结构拓扑优化方法研究及其在螺旋锥齿轮机床中的应用
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摘要
摘要：螺旋锥齿轮是机械动力设备的关键零件。用于加工螺旋锥齿轮的装备的设计和制造,在工业发展中具有重要的地位。在数控螺旋锥齿轮机床设计和制造中,机床结构的设计布局始终是高性能机床创新的关键。目前,结构拓扑优化方法是提高结构性能的重要手段。为了解决复杂结构拓扑优化低的计算效率等问题,本文研究了拓扑优化方法,并将其应用到数控螺旋锥齿轮机床的优化中,进一步改进了机床的性能。
针对机床刚度的要求,提出了柔顺度最小化和给定位移约束的条件下的结构拓扑优化方法,解决了机床大件的静刚度优化；然后,研究了动刚度的拓扑优化问题,有效的解决了指定动响应约束的优化；最后,基于节点密度法,研究了结构稳定性的拓扑优化问题,为保证结构设计的稳定性提供了有效手段。针对上面提到的研究问题,完成了如下研究工作：
(1)为了提高现有拓扑优化方法解决实际问题的能力,研究了静刚度相关的拓扑优化方法。针对刚度最大化的设计要求,结合设计空间调整策略,体积相关的灵敏度过滤算法,以及变约束限的体积控制方式,提出了一种新的考虑柔顺度要求的结构拓扑优化方法,并采用准则优化法求解了优化模型,提高了方法的计算效率；针对类似板梁组合,同时考虑拓扑和尺寸的优化问题,运用材料属性的有理近似模型,结合归一化处理方法,以及对偶优化原理和二次规划法,提出了一种考虑位移约束的组合结构优化方法。
(2)为了控制随机激励力作用下结构的响应,提出了考虑动响应约束的结构拓扑优化方法。采用材料属性有理近似模型建立了单元伪密度与单元材料属性之间的关系,同时,引入一种删除低密度单元策略,共同作用消除了动力学设计中易出现的局部模态现象。采用变动响应约束限的方式,使得动响应约束的显式表达式近似有效；通过单元增删策略实现了设计空间的自动调整。最后,基于对偶优化和二次规划法求解优化模型。
(3)针对连续体结构多工况和多约束的状态,提出了节点密度的拓扑优化方法。以此为基础,研究了以位移或屈曲特征值为约束的拓扑优化方法。在优化方法中,以有限元网格的节点密度作为设计变量。设计域内任意点的密度通过Shepard插值方法和Heaviside函数插值得到。没有采用其它的处理方式,比如过滤技术,得到了网格独立的、无棋盘格模式的、清晰的优化拓扑结构。基于共享内存的并行计算技术,采用OpenMP编程方法编写了优化程序,极大的提高了本方法的计算效率。
(4)基于Abaqus平台,采用Fortran语言开发了拓扑优化软件,并通过简单算例完成了提出的优化方法考核。而后,基于经过试验验证正确的数控螺旋锥齿轮机床有限元模型,对机床床身、立柱、滑台和刀具箱进行了优化设计,实现了数控螺旋锥齿轮机床的拓扑优化设计,提高了机床的刚度和稳定性。图82幅,表11个,参考文献222篇
Abstract:Spiral bevel gears are the key components of mechanical power equipment. The design and manufacture of the equipment for machining spiral bevel gears are very important in the industrial development. In the design and manufacture of the CNC spiral bevel gear machine tool, the structure design of the machine tool is always crucial to the innovation of the high performance machine tool. Currently, the structural topology optimization method is very important tool to improve the structural performance. In this dissertation, to overcome the difficulties of the low computational efficiency of the complex structural topology optimization and so on, the topology optimization methods are investigated. Then, the methods are applied to the optimization of the machine tool components and the machine tool performance is improved further.
For the requirement of the machine tool stiffness, the topology optimization methods of minimum compliance or displacement constraints are proposed, the stiffness optimizations of the big machine tool parts are settled; then, the topology optimization method of considering dynamic stiffness is studied, the topology optimization method subject to dynamic response is solved efficiently; lastly, based on the nodal density, the topology optimization method with structural stability constraints is investigated, a valid method is formulated for guaranteeing the stability of the designed structure. As a result, the following innovative works are completed:
(1) To improve the ability of the existing topology optimization to dispose of practical problems, the topology optimization methods relative to static stiffness are researched. For the design requirement of maximum stiffness, incorporating the design space adjustment strategy, sensitivity filtering algorithm relative to volumes, the varying volume constraints, a new structural topology optimization method of minimum compliance is proposed. The model is solved by optimality criteria, and the computational efficiency is improved; For the optimization of the structures which consist of planes and beams, the topology and size optimization are considered at the same time, the RAMP material model and the normalization method are used, a combinatorial optimization method subject to displacement constraints is presented based on the dual optimization and the quadratic programming.
(2) The topology optimization considering dynamic response is proposed to control the response under the random excitation. Based on RAMP method, the relations of element pseudo-density and material property are built, at the same time, the scheme of removing the lowest density elements is introduced to avoid the occurrence of localized mode. In order to make the explicit approximation of dynamic response constraints be effective, a new dynamic response constraints with varying constraint limits are introduced. Moreover, structural optimization strategies of the element deletion and addition criterion are given to automatically adjust the design space. Finally, the optimization problem is solved by the dual optimization and the quadratic programming.
(3) For the multiple load cases and constraints state of continuum structure, a topology optimization method based on the nodal density is developed. Based on the method, the topology optimization methods subject to displacement or buckling eigenvalue constraints are carried out. Nodal densities of the finite element mesh are used as the design variables. The nodal densities are interpolated into any point in the design domain by the Shepard interpolation scheme and the Heaviside function. Without using additional constraints (such as the filtering technique), mesh-independent, checkerboard-free, distinct optimal topology can be obtained. The computational efficiency is greatly improved by multithread parallel computing with OpenMP which is portable shared memory parallel programming.
(4) Base on Abaqus software, the topology optimization methods are programmed using FORTRAN programming language. The validity of proposed topology optimization methods is verified by some numerical examples. Then, the bed, column, slide and cuter-box of the machine tool, which are tested and verified, are optimized based on their finite element models, respectively. An innovative design of CNC spiral bevel gear machine tools is obtained, incorporating the component topology optimization results. The stiffness and stability of the machine tool are improved.
针对机床刚度的要求,提出了柔顺度最小化和给定位移约束的条件下的结构拓扑优化方法,解决了机床大件的静刚度优化；然后,研究了动刚度的拓扑优化问题,有效的解决了指定动响应约束的优化；最后,基于节点密度法,研究了结构稳定性的拓扑优化问题,为保证结构设计的稳定性提供了有效手段。针对上面提到的研究问题,完成了如下研究工作：
(1)为了提高现有拓扑优化方法解决实际问题的能力,研究了静刚度相关的拓扑优化方法。针对刚度最大化的设计要求,结合设计空间调整策略,体积相关的灵敏度过滤算法,以及变约束限的体积控制方式,提出了一种新的考虑柔顺度要求的结构拓扑优化方法,并采用准则优化法求解了优化模型,提高了方法的计算效率；针对类似板梁组合,同时考虑拓扑和尺寸的优化问题,运用材料属性的有理近似模型,结合归一化处理方法,以及对偶优化原理和二次规划法,提出了一种考虑位移约束的组合结构优化方法。
(2)为了控制随机激励力作用下结构的响应,提出了考虑动响应约束的结构拓扑优化方法。采用材料属性有理近似模型建立了单元伪密度与单元材料属性之间的关系,同时,引入一种删除低密度单元策略,共同作用消除了动力学设计中易出现的局部模态现象。采用变动响应约束限的方式,使得动响应约束的显式表达式近似有效；通过单元增删策略实现了设计空间的自动调整。最后,基于对偶优化和二次规划法求解优化模型。
(3)针对连续体结构多工况和多约束的状态,提出了节点密度的拓扑优化方法。以此为基础,研究了以位移或屈曲特征值为约束的拓扑优化方法。在优化方法中,以有限元网格的节点密度作为设计变量。设计域内任意点的密度通过Shepard插值方法和Heaviside函数插值得到。没有采用其它的处理方式,比如过滤技术,得到了网格独立的、无棋盘格模式的、清晰的优化拓扑结构。基于共享内存的并行计算技术,采用OpenMP编程方法编写了优化程序,极大的提高了本方法的计算效率。
(4)基于Abaqus平台,采用Fortran语言开发了拓扑优化软件,并通过简单算例完成了提出的优化方法考核。而后,基于经过试验验证正确的数控螺旋锥齿轮机床有限元模型,对机床床身、立柱、滑台和刀具箱进行了优化设计,实现了数控螺旋锥齿轮机床的拓扑优化设计,提高了机床的刚度和稳定性。图82幅,表11个,参考文献222篇
Abstract:Spiral bevel gears are the key components of mechanical power equipment. The design and manufacture of the equipment for machining spiral bevel gears are very important in the industrial development. In the design and manufacture of the CNC spiral bevel gear machine tool, the structure design of the machine tool is always crucial to the innovation of the high performance machine tool. Currently, the structural topology optimization method is very important tool to improve the structural performance. In this dissertation, to overcome the difficulties of the low computational efficiency of the complex structural topology optimization and so on, the topology optimization methods are investigated. Then, the methods are applied to the optimization of the machine tool components and the machine tool performance is improved further.
For the requirement of the machine tool stiffness, the topology optimization methods of minimum compliance or displacement constraints are proposed, the stiffness optimizations of the big machine tool parts are settled; then, the topology optimization method of considering dynamic stiffness is studied, the topology optimization method subject to dynamic response is solved efficiently; lastly, based on the nodal density, the topology optimization method with structural stability constraints is investigated, a valid method is formulated for guaranteeing the stability of the designed structure. As a result, the following innovative works are completed:
(1) To improve the ability of the existing topology optimization to dispose of practical problems, the topology optimization methods relative to static stiffness are researched. For the design requirement of maximum stiffness, incorporating the design space adjustment strategy, sensitivity filtering algorithm relative to volumes, the varying volume constraints, a new structural topology optimization method of minimum compliance is proposed. The model is solved by optimality criteria, and the computational efficiency is improved; For the optimization of the structures which consist of planes and beams, the topology and size optimization are considered at the same time, the RAMP material model and the normalization method are used, a combinatorial optimization method subject to displacement constraints is presented based on the dual optimization and the quadratic programming.
