结构优化设计的遗传演化算法研究
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摘要
结构优化设计目前正处于蓬勃发展时期,工程师们越来越多地意识到合理设计方法的重要性,追求以最小的代价获取最大的利益。过去几十年间连续结构的拓扑优化方法有很大的发展,应用范围不断拓广,特别是在航空航天、机械制造等领域重量的减轻对设计相当重要,优化方法的应用已颇为成熟。与其它领域相比,工程结构优化设计理论和应用的重要性尚未得到足够的重视,发展相对缓慢,结构设计仍停留在依靠工程师个人经验分析试算的阶段。目前我国仍处在建设高潮期,据建设部的预测,至2020年建筑工程面积将达到300亿平米,这种情况对于我们这样一个资源相对匮乏的国家,在建筑工程领域发展和应用最优设计方法已迫在眉睫,为此本文针对结构优化设计中较难的拓扑优化问题展开研究。
目前求解拓扑优化问题能力较强的算法有演化结构优化算法(ESO算法)、均匀化方法、密度函数法、仿生学算法等,其中演化结构优化算法(ESO算法)以其概念清晰、规则简单、计算方便的优点受到很多学者的青睐,但同时又因不能保证所得解为最优解受到一些学者的质疑。遗传算法(GA)属于仿生学算法,近年来的研究表明它具备找到最优解的长处。但遗传算法本身并不具有明确的结构优化设计的背景,虽然在计算数学层面上它能有效地解决组合优化的一些问题,具体应用于结构工程时,由于计算太耗时仅限于桁架结构的优化,无法进一步拓展到平面或空间结构。本文在这两种算法的基础上首次提出了一种新的拓扑优化方法“遗传演化算法(GESO算法)”,遗传演化算法继承了演化结构优化算法的灵敏度计算方法和基本步骤,从一个包含了最优解的初始设计域出发,根据遗传算法优胜劣汰的思想,以选择、变异、杂交等遗传算子决定单元的取舍,避免了错误的单元舍去,保证了算法所得解的最优性。新的遗传演化算法所得解的质量均好于原有的演化结构优化算法,同时计算效率又高于遗传算法。本文在提出新算法后,详细研究了算法中各计算参数对算法性能的影响,结果表明新算法鲁棒性好,最优解对计算参数不敏感,采用不同计算参数时均能找到最优解。
本文所做的另一项具有创新性的工作是利用Michell准则与遗传演化算法联合寻优。Michell准则是澳大利亚工程师Michell 1904年用数学解析方法研究的最优结构应满足的条件,符合这一条件的结构称为Michell桁架。然而直接由Michell准则建立Michell桁架十分难,目前缺乏有效的求解方法。本文一方面采用已有的多个符合Michell准则的虚应变场检验校核了遗传演化算法所得解的最优性,另一方面通过遗传演化算法得到了新的符合Michell准则的实际结构,为进一步获得不Ⅱ同类型Michell桁架打下基础。本文重点研究了两点或多点铰支条件下平面结构在各种荷载作用下的优化拓扑,根据Michell准则和遗传演化算法计算结果,归纳并发现这类结构的优化拓扑所应遵循的规律。本文作者认为两点或多点铰支条件下的平面结构的优化拓扑主要分为两类,一类由直线和圆组成,一类结构由曲线组成,曲线形状大多类似于理论推导过的“Michell悬臂”曲线或圆滚线,这些结论对工程结构优化的概念设计具有重要意义。
遗传演化算法被证明能方便地用于钢筋混凝土结构的配筋优化,钢筋混凝土结构由两种不同的材料组成,而且由于混凝土开裂和钢筋的屈服,材料表现出非线性特性,配筋优化计算时是否考虑两种材料的区别,是否考虑材料的非线性成为焦点。为此,本文设计了三套方案进行配筋优化,运算表明,不区分两种材料的差别,不考虑材料的非线性的方案是最适合遗传演化算法的方案,计算简单,结果实用。因为遗传演化算法擅长建立桁架模型,构件达到承载力极限状态时,荷载可认为是通过由钢筋拉杆和未开裂的混凝土压杆组成的拉压模型传递,构件的破坏是由于传力路径的破坏造成的。所以遗传演化算法配筋优化的过程是先求得构件的拉压杆模型,再在拉杆位置布置钢筋。本文利用遗传演化算法完成了不同跨高比的简支梁、深受弯构件、牛腿、框架节点等构件的配筋优化,提出了配筋优化方案,具有实际意义。
The subject of optimum structures is currently undergoing something of a boom. Engineers have realized the importance of the gains that can be made by a rational approach to design, which quantifies a principle of minimum cost. Topology optimization methods for continuum structures have achieved significant progress in the last two decades. These techniques have increasingly been used in aeronautical, mechanical and automotive industries in which the weight reduction of structures is very important. However, the potential of structural optimization techniques has not been realized by civil engineering industry and most works are designed by trial and error based on individual engineering experiences. As officers of the Ministry of Construction estimated, civil engineering industry is promising in China and the total of 30 billions m2 buildings will be constructed until 2020. As the relative lack of resources in our country the development and application of optimal theory is very important task for construction industry. On this background this thesis focuses on the structural topology optimization.
Topology optimization methods include evolutionary structural optimization (ESO), homogenization, density function, genetic algorithm (GA), et al, among which evolutionary structural optimization attained more attention for its simple concepts and easy programming. But ESO often came under some suspicious remarks for no guarantee of optimum structures. GA is excellent in optimization by using Darwinian’s theory of survival of the fittest though having no definite mathematic background. The time-consuming characteristics of GA in engineering limited its application to truss structures instead of plane or space structures. However, a new method deal with topology optimization of plane structures was proposed in this dissertation on the basis of ESO and GA. The new algorithm is named genetic evolutionary structural optimization (GESO) which starts from an original design field consist of all potential optimum structures and computed sensitivity numbers as done in ESO, selected and deleted elements from a ground structure by genetic operation such as mutation and crossover. Combination of the genetic operations and the process of ESO decrease the possibility of improper element being deleted. Compared with the results acquired by ESO, the topologies obtained by the new GESO are more exact while the time consumed is lesser than that of GA. It is shown in succeeding studies that the acquired optimums are insensitive to the value of parameters, so the robustness of the new method is proved.
Another innovative work in the dissertation is connecting GESO with Michell theory. The Michell theory of least-weight trusses for one load condition and a stress constraint was established in a milestone contribution by Michell (1904, Australia). Michell theory plays an important role in structural topology optimization. In general, Michell truss is a kind of framework with continuous distributions of members. It is often called“truss-like continuum”which is difficult to be deduced by mathematics analysis. In the dissertation the numerical studies of the new method use the classical Michell trusses for verifying their results. On the other hand, with the numerical calculations by GESO new“Michell type”structures are presented. Two or more pin-point supported plane structures under different loadings attained more attentions in the studies of discovering their optimal principle of topology. A large number of numerical experiments gained two main kinds of optimums for pin-point plane structures that is the structure composed of straight lines and circles and the structure made up of“Michell cantilever”and cycloids. These conclusions are of significant importance to structural concept design.
