizes [J]. Engineering Optimization,1986,9(4):303-312.
[177]Lee K S, Geem Z W, Lee S, et al. The harmony search heuristic algorithm for discrete structural optimization [J]. Engineering Optimization,2005,37(7):663-684.
[178]Li L J, Huang Z B, Liu F. A heuristic particle swarm optimization method for truss structures with discrete variables [J].Computers & Structures,2009,87(7-8):435-443.
[179]Dede T, Bekiroglu S, Ayvaz Y. Weight minimization of trusses with genetic algorithm [J]. Applied Soft Computing,2011,11(2):2565-2575.
[180]Ghasemi M R, Hinton E, Wood R D. Optimization of trusses using genetic algorithms for discrete and continuous variables [J]. Engineering Computations,1999,16(3):272-303.
[181]Cai J, Thierauf G. Discrete structural optimization using evolution strategies [J]. In:Neural Networks and Combinatorial Optimization in Civil and Structural Engineering, Edinburg. Civil-Comp,95-100.
[182]Saka M P. Optimum design of steel frames with tapered members [J]. Computers & Structures, 1997,63(4):797-811.
[183]Huang M W, Arora J S. Optimal design of steel structures using standard sections [J]. Structural and Multidisciplinary Optimization,1997,14(1):24-35.
[184]Torregosa R F, Kanok-Nukulchai W. Weight optimization of steel frames using genetic algorithm [J]. Advances in Structural Engineering,2002,5(2):99-111.
[185]GB50017-2003,刚结构设计规范[S].北京：中国计划出版社,2003.
[186]钱令希.工程结构优化设计[M].水利电力出版社,1983.
[187]Beckers M. Topology optimization using a dual method with discrete variables [J]. Structural and Multidisciplinary Optimization,1999,17(1):14-24.
[188]Olsen G R, Vanderplaats G N. Method for nonlinear optimization with discrete design variables [J]. AIAA Journal,1989,27(11):1584-1589.
[189]Teller E, Metropolis N, Rosenbluth A. Equation of state calculations by fast computing machines [J]. Journal of Chemical Physics,1953,21(13):1087-1092.