(2) The topology optimization considering dynamic response is proposed to control the response under the random excitation. Based on RAMP method, the relations of element pseudo-density and material property are built, at the same time, the scheme of removing the lowest density elements is introduced to avoid the occurrence of localized mode. In order to make the explicit approximation of dynamic response constraints be effective, a new dynamic response constraints with varying constraint limits are introduced. Moreover, structural optimization strategies of the element deletion and addition criterion are given to automatically adjust the design space. Finally, the optimization problem is solved by the dual optimization and the quadratic programming.
(3) For the multiple load cases and constraints state of continuum structure, a topology optimization method based on the nodal density is developed. Based on the method, the topology optimization methods subject to displacement or buckling eigenvalue constraints are carried out. Nodal densities of the finite element mesh are used as the design variables. The nodal densities are interpolated into any point in the design domain by the Shepard interpolation scheme and the Heaviside function. Without using additional constraints (such as the filtering technique), mesh-independent, checkerboard-free, distinct optimal topology can be obtained. The computational efficiency is greatly improved by multithread parallel computing with OpenMP which is portable shared memory parallel programming.
(4) Base on Abaqus software, the topology optimization methods are programmed using FORTRAN programming language. The validity of proposed topology optimization methods is verified by some numerical examples. Then, the bed, column, slide and cuter-box of the machine tool, which are tested and verified, are optimized based on their finite element models, respectively. An innovative design of CNC spiral bevel gear machine tools is obtained, incorporating the component topology optimization results. The stiffness and stability of the machine tool are improved.
引文
[1]良辰.我国高档数控装备发展中应重点关注的问题——访全国高校先进制造技术与机床研究学会秘书长梅雪松[J].航空制造技术,2013,(1-2)：65-67.
[2]钱令希,钟万勰,隋允康,等.工程结构优化设计[J].工程机械,1988,(5)：22-26+55.
[3]辛志杰.面向产品族的机床结构可适应动态设计理论、方法与应用[D].天津：天津大学,2008.
[4]阎楚良,杨方飞.机械数字化设计新技术[M].机械工业出版社,2007
[5]Winner Robert I. The role of concurrent engineering in weapons systems acquisition[M]. Institute for Defence Analyses,1988
[6]Tseng Mitchell M, Jiao Jianxin, Merchant M Eugene. Design for mass customization[J]. CIRP Annals-Manufacturing Technology,1996,45(1):153-156.
[7]Koenigsberger Franz, Tlusty Jiri. Machine tool structures [M]. Pergamon Press Oxford, England,1970
[8]徐燕申,张兴朝,牛占文,等.基于元结构和框架优选的数控机床床身结构动态设计研究[J].机械强度,2001,23(1)：1-3+110.
[9]陈新,陈新度,秦叶,等.机械结构动态设计若干关键技术[J].中国机械工程,1997,8(5)：104-108.
[10]郭春星,丁晓红,郭媛美,等.磨床床身结构优化设计[J].机械设计与研究,2009,25(5)：104-107.
[11]陈新,何杰,毛海军,等.基于动力学特征的磨床床身结构优化布局设计[J].制造技术与机床,2001,(2)：25-26+23.
[12]Zhao Ling, Ma Jianfeng, Wang Ting, et al. Lightweight design of mechanical structures based on structural bionic methodology[J]. Journal of Bionic Engineering,2010,7(S): S224-S231.
[13]Ma Jian-feng, Chen Wu-yi, Zhao Ling, et al. Elastic buckling of bionic cylindrical shells based on bamboo[J]. Journal of Bionic Engineering,2008,5(3):231-238.
[14]Liu Shihao, Ye Wenhua, Lou Peihuang, et al. Bionic design for column of gantry machining center to improve the static and dynamic performance [J]. Shock and Vibration,2012,19(4):493-504.
[15]肖利利,陈蔚芳,叶文华,等.“箱中箱”结构立滑板的有限元分析及其优化设计[J].山东大学学报(工学版),2010,40(1)：78-83.
[16]王琦.采用重心驱动(DCG)的机床[J].制造技术与机床,2005,(2)：21-22.
[17]卢天健,张钱城,王春野,等.轻质材料和结构在机床上的应用[J].力学与实践,2007,29(6)：1-8.
[18]Kim D I, Jung S C, Lee J E, et al. Parametric study on design of composite-foam-resin concrete sandwich structures for precision machine tool structures [J]. Composite Structures,2006,75(1):408-414.
[19]Chang Seung Hwan, Kim Po Jin, Lee Dai Gil, et al.Steel-composite hybrid headstock for high-precision grinding machines[J]. Composite structures,2001,53(1):1-8.
[20]Lee Dai Gil, Chang Seung Hwan, Kim Hyun Surk. Damping improvement of machine tool columns with polymer matrix fiber composite material[J]. Composite Structures, 1998,43(2):155-163.
[21]Rahman M, Mansur M A, Feng Zhou. Design, fabrication and evaluation of a steel fibre reinforced concrete column for grinding machines[J]. Materials & Design,1995,16(4): 205-209.-
[22]王艳辉,伍建国,缪建成.精密机床床身的模态分析与结构优选[J].机械设计与制造,2005,3(3)：76-77.
[23]米成秋,孙靖民.机床结构优化方法初探[J].哈尔滨工业大学学报,1983,(3)：8.
[24]Tandon V, EI-Mounayri H, Kishawy H. NC end milling optimization using evolutionary computation[J]. International Journal of Machine Tools and Manufacture,2002,42(5): 595-605.
[25]汪列隆,朱壮瑞,张建润.基于APDL的机床大件筋板结构尺寸的优化设计[J].轻工机械,2006,24(4)：60-62.
[26]戴磊,关振群,单菊林,等.机床结构三维参数化形状优化设计[J].机械工程学报,2008,44(5)：152-159.
[27]汤文成,易红,唐寅.机床大件结构的拓扑优化设计[J].东南大学学报,1996,(5)：24-28.
[28]饶柳生,侯亮,潘勇军.基于拓扑优化的机床立柱筋板改进[J].机械设计与研究,2010,26(1)：87-92.
[29]张永存,崔雷,周玲丰,等.基于拓扑优化的机床床鞍创新构型设计[J].固体力学学报,2011,32(S)：335-342.
[30]童水光,刘或,张健,等.基于拓扑优化的卧式旋压机床身加强肋布局优化设计方法[J].机械设计,2012,29(6)：31-35.
[31]张向宇,熊计,郝锌,等.基于ANSYS的加工中心滑座拓扑优化设计[J].制造技术与机床,2008,(6)：68-70.
[32]刘舜尧,李进,贺浩.基于渐进结构优化的取料梁腹板拓扑优化[J].工程设计学报,2011,18(3)：174-177.
[33]涂星.YDG型堆料机臂架形变分析及拓扑结构优化设计[D].长沙：中南大学,2011.
[34]徐岩.汽车零部件结构的拓扑优化设计[D].长春：吉林大学,2009.
[35]郭立群.商用车车架拓扑优化轻量化设计方法研究[D].长春：吉林大学,2011.
[36]杜春江.连续体结构拓扑优化理论及其在炮塔结构设计中的应用研究[D].南京：南京理工大学,2008.
[37]Michell A G M. The limits of economy of material in frame structures [J]. Philosophical Magazine,1904,8(6):589-597.
[38]Rozvany G I N. Grillages of maximum strength and maximum stiffness[J]. International Journal of Mechanical Sciences,1972,14(10):1217-1222.
[39]Dorn W S, Gomory R E, Greenberg H J. Automatic design of optimal structures[J]. Journal de Mecanique,1964,3(6):25-52.
[40]Rossow M P, Taylor J E. A finite element method for the optimal design of variable thickness sheets[J]. AIAA Journal,1973,11 (11):1566-1569.
[41]Bendsoe Martin Philip, Kikuchi Noboru. Generating optimal topologies in structural design using a homogenization methodfJ]. Computer Methods in Applied Mechanics and Engineering,1988,71(2):197-224.
[42]Guedes J M, Kikuchi N. Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[J]. Computer Methods in Applied Mechanics and Engineering,1990,83(2):143-198.
[43]Suzuki Katsuyuki, Kikuchi Noboru.A homogenization method for shape and topology optimization[J]. Computer Methods in Applied Mechanics and Engineering,1991,93 (3):291-318.
[44]Diaz A R, Bendsoe M P. Shape optimization of structures for multiple loading conditions using a homogenization method [J]. Structural and Multidisciplinary Optimization,1992, 4(1):17-22.
[45]Michel J C, Moulinec H, Suquet P. Effective properties of composite materials with periodic microstructure:a computational approach[J]. Computer Methods in Applied Mechanics and Engineering,1999,172(1-4):109-143.
[46]Hassani B, Hinton E. A review of homogenization and topology I-homogenization theory for media with periodic structure[J]. Coputers and Structures,1998,69(6):707-717.
[47]Hassani B, Hinton E. A review of homogenization and topology opimization Ⅱ— analytical and numerical solution of homogenization equations [J]. Coputers and Structures,1998,69(6):719-738.
[48]Hassani B, Hinton E. A review of homogenization and topology Ⅲ-topology optimization using optimality criteria[J]. Coputers and Structures,1998,69(6):739-756.
[49]刘书田,程耿东.复合材料应力分析的均匀化方法[J].力学学报,1997,29(3)：306- 313.
[50]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[J].中国机械工程,1999,10(11)：1264-1267.
[51]袁振,吴长春.复合材料周期性线弹性微结构的拓扑优化设计[J].固体力学学报,2003,24(1)：40-45.
[52]王书亭,左孔天.基于均匀化理论的拓扑优化算法研究[J].华中科技大学学报,2004,32(10)：25-30.
[53]谢先海,廖道训.基于均匀化方法的柔顺机构的设计[J].中国机械工程,2003,14(11)：953-956.
[54]Rozvany G I N, Zhou M, Birker T. Generalized shape optimization without homogenization[J].Structural and Multidisciplinary Optimization,1992,4(3):250-252.
[55]Bendsoe M P, Sigmund O. Topology optimization:theory, methods, and applications [M]. Springer-Verlag,2003
[56]Bendsoe M P, Sigmund O. Material interpolation schemes in topology optimization[J]. Archive of Applied Mechanics,1999,69(9-10):635-654.
[57]Luo Z, Chen L, Yang J, et al. Compliant mechanism design using multi-objective topology optimization scheme of continuum structures[J]. Structural and Multidisciplinary Optimization,2005,30(2):142-154.
[58]罗震,陈立平,张云清,等.多工况下连续体结构的多刚度拓扑优化设计和二重敏度过滤技术[J].固体力学学报,2005,26(1)：29-36.