GESO can be conveniently applied to reinforcement layout optimization in reinforced concrete structure. Reinforced concrete is a composite material and the nonlinear behavior of reinforced concrete is characterized by the cracking of concrete and the yielding of steel reinforcement. The key problem for which solutions are obtained are whether the stiffness of reinforcing steel and the nonlinear behavior of reinforced concrete can be considered in the finite element model. Three schemes were then designed for the question. The answer is that the stiffness of reinforcing steel and the nonlinear behavior of reinforced concrete cannot be taken into account in the finite element model. After extensive cracking of concrete, the loads applied to a reinforced concrete member are mainly carried by the concrete struts and steel reinforcement. The failure of a reinforced concrete member is mainly caused by the breakdown of the load transfer mechanism, rather by that the tensile stress attains the tensile strength of concrete. The design task is to develop an appropriate strut-and-tie model for the structural concrete member in order to reinforce it. Simple supported beams with different span-to-depth ratios, deep beams, corbels and beam-column connections are discussed for acquiring optimal reinforcement layout which can be used in engineering structures.
目前求解拓扑优化问题能力较强的算法有演化结构优化算法(ESO算法)、均匀化方法、密度函数法、仿生学算法等,其中演化结构优化算法(ESO算法)以其概念清晰、规则简单、计算方便的优点受到很多学者的青睐,但同时又因不能保证所得解为最优解受到一些学者的质疑。遗传算法(GA)属于仿生学算法,近年来的研究表明它具备找到最优解的长处。但遗传算法本身并不具有明确的结构优化设计的背景,虽然在计算数学层面上它能有效地解决组合优化的一些问题,具体应用于结构工程时,由于计算太耗时仅限于桁架结构的优化,无法进一步拓展到平面或空间结构。本文在这两种算法的基础上首次提出了一种新的拓扑优化方法“遗传演化算法(GESO算法)”,遗传演化算法继承了演化结构优化算法的灵敏度计算方法和基本步骤,从一个包含了最优解的初始设计域出发,根据遗传算法优胜劣汰的思想,以选择、变异、杂交等遗传算子决定单元的取舍,避免了错误的单元舍去,保证了算法所得解的最优性。新的遗传演化算法所得解的质量均好于原有的演化结构优化算法,同时计算效率又高于遗传算法。本文在提出新算法后,详细研究了算法中各计算参数对算法性能的影响,结果表明新算法鲁棒性好,最优解对计算参数不敏感,采用不同计算参数时均能找到最优解。
本文所做的另一项具有创新性的工作是利用Michell准则与遗传演化算法联合寻优。Michell准则是澳大利亚工程师Michell 1904年用数学解析方法研究的最优结构应满足的条件,符合这一条件的结构称为Michell桁架。然而直接由Michell准则建立Michell桁架十分难,目前缺乏有效的求解方法。本文一方面采用已有的多个符合Michell准则的虚应变场检验校核了遗传演化算法所得解的最优性,另一方面通过遗传演化算法得到了新的符合Michell准则的实际结构,为进一步获得不Ⅱ同类型Michell桁架打下基础。本文重点研究了两点或多点铰支条件下平面结构在各种荷载作用下的优化拓扑,根据Michell准则和遗传演化算法计算结果,归纳并发现这类结构的优化拓扑所应遵循的规律。本文作者认为两点或多点铰支条件下的平面结构的优化拓扑主要分为两类,一类由直线和圆组成,一类结构由曲线组成,曲线形状大多类似于理论推导过的“Michell悬臂”曲线或圆滚线,这些结论对工程结构优化的概念设计具有重要意义。
遗传演化算法被证明能方便地用于钢筋混凝土结构的配筋优化,钢筋混凝土结构由两种不同的材料组成,而且由于混凝土开裂和钢筋的屈服,材料表现出非线性特性,配筋优化计算时是否考虑两种材料的区别,是否考虑材料的非线性成为焦点。为此,本文设计了三套方案进行配筋优化,运算表明,不区分两种材料的差别,不考虑材料的非线性的方案是最适合遗传演化算法的方案,计算简单,结果实用。因为遗传演化算法擅长建立桁架模型,构件达到承载力极限状态时,荷载可认为是通过由钢筋拉杆和未开裂的混凝土压杆组成的拉压模型传递,构件的破坏是由于传力路径的破坏造成的。所以遗传演化算法配筋优化的过程是先求得构件的拉压杆模型,再在拉杆位置布置钢筋。本文利用遗传演化算法完成了不同跨高比的简支梁、深受弯构件、牛腿、框架节点等构件的配筋优化,提出了配筋优化方案,具有实际意义。
The subject of optimum structures is currently undergoing something of a boom. Engineers have realized the importance of the gains that can be made by a rational approach to design, which quantifies a principle of minimum cost. Topology optimization methods for continuum structures have achieved significant progress in the last two decades. These techniques have increasingly been used in aeronautical, mechanical and automotive industries in which the weight reduction of structures is very important. However, the potential of structural optimization techniques has not been realized by civil engineering industry and most works are designed by trial and error based on individual engineering experiences. As officers of the Ministry of Construction estimated, civil engineering industry is promising in China and the total of 30 billions m2 buildings will be constructed until 2020. As the relative lack of resources in our country the development and application of optimal theory is very important task for construction industry. On this background this thesis focuses on the structural topology optimization.
Topology optimization methods include evolutionary structural optimization (ESO), homogenization, density function, genetic algorithm (GA), et al, among which evolutionary structural optimization attained more attention for its simple concepts and easy programming. But ESO often came under some suspicious remarks for no guarantee of optimum structures. GA is excellent in optimization by using Darwinian’s theory of survival of the fittest though having no definite mathematic background. The time-consuming characteristics of GA in engineering limited its application to truss structures instead of plane or space structures. However, a new method deal with topology optimization of plane structures was proposed in this dissertation on the basis of ESO and GA. The new algorithm is named genetic evolutionary structural optimization (GESO) which starts from an original design field consist of all potential optimum structures and computed sensitivity numbers as done in ESO, selected and deleted elements from a ground structure by genetic operation such as mutation and crossover. Combination of the genetic operations and the process of ESO decrease the possibility of improper element being deleted. Compared with the results acquired by ESO, the topologies obtained by the new GESO are more exact while the time consumed is lesser than that of GA. It is shown in succeeding studies that the acquired optimums are insensitive to the value of parameters, so the robustness of the new method is proved.
Another innovative work in the dissertation is connecting GESO with Michell theory. The Michell theory of least-weight trusses for one load condition and a stress constraint was established in a milestone contribution by Michell (1904, Australia). Michell theory plays an important role in structural topology optimization. In general, Michell truss is a kind of framework with continuous distributions of members. It is often called“truss-like continuum”which is difficult to be deduced by mathematics analysis. In the dissertation the numerical studies of the new method use the classical Michell trusses for verifying their results. On the other hand, with the numerical calculations by GESO new“Michell type”structures are presented. Two or more pin-point supported plane structures under different loadings attained more attentions in the studies of discovering their optimal principle of topology. A large number of numerical experiments gained two main kinds of optimums for pin-point plane structures that is the structure composed of straight lines and circles and the structure made up of“Michell cantilever”and cycloids. These conclusions are of significant importance to structural concept design.
