基于水平集的多材料结构拓扑优化设计方法与应用
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摘要
结构的形状和拓扑优化设计是随着计算机软、硬件的发展,利用数学和力学理论寻找工程问题中某个目标最优的过程。根据问题复杂度的不同,结构优化可分为三个层次:尺寸优化、形状优化和拓扑优化。其中,尺寸优化是最简单的,其研究工作已经基本成熟。在形状优化中,结构的边界形状可以变化,然而,结构的拓扑是固定的。与尺寸优化和形状优化相比,拓扑优化在设计前没有任何关于结构的几何形状与拓扑信息。因此,拓扑优化在结构优化设计中最难、最具有挑战性。
目前结构拓扑优化设计理论存在的主要问题包括结构形状和拓扑控制方法尚未成熟,难以实现对求解过程中的形状和拓扑进行描述和控制。目前采用的依赖不同坐标系下单元映射方法来控制边界形状,不可能在形状和拓扑设计方面产生质的飞跃。它只能改变现有结构的几何形状,而不能对结构进行自然的几何拓扑优化。其次,目前结构拓扑优化的对象主要由单一的均质材料构成,随着对结构综合性能要求的提高,由多种不同特性的材料组成的多功能材料,在工程实际中具有日益广泛的应用前景。因此,如何优化结构的形状拓扑关系,将不同性能和功能的材料配置在一起,以达到设计所要求的性能的研究工作就变得日益紧迫。
针对上述结构拓扑优化领域中存在的问题,开展了深入的探索和研究,提出基于水平集的散热结构与弹性结构拓扑优化设计及其改进算法,通过数值仿真验证算法的可行性与有效性,主要的研究工作及创新性成果总结如下:
一、介绍了基于水平集方法的动态界面处理技术、分析了水平集方法计算和分析动态界面的优点。然后,研究了基于水平集方法的优化原理——变分水平集方法。针对以物理域的形状为变量的能量泛函变分的计算,给出了形状灵敏度分析的两个重要形状导数引理。最后,论述了利用边界推进技术与形状导数理论进行拓扑优化的基本过程。
二、研究基于水平集的隐式描述材料边界方法,以温度梯度范数平方的积分为目标函数,利用形状导数理论和伴随变量方法进行形状灵敏度分析,构造基于水平集的最优化条件。针对拓扑优化结果对初始拓扑猜测依赖性问题,引入以椭圆型偏微分方程为约束的拓扑导数理论,在一定程度上抑制了初始拓扑猜测的依赖性。通过与SIMP方法的数值算例比较,证明方法的可行性与正确性。
三、对于弹性结构拓扑优化问题,提出根据Von Mises应力分布情况进行开孔的策略,弥补了水平集方法本身不能开孔的缺陷。提出基于极限窄带单元推进的结构拓扑优化方法,在保持优化过程中自然处理拓扑变化优点的同时,不用求解Hamilton-Jacobi系统的偏微分方程实现隐式边界推进,避免了求解重新初始化、速度扩展偏微分方程与考虑重新初始化频率的问题,提高了拓扑优化过程的计算效率和收敛速度。
四、研究了矢量水平集在多材料设计中的应用,构造了基于水平集的多材料散热结构拓扑优化设计的模型,推导使目标函数下降的速度场,通过求解多个水平集方程实现多材料散热结构拓扑优化设计。对于弹性结构拓扑优化设计,根据各材料Von Mises应力分布情况,制定各材料替换策略,产生新的拓扑,然后综合利用基于水平集的形状优化理论实现弹性结构的多材料拓扑优化设计,从而抑制多材料的弹性结构拓扑优化设计对初始拓扑猜测的依赖性。
五、针对工程实际中的三维空间问题,研究了基于SIMP方法三维结构拓扑优化的密度惩罚项和滤波半径对最终拓扑优化结果的影响及其合理的取值范围。通过与基于水平集方法的三维弹性结构拓扑优化结果比较,得出基于水平集的三维结构拓扑优化设计不存在初始拓扑猜测依赖性问题。
Structural shape and topology optimization uses the theory of mathematics and mechanics to find the optimal result of engineering problems and is prospering with the progress of computer software and hardware. The structural optimization problem is specified as three types, the size optimization, the shape optimization and the topology optimization. The size optimization problem is simple and almost mature. For shape optimization, the topology of structure is fixed before design and only the shape of structure changes during the optimization process. However, the topology optimization has no information of structural shape and topology before the optimization design. So the topology optimization is the most difficult and challenging for structural optimization design.