[59]Rozany G I N. Aims scope method history and unified terminology of computer-aided topology optimization in structure mechanics [J]. Structural and Multidisciplinary Optimization,21:90-108.
[60]袁振,吴长春,庄守兵.基于杂交元和变密度法的连续体结构拓扑优化设计[J].中国科学技术大学学报,2001,31(6)：694-699.
[61]Martinez J M. A note on the theoretical convergence properties of the SIMP method [J]. Structural and Multidisciplinary Optimization,2004,29(4):319-323
[62]邓扬晨,张卫红,刘晓欧,等.拓扑优化设计中刚度与密度关系的一种力学模型[J].航空计算技术,2004(01)：1-4.
[63]左孔天.连续体结构拓扑优化理论与应用研究[D].武汉：华中科技大学,2004.
[64]左孔天,钱勤,赵雨东,等.热固耦合结构的拓扑优化设计研究[J].固体力学学报,2005,26(4)：447-452.
[65]隋允康,彭细荣.结构拓扑优化ICM方法的改善[J].力学学报,2005,37(2)：190-198.
[66]隋允康,任旭春,龙连春,等.基于ICM方法的刚架拓扑优化[J].计算力学学报, 2003,20(3)：286-289.
[67]隋允康,杨德庆,孙焕纯.统一骨架与连续体德结构拓扑优化的ICM理论与方法[J].计算力学学报,2000,17(1)：28-33.
[68]隋允康,叶红玲,杜家政.结构拓扑优化的发展及其模型转化为独立层次的迫切性[J].工程力学,2005,22(S)：107-118.
[69]叶红玲,隋允康.应力约束下连续体结构拓扑优化的ICM方法[J].北京工业大学学报,2005,31(S1)：1-5.
[70]彭细荣,隋允康.用ICM法拓扑优化静位移及频率约束下连续体结构[J].计算力学学报,2006,23(4)：391-396.
[71]朱润,隋允康.基于ICM方法的多工况位移约束下板壳结构拓扑优化设计[J].固体力学学报,2012,33(1)：81-90.
[72]Osher S, Sethian J A. Fronts propagating with curvature-dependent speed:algorithms based on Hamilton-Jacobi formulations [J]. Journal of Computational Physics,1988,79 (1):12-49.
[73]Sethian J A. Level set methods and fast marching methods:evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science [M]. Cambridge University Press,1999
[74]Osher S, Fedkiw R P. Level set methods:an overview and some recent results[J]. Journal of Computational Physics,2001,169(2):463-502.
[75]Osher S J, Santosa F. Level set methods for optimization problems involying geometry and constraints:I. Frequencies of a two-density inhomogeneous drum[J]. Journal of Computational Physics,2001,171(1):272-288.
[76]Sethian J A, Wiegmann Andreas. Structural bondary design via level set and immersed interface methods[J]. Journal of Computational Physics,2000,163(2):489-528.
[77]梅玉林.拓扑优化的水平集法及其在刚性结构、柔性结构和材料设计种的应用[D].大连理工大学,2004.
[78]Wang M Y, Wang X, Guo D. A level set method for structural topology optimization[J]. Computer Methods in Applied Mechanics and Engineering,2003,192(1):227-246.
[79]Wang X, Wang M Y, Guo D. Structural shape and topology optimization in a level-set-based framework of region representation[J]. Structural and Multidisciplinary Optimization,2004,27(1):1-19.
[80]庄春刚,熊振华,丁汉.三维线弹性连续体的结构拓扑优化设计[J].中国机械工程,2007,18(14)：1703-1708.
[81]Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization[J]. Computers & Structures,1993,49(5):885-896.
[82]Xie Y M, Steven G P. Evolutionary structural optimization[M]. Springer,1997
[83]Xie Y M, Steven G P. Evolutionary structural optimization for dynamic problems[J]. Computers and Structures,1996,58(6):1067-1073.
[84]Chu D Nha, Xie Y M, Hira A, et al. Evolutionary structural optimization for problems with stiffness constraints[J]. Finite Elements in Analysis and Design,1996, 21(4):239-251.
[85]Chu D Nha, Xie Y M, Steven G P. An evolutionary structural optimization method for sizing problems with discrete design variables[J]. Computers and Structures,1998,68: 419-431.
[86]Kim H, Querin O M, Steven G P, et al. A method for varying the number of cavities in an optimized topology using evolutionary structural optimization[J]. Structural and Multidisciplinary Optimization,2000,19(2):140-147.
[87]Kim H, Querin O M, Steven G P, et al. Improving efficiency of evolutionary structural optimization by implementing fixed grid mesh[J]. Structural and Multidisciplinary Optimization,2003,24(6):441-448.
[88]Zhao C B, Steven G P, Xie Y M. Evolutionary optimization of maximizing the difference between two natural frequencies of a vibrating structure[J]. Structural Optimization, 1997,13(2-3):148-154.
[89]Steven G P, Li Q, Xie Y M. Multicriteria optimization that minimizes maximum stress and maximizes stiffness[J]. Computers and Structures,2002,80(27-30):2433-2448.
[90]Tanskanen Pasi. The evolutionary sturctural optimization method:theoretical aspects[J]. Computer Methods in Applied Mechanics and Engineering,2002,191(47-48):5485-5498.
[91]Querin O M, Steven G P, Xie Y M. Evolutionary structural optimisation (ESO) using a bidirectional algorithm[J]. Engineering Computations,1998,15(8):1031-1048.
[92]Querin O M. Evolutionary structural optimization:stress based formulation and implementation[D]. Sydney:University of Sydney Australia,1997.
[93]Young V, Querin O M, Steven G P, et al.3D bi-directional evolutionary structural optimization(BESO)[C]. Proceedings of the Austalian Conference on Structural Optimization, Sydney,1998:275-282.
[94]Zhou M, Rozvany G I N. On the validity of ESO type methods in topology optimization [J]. Structural and Multidisciplinary Optimization,2001,21(1):80-83.
[95]罗志凡,荣见华,杜海珍.一种基于主应力的双方向渐进结构拓扑优化方法[J].应用基础与工程科学学报,2003,11(1)：98-105.
[96]荣见华,姜节胜,徐飞鸿,等.一种基于应力的双方向结构拓扑优化算法[J].计 算力学学报,2004,21(3)：322-329.
[97]荣见华,姜节胜,颜东煌,等.基于人工材料的结构拓扑渐进优化设计[J].工程力学,2004,21(5)：64-71.
[98]郭中泽,陈裕泽,张卫红,等.渐进优化法的一种高阶棋盘格式抑制方法[J].机械设计,2006,23(5)：1-4.
[99]郭中泽,陈裕泽,张卫红,等.基于单元材料属性更改的结构渐进拓扑优化方法[J].机械科学与技术,2006,25(8)：928-931.
[100]蔡坤,张洪武,罗阳军,等.三维连续体结构仿生拓扑优化新方法[J].工程力学,2007,24(2)：15-21.
[101]Holland J H. Adaptation in natural and artificial systems[M]. University of Michigan Press,1975
[102]陈国良,王煦法,庄镇泉,等.遗传算法及其应用[M].人民邮电出版社,2001
[103]Ohsaki M. Genetic algorithm for topology optimization of trusses [J]. Computers & Structures,1995,57(2):219-225.
[104]Hajela P, Lee E. Genetic algorithms in truss topological optimization[J]. Solids Structures,1995,132(22):3341-3357.
[105]Kita E, Tanie H. Topology and shape optimization of continuum structures using GA andBEM[J]. Structural Optimization,1999,17(2-3):130-139.
[106]Chapman C D, Jakiela M J. Genetic algorithm-based structural topology design with compliance and topology simplification considerations [J]. Journal o Mechanical Design,1996,118:89-98.
[107]Chen Ting-Yu, Lin Chia-Yang. Determination of optimum design spaces for topology optimization[J]. Finite Elements in Analysis and Design,2000,36(1):1-16.
[108]Aydin Zekeriya, Ayvaz Yusuf. Optimum topology and shape design of prestressed concrete bridge girders using a genetic algorithm[J]. Structural and Multidisciplinary Optimization,2010,41(1):151-162.
[109]Nakanishi Yasuhiko. Application of homology theory to topology optimization of three-dimensional structures using genetic algorithm[J]. Computer Methods in Applied Mechanics and Engineering,2001,190(29-30):3849-3863.
[110]Metropolis N, Rosenbluth A, Rosenbluth M, et al. Simulated annealing[J]. Journal of Chemical Physics,1953,21:1087-1092.
[111]Kirkpatrick S, Gelatt Jr C D, Vecchi M P. Optimization by simulated annealing[J]. Science,1983,220(4598):671-680.
[112]蔡文学,程耿东.桁架结构拓扑优化设计的模拟退火算法[J].华南理工大学学报,1998,26(9)：78-83.
[113]王中华,温卫东.基于模拟退火遗传算法的平面连续体结构的拓扑优化[J].航空动力学报,2004,19(4)：495-498.
[114]王中华,温卫东.基于自适应遗传算法的连续体结构的拓扑优化[J].机械科学与技术,2004,23(11)：1301-1303.
[115]Luh Guan-Chun, Lin Chun-Yi, Lin Yu-Shu. A binary particle swarm optimization for continuum structural topology optimization [J]. Applied Soft Computing,2011,11(2): 2833-2844.
[116]Penninger C L, Watson L T, Tovar A, et al. Convergence analysis of hybrid cellular automata for topology optimization[J]. Structural and Multidisciplinary Optimization, 2010,40(1-6):271-282.
[117]Garcia-Lopez N P, Sanchez-Silva M, Medaglia A L, et al. A hybrid topology optimization methodology combining simulated annealing and SIMP[J]. Computers & Structures,2011,89(15):1512-1522.
[118]周永良.数控车床精度分析及提高精度保持性措施[J].中国制造业信息化,2010,39(9)：78-80+84.
[119]鲁恒.超精密机床动力学建模及分析[D].哈尔滨：哈尔滨工业大学,2007.
[120]杨国哲,许锋,巩亚东,等.虚拟车床动力响应系统的建模与实验[J].东北大学学报(自然科学版),2004,25(1)：70-73
[121]卢秉恒.机械制造技术基础[M].机械工业出版社,2008
[122]毛海军,孙庆鸿,陈南.基于子结构阻抗匹配与优化拟合的高精度磨床整机动力学建模研究[J].中国机械工程,2002,13(5)：376-378.
[123]刘春时,孙伟,李小彭,等.数控机床整机振动测试方法研究[J].中国工程机械学报,2009,7(3)：330-335
[124]郭志全.基于有限元分析的机械产品结构动态设计及其工程应用[D].天津大学,2006.