GESO can be conveniently applied to reinforcement layout optimization in reinforced concrete structure. Reinforced concrete is a composite material and the nonlinear behavior of reinforced concrete is characterized by the cracking of concrete and the yielding of steel reinforcement. The key problem for which solutions are obtained are whether the stiffness of reinforcing steel and the nonlinear behavior of reinforced concrete can be considered in the finite element model. Three schemes were then designed for the question. The answer is that the stiffness of reinforcing steel and the nonlinear behavior of reinforced concrete cannot be taken into account in the finite element model. After extensive cracking of concrete, the loads applied to a reinforced concrete member are mainly carried by the concrete struts and steel reinforcement. The failure of a reinforced concrete member is mainly caused by the breakdown of the load transfer mechanism, rather by that the tensile stress attains the tensile strength of concrete. The design task is to develop an appropriate strut-and-tie model for the structural concrete member in order to reinforce it. Simple supported beams with different span-to-depth ratios, deep beams, corbels and beam-column connections are discussed for acquiring optimal reinforcement layout which can be used in engineering structures.
引文
[1] 孙焕纯,柴山,王跃方. 离散变量结构优化设计.第 1 版.大连:大连理工大学出版社,1995: 1-13.
[2] 张炳华,侯昶. 土建结构优化设计.第 2 版.上海:同济大学出版社,1998: 1-206.
[3] Narayanan S., Azarm S.. On improving multiobjective genetic algorithms for design optimization. Structural Optimization. 1999, 18(2-3): 146 – 155.
[4] 李芳,凌道盛. 工程结构优化设计发展综述. 工程设计学报,2002,9(5): 229-235.
[5] Michell A.G.M.. The Limits of Economy of Materials in Frame-structures. Philosophical Magazine, 1904, 8(17): 589-597.
[6] Yang X.Y. Bi-directional Evolutionary method for stiffness and displacement optimisation: [Degree of Master of Engineering to Victoria University of Technology]. Melbourne, Australia: Victoria University of Technology, 1999:1-140.
[7] Topping, B.H.V.. Topology design of discrete structures. In: Bends?e, M.P. and Mota Soares, C.A. (eds.). Topology Design of Structure, Netherlands: Kluwer Academic Publishers, 1993: 56-78.
[8] Bends?e, M.P., Kikuchi, N. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanical Engineering,1988,71: 197-224.
[9] Yang,R.J., Chuang,C.H.. Optimal topology design using linear programming. Computers and Structures, 1994, 52: 265-275.
[10] Kikuchin, Suzukik. A homogenization method for shape and topology optimization. Computer Methods in Applied Mechanical Engineering, 1991, 93: 291- 318.
[11] Allaire G., Jouve F., Maillot H.. Topology optimization for minimum stress design with the homogenization method. Structural and Multidisciplinary Optimization, 2004, 28(2-3): 87-98.
[12] 刘书田,程耿东. 基于均匀化理论的梯度功能材料优化设计方法. 宇航材料工艺, 1995, 6: 21-27.
[13] 王晓明,刘震宇,郭东明. 基于均匀化理论的微小型柔性结构拓扑优化的敏度分析. 中国机械工程, 1999, 10 (11) : 1264-1266.
[14] Yang R. J., Chuang C.H.. Optimal topology design using linear programming.Computers & Structures, 1994, 52: 265- 275.
[15] Hassani B., Hinton E.. A review of homogenization and topology optimization I--homogenization theory for media with periodic structure. Computers & Structures,1998, 69: 707-717.
[16] 汪树玉,刘国华,包志仁. 结构优化设计的现状与进展. 基建优化,1999,20(4):3-14.
[17] 姚侗,何淦瞳. 线性规划的一种新算法——直接搜索迭代法. 高校应用数学学报,1997, 12 (1): 73-84.
[18] 汪树玉,刘国华,包志仁. 结构优化设计的现状与进展(续). 基建优化,1999,20(5):3-7.
[19] 王小平,曹立明. 遗传算法——理论、应用与软件实现. 第 1 版. 西安:西安交通大学出版社,2002: 1-50.
[20] Chakraborty S., Mukhopadhyay M., Sha O.P.. Determination of Physical Parameters of Stiffened Plates using Genetic Algorithm. Journal of Computing in Civil Engineering, 2002, 16(3): 206-221.
[21] Pezeshk S., Camp C.V., Chen D,. Design of Nonlinear Framed Structures Using Genetic Optimization. Journal of Structural Engineering, 2000, 126(3): 382-388.
[22] Hong H., Yong X.. Vibration-based Damage Detection of Structures by Genetic Algorithm. Journal of Computing in Civil Engineering, 2002, 16(3): 222-229.
[23] Narayanan S., Azarm S.. On improving multiobjective genetic algorithms for design optimization. Structural Optimization. 1999, 18(2-3): 146 – 155.
[24] Krishnamoorthy C.S., Venkatesh P. P., Sudarshan R.. Object-Oriented Framework for Genetic Algorithms with Application to Space Truss Optimization. Journal of Computing in Civil Engineering, 2002, 16(1): 66-75.
[25] 刘勇,康立山,陈毓屏. 非数值并行算法(第 2 册)——遗传算法. 第 1 版. 北京:科学出版社,1995: 1-60.
[26] 潘正军,康立山,陈毓屏. 演化计算. 第 1 版. 北京:清华大学出版社, 1998: 1-36.
[27] 张文修,梁怡. 遗传算法的数学基础. 第 1 版. 西安:西安交通大学出版社,2000: 1-12.
[28] 周明,孙树栋. 遗传算法原理及应用. 第 1 版. 北京:国防工业出版社,1999: 1-45.
[29] 李敏强,寇纪淞,林丹等. 遗传算法的基本理论与应用. 第 1 版. 北京:科学出版社,2002: 1-70.
[30] 马光文,王黎. 遗传算法在桁架结构优化设计中的应用. 工程力学,1998,15(2): 38-44.
[31] 高峰,王德俊,胡俏等. 离散结构的遗传形状优化设计. 计算力学学报,1998, 15(4): 485-489.
[32] Camp C., Pezeshk S., Cao G. Z.. Optimized design of two-dimensional structures using a genetic algorithm. Journal of Structural Engineering, 1998, 124(5): 551-559.
[33] Rajeev S., Krishnamoorthy C.S.. Genetic algorithms-based methodologies for design optimization of trusses. Journal of Structural Engineering, 1997, 123(3): 350-358.
[34] Hajela P., Lee E., Cho H.. Genetic algorithms in topologic design of grillage structures. Computer-Aided Civil and Infrastructure Engineering, 1998, 13: 13-22.
[35] Kamal C. S., Adeli H.. Fuzzy genetic algorithm for optimization of steel structures. Journal of Structural Engineering, 2000, 126(5): 596-604.
[36] Soh C.K., Yang J.P.. Optimal layout of bridge trusses by genetic algorithms. Computer-Aided Civil and Infrastructure Engineering, 1998, 13: 247-254.
[37] Koumousis V.K., Arsenis S.J.. Genetic algorithms in optimal detailed design of reinforced concrete members. Computer-Aided Civil and Infrastructure Engineering, 1998, 13: 43-52.
[38] Rjeev S., Krishamoorthy C.S.. Genetic algorithm-based methodology for design optimization of reinforced concrete frames. Computer-Aided Civil and infrastructure Engineering, 1998, 13: 63-74.