One of the difficult problems in the structural topology optimization field is the representation method of structural shape and topology. The usual representation of structural shape is realized by the finite element map, which can not generate new breakthrough for the design of geometric shape and topology in essence. The finite element map method can not naturally handle topological changes during the topology optimization process. Next, the current structural design focuses on single material. The multi-function material becomes more and more important with the enhancement of structural performance requirement in the engineering field. So the research of structural shape and topology with multi-phase materials for specific performance becomes more urgent.
In order to overcome the problems above, we investigate the dissipated and elastic structural topology optimization based on the level set method and the corresponding improved algorithms. Numerical simulations are performed to demonstrate the feasibility and the validity of the proposed method. The main researches and the novel contributions are listed as follows:
Firstly, the implicit dynamic interface technique is introduced and the advantages of computation and analysis for the propagation interface based on the level set method are listed. Then, the level set model based optimization theory, namely variational level set method, is presented. The two important lemmas of shape derivative are presented for the shape sensitivity analysis which is crucial when the shape of material region is considered as the design variable. At last, the process of optimization based on the boundary propagation technique and the shape derivative is given. Secondly, the implicit representation approach based on the level set model of material boundaries is discussed. For the dissipated structural topology optimization problem, the integral of square of the temperature gradient is taken as the optimization objective. The shape sensitivity analysis is implemented by the shape derivative theory and the adjoint variable method. Then, the optimization condition for the level set equation is constructed. The topology derivative theory of the elliptic partial differential equation is introduced to restrain the dependence of initial topology guess.
Thirdly, the strategy of new holes generation is presented according to the Von Mises stress distribution, which can suppress the dependence of initial topology guess for the elastic structural topology optimization. At the same time, the new strategy can make up the disadvantage that the level set method can not generate new holes in the material region. The element propagation of extreme narrow band is applied to the structural topology optimization. This method not only can deal with topological changes naturally, but also aovid solving the level set equation, the reinitialization partial differential equation and the velocity extension partial differential equation. So the element propagation of extreme narrow band based topology optimization improves the computational efficiency and the convergent process.
Fourthly, the vector level set model is introduced for the structural topology optimization with multi-material. The vector level set function is incorporated with the topology optimization model for the heat conduction problem. And then, the velocity field of the vector level set function is constructed which makes the objective function descent. The final result of multi-material structural topology optimization is obtained by solving the multiple level set equations. For the elasticity problem, we establish the strategy of material substitution according to the distribution of Von Mises stress when the current material volumes exceed the volume constraints, which can suppress the dependence of initialization for multi-material design to some extent.
Finally, the SIMP method based the penalization power and the filter radius of three-dimensional structural topology optimization for the practical engineering application is explored. Then, we discuss the rational range of the penalization power and the filter radius for three-dimensional structural topology optimization. The three-dimensional structural optimization result using the SIMP method is compared with that based on the level set model. The three-dimensional structural topology optimization based on the level set model has no problem of dependence of initial topological guess.
目前结构拓扑优化设计理论存在的主要问题包括结构形状和拓扑控制方法尚未成熟,难以实现对求解过程中的形状和拓扑进行描述和控制。目前采用的依赖不同坐标系下单元映射方法来控制边界形状,不可能在形状和拓扑设计方面产生质的飞跃。它只能改变现有结构的几何形状,而不能对结构进行自然的几何拓扑优化。其次,目前结构拓扑优化的对象主要由单一的均质材料构成,随着对结构综合性能要求的提高,由多种不同特性的材料组成的多功能材料,在工程实际中具有日益广泛的应用前景。因此,如何优化结构的形状拓扑关系,将不同性能和功能的材料配置在一起,以达到设计所要求的性能的研究工作就变得日益紧迫。
针对上述结构拓扑优化领域中存在的问题,开展了深入的探索和研究,提出基于水平集的散热结构与弹性结构拓扑优化设计及其改进算法,通过数值仿真验证算法的可行性与有效性,主要的研究工作及创新性成果总结如下:
一、介绍了基于水平集方法的动态界面处理技术、分析了水平集方法计算和分析动态界面的优点。然后,研究了基于水平集方法的优化原理——变分水平集方法。针对以物理域的形状为变量的能量泛函变分的计算,给出了形状灵敏度分析的两个重要形状导数引理。最后,论述了利用边界推进技术与形状导数理论进行拓扑优化的基本过程。
二、研究基于水平集的隐式描述材料边界方法,以温度梯度范数平方的积分为目标函数,利用形状导数理论和伴随变量方法进行形状灵敏度分析,构造基于水平集的最优化条件。针对拓扑优化结果对初始拓扑猜测依赖性问题,引入以椭圆型偏微分方程为约束的拓扑导数理论,在一定程度上抑制了初始拓扑猜测的依赖性。通过与SIMP方法的数值算例比较,证明方法的可行性与正确性。