[125]姚屏,姚宏,王晓军,等.ANSYS下数控车床床身拓扑优化设计[J].机床与液压,2009,37(11)：208-210.
[126]Cheng Keng-Tung, Olhoff Niels. An investigation concerning optimal design of solid elastic plates[J]. International Journal of Solids and Structures,1981,17(3):305-323.
[127]Bends(?)e M P, Kikuchi N. Generating Optimal Topology in structural Design Using a Homogenization Method[J]. Computer Methods in Applied Mechanics and Engineering,1988,71(2):197-224.
[128]Belblidia F, Lee J E B, Rechak S, et al. Topology optimization of plate struetures using a single-or three-layered artificial material model [J]. Advanees in Engineering Software,2001,32(2):159-168
[129]Giuseppe C A. Hierarchical solution of large-scale three-dimensional topology optimization problems[D]. Miehigan:Miehigan State University,1996.
[130]荣见华,唐国金,罗银燕,等.考虑位移要求的大型三维连续体结构拓扑优化数值技术研究[J].工程力学,2007,24(3)：20-27.
[131]荣见华,姜节胜,胡德文,等.基于应力及其灵敏度的结构拓扑渐进优化方法[J].力学学报,2003,35(5)：584-591.
[132]Wang X, Wang M Y, Guo D. Structural shape and topology optimization in a level-set-based framework of region representation [J]. Structural and Multidisciplinary Optimization,2004,27(1-2):1-19.
[133]欧阳高飞,张宪民.结构拓扑优化中水平集方法的快速初始化研究[J].应用基础与工程科学学报,2008,16(1)：136-143.
[134]Rong Jian Hua, Liang Qing Quan. A level set method for topology optimization of continuum structures with bounded design domains[J]. Computer Methods in Applied Mechanics and Engineering,2008,197(17-18):1447-1465
[135]Sui Y K, Yang D Q. A new method for structural topological optimization based on the concept of independent continuous variables and smooth model[J]. Acta Mechanica Sinica,1998,18(2):179-185.
[136]隋允康,叶红玲,刘建信,等.追究根基的结构拓扑优化方法[J].工程力学,2008,25(S2)：7-19.
[137]郭旭,赵康.基于拓扑描述函数的连续体结构拓扑优化方法[J].力学学报,2004,36(5)：522-526.
[138]程耿东.工程结构优化设计基础[M].水利电力出版社,1984
[139]Jang I G, Kwak B M. Design space optimization using design space adjustment and refinement[J]. Structural and Multidisciplinary Optimization,2008,35(1):41-54.
[140]Kim I Y, Kwak B M. Design space optimization using a numerical design continuation method[J]. International Journal for Numerical Methods in Engineering,2002,53(8): 1979-2002.
[141]Stolpe M, Svanberg K. An alternative interpolation scheme for minimum compliance topology optimization[J]. Structural and Multidisciplinary Optimization,2001,22(2): 116-124.
[142]Pedersen Niels L. Maximization of eigenvalues using topology optimization[J]. Structural and Multidisciplinary Optimization,2000,20(1):2-11.
[143]高彤,张卫红,朱继宏.惯性载荷作用下结构拓扑优化[J],2009,41(4)：530-541.
[144]Huang X, Xie Y M. A revised BESO method for structures with design-dependent gravity loads[C]. The Fifth International Structual Engineering and Construction Conference (ISEC-5), London, Taylor and Francis Group,2010:393-398.
[145]高彤,张卫红,朱继宏,等.循环对称结构静力学渐进拓扑优化设计[J].机械工程学报,2008,44(3)：166-172.
[146]Sigmund O. A 99 line topology optimization code written in Matlab[J]. Structural and Multidisciplinary Optimization,2001,21(2):120-127.
[147]Liang Q Q, Steven G P. A performance-based optimization method for topology design of continuum structures with mean compliance constraints [J]. Computer Methods in Applied Mechanics and Engineering,2002,191(13-14):1471-1489.
[148]Suzuki K, Kikuchi N. A homogenization Method for Shape and Topology Optimization[J]. Computer Methods in Applied Mechanics and Engineering,1991, 93(3):291-318.
[149]Rozvany G I N, Zhou M. Advances in overcoming computational pitfalls in topology optimization[C].6th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Reston,1996:1122-1132.
[150]Zhou M, Pagaldipti N, Thomas H L, et al. An integrated approach to topology, sizing and shape optimization[J]. Structural and Multidisciplinary Optimization,2004,26(5): 308-317.
[151]Steven Grant, Querin Osvaldo, Xie Mike. Evolutionary structural optimisation (ESO) for combined topology and size optimisation of discrete structures[J]. Computer Methods in Applied Mechanics and Engineering,2000,188(4):743-754.
[152]刘涛,邓子辰.桁架结构尺寸和形状、拓扑的渐进优化方法[J].西北工业大学学报,2004,22(6)：739-743.
[153]Rajan S D. Sizing, shape, and topology design optimization of trusses using genetic algorithm[J]. Journal of Structural Engineering,1995,121(10):1480-1487.
[154]Chan Chun-Man, Wong Kin-Ming. Structural topology and element sizing design optimisation of tall steel frameworks using a hybrid OC-GA method[J]. Structural and Multidisciplinary Optimization,2008,35(5):473-488.
[155]刘军伟,姜节胜.桁架动力学形状优化的统一设计变量方法[J].振动工程学报,2000,13(1)：84-88.
[156]Afonso Silvana B, Belblidia Fawzi, Hinton Ernest, et al. A combined structural topology and sizing optimization procedure for optimum plates design[C]. European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, 2000:125-131.
[157]王伟,杨伟,赵美英.大展弦比飞翼结构拓扑、形状与尺寸综合优化设计[J].机械强度,2008,30(4)：596-600.
[158]Rong Jian Hua, Jiang Jie Sheng, Xie Yi Min. Evolutionary structural topology optimization for continuum structures with structural size and topology variables[J]. Advances in Structural Engineering,2007,10(6):681-695.
[159]荣见华,邢晓娟,邓果.一种变位移约束限的结构拓扑优化方法[J].力学学报,2009,41(3)：431-439.
[160]荣见华,张强,葛森,等.基于设计空间调整的结构拓扑优化方法[J].力学学报,2010,42(2)：256-267.
[161]Huang X, Xie Y M. Optimal design of periodic structures using evolutionary topology optimization[J]. Structural and Multidisciplinary Optimization,2008,36(6):597-606.
[162]Diaaz Alejandro R, Kikuchi Noboru. Solutions to shape and topology eigenvalue optimization problems using a homogenization method [J]. International Journal for Numerical Methods in Engineering,1992,35(7):1487-1502.
[163]Yoon Gil Ho. Maximizing the fundamental eigenfrequency of geometrically nonlinear structures by topology optimization based on element connectivity parameterization[J]. Computers& Structures,2010,88(1):120-133.
[164]Ma Zheng-Dong, Kikuchi Noboru, Cheng Hsien-Chie. Topological design for vibrating structures [J]. Computer Methods in Applied Mechanics and Engineering,1995,121(1): 259-280.
[165]Jensen Jakob S. Topology optimization of dynamics problems with Pade approximants [J]. International Journal for Numerical Methods in Engineering,2007, 72(13):1605-1630.
[166]Yoon Gil Ho. Structural topology optimization for frequency response problem using model reduction schemes[J]. Computer Methods in Applied Mechanics and Engineering,2010,199(25):1744-1763.
[167]Rong J H, Xie Y M, Yang X Y, et al. Topology optimization of structures under dynamic response constraints [J]. Journal of Sound and Vibration,2000,234(2):177-189.
[168]杨振兴,荣见华,傅建林.窄带随机激励下的结构动力学拓扑优化设计[J].振动与冲击,2005,24(6)：85-87+94+140.
[169]Qiao Zhang, Weihong Zhang, Jihong Zhu, et al. Layout optimization of multi-component structures under static loads and random excitations[J]. Engineering Structures,2012,43:120-128.
[170]唐治丽.随机动响应约束的结构拓扑优化方法研究与软件开发[D].长沙：长沙理工大学,2012.
[171]潘晋,王德禹.动力响应约束下的桁架结构拓扑优化[J].振动与冲击,2006,25(4)：8-12+171.
[172]方同.工程随机振动[M].国防工业出版社,1995
[173]Bruggi Matteo, Taliercio Alberto. Maximization of the fundamental eigenfrequency of micropolar solids through topology optimization[J]. Structural and Multidisciplinary Optimization,2012,46(4):549-560.
[174]张桥,张卫红,朱继宏.动力响应约束下的结构拓扑优化设计[J],2010
[175]Jang I G, Kwak B M. Evolutionary topology optimization using design space adjustment based on fixed grid[J]. International Journal for Numerical Methods in Engineering,2006,66(11):1817-1840.
[176]Adelman Howard M, Haftka Raphael T. Sensitivity analysis of discrete structural systems[J]. AIAA Journal,1986,24(5):823-832.
[177]Baldwin John F, Hutton Stanley G. Natural modes of modified structures[J]. AIAA Journal,1985,23(11):1737-1743.
[178]Lim K. B, Junkins J L, Wang B P. Re-examination of eigenvector derivatives [J]. Journal of Guidance, Control, and Dynamics,1987,10(6):581-587.
[179]Liu Zhong-sheng, Chen Su-huan, Zhao You-qun. An accurate method for computing eigenvector derivatives for free-free structures [J]. Computers & Structures,1994,52 (6):1135-1143.
[180]Fox R L, Kapoor M P. Rates of change of eigenvalues and eigenvectors[J]. AIAA Journal,1968,6(12):2426-2429.
[181]Nelson Richard B. Simplified calculation of eigenvector derivatives [J]. AIAA Journal, 1976,14(9):1201-1205.
[182]Chen Ting-Yu. Design sensitivity analysis for repeated eigenvalues in structural design[J]. AIAA Journal,1993,31(12):2347-2350.
[183]Alvin Kenneth F. Efficient computation of eigenvector sensitivities for structural dynamics[J]. AIAA Journal,1997,35(11):1760-1766.
[184]Diaz A, Sigmund O. Checkerboard patterns in layout optimization[J]. Structural and Multidisciplinary Optimization,1995,10(1):40-45.
[185]Sigmund O, Petersson J. Numerical instabilities in topology optimization:a survey on procedures dealing with checkerboards, mesh-dependencies and local minima[J]. Structural Optimization,1998,16(1):68-75.
[186]Swan C C, Kosaka I. Voigt-Reuss topology optimization for structures with linear elastic material behaviours[J].International Journal for Numerical Methods in Engineering,1997,40(16):3033-3057.