[39] Ceranic B., Fryer C.. A genetic approach to the minimum cost design of reinforced concrete flanged beams under multiple loading conditions. ACSO, 1998: 71-78.
[40] Rafig M.Y., Southcombe C.. Genetic algorithms in optimal design and detailing of reinforced concrete biaxial columns supported by a declarative approach for capacity checking. Computers & Structures, 1998, 69(4): 443-457.
[41] Woon S.Y., Querin O.M., G.P. Steven. Structural application of a shape optimization method based on a genetic algorithm. Structural and Multidisciplinary Optimization, 2001, 22: 57-64.
[42] Hayalioglu M.S.. Optimum load and resistance factor design of steel space frames using genetic algorithm. Structural and Multidisciplinary Optimization, 2001, 21: 292-299.
[43] Concei??o Ant?nio C.A.. A multilevel genetic algorithm for optimization ofgeometrically nonlinear stiffened composite structures. Structural and Multidisciplinary Optimization, 2002, 24: 372-386.
[44] 陈存恩,李文雄. 结合灵敏度修正的遗传算法的结构损伤诊断. 地震工程与工程振动, 2006, 26(5): 172-176.
[45] 马祥森,史治宇. 基于 GA-BP 神经网络的结构损伤位置识别. 振动工程学报,2004, 17(4): 453-456.
[46] 易伟建,刘霞. 结构损伤诊断的遗传算法研究. 系统工程理论与实践,2001, (5): 114-118.
[47] 易伟建,刘霞. 混凝土梁板类构件边界条件识别与研究. 湖南大学学报,2000, 27(4): 81-87.
[48] 易伟建,刘霞. 基于遗传算法的结构损伤诊断研究. 工程力学,2001, 18(2): 64-71.
[49] Dumn S.A.. The Use of Genetic Algorithms and Stochastic Hill-Climbing in Dynamic Finite Element Model Identification. Computers & Structures, 1998, 66(4): 489-497.
[50] Friswell M.I., Penny J.E.T., Garvey S.D.. A combined genetic and eigensensitivity algorithm for the location of damage in structures. Computers & Structures, 1998, 69(5): 547-556.
[51] Bishop J.A., Striz A.G.. On using genetic algorithms for optimum damper placement in space trusses. Structural and Multidisciplinary Optimization, 2004, 28(2-3): 136-145.
[52] Salajegheh E., Heidari A.. Optimum design of structures against earthquake by adaptive genetic algorithm using wavelet networks. Structural and Multidisciplinary Optimization, 2004, 28(4): 277-285.
[53] 田军,寇纪松,李敏强. 利用遗传算法优化施工网络计划. 系统工程理论与实践,1999, 19(5): 78-81.
[54] 康立山,谢云,尤矢勇等. 非数值并行算法——模拟退火算法. 第 1 版. 北京:科学出版社,1998: 1-55.
[55] 邢文训,谢金星. 现代优化计算方法. 第 1 版. 北京:清华大学出版社,1999: 1-36.
[56] 吴剑国,赵莉萍,王建华. 工程结构混合离散变量优化的模拟退火方法. 工程力学, 1997, 14(3): 138-143.
[57] Patrick Y. S., Manoochehri S.. Generating Optimal Configurations in Structural Design Using Simulated Annealing. International Journal for Numerical Methods in Engineering, 1997, 40: 1053-1069.
[58] Shyh-Rong T., Chris P.P.. Annealing strategy for optimal structural design. Journal of Structural Engineering, 1996, 122(7): 815-827.
[59] Barski M., Kru˙zelecki J.. Optimal design of shells against buckling by means of the simulated annealing method. Structural and Multidisciplinary Optimization, 2004, 29(1): 61-72.
[60] 刘先珊,周创兵,张立君. 基于模拟退火的 Gauss-Newton 算法的神经网络在渗流反分析中的应用. 岩土力学, 2005, 26(3): 404-414.
[61] 李红芳,侯朝胜. 模拟退火遗传算法与结构优化设计. 天津理工学院学报,2004, 20(4): 96-98.
[62] 邹广电,陈永平. 滑坡和边坡稳定性分析的模拟退火算法-随机搜索耦合算法. 岩石力学与工程学报, 2004, 23(12): 2032-2037.
[63] 吴新生,谢益民. 基于人工神经网络优化抄纸浆料配比. 中国造纸,2005, 24(5): 44-46.
[64] 宋小莉,牛欣,司银楚. 基于 BP 神经网络的半夏、生姜、甘草三泻心汤配伍研究. 中国临床药理学与治疗学, 2005, 10(5): 527-531.
[65] 郭一楠,巩敦卫,程健. 基于分布式神经网络的焦炭质量预测模型. 中国矿业大学学报,2005, 34(4): 514-517.
[66] 荣辉,张济世. 人工神经网络及其现状与展望. 电子技术应用,1995, 10: 4-5.
[67] 王科俊,王克成. 神经网络建模预报与控制. 第 1 版. 哈尔滨:哈尔滨工程大学出版社,1996: 1-47.
[68] 赵振宇,徐用懋. 模糊理论和神经网络的基础与应用. 第 1 版. 北京:清华大学出版社,广西科学技术出版社, 1996: 1-13.
[69] 吴剑国,赵丽萍. 工程结构优化的神经网络方法. 计算力学学报,1998, 15(1): 69-74.
[70] 姜绍飞,刘之洋. 人工神经网络在外包钢混凝土框架柱设计参数预报中的适应性. 土木工程学报,1997, 30(1): 73-77.
[71] 郭国会,易伟建. 基于神经网络的结构边界条件识别和损伤诊断. 湖南大学学报,1998, 25(1): 71-76.
[72] 李海岭,霍达,王志忠. 考虑土-结构动力相互作用影响的结构神经网络控制. 土木工程学报,2002, 35(3): 61-65.
[73] 姜绍飞,钟善桐. 神经网络在结构工程中的应用. 哈尔滨建筑大学学报,1998,31(6): 129-134.
[74] 李士勇,陈永强,李研. 蚁群算法及其应用. 第 1 版. 哈尔滨:哈尔滨工业大学出版社,2004: 1-40.
[75] Dorigo M., Gambardella L M. Ant colonies for the traveling salesman problem.BioSysterms, 1997, 43: 73-81.
[76] Dorigo M., Caro G. D.. Ant algorithms for discrete optimization. Artificial Life, 1999, 5(3): 137-172.
[77] Xie Y.M., Steven G.P.. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885-896.
[78] Xie Y.M., Steven G.P.. A simple approach to structural frequency optimization. Computers & Structures, 1994, 53(6): 1487-1491.
[79] Xie Y.M., Steven G.P.. Evolutionary structural optimization for dynamic problems. Computers & Structures, 1996, 58(6): 1067-1073.
[80] Zhao C. B., Steven G. P., Xie Y. M.. Effect of initial nondesign domain on optimal topologies of structures during natural frequency optimization. Computers & Structures, 1997, 62(1): 119-131.
[81] Zhao C.B., Steven G. P., Xie Y. M.. A generalized evolutionary method for natural frequency optimization of membrane vibration problems in finite element analysis. Computers & Structures, 1998, 66(2-3): 353-364.