三、对于弹性结构拓扑优化问题,提出根据Von Mises应力分布情况进行开孔的策略,弥补了水平集方法本身不能开孔的缺陷。提出基于极限窄带单元推进的结构拓扑优化方法,在保持优化过程中自然处理拓扑变化优点的同时,不用求解Hamilton-Jacobi系统的偏微分方程实现隐式边界推进,避免了求解重新初始化、速度扩展偏微分方程与考虑重新初始化频率的问题,提高了拓扑优化过程的计算效率和收敛速度。
四、研究了矢量水平集在多材料设计中的应用,构造了基于水平集的多材料散热结构拓扑优化设计的模型,推导使目标函数下降的速度场,通过求解多个水平集方程实现多材料散热结构拓扑优化设计。对于弹性结构拓扑优化设计,根据各材料Von Mises应力分布情况,制定各材料替换策略,产生新的拓扑,然后综合利用基于水平集的形状优化理论实现弹性结构的多材料拓扑优化设计,从而抑制多材料的弹性结构拓扑优化设计对初始拓扑猜测的依赖性。
五、针对工程实际中的三维空间问题,研究了基于SIMP方法三维结构拓扑优化的密度惩罚项和滤波半径对最终拓扑优化结果的影响及其合理的取值范围。通过与基于水平集方法的三维弹性结构拓扑优化结果比较,得出基于水平集的三维结构拓扑优化设计不存在初始拓扑猜测依赖性问题。
Structural shape and topology optimization uses the theory of mathematics and mechanics to find the optimal result of engineering problems and is prospering with the progress of computer software and hardware. The structural optimization problem is specified as three types, the size optimization, the shape optimization and the topology optimization. The size optimization problem is simple and almost mature. For shape optimization, the topology of structure is fixed before design and only the shape of structure changes during the optimization process. However, the topology optimization has no information of structural shape and topology before the optimization design. So the topology optimization is the most difficult and challenging for structural optimization design.
One of the difficult problems in the structural topology optimization field is the representation method of structural shape and topology. The usual representation of structural shape is realized by the finite element map, which can not generate new breakthrough for the design of geometric shape and topology in essence. The finite element map method can not naturally handle topological changes during the topology optimization process. Next, the current structural design focuses on single material. The multi-function material becomes more and more important with the enhancement of structural performance requirement in the engineering field. So the research of structural shape and topology with multi-phase materials for specific performance becomes more urgent.
In order to overcome the problems above, we investigate the dissipated and elastic structural topology optimization based on the level set method and the corresponding improved algorithms. Numerical simulations are performed to demonstrate the feasibility and the validity of the proposed method. The main researches and the novel contributions are listed as follows:
Firstly, the implicit dynamic interface technique is introduced and the advantages of computation and analysis for the propagation interface based on the level set method are listed. Then, the level set model based optimization theory, namely variational level set method, is presented. The two important lemmas of shape derivative are presented for the shape sensitivity analysis which is crucial when the shape of material region is considered as the design variable. At last, the process of optimization based on the boundary propagation technique and the shape derivative is given. Secondly, the implicit representation approach based on the level set model of material boundaries is discussed. For the dissipated structural topology optimization problem, the integral of square of the temperature gradient is taken as the optimization objective. The shape sensitivity analysis is implemented by the shape derivative theory and the adjoint variable method. Then, the optimization condition for the level set equation is constructed. The topology derivative theory of the elliptic partial differential equation is introduced to restrain the dependence of initial topology guess.
Thirdly, the strategy of new holes generation is presented according to the Von Mises stress distribution, which can suppress the dependence of initial topology guess for the elastic structural topology optimization. At the same time, the new strategy can make up the disadvantage that the level set method can not generate new holes in the material region. The element propagation of extreme narrow band is applied to the structural topology optimization. This method not only can deal with topological changes naturally, but also aovid solving the level set equation, the reinitialization partial differential equation and the velocity extension partial differential equation. So the element propagation of extreme narrow band based topology optimization improves the computational efficiency and the convergent process.
Fourthly, the vector level set model is introduced for the structural topology optimization with multi-material. The vector level set function is incorporated with the topology optimization model for the heat conduction problem. And then, the velocity field of the vector level set function is constructed which makes the objective function descent. The final result of multi-material structural topology optimization is obtained by solving the multiple level set equations. For the elasticity problem, we establish the strategy of material substitution according to the distribution of Von Mises stress when the current material volumes exceed the volume constraints, which can suppress the dependence of initialization for multi-material design to some extent.