[187]Huang X, Xie Y M. Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method [J]. Finite Elements in Analysis and Design,2007,43(14):1039-1049.
[188]Carbonari R C, Silva E C N, Nishiwaki S. Topology optimization applied to the design of multi-actuated piezoelectric micro-tools[C]. Smart Structures and Materials 2004: Modeling, Signal Processing, and Control (Proceedings of SPIE), San Diego,2004: 277-288.
[189]Matsui K, Terada K. Continuous approximation of material distribution for topology optimization[J]. International Journal for Numerical Methods in Engineering,2004,59 (14):1925-1944.
[190]Rahmatalla S F, Swan C C. A Q4/Q4 continuum structural topology optimization implementation[J]. Structural and Multidisciplinary Optimization,2004,27(1):130-135.
[191]Paulino G H, Le C H. A modified Q4/Q4 element for topology optimization[J]. Structural and Multidisciplinary Optimization,2009,37(3):255-264.
[192]Poulsen T A. Topology optimization in wavelet space [J]. International Journal for Numerical Methods in Engineering,2002,53(3):567-582.
[193]Jog C S, Haber R B. Stability of finite elements models for distributed-parameter optimization and topology design[J]. Computer Methods in Applied Mechanics and Engineering,1996,130(3-4):203-226.
[194]Kang Z, Wang Y Q. Structural topology optimization based on non-local Shepard interpolation of density field [J]. Computer Methods in Applied Mechanics and Engineering,2011,200(49):3515-3525.
[195]Guest J K, Prevost J H, Belytschko T. Achieving minimum length scale in topology optimization using nodal design variables and projection functions[J]. International Journal for Numerical Methods in Engineering,2004,61(2):238-254.
[196]Shepard Donald. A two-dimensional interpolation function for irregularly-spaced data[C]. Proceedings of the 1968 23rd ACM National Conference, New York, 1968:517-524.
[197]Brodlie K W, Asim M R, Unsworth K. Constrained visualization using the Shepard interpolation family[J]. Computer Graphics Forum,2005,24(4):809-820.
[198]Bends(?)e M P. Optimal shape design as a material distribution problem[J]. Structural and Multidisciplinary Optimization,1989,1(4):193-202.
[199]Yang R J, Chuang C H. Optimal topology design using linear programming[J]. Computers & Structures,1994,52(2):265-275.
[200]Rozvany G I N. Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics[J]. Structural and Multidisciplinary Optimization,2001,21(2):90-108.
[201]Beckers M. Topology optimization using a dual method with discrete variables[J]. Structural and Multidisciplinary Optimization,1999,17(1):14-24.
[202]Rong Jian Hua, Li Wei Xiang, Feng Bing. A structural topological optimization method based on varying displacement limits and design space adjustments[J], Advanced Materials Research,2010,97-101:3609-3616.
[203]Chapman B, Jost G, Pas R V D. Using OpenMP:Portable Shared Memory Parallel Programming[M]. The MIT Press,2007
[204]Colominas I, Paris J, Navarrina F, et al. High performance parallel computing in structural topology optimization[C]. BHV Topping LF Costa Neves, RC Barros. Proceedings of the 12th International Conference on Civil, Structural and Environmental Engineering Computing, London,2009
[205]Manickarajah D, Xie Y M, Steven G P. An evolutionary method for optimization of plate buckling resistance[J]. Finite Elements in Analysis and Design,1998,29(3):205-230.
[206]Rong J H, Xie Y M, Yang X Y. An improved method for evolutionary structural optimization against buckling[J]. Computers & Structures,2001,79(3):253-263.
[207]俞焕然,王璠.具有初始缺陷变厚度圆板在复合载荷作用下的稳定性及优化[J].福州大学学报(自然科学版),1994,22(4)：90-95
[208]LIU Tao, XU Qi-nan, PEI Jun-hou. Buckling analysis and optimization design of composite cylindrical shells[J]. Journal of Ship Mechanics,2000,6(4):39-50.
[209]唐文艳,袁清珂.屈曲约束桁架拓扑优化的遗传算法[J].机械科学与技术,2009,(4)：460-463.
[210]梁斌,高笑娟,俞焕然.几何约束下圆柱壳的屈曲优化设计[J].兰州大学学报：自然科学版,2003,39(1)：24-27.
[211]吴浩,燕瑛.基于可靠性的复合材料结构稳定性约束优化设计[J].复合材料学报,2007,24(5)：149-153.
[212]Zhou M. Topology optimization for shell structures with linear buckling responses [J]. WCCM VI, Beijing, China,2004
[213]Guo X, Cheng G D, Olhoff N. Optimum design of truss topology under buckling constraints[J]. Structural and Multidisciplinary Optimization,2005,30(3):169-180.
[214]隋允康,边炳传.屈曲与应力约束下连续体结构的拓扑优化[J].工程力学,2008,(8)：6-12.
[215]隋允康,边炳传,叶红玲.连续体结构屈曲约束的ICM方法拓扑优化[J].计算力学学报,2008,25(3)：345-351.
[216]李洋,龙连春.考虑整体屈曲薄壁加筋板加筋布局优化[C].北京力学会.北京力学会第19届学术年会,中国北京,2013：2.
[217]韩强,张善元,杨桂通.结构静动力屈曲问题研究进展[J].力学进展,1998,28(3)：349-360.
[218]Yi Jijun, Rong Jianhua, Zeng Tao, et al. A topology optimization method for multiple load cases and constraints based on element independent nodal density[J]. Structural Engineering and Mechanics,2013,45(6):759-777.
[219]张学玲.基于广义模块化设计的机械结构静、动态特性分析及优化设计[D].天津：天津大学,2004.
[220]Pahl G, Beitz Wolfgong. Engineering design:a systematic approach[M]. Springer-Verlag, London,1988
[221]Gu P, Hashemian M, Nee A Y C. Adaptable design[J]. CIRP Annals-Manufacturing Technology,2004,53(2):539-557.
[222]高卫国,徐燕申,陈永亮,等.广义模块化设计原理及方法[J].机械工程学报,2007,43(6)：48-54.
[2]钱令希,钟万勰,隋允康,等.工程结构优化设计[J].工程机械,1988,(5)：22-26+55.
[3]辛志杰.面向产品族的机床结构可适应动态设计理论、方法与应用[D].天津：天津大学,2008.
[4]阎楚良,杨方飞.机械数字化设计新技术[M].机械工业出版社,2007
[5]Winner Robert I. The role of concurrent engineering in weapons systems acquisition[M]. Institute for Defence Analyses,1988
[6]Tseng Mitchell M, Jiao Jianxin, Merchant M Eugene. Design for mass customization[J]. CIRP Annals-Manufacturing Technology,1996,45(1):153-156.
[7]Koenigsberger Franz, Tlusty Jiri. Machine tool structures [M]. Pergamon Press Oxford, England,1970
[8]徐燕申,张兴朝,牛占文,等.基于元结构和框架优选的数控机床床身结构动态设计研究[J].机械强度,2001,23(1)：1-3+110.
[9]陈新,陈新度,秦叶,等.机械结构动态设计若干关键技术[J].中国机械工程,1997,8(5)：104-108.
[10]郭春星,丁晓红,郭媛美,等.磨床床身结构优化设计[J].机械设计与研究,2009,25(5)：104-107.
[11]陈新,何杰,毛海军,等.基于动力学特征的磨床床身结构优化布局设计[J].制造技术与机床,2001,(2)：25-26+23.
[12]Zhao Ling, Ma Jianfeng, Wang Ting, et al. Lightweight design of mechanical structures based on structural bionic methodology[J]. Journal of Bionic Engineering,2010,7(S): S224-S231.
[13]Ma Jian-feng, Chen Wu-yi, Zhao Ling, et al. Elastic buckling of bionic cylindrical shells based on bamboo[J]. Journal of Bionic Engineering,2008,5(3):231-238.
[14]Liu Shihao, Ye Wenhua, Lou Peihuang, et al. Bionic design for column of gantry machining center to improve the static and dynamic performance [J]. Shock and Vibration,2012,19(4):493-504.
[15]肖利利,陈蔚芳,叶文华,等.“箱中箱”结构立滑板的有限元分析及其优化设计[J].山东大学学报(工学版),2010,40(1)：78-83.
[16]王琦.采用重心驱动(DCG)的机床[J].制造技术与机床,2005,(2)：21-22.
[17]卢天健,张钱城,王春野,等.轻质材料和结构在机床上的应用[J].力学与实践,2007,29(6)：1-8.
[18]Kim D I, Jung S C, Lee J E, et al. Parametric study on design of composite-foam-resin concrete sandwich structures for precision machine tool structures [J]. Composite Structures,2006,75(1):408-414.
[19]Chang Seung Hwan, Kim Po Jin, Lee Dai Gil, et al.Steel-composite hybrid headstock for high-precision grinding machines[J]. Composite structures,2001,53(1):1-8.
[20]Lee Dai Gil, Chang Seung Hwan, Kim Hyun Surk. Damping improvement of machine tool columns with polymer matrix fiber composite material[J]. Composite Structures, 1998,43(2):155-163.
[21]Rahman M, Mansur M A, Feng Zhou. Design, fabrication and evaluation of a steel fibre reinforced concrete column for grinding machines[J]. Materials & Design,1995,16(4): 205-209.-
[22]王艳辉,伍建国,缪建成.精密机床床身的模态分析与结构优选[J].机械设计与制造,2005,3(3)：76-77.
[23]米成秋,孙靖民.机床结构优化方法初探[J].哈尔滨工业大学学报,1983,(3)：8.
[24]Tandon V, EI-Mounayri H, Kishawy H. NC end milling optimization using evolutionary computation[J]. International Journal of Machine Tools and Manufacture,2002,42(5): 595-605.
[25]汪列隆,朱壮瑞,张建润.基于APDL的机床大件筋板结构尺寸的优化设计[J].轻工机械,2006,24(4)：60-62.
[26]戴磊,关振群,单菊林,等.机床结构三维参数化形状优化设计[J].机械工程学报,2008,44(5)：152-159.
[27]汤文成,易红,唐寅.机床大件结构的拓扑优化设计[J].东南大学学报,1996,(5)：24-28.
[28]饶柳生,侯亮,潘勇军.基于拓扑优化的机床立柱筋板改进[J].机械设计与研究,2010,26(1)：87-92.
[29]张永存,崔雷,周玲丰,等.基于拓扑优化的机床床鞍创新构型设计[J].固体力学学报,2011,32(S)：335-342.