[82] Chu D.N., Xie Y.M., Hira A., et.al.. Evolutionary structural optimization for problems with stiffness constrains. Finite Elements in Analysis and Design, 1996, 21: 239-251.
[83] Chu D.N., Xie Y.M., Hira A., et.al.. On various aspects of evolutionary structural optimization for problems with stiffness constraints. Finite Elements in Analysis and Design, 1997, 24: 197-212.
[84] Chu D. N., Xie Y. M., Steven G. P.. An evolutionary structural optimization method for sizing problems with discrete design variables. Computers & Structures, 1998, 68: 419-431.
[85] Li Q., Steven G. P., Xie Y. M.. Evolutionary structural optimization for stress minimization problems by discrete thickness design. Computers & Structures, 2000, 78: 769-780.
[86] Manickarajah D, Xie Y. M., Steven G. P.. Optimum design of frames with multiple constraints using an evolutionary method. Computers & Structures, 2000, 74: 731-741.
[87] Querin O.M., Steven G.P., Xie Y.M.. Evolutionary Structural Optimisation Using an Additive Algorithm. Finite Elements in Analysis and Design, 2000, 34: 291-308.
[88] Kurpati A., Azarm S., Wu J.. Constraint Handling Improvements for Multiobjective Genetic Algorithms. Structural and MultidisciplinaryOptimization, 2002, 23: 204-213.
[89] Querin O.M., Young V., Steven G.P., et al. Computational Efficiency and Validation of Bi-directional evolutionary Structural optimisation. Computer Methods in Applied Mechanics and Engineering, 2000, 189: 559-573.
[90] Manickarajah D., Xie Y. M., Steven G. P.. An evolutionary method for optimization of plate buckling resistance. Finite Elements in Analysis and Design, 1998, 29: 205-230.
[91] Li Q., Steven G. P., Querin O. M., et al. Shape and Topology Design for Heat Conduction by Evolutionary Structural Optimization. International Journal of Heat and Mass Transfer, 1999, 42:3361-3371.
[92] Li Q., Steven G. P., Xie Y. M.. Displacement Minimization of Thermoelastic Structures by Evolutionary Thickness Design. Computer Methods in Applied Mechanics and Engineering, 1999, 179: 361-378.
[93] Das R., Jones R., Xie Y.M.. Design of Structures for Optimal Static Strength Using ESO. Engineering Failure Analysis, 2005, 12: 61-80.
[94] Steven G. P., Li Q., Xie Y. M.. Multicriteria optimization that minimizes maximum stress and maximizes stiffness. Computers & Structures, 2002, 80: 2433-2448.
[95] Li W., Li Q., Steven G. P., et al. An Evolutionary Approach to Elastic Contact Optimization of Frame Structures. Finite Elements in Analysis and Design, 2003, 40: 61-81.
[96] Li Q., Steven G. P., Querin O. M., et al. Stress Based Optimization of Torsional Shafts Using an Evolutionary Procedure. International Journal of Solids and Structures, 2001, 38: 5661-5677.
[97] Rozvany G.I.N.. Stress ration and compliance based methods in topology optimization—a critical review. Structural and Multidisciplinary Optimization, 2001, 21: 109-119.
[98] Liang Q. Q., Xie Y. M., Steven G. P.. Optimal topology selection of continuum structures with displacement constraints. Computers & Structures, 2000, 77: 635-644.
[99] Li Q., Steven G. P., Querin O. M., et.al. Evolutionary shape optimization for stress minimization. Mechanics Research Communications, 1999, 26(6): 657-664.
[100] 荣见华,郑健龙,徐飞鸿. 结构动力修改及优化设计. 第一版. 北京:人民交通出版社,2002: 249-335.
[101] Tanskanen P.. The evolutionary structural optimization method: theoretical aspects. Computer methods in applied mechanics and engineering, 2002, 191: 5485-5498.
[102] Cox H.L.. The Design of Structures of Least Weight. Oxford: Pergamon Press, 1965: 80-114.
[103] Zhou M., Rozvany G. I. N.. On the validity of ESO type methods in topology optimization. Structural and Multidisciplinary Optimization, 2001, 21: 80-83.
[104] Cheng F. Y.,Li D.. Multiobjective Optimization Design with Pareto Genetic Algorithm. Journal of Structural Engineering, 1997, 123(9): 1252-1261.
[105] Lin C.Y., Chou J.N.. A Two-stage Approach for Structural Topology Optimization. Advances in Engineering Software, 1999, 30: 261-271.
[106] 周克民,李俊峰,李霞. 结构拓扑优化研究方法综述. 力学进展,2005,35(1):69-76.
[107] Rozvany G.I.N.. Aims, Scope, Methods, History and Unified Terminology of Computer-aided Topology Optimization in Structural Mechanics. Structural and Multidisciplinary Optimization, 2001, 21: 90-108.
[108] Liang Q. Q., Xie Y. M., Steven G. P.. Generating Optimal Strut-and-Tie Models in Prestressed Concrete Beams by Performance-Based Optimization. ACI Structural Journal, 2001, 98(2): 226-232.
[109] Liang Q. Q., Xie Y. M., Steven G. P.. Topology Optimization of Strut-and-Tie Models in Reinforced Concrete Structures Using an Evolutionary Procedure. ACI Structural Journal, 2000, 97(2): 322-332.
[110] Kong F.K., Sharp G.R.. Structural Idealization for Deep Beams with Web Openings. Magazine of Concrete Research, 1977, 29:81-91.
[111] Rozvany G.I.N.. Exact analytical solutions for some popular benchmark problems in topology optimization. Structural Optimization, 1998, 15: 42-48.
[112] 周克民,胡云昌. 利用有限元构造 Michell 桁架的一种方法. 力学学报,2002, 34(6): 935-940.
[113] Hemp W. S.. Optimum Structure. Oxford: Clarendon Press, 1973: 70-101.
[114] Rozvany G.I.N.. Partial relaxation of the orthogonality requirement for classical Michell trusses. Structural Optimization, 1997, 13: 271-274.
[115] Rozvany G.I.N., Gollub W.. Michell Layouts for Various Combinations of Line Supports—I. International Journal of Mechanical Sciences, 1990, 32(12): 1021-1043.
[116] Rozvany G.I.N., Gollub W., Zhou M.. Exact Michell Layouts for VariousCombinations of Line Supports—Part II. Structural Opitimization, 1997, 14(2~3): 138-149.
[117] 周履. 结构混凝土通向协调设计的压杆-拉杆模型. 国外桥梁,2001,(4):25-35.
[118] 王田友,苏小卒. 钢筋混凝土结构的拉压杆模型设计方法及现状. 四川建筑科学研究,2004,30(3): 18-20.
[119] 张元元,李继袢,刘建军等. 钢筋混凝土拉压杆理论研究与应用探讨. 武汉工业学院学报,2005,24(3): 72-77.
[120] Liang Q.Q.. Performance-based Optimization of Structures. 1. London and New York: Spon Press, 2005: 134-254.
[121] 丁大钧. 结构机理学(8)——深梁. 工业建筑,1995, 25(3): 41-46.