[30]童水光,刘或,张健,等.基于拓扑优化的卧式旋压机床身加强肋布局优化设计方法[J].机械设计,2012,29(6)：31-35.
[31]张向宇,熊计,郝锌,等.基于ANSYS的加工中心滑座拓扑优化设计[J].制造技术与机床,2008,(6)：68-70.
[32]刘舜尧,李进,贺浩.基于渐进结构优化的取料梁腹板拓扑优化[J].工程设计学报,2011,18(3)：174-177.
[33]涂星.YDG型堆料机臂架形变分析及拓扑结构优化设计[D].长沙：中南大学,2011.
[34]徐岩.汽车零部件结构的拓扑优化设计[D].长春：吉林大学,2009.
[35]郭立群.商用车车架拓扑优化轻量化设计方法研究[D].长春：吉林大学,2011.
[36]杜春江.连续体结构拓扑优化理论及其在炮塔结构设计中的应用研究[D].南京：南京理工大学,2008.
[37]Michell A G M. The limits of economy of material in frame structures [J]. Philosophical Magazine,1904,8(6):589-597.
[38]Rozvany G I N. Grillages of maximum strength and maximum stiffness[J]. International Journal of Mechanical Sciences,1972,14(10):1217-1222.
[39]Dorn W S, Gomory R E, Greenberg H J. Automatic design of optimal structures[J]. Journal de Mecanique,1964,3(6):25-52.
[40]Rossow M P, Taylor J E. A finite element method for the optimal design of variable thickness sheets[J]. AIAA Journal,1973,11 (11):1566-1569.
[41]Bendsoe Martin Philip, Kikuchi Noboru. Generating optimal topologies in structural design using a homogenization methodfJ]. Computer Methods in Applied Mechanics and Engineering,1988,71(2):197-224.
[42]Guedes J M, Kikuchi N. Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[J]. Computer Methods in Applied Mechanics and Engineering,1990,83(2):143-198.
[43]Suzuki Katsuyuki, Kikuchi Noboru.A homogenization method for shape and topology optimization[J]. Computer Methods in Applied Mechanics and Engineering,1991,93 (3):291-318.
[44]Diaz A R, Bendsoe M P. Shape optimization of structures for multiple loading conditions using a homogenization method [J]. Structural and Multidisciplinary Optimization,1992, 4(1):17-22.
[45]Michel J C, Moulinec H, Suquet P. Effective properties of composite materials with periodic microstructure:a computational approach[J]. Computer Methods in Applied Mechanics and Engineering,1999,172(1-4):109-143.
[46]Hassani B, Hinton E. A review of homogenization and topology I-homogenization theory for media with periodic structure[J]. Coputers and Structures,1998,69(6):707-717.
[47]Hassani B, Hinton E. A review of homogenization and topology opimization Ⅱ— analytical and numerical solution of homogenization equations [J]. Coputers and Structures,1998,69(6):719-738.
[48]Hassani B, Hinton E. A review of homogenization and topology Ⅲ-topology optimization using optimality criteria[J]. Coputers and Structures,1998,69(6):739-756.
[49]刘书田,程耿东.复合材料应力分析的均匀化方法[J].力学学报,1997,29(3)：306- 313.
[50]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[J].中国机械工程,1999,10(11)：1264-1267.
[51]袁振,吴长春.复合材料周期性线弹性微结构的拓扑优化设计[J].固体力学学报,2003,24(1)：40-45.
[52]王书亭,左孔天.基于均匀化理论的拓扑优化算法研究[J].华中科技大学学报,2004,32(10)：25-30.
[53]谢先海,廖道训.基于均匀化方法的柔顺机构的设计[J].中国机械工程,2003,14(11)：953-956.
[54]Rozvany G I N, Zhou M, Birker T. Generalized shape optimization without homogenization[J].Structural and Multidisciplinary Optimization,1992,4(3):250-252.
[55]Bendsoe M P, Sigmund O. Topology optimization:theory, methods, and applications [M]. Springer-Verlag,2003
[56]Bendsoe M P, Sigmund O. Material interpolation schemes in topology optimization[J]. Archive of Applied Mechanics,1999,69(9-10):635-654.
[57]Luo Z, Chen L, Yang J, et al. Compliant mechanism design using multi-objective topology optimization scheme of continuum structures[J]. Structural and Multidisciplinary Optimization,2005,30(2):142-154.
[58]罗震,陈立平,张云清,等.多工况下连续体结构的多刚度拓扑优化设计和二重敏度过滤技术[J].固体力学学报,2005,26(1)：29-36.
[59]Rozany G I N. Aims scope method history and unified terminology of computer-aided topology optimization in structure mechanics [J]. Structural and Multidisciplinary Optimization,21:90-108.
[60]袁振,吴长春,庄守兵.基于杂交元和变密度法的连续体结构拓扑优化设计[J].中国科学技术大学学报,2001,31(6)：694-699.
[61]Martinez J M. A note on the theoretical convergence properties of the SIMP method [J]. Structural and Multidisciplinary Optimization,2004,29(4):319-323
[62]邓扬晨,张卫红,刘晓欧,等.拓扑优化设计中刚度与密度关系的一种力学模型[J].航空计算技术,2004(01)：1-4.
[63]左孔天.连续体结构拓扑优化理论与应用研究[D].武汉：华中科技大学,2004.
[64]左孔天,钱勤,赵雨东,等.热固耦合结构的拓扑优化设计研究[J].固体力学学报,2005,26(4)：447-452.
[65]隋允康,彭细荣.结构拓扑优化ICM方法的改善[J].力学学报,2005,37(2)：190-198.
[66]隋允康,任旭春,龙连春,等.基于ICM方法的刚架拓扑优化[J].计算力学学报, 2003,20(3)：286-289.
[67]隋允康,杨德庆,孙焕纯.统一骨架与连续体德结构拓扑优化的ICM理论与方法[J].计算力学学报,2000,17(1)：28-33.
[68]隋允康,叶红玲,杜家政.结构拓扑优化的发展及其模型转化为独立层次的迫切性[J].工程力学,2005,22(S)：107-118.
[69]叶红玲,隋允康.应力约束下连续体结构拓扑优化的ICM方法[J].北京工业大学学报,2005,31(S1)：1-5.
[70]彭细荣,隋允康.用ICM法拓扑优化静位移及频率约束下连续体结构[J].计算力学学报,2006,23(4)：391-396.
[71]朱润,隋允康.基于ICM方法的多工况位移约束下板壳结构拓扑优化设计[J].固体力学学报,2012,33(1)：81-90.
[72]Osher S, Sethian J A. Fronts propagating with curvature-dependent speed:algorithms based on Hamilton-Jacobi formulations [J]. Journal of Computational Physics,1988,79 (1):12-49.
[73]Sethian J A. Level set methods and fast marching methods:evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science [M]. Cambridge University Press,1999
[74]Osher S, Fedkiw R P. Level set methods:an overview and some recent results[J]. Journal of Computational Physics,2001,169(2):463-502.
[75]Osher S J, Santosa F. Level set methods for optimization problems involying geometry and constraints:I. Frequencies of a two-density inhomogeneous drum[J]. Journal of Computational Physics,2001,171(1):272-288.
[76]Sethian J A, Wiegmann Andreas. Structural bondary design via level set and immersed interface methods[J]. Journal of Computational Physics,2000,163(2):489-528.
[77]梅玉林.拓扑优化的水平集法及其在刚性结构、柔性结构和材料设计种的应用[D].大连理工大学,2004.
[78]Wang M Y, Wang X, Guo D. A level set method for structural topology optimization[J]. Computer Methods in Applied Mechanics and Engineering,2003,192(1):227-246.
[79]Wang X, Wang M Y, Guo D. Structural shape and topology optimization in a level-set-based framework of region representation[J]. Structural and Multidisciplinary Optimization,2004,27(1):1-19.
[80]庄春刚,熊振华,丁汉.三维线弹性连续体的结构拓扑优化设计[J].中国机械工程,2007,18(14)：1703-1708.
[81]Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization[J]. Computers & Structures,1993,49(5):885-896.
[82]Xie Y M, Steven G P. Evolutionary structural optimization[M]. Springer,1997
[83]Xie Y M, Steven G P. Evolutionary structural optimization for dynamic problems[J]. Computers and Structures,1996,58(6):1067-1073.
[84]Chu D Nha, Xie Y M, Hira A, et al. Evolutionary structural optimization for problems with stiffness constraints[J]. Finite Elements in Analysis and Design,1996, 21(4):239-251.
[85]Chu D Nha, Xie Y M, Steven G P. An evolutionary structural optimization method for sizing problems with discrete design variables[J]. Computers and Structures,1998,68: 419-431.
[86]Kim H, Querin O M, Steven G P, et al. A method for varying the number of cavities in an optimized topology using evolutionary structural optimization[J]. Structural and Multidisciplinary Optimization,2000,19(2):140-147.
[87]Kim H, Querin O M, Steven G P, et al. Improving efficiency of evolutionary structural optimization by implementing fixed grid mesh[J]. Structural and Multidisciplinary Optimization,2003,24(6):441-448.
[88]Zhao C B, Steven G P, Xie Y M. Evolutionary optimization of maximizing the difference between two natural frequencies of a vibrating structure[J]. Structural Optimization, 1997,13(2-3):148-154.
[89]Steven G P, Li Q, Xie Y M. Multicriteria optimization that minimizes maximum stress and maximizes stiffness[J]. Computers and Structures,2002,80(27-30):2433-2448.
[90]Tanskanen Pasi. The evolutionary sturctural optimization method:theoretical aspects[J]. Computer Methods in Applied Mechanics and Engineering,2002,191(47-48):5485-5498.
[91]Querin O M, Steven G P, Xie Y M. Evolutionary structural optimisation (ESO) using a bidirectional algorithm[J]. Engineering Computations,1998,15(8):1031-1048.
[92]Querin O M. Evolutionary structural optimization:stress based formulation and implementation[D]. Sydney:University of Sydney Australia,1997.
[93]Young V, Querin O M, Steven G P, et al.3D bi-directional evolutionary structural optimization(BESO)[C]. Proceedings of the Austalian Conference on Structural Optimization, Sydney,1998:275-282.
[94]Zhou M, Rozvany G I N. On the validity of ESO type methods in topology optimization [J]. Structural and Multidisciplinary Optimization,2001,21(1):80-83.
[95]罗志凡,荣见华,杜海珍.一种基于主应力的双方向渐进结构拓扑优化方法[J].应用基础与工程科学学报,2003,11(1)：98-105.