[122] 刘立新. 钢筋混凝土深梁、短梁和浅梁受剪承载力的统一计算方法. 建筑结构学报,1995, 16(4): 13-21.
[123] 王新玲,刘立新. 钢筋混凝土开洞连续深梁在集中荷载作用下抗剪性能的试验研究. 建筑结构学报,1994, 15(4): 11-19.
[124] 朱伯芳.有限单元法原理与应用. 北京:中国水利水电出版社,1998: 530-563.
[125] Thomas T. C. Hsu,Unified approach to shear analysis and design,Cement and Concrete Composites, 1998, 20: 419-435.
[2] 张炳华,侯昶. 土建结构优化设计.第 2 版.上海:同济大学出版社,1998: 1-206.
[3] Narayanan S., Azarm S.. On improving multiobjective genetic algorithms for design optimization. Structural Optimization. 1999, 18(2-3): 146 – 155.
[4] 李芳,凌道盛. 工程结构优化设计发展综述. 工程设计学报,2002,9(5): 229-235.
[5] Michell A.G.M.. The Limits of Economy of Materials in Frame-structures. Philosophical Magazine, 1904, 8(17): 589-597.
[6] Yang X.Y. Bi-directional Evolutionary method for stiffness and displacement optimisation: [Degree of Master of Engineering to Victoria University of Technology]. Melbourne, Australia: Victoria University of Technology, 1999:1-140.
[7] Topping, B.H.V.. Topology design of discrete structures. In: Bends?e, M.P. and Mota Soares, C.A. (eds.). Topology Design of Structure, Netherlands: Kluwer Academic Publishers, 1993: 56-78.
[8] Bends?e, M.P., Kikuchi, N. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanical Engineering,1988,71: 197-224.
[9] Yang,R.J., Chuang,C.H.. Optimal topology design using linear programming. Computers and Structures, 1994, 52: 265-275.
[10] Kikuchin, Suzukik. A homogenization method for shape and topology optimization. Computer Methods in Applied Mechanical Engineering, 1991, 93: 291- 318.
[11] Allaire G., Jouve F., Maillot H.. Topology optimization for minimum stress design with the homogenization method. Structural and Multidisciplinary Optimization, 2004, 28(2-3): 87-98.
[12] 刘书田,程耿东. 基于均匀化理论的梯度功能材料优化设计方法. 宇航材料工艺, 1995, 6: 21-27.
[13] 王晓明,刘震宇,郭东明. 基于均匀化理论的微小型柔性结构拓扑优化的敏度分析. 中国机械工程, 1999, 10 (11) : 1264-1266.
[14] Yang R. J., Chuang C.H.. Optimal topology design using linear programming.Computers & Structures, 1994, 52: 265- 275.
[15] Hassani B., Hinton E.. A review of homogenization and topology optimization I--homogenization theory for media with periodic structure. Computers & Structures,1998, 69: 707-717.
[16] 汪树玉,刘国华,包志仁. 结构优化设计的现状与进展. 基建优化,1999,20(4):3-14.
[17] 姚侗,何淦瞳. 线性规划的一种新算法——直接搜索迭代法. 高校应用数学学报,1997, 12 (1): 73-84.
[18] 汪树玉,刘国华,包志仁. 结构优化设计的现状与进展(续). 基建优化,1999,20(5):3-7.
[19] 王小平,曹立明. 遗传算法——理论、应用与软件实现. 第 1 版. 西安:西安交通大学出版社,2002: 1-50.
[20] Chakraborty S., Mukhopadhyay M., Sha O.P.. Determination of Physical Parameters of Stiffened Plates using Genetic Algorithm. Journal of Computing in Civil Engineering, 2002, 16(3): 206-221.
[21] Pezeshk S., Camp C.V., Chen D,. Design of Nonlinear Framed Structures Using Genetic Optimization. Journal of Structural Engineering, 2000, 126(3): 382-388.
[22] Hong H., Yong X.. Vibration-based Damage Detection of Structures by Genetic Algorithm. Journal of Computing in Civil Engineering, 2002, 16(3): 222-229.
[23] Narayanan S., Azarm S.. On improving multiobjective genetic algorithms for design optimization. Structural Optimization. 1999, 18(2-3): 146 – 155.
[24] Krishnamoorthy C.S., Venkatesh P. P., Sudarshan R.. Object-Oriented Framework for Genetic Algorithms with Application to Space Truss Optimization. Journal of Computing in Civil Engineering, 2002, 16(1): 66-75.
[25] 刘勇,康立山,陈毓屏. 非数值并行算法(第 2 册)——遗传算法. 第 1 版. 北京:科学出版社,1995: 1-60.
[26] 潘正军,康立山,陈毓屏. 演化计算. 第 1 版. 北京:清华大学出版社, 1998: 1-36.
[27] 张文修,梁怡. 遗传算法的数学基础. 第 1 版. 西安:西安交通大学出版社,2000: 1-12.
[28] 周明,孙树栋. 遗传算法原理及应用. 第 1 版. 北京:国防工业出版社,1999: 1-45.
[29] 李敏强,寇纪淞,林丹等. 遗传算法的基本理论与应用. 第 1 版. 北京:科学出版社,2002: 1-70.
[30] 马光文,王黎. 遗传算法在桁架结构优化设计中的应用. 工程力学,1998,15(2): 38-44.
[31] 高峰,王德俊,胡俏等. 离散结构的遗传形状优化设计. 计算力学学报,1998, 15(4): 485-489.
[32] Camp C., Pezeshk S., Cao G. Z.. Optimized design of two-dimensional structures using a genetic algorithm. Journal of Structural Engineering, 1998, 124(5): 551-559.
[33] Rajeev S., Krishnamoorthy C.S.. Genetic algorithms-based methodologies for design optimization of trusses. Journal of Structural Engineering, 1997, 123(3): 350-358.
[34] Hajela P., Lee E., Cho H.. Genetic algorithms in topologic design of grillage structures. Computer-Aided Civil and Infrastructure Engineering, 1998, 13: 13-22.
[35] Kamal C. S., Adeli H.. Fuzzy genetic algorithm for optimization of steel structures. Journal of Structural Engineering, 2000, 126(5): 596-604.
[36] Soh C.K., Yang J.P.. Optimal layout of bridge trusses by genetic algorithms. Computer-Aided Civil and Infrastructure Engineering, 1998, 13: 247-254.
[37] Koumousis V.K., Arsenis S.J.. Genetic algorithms in optimal detailed design of reinforced concrete members. Computer-Aided Civil and Infrastructure Engineering, 1998, 13: 43-52.
[38] Rjeev S., Krishamoorthy C.S.. Genetic algorithm-based methodology for design optimization of reinforced concrete frames. Computer-Aided Civil and infrastructure Engineering, 1998, 13: 63-74.
[39] Ceranic B., Fryer C.. A genetic approach to the minimum cost design of reinforced concrete flanged beams under multiple loading conditions. ACSO, 1998: 71-78.
[40] Rafig M.Y., Southcombe C.. Genetic algorithms in optimal design and detailing of reinforced concrete biaxial columns supported by a declarative approach for capacity checking. Computers & Structures, 1998, 69(4): 443-457.
[41] Woon S.Y., Querin O.M., G.P. Steven. Structural application of a shape optimization method based on a genetic algorithm. Structural and Multidisciplinary Optimization, 2001, 22: 57-64.
[42] Hayalioglu M.S.. Optimum load and resistance factor design of steel space frames using genetic algorithm. Structural and Multidisciplinary Optimization, 2001, 21: 292-299.
[43] Concei??o Ant?nio C.A.. A multilevel genetic algorithm for optimization ofgeometrically nonlinear stiffened composite structures. Structural and Multidisciplinary Optimization, 2002, 24: 372-386.
[44] 陈存恩,李文雄. 结合灵敏度修正的遗传算法的结构损伤诊断. 地震工程与工程振动, 2006, 26(5): 172-176.
[45] 马祥森,史治宇. 基于 GA-BP 神经网络的结构损伤位置识别. 振动工程学报,2004, 17(4): 453-456.
[46] 易伟建,刘霞. 结构损伤诊断的遗传算法研究. 系统工程理论与实践,2001, (5): 114-118.
[47] 易伟建,刘霞. 混凝土梁板类构件边界条件识别与研究. 湖南大学学报,2000, 27(4): 81-87.
[48] 易伟建,刘霞. 基于遗传算法的结构损伤诊断研究. 工程力学,2001, 18(2): 64-71.
[49] Dumn S.A.. The Use of Genetic Algorithms and Stochastic Hill-Climbing in Dynamic Finite Element Model Identification. Computers & Structures, 1998, 66(4): 489-497.
[50] Friswell M.I., Penny J.E.T., Garvey S.D.. A combined genetic and eigensensitivity algorithm for the location of damage in structures. Computers & Structures, 1998, 69(5): 547-556.
[51] Bishop J.A., Striz A.G.. On using genetic algorithms for optimum damper placement in space trusses. Structural and Multidisciplinary Optimization, 2004, 28(2-3): 136-145.
[52] Salajegheh E., Heidari A.. Optimum design of structures against earthquake by adaptive genetic algorithm using wavelet networks. Structural and Multidisciplinary Optimization, 2004, 28(4): 277-285.
[53] 田军,寇纪松,李敏强. 利用遗传算法优化施工网络计划. 系统工程理论与实践,1999, 19(5): 78-81.
[54] 康立山,谢云,尤矢勇等. 非数值并行算法——模拟退火算法. 第 1 版. 北京:科学出版社,1998: 1-55.
[55] 邢文训,谢金星. 现代优化计算方法. 第 1 版. 北京:清华大学出版社,1999: 1-36.
[56] 吴剑国,赵莉萍,王建华. 工程结构混合离散变量优化的模拟退火方法. 工程力学, 1997, 14(3): 138-143.
[57] Patrick Y. S., Manoochehri S.. Generating Optimal Configurations in Structural Design Using Simulated Annealing. International Journal for Numerical Methods in Engineering, 1997, 40: 1053-1069.
[58] Shyh-Rong T., Chris P.P.. Annealing strategy for optimal structural design. Journal of Structural Engineering, 1996, 122(7): 815-827.
[59] Barski M., Kru˙zelecki J.. Optimal design of shells against buckling by means of the simulated annealing method. Structural and Multidisciplinary Optimization, 2004, 29(1): 61-72.
[60] 刘先珊,周创兵,张立君. 基于模拟退火的 Gauss-Newton 算法的神经网络在渗流反分析中的应用. 岩土力学, 2005, 26(3): 404-414.
[61] 李红芳,侯朝胜. 模拟退火遗传算法与结构优化设计. 天津理工学院学报,2004, 20(4): 96-98.
[62] 邹广电,陈永平. 滑坡和边坡稳定性分析的模拟退火算法-随机搜索耦合算法. 岩石力学与工程学报, 2004, 23(12): 2032-2037.
[63] 吴新生,谢益民. 基于人工神经网络优化抄纸浆料配比. 中国造纸,2005, 24(5): 44-46.
[64] 宋小莉,牛欣,司银楚. 基于 BP 神经网络的半夏、生姜、甘草三泻心汤配伍研究. 中国临床药理学与治疗学, 2005, 10(5): 527-531.
[65] 郭一楠,巩敦卫,程健. 基于分布式神经网络的焦炭质量预测模型. 中国矿业大学学报,2005, 34(4): 514-517.
[66] 荣辉,张济世. 人工神经网络及其现状与展望. 电子技术应用,1995, 10: 4-5.
[67] 王科俊,王克成. 神经网络建模预报与控制. 第 1 版. 哈尔滨:哈尔滨工程大学出版社,1996: 1-47.
[68] 赵振宇,徐用懋. 模糊理论和神经网络的基础与应用. 第 1 版. 北京:清华大学出版社,广西科学技术出版社, 1996: 1-13.
[69] 吴剑国,赵丽萍. 工程结构优化的神经网络方法. 计算力学学报,1998, 15(1): 69-74.
[70] 姜绍飞,刘之洋. 人工神经网络在外包钢混凝土框架柱设计参数预报中的适应性. 土木工程学报,1997, 30(1): 73-77.
[71] 郭国会,易伟建. 基于神经网络的结构边界条件识别和损伤诊断. 湖南大学学报,1998, 25(1): 71-76.
[72] 李海岭,霍达,王志忠. 考虑土-结构动力相互作用影响的结构神经网络控制. 土木工程学报,2002, 35(3): 61-65.
[73] 姜绍飞,钟善桐. 神经网络在结构工程中的应用. 哈尔滨建筑大学学报,1998,31(6): 129-134.
[74] 李士勇,陈永强,李研. 蚁群算法及其应用. 第 1 版. 哈尔滨:哈尔滨工业大学出版社,2004: 1-40.
[75] Dorigo M., Gambardella L M. Ant colonies for the traveling salesman problem.BioSysterms, 1997, 43: 73-81.
[76] Dorigo M., Caro G. D.. Ant algorithms for discrete optimization. Artificial Life, 1999, 5(3): 137-172.
[77] Xie Y.M., Steven G.P.. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885-896.
[78] Xie Y.M., Steven G.P.. A simple approach to structural frequency optimization. Computers & Structures, 1994, 53(6): 1487-1491.
[79] Xie Y.M., Steven G.P.. Evolutionary structural optimization for dynamic problems. Computers & Structures, 1996, 58(6): 1067-1073.
[80] Zhao C. B., Steven G. P., Xie Y. M.. Effect of initial nondesign domain on optimal topologies of structures during natural frequency optimization. Computers & Structures, 1997, 62(1): 119-131.
[81] Zhao C.B., Steven G. P., Xie Y. M.. A generalized evolutionary method for natural frequency optimization of membrane vibration problems in finite element analysis. Computers & Structures, 1998, 66(2-3): 353-364.
[82] Chu D.N., Xie Y.M., Hira A., et.al.. Evolutionary structural optimization for problems with stiffness constrains. Finite Elements in Analysis and Design, 1996, 21: 239-251.
[83] Chu D.N., Xie Y.M., Hira A., et.al.. On various aspects of evolutionary structural optimization for problems with stiffness constraints. Finite Elements in Analysis and Design, 1997, 24: 197-212.