[96]荣见华,姜节胜,徐飞鸿,等.一种基于应力的双方向结构拓扑优化算法[J].计 算力学学报,2004,21(3)：322-329.
[97]荣见华,姜节胜,颜东煌,等.基于人工材料的结构拓扑渐进优化设计[J].工程力学,2004,21(5)：64-71.
[98]郭中泽,陈裕泽,张卫红,等.渐进优化法的一种高阶棋盘格式抑制方法[J].机械设计,2006,23(5)：1-4.
[99]郭中泽,陈裕泽,张卫红,等.基于单元材料属性更改的结构渐进拓扑优化方法[J].机械科学与技术,2006,25(8)：928-931.
[100]蔡坤,张洪武,罗阳军,等.三维连续体结构仿生拓扑优化新方法[J].工程力学,2007,24(2)：15-21.
[101]Holland J H. Adaptation in natural and artificial systems[M]. University of Michigan Press,1975
[102]陈国良,王煦法,庄镇泉,等.遗传算法及其应用[M].人民邮电出版社,2001
[103]Ohsaki M. Genetic algorithm for topology optimization of trusses [J]. Computers & Structures,1995,57(2):219-225.
[104]Hajela P, Lee E. Genetic algorithms in truss topological optimization[J]. Solids Structures,1995,132(22):3341-3357.
[105]Kita E, Tanie H. Topology and shape optimization of continuum structures using GA andBEM[J]. Structural Optimization,1999,17(2-3):130-139.
[106]Chapman C D, Jakiela M J. Genetic algorithm-based structural topology design with compliance and topology simplification considerations [J]. Journal o Mechanical Design,1996,118:89-98.
[107]Chen Ting-Yu, Lin Chia-Yang. Determination of optimum design spaces for topology optimization[J]. Finite Elements in Analysis and Design,2000,36(1):1-16.
[108]Aydin Zekeriya, Ayvaz Yusuf. Optimum topology and shape design of prestressed concrete bridge girders using a genetic algorithm[J]. Structural and Multidisciplinary Optimization,2010,41(1):151-162.
[109]Nakanishi Yasuhiko. Application of homology theory to topology optimization of three-dimensional structures using genetic algorithm[J]. Computer Methods in Applied Mechanics and Engineering,2001,190(29-30):3849-3863.
[110]Metropolis N, Rosenbluth A, Rosenbluth M, et al. Simulated annealing[J]. Journal of Chemical Physics,1953,21:1087-1092.
[111]Kirkpatrick S, Gelatt Jr C D, Vecchi M P. Optimization by simulated annealing[J]. Science,1983,220(4598):671-680.
[112]蔡文学,程耿东.桁架结构拓扑优化设计的模拟退火算法[J].华南理工大学学报,1998,26(9)：78-83.
[113]王中华,温卫东.基于模拟退火遗传算法的平面连续体结构的拓扑优化[J].航空动力学报,2004,19(4)：495-498.
[114]王中华,温卫东.基于自适应遗传算法的连续体结构的拓扑优化[J].机械科学与技术,2004,23(11)：1301-1303.
[115]Luh Guan-Chun, Lin Chun-Yi, Lin Yu-Shu. A binary particle swarm optimization for continuum structural topology optimization [J]. Applied Soft Computing,2011,11(2): 2833-2844.
[116]Penninger C L, Watson L T, Tovar A, et al. Convergence analysis of hybrid cellular automata for topology optimization[J]. Structural and Multidisciplinary Optimization, 2010,40(1-6):271-282.
[117]Garcia-Lopez N P, Sanchez-Silva M, Medaglia A L, et al. A hybrid topology optimization methodology combining simulated annealing and SIMP[J]. Computers & Structures,2011,89(15):1512-1522.
[118]周永良.数控车床精度分析及提高精度保持性措施[J].中国制造业信息化,2010,39(9)：78-80+84.
[119]鲁恒.超精密机床动力学建模及分析[D].哈尔滨：哈尔滨工业大学,2007.
[120]杨国哲,许锋,巩亚东,等.虚拟车床动力响应系统的建模与实验[J].东北大学学报(自然科学版),2004,25(1)：70-73
[121]卢秉恒.机械制造技术基础[M].机械工业出版社,2008
[122]毛海军,孙庆鸿,陈南.基于子结构阻抗匹配与优化拟合的高精度磨床整机动力学建模研究[J].中国机械工程,2002,13(5)：376-378.
[123]刘春时,孙伟,李小彭,等.数控机床整机振动测试方法研究[J].中国工程机械学报,2009,7(3)：330-335
[124]郭志全.基于有限元分析的机械产品结构动态设计及其工程应用[D].天津大学,2006.
[125]姚屏,姚宏,王晓军,等.ANSYS下数控车床床身拓扑优化设计[J].机床与液压,2009,37(11)：208-210.
[126]Cheng Keng-Tung, Olhoff Niels. An investigation concerning optimal design of solid elastic plates[J]. International Journal of Solids and Structures,1981,17(3):305-323.
[127]Bends(?)e M P, Kikuchi N. Generating Optimal Topology in structural Design Using a Homogenization Method[J]. Computer Methods in Applied Mechanics and Engineering,1988,71(2):197-224.
[128]Belblidia F, Lee J E B, Rechak S, et al. Topology optimization of plate struetures using a single-or three-layered artificial material model [J]. Advanees in Engineering Software,2001,32(2):159-168
[129]Giuseppe C A. Hierarchical solution of large-scale three-dimensional topology optimization problems[D]. Miehigan:Miehigan State University,1996.
[130]荣见华,唐国金,罗银燕,等.考虑位移要求的大型三维连续体结构拓扑优化数值技术研究[J].工程力学,2007,24(3)：20-27.
[131]荣见华,姜节胜,胡德文,等.基于应力及其灵敏度的结构拓扑渐进优化方法[J].力学学报,2003,35(5)：584-591.
[132]Wang X, Wang M Y, Guo D. Structural shape and topology optimization in a level-set-based framework of region representation [J]. Structural and Multidisciplinary Optimization,2004,27(1-2):1-19.
[133]欧阳高飞,张宪民.结构拓扑优化中水平集方法的快速初始化研究[J].应用基础与工程科学学报,2008,16(1)：136-143.
[134]Rong Jian Hua, Liang Qing Quan. A level set method for topology optimization of continuum structures with bounded design domains[J]. Computer Methods in Applied Mechanics and Engineering,2008,197(17-18):1447-1465
[135]Sui Y K, Yang D Q. A new method for structural topological optimization based on the concept of independent continuous variables and smooth model[J]. Acta Mechanica Sinica,1998,18(2):179-185.
[136]隋允康,叶红玲,刘建信,等.追究根基的结构拓扑优化方法[J].工程力学,2008,25(S2)：7-19.
[137]郭旭,赵康.基于拓扑描述函数的连续体结构拓扑优化方法[J].力学学报,2004,36(5)：522-526.
[138]程耿东.工程结构优化设计基础[M].水利电力出版社,1984
[139]Jang I G, Kwak B M. Design space optimization using design space adjustment and refinement[J]. Structural and Multidisciplinary Optimization,2008,35(1):41-54.
[140]Kim I Y, Kwak B M. Design space optimization using a numerical design continuation method[J]. International Journal for Numerical Methods in Engineering,2002,53(8): 1979-2002.
[141]Stolpe M, Svanberg K. An alternative interpolation scheme for minimum compliance topology optimization[J]. Structural and Multidisciplinary Optimization,2001,22(2): 116-124.
[142]Pedersen Niels L. Maximization of eigenvalues using topology optimization[J]. Structural and Multidisciplinary Optimization,2000,20(1):2-11.
[143]高彤,张卫红,朱继宏.惯性载荷作用下结构拓扑优化[J],2009,41(4)：530-541.
[144]Huang X, Xie Y M. A revised BESO method for structures with design-dependent gravity loads[C]. The Fifth International Structual Engineering and Construction Conference (ISEC-5), London, Taylor and Francis Group,2010:393-398.
[145]高彤,张卫红,朱继宏,等.循环对称结构静力学渐进拓扑优化设计[J].机械工程学报,2008,44(3)：166-172.
[146]Sigmund O. A 99 line topology optimization code written in Matlab[J]. Structural and Multidisciplinary Optimization,2001,21(2):120-127.
[147]Liang Q Q, Steven G P. A performance-based optimization method for topology design of continuum structures with mean compliance constraints [J]. Computer Methods in Applied Mechanics and Engineering,2002,191(13-14):1471-1489.
[148]Suzuki K, Kikuchi N. A homogenization Method for Shape and Topology Optimization[J]. Computer Methods in Applied Mechanics and Engineering,1991, 93(3):291-318.
[149]Rozvany G I N, Zhou M. Advances in overcoming computational pitfalls in topology optimization[C].6th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Reston,1996:1122-1132.
[150]Zhou M, Pagaldipti N, Thomas H L, et al. An integrated approach to topology, sizing and shape optimization[J]. Structural and Multidisciplinary Optimization,2004,26(5): 308-317.
[151]Steven Grant, Querin Osvaldo, Xie Mike. Evolutionary structural optimisation (ESO) for combined topology and size optimisation of discrete structures[J]. Computer Methods in Applied Mechanics and Engineering,2000,188(4):743-754.
[152]刘涛,邓子辰.桁架结构尺寸和形状、拓扑的渐进优化方法[J].西北工业大学学报,2004,22(6)：739-743.
[153]Rajan S D. Sizing, shape, and topology design optimization of trusses using genetic algorithm[J]. Journal of Structural Engineering,1995,121(10):1480-1487.
[154]Chan Chun-Man, Wong Kin-Ming. Structural topology and element sizing design optimisation of tall steel frameworks using a hybrid OC-GA method[J]. Structural and Multidisciplinary Optimization,2008,35(5):473-488.
[155]刘军伟,姜节胜.桁架动力学形状优化的统一设计变量方法[J].振动工程学报,2000,13(1)：84-88.
[156]Afonso Silvana B, Belblidia Fawzi, Hinton Ernest, et al. A combined structural topology and sizing optimization procedure for optimum plates design[C]. European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, 2000:125-131.
[157]王伟,杨伟,赵美英.大展弦比飞翼结构拓扑、形状与尺寸综合优化设计[J].机械强度,2008,30(4)：596-600.
[158]Rong Jian Hua, Jiang Jie Sheng, Xie Yi Min. Evolutionary structural topology optimization for continuum structures with structural size and topology variables[J]. Advances in Structural Engineering,2007,10(6):681-695.
[159]荣见华,邢晓娟,邓果.一种变位移约束限的结构拓扑优化方法[J].力学学报,2009,41(3)：431-439.