[84] Chu D. N., Xie Y. M., Steven G. P.. An evolutionary structural optimization method for sizing problems with discrete design variables. Computers & Structures, 1998, 68: 419-431.
[85] Li Q., Steven G. P., Xie Y. M.. Evolutionary structural optimization for stress minimization problems by discrete thickness design. Computers & Structures, 2000, 78: 769-780.
[86] Manickarajah D, Xie Y. M., Steven G. P.. Optimum design of frames with multiple constraints using an evolutionary method. Computers & Structures, 2000, 74: 731-741.
[87] Querin O.M., Steven G.P., Xie Y.M.. Evolutionary Structural Optimisation Using an Additive Algorithm. Finite Elements in Analysis and Design, 2000, 34: 291-308.
[88] Kurpati A., Azarm S., Wu J.. Constraint Handling Improvements for Multiobjective Genetic Algorithms. Structural and MultidisciplinaryOptimization, 2002, 23: 204-213.
[89] Querin O.M., Young V., Steven G.P., et al. Computational Efficiency and Validation of Bi-directional evolutionary Structural optimisation. Computer Methods in Applied Mechanics and Engineering, 2000, 189: 559-573.
[90] Manickarajah D., Xie Y. M., Steven G. P.. An evolutionary method for optimization of plate buckling resistance. Finite Elements in Analysis and Design, 1998, 29: 205-230.
[91] Li Q., Steven G. P., Querin O. M., et al. Shape and Topology Design for Heat Conduction by Evolutionary Structural Optimization. International Journal of Heat and Mass Transfer, 1999, 42:3361-3371.
[92] Li Q., Steven G. P., Xie Y. M.. Displacement Minimization of Thermoelastic Structures by Evolutionary Thickness Design. Computer Methods in Applied Mechanics and Engineering, 1999, 179: 361-378.
[93] Das R., Jones R., Xie Y.M.. Design of Structures for Optimal Static Strength Using ESO. Engineering Failure Analysis, 2005, 12: 61-80.
[94] Steven G. P., Li Q., Xie Y. M.. Multicriteria optimization that minimizes maximum stress and maximizes stiffness. Computers & Structures, 2002, 80: 2433-2448.
[95] Li W., Li Q., Steven G. P., et al. An Evolutionary Approach to Elastic Contact Optimization of Frame Structures. Finite Elements in Analysis and Design, 2003, 40: 61-81.
[96] Li Q., Steven G. P., Querin O. M., et al. Stress Based Optimization of Torsional Shafts Using an Evolutionary Procedure. International Journal of Solids and Structures, 2001, 38: 5661-5677.
[97] Rozvany G.I.N.. Stress ration and compliance based methods in topology optimization—a critical review. Structural and Multidisciplinary Optimization, 2001, 21: 109-119.
[98] Liang Q. Q., Xie Y. M., Steven G. P.. Optimal topology selection of continuum structures with displacement constraints. Computers & Structures, 2000, 77: 635-644.
[99] Li Q., Steven G. P., Querin O. M., et.al. Evolutionary shape optimization for stress minimization. Mechanics Research Communications, 1999, 26(6): 657-664.
[100] 荣见华,郑健龙,徐飞鸿. 结构动力修改及优化设计. 第一版. 北京:人民交通出版社,2002: 249-335.
[101] Tanskanen P.. The evolutionary structural optimization method: theoretical aspects. Computer methods in applied mechanics and engineering, 2002, 191: 5485-5498.
[102] Cox H.L.. The Design of Structures of Least Weight. Oxford: Pergamon Press, 1965: 80-114.
[103] Zhou M., Rozvany G. I. N.. On the validity of ESO type methods in topology optimization. Structural and Multidisciplinary Optimization, 2001, 21: 80-83.
[104] Cheng F. Y.,Li D.. Multiobjective Optimization Design with Pareto Genetic Algorithm. Journal of Structural Engineering, 1997, 123(9): 1252-1261.
[105] Lin C.Y., Chou J.N.. A Two-stage Approach for Structural Topology Optimization. Advances in Engineering Software, 1999, 30: 261-271.
[106] 周克民,李俊峰,李霞. 结构拓扑优化研究方法综述. 力学进展,2005,35(1):69-76.
[107] Rozvany G.I.N.. Aims, Scope, Methods, History and Unified Terminology of Computer-aided Topology Optimization in Structural Mechanics. Structural and Multidisciplinary Optimization, 2001, 21: 90-108.
[108] Liang Q. Q., Xie Y. M., Steven G. P.. Generating Optimal Strut-and-Tie Models in Prestressed Concrete Beams by Performance-Based Optimization. ACI Structural Journal, 2001, 98(2): 226-232.
[109] Liang Q. Q., Xie Y. M., Steven G. P.. Topology Optimization of Strut-and-Tie Models in Reinforced Concrete Structures Using an Evolutionary Procedure. ACI Structural Journal, 2000, 97(2): 322-332.
[110] Kong F.K., Sharp G.R.. Structural Idealization for Deep Beams with Web Openings. Magazine of Concrete Research, 1977, 29:81-91.
[111] Rozvany G.I.N.. Exact analytical solutions for some popular benchmark problems in topology optimization. Structural Optimization, 1998, 15: 42-48.
[112] 周克民,胡云昌. 利用有限元构造 Michell 桁架的一种方法. 力学学报,2002, 34(6): 935-940.
[113] Hemp W. S.. Optimum Structure. Oxford: Clarendon Press, 1973: 70-101.
[114] Rozvany G.I.N.. Partial relaxation of the orthogonality requirement for classical Michell trusses. Structural Optimization, 1997, 13: 271-274.
[115] Rozvany G.I.N., Gollub W.. Michell Layouts for Various Combinations of Line Supports—I. International Journal of Mechanical Sciences, 1990, 32(12): 1021-1043.
[116] Rozvany G.I.N., Gollub W., Zhou M.. Exact Michell Layouts for VariousCombinations of Line Supports—Part II. Structural Opitimization, 1997, 14(2~3): 138-149.
[117] 周履. 结构混凝土通向协调设计的压杆-拉杆模型. 国外桥梁,2001,(4):25-35.
[118] 王田友,苏小卒. 钢筋混凝土结构的拉压杆模型设计方法及现状. 四川建筑科学研究,2004,30(3): 18-20.
[119] 张元元,李继袢,刘建军等. 钢筋混凝土拉压杆理论研究与应用探讨. 武汉工业学院学报,2005,24(3): 72-77.
[120] Liang Q.Q.. Performance-based Optimization of Structures. 1. London and New York: Spon Press, 2005: 134-254.
[121] 丁大钧. 结构机理学(8)——深梁. 工业建筑,1995, 25(3): 41-46.
[122] 刘立新. 钢筋混凝土深梁、短梁和浅梁受剪承载力的统一计算方法. 建筑结构学报,1995, 16(4): 13-21.
[123] 王新玲,刘立新. 钢筋混凝土开洞连续深梁在集中荷载作用下抗剪性能的试验研究. 建筑结构学报,1994, 15(4): 11-19.
[124] 朱伯芳.有限单元法原理与应用. 北京:中国水利水电出版社,1998: 530-563.
[125] Thomas T. C. Hsu,Unified approach to shear analysis and design,Cement and Concrete Composites, 1998, 20: 419-435.