[160]荣见华,张强,葛森,等.基于设计空间调整的结构拓扑优化方法[J].力学学报,2010,42(2)：256-267.
[161]Huang X, Xie Y M. Optimal design of periodic structures using evolutionary topology optimization[J]. Structural and Multidisciplinary Optimization,2008,36(6):597-606.
[162]Diaaz Alejandro R, Kikuchi Noboru. Solutions to shape and topology eigenvalue optimization problems using a homogenization method [J]. International Journal for Numerical Methods in Engineering,1992,35(7):1487-1502.
[163]Yoon Gil Ho. Maximizing the fundamental eigenfrequency of geometrically nonlinear structures by topology optimization based on element connectivity parameterization[J]. Computers& Structures,2010,88(1):120-133.
[164]Ma Zheng-Dong, Kikuchi Noboru, Cheng Hsien-Chie. Topological design for vibrating structures [J]. Computer Methods in Applied Mechanics and Engineering,1995,121(1): 259-280.
[165]Jensen Jakob S. Topology optimization of dynamics problems with Pade approximants [J]. International Journal for Numerical Methods in Engineering,2007, 72(13):1605-1630.
[166]Yoon Gil Ho. Structural topology optimization for frequency response problem using model reduction schemes[J]. Computer Methods in Applied Mechanics and Engineering,2010,199(25):1744-1763.
[167]Rong J H, Xie Y M, Yang X Y, et al. Topology optimization of structures under dynamic response constraints [J]. Journal of Sound and Vibration,2000,234(2):177-189.
[168]杨振兴,荣见华,傅建林.窄带随机激励下的结构动力学拓扑优化设计[J].振动与冲击,2005,24(6)：85-87+94+140.
[169]Qiao Zhang, Weihong Zhang, Jihong Zhu, et al. Layout optimization of multi-component structures under static loads and random excitations[J]. Engineering Structures,2012,43:120-128.
[170]唐治丽.随机动响应约束的结构拓扑优化方法研究与软件开发[D].长沙：长沙理工大学,2012.
[171]潘晋,王德禹.动力响应约束下的桁架结构拓扑优化[J].振动与冲击,2006,25(4)：8-12+171.
[172]方同.工程随机振动[M].国防工业出版社,1995
[173]Bruggi Matteo, Taliercio Alberto. Maximization of the fundamental eigenfrequency of micropolar solids through topology optimization[J]. Structural and Multidisciplinary Optimization,2012,46(4):549-560.
[174]张桥,张卫红,朱继宏.动力响应约束下的结构拓扑优化设计[J],2010
[175]Jang I G, Kwak B M. Evolutionary topology optimization using design space adjustment based on fixed grid[J]. International Journal for Numerical Methods in Engineering,2006,66(11):1817-1840.
[176]Adelman Howard M, Haftka Raphael T. Sensitivity analysis of discrete structural systems[J]. AIAA Journal,1986,24(5):823-832.
[177]Baldwin John F, Hutton Stanley G. Natural modes of modified structures[J]. AIAA Journal,1985,23(11):1737-1743.
[178]Lim K. B, Junkins J L, Wang B P. Re-examination of eigenvector derivatives [J]. Journal of Guidance, Control, and Dynamics,1987,10(6):581-587.
[179]Liu Zhong-sheng, Chen Su-huan, Zhao You-qun. An accurate method for computing eigenvector derivatives for free-free structures [J]. Computers & Structures,1994,52 (6):1135-1143.
[180]Fox R L, Kapoor M P. Rates of change of eigenvalues and eigenvectors[J]. AIAA Journal,1968,6(12):2426-2429.
[181]Nelson Richard B. Simplified calculation of eigenvector derivatives [J]. AIAA Journal, 1976,14(9):1201-1205.
[182]Chen Ting-Yu. Design sensitivity analysis for repeated eigenvalues in structural design[J]. AIAA Journal,1993,31(12):2347-2350.
[183]Alvin Kenneth F. Efficient computation of eigenvector sensitivities for structural dynamics[J]. AIAA Journal,1997,35(11):1760-1766.
[184]Diaz A, Sigmund O. Checkerboard patterns in layout optimization[J]. Structural and Multidisciplinary Optimization,1995,10(1):40-45.
[185]Sigmund O, Petersson J. Numerical instabilities in topology optimization:a survey on procedures dealing with checkerboards, mesh-dependencies and local minima[J]. Structural Optimization,1998,16(1):68-75.
[186]Swan C C, Kosaka I. Voigt-Reuss topology optimization for structures with linear elastic material behaviours[J].International Journal for Numerical Methods in Engineering,1997,40(16):3033-3057.
[187]Huang X, Xie Y M. Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method [J]. Finite Elements in Analysis and Design,2007,43(14):1039-1049.
[188]Carbonari R C, Silva E C N, Nishiwaki S. Topology optimization applied to the design of multi-actuated piezoelectric micro-tools[C]. Smart Structures and Materials 2004: Modeling, Signal Processing, and Control (Proceedings of SPIE), San Diego,2004: 277-288.
[189]Matsui K, Terada K. Continuous approximation of material distribution for topology optimization[J]. International Journal for Numerical Methods in Engineering,2004,59 (14):1925-1944.
[190]Rahmatalla S F, Swan C C. A Q4/Q4 continuum structural topology optimization implementation[J]. Structural and Multidisciplinary Optimization,2004,27(1):130-135.
[191]Paulino G H, Le C H. A modified Q4/Q4 element for topology optimization[J]. Structural and Multidisciplinary Optimization,2009,37(3):255-264.
[192]Poulsen T A. Topology optimization in wavelet space [J]. International Journal for Numerical Methods in Engineering,2002,53(3):567-582.
[193]Jog C S, Haber R B. Stability of finite elements models for distributed-parameter optimization and topology design[J]. Computer Methods in Applied Mechanics and Engineering,1996,130(3-4):203-226.
[194]Kang Z, Wang Y Q. Structural topology optimization based on non-local Shepard interpolation of density field [J]. Computer Methods in Applied Mechanics and Engineering,2011,200(49):3515-3525.
[195]Guest J K, Prevost J H, Belytschko T. Achieving minimum length scale in topology optimization using nodal design variables and projection functions[J]. International Journal for Numerical Methods in Engineering,2004,61(2):238-254.
[196]Shepard Donald. A two-dimensional interpolation function for irregularly-spaced data[C]. Proceedings of the 1968 23rd ACM National Conference, New York, 1968:517-524.
[197]Brodlie K W, Asim M R, Unsworth K. Constrained visualization using the Shepard interpolation family[J]. Computer Graphics Forum,2005,24(4):809-820.
[198]Bends(?)e M P. Optimal shape design as a material distribution problem[J]. Structural and Multidisciplinary Optimization,1989,1(4):193-202.
[199]Yang R J, Chuang C H. Optimal topology design using linear programming[J]. Computers & Structures,1994,52(2):265-275.
[200]Rozvany G I N. Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics[J]. Structural and Multidisciplinary Optimization,2001,21(2):90-108.
[201]Beckers M. Topology optimization using a dual method with discrete variables[J]. Structural and Multidisciplinary Optimization,1999,17(1):14-24.
[202]Rong Jian Hua, Li Wei Xiang, Feng Bing. A structural topological optimization method based on varying displacement limits and design space adjustments[J], Advanced Materials Research,2010,97-101:3609-3616.
[203]Chapman B, Jost G, Pas R V D. Using OpenMP:Portable Shared Memory Parallel Programming[M]. The MIT Press,2007
[204]Colominas I, Paris J, Navarrina F, et al. High performance parallel computing in structural topology optimization[C]. BHV Topping LF Costa Neves, RC Barros. Proceedings of the 12th International Conference on Civil, Structural and Environmental Engineering Computing, London,2009
[205]Manickarajah D, Xie Y M, Steven G P. An evolutionary method for optimization of plate buckling resistance[J]. Finite Elements in Analysis and Design,1998,29(3):205-230.
[206]Rong J H, Xie Y M, Yang X Y. An improved method for evolutionary structural optimization against buckling[J]. Computers & Structures,2001,79(3):253-263.
[207]俞焕然,王璠.具有初始缺陷变厚度圆板在复合载荷作用下的稳定性及优化[J].福州大学学报(自然科学版),1994,22(4)：90-95
[208]LIU Tao, XU Qi-nan, PEI Jun-hou. Buckling analysis and optimization design of composite cylindrical shells[J]. Journal of Ship Mechanics,2000,6(4):39-50.
[209]唐文艳,袁清珂.屈曲约束桁架拓扑优化的遗传算法[J].机械科学与技术,2009,(4)：460-463.
[210]梁斌,高笑娟,俞焕然.几何约束下圆柱壳的屈曲优化设计[J].兰州大学学报：自然科学版,2003,39(1)：24-27.
[211]吴浩,燕瑛.基于可靠性的复合材料结构稳定性约束优化设计[J].复合材料学报,2007,24(5)：149-153.
[212]Zhou M. Topology optimization for shell structures with linear buckling responses [J]. WCCM VI, Beijing, China,2004
[213]Guo X, Cheng G D, Olhoff N. Optimum design of truss topology under buckling constraints[J]. Structural and Multidisciplinary Optimization,2005,30(3):169-180.
[214]隋允康,边炳传.屈曲与应力约束下连续体结构的拓扑优化[J].工程力学,2008,(8)：6-12.
[215]隋允康,边炳传,叶红玲.连续体结构屈曲约束的ICM方法拓扑优化[J].计算力学学报,2008,25(3)：345-351.
[216]李洋,龙连春.考虑整体屈曲薄壁加筋板加筋布局优化[C].北京力学会.北京力学会第19届学术年会,中国北京,2013：2.
[217]韩强,张善元,杨桂通.结构静动力屈曲问题研究进展[J].力学进展,1998,28(3)：349-360.
[218]Yi Jijun, Rong Jianhua, Zeng Tao, et al. A topology optimization method for multiple load cases and constraints based on element independent nodal density[J]. Structural Engineering and Mechanics,2013,45(6):759-777.
[219]张学玲.基于广义模块化设计的机械结构静、动态特性分析及优化设计[D].天津：天津大学,2004.
[220]Pahl G, Beitz Wolfgong. Engineering design:a systematic approach[M]. Springer-Verlag, London,1988
[221]Gu P, Hashemian M, Nee A Y C. Adaptable design[J]. CIRP Annals-Manufacturing Technology,2004,53(2):539-557.
[222]高卫国,徐燕申,陈永亮,等.广义模块化设计原理及方法[J].机械工程学报,2007,43(6)：48-54.
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