Finally, the SIMP method based the penalization power and the filter radius of three-dimensional structural topology optimization for the practical engineering application is explored. Then, we discuss the rational range of the penalization power and the filter radius for three-dimensional structural topology optimization. The three-dimensional structural optimization result using the SIMP method is compared with that based on the level set model. The three-dimensional structural topology optimization based on the level set model has no problem of dependence of initial topological guess.
引文
[1] 钱令希,工程结构优化设计,北京,水利电力出版社,1983.
[2] 程耿东,工程结构优化设计基础,北京,水利电力出版社,1984.
[3] 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,工程力学,2005,22:107-118.
[4] Optimization in automotive engineering, http://www.fe-design.de.
[5] 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):3-149.
[6] Haber R B, Bends?e M P, Problem formulation, solution procedure and geometric modeling: Key issues in variable-topology optimization, Proceedings 7th AIAA /USAF /NASA /ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, MI, 1998, 1864-1873.
[7] Eschenauer H A,Olhoff N, Topology optimization of continuum structures: A review. Applied Mechanics Reviews, 2001, 54(4): 331-390.
[8] Michell A G M, The limits economy in frame structure, Philosophical Magazine, Series, 1904, 6(8): 589-597.
[9] 程耿东,连续体结构优化设计,大连,大连理工大学出版社,1989.
[10]程耿东,张东旭,受应力约束的平面弹性体的拓扑优化,大连理工 大学学报,1995,35:1-9.
[11]王 健 , 程 耿 东 , 应 力 约束 下 薄 板 结 构的 拓 扑 优化 , 固 体 力 学 学报,1997,18(4):317-322.
[12]叶红玲,隋允康,连续体结构静力拓扑优化方法与软件开发,[学位论文],北京,北京工业大学,2005.
[13]杨德庆,刘正兴,隋允康,连续体结构拓扑优化设计的ICM方法,上海交通大学学报,1999,33(6):734-736.
[14]杨德庆,隋允康,刘正兴等,应力和位移约束下连续体结构拓扑优化,应用数学和力学,2000,21(1):17-24.
[15]隋允康,杨德庆,孙焕纯,统一骨架与连续体的结构拓扑优化的ICM理论与方法,计算力学学报,2000,17(1):28-33.
[16]隋允康,任旭春,龙连春等,基于ICM 方法的刚架拓扑优化,计算力学学报,2003,20(3):286-289.
[17]隋 允 康 , 彭 细 荣 , 结 构 拓 扑 优 化 ICM 方 法 的 改 善 , 力 学 学报,2005,37(2):190-198.
[18]隋允康,叶红玲,彭细荣,应力约束全局化策略下的连续体结构拓扑优化,力学学报,2006,38(3):364-370.
[19]隋允康,彭细荣,连续体结构考虑离散性目标的ICM方法,计算力学学报,2006,23(2):163-168.
[20]叶红玲,隋允康,应力约束下连续体结构的拓扑优化,北京工业大学学报,2006,32(4):301-305.
[21]隋允康,张学胜,龙连春等,位移约束集成化处理的连续体结构拓扑优化,固体力学学报,2006,27(1):102-107.
[22]隋允康,彭细荣,叶红玲,应力约束全局化处理的连续体结构ICM拓扑优化方法,工程力学,2006,23(7):1-7.
[23]袁 振 , 吴 长 春 , 采 用 非 协 调 元 的 连 续体 拓 扑 优 化 设计 , 力 学 学报,2003,35(2):176-180.
[24]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,中国科学技术大学学报,2001,31(6):694-699.
[25]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,固体力学学报,2003,24(1):40-45.
[26]袁振 , 吴长春 , 复 合材料 扭 转 轴截 面 微 结构拓扑优化设计 , 力学学报,2003,35(1):39-42.
[27]荣见华,谢忆民,姜节胜,渐进结构优化设计的现状与进展,长沙交通学院学报,2001,17(3):16-23.
[28]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,2003,35(5):584-591.
[29]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,工程力学,2004,21(5):64-71.
[30]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,计算力学学报,2004,21(3):322-329.
[31]罗志凡,荣见华,杜海珍,一种基于主应力的双向渐进结构拓扑优化方法,应用基础与工程科学学报,2003,11(1):98-105.
[32]罗志凡,卢耀祖,荣见华等,基于一种新的应力准则的渐进结构优化方法,同济大学学报,2005,33(3):372-375.
[33]杜海珍,荣见华,傅建林等,基于应变能的双向结构渐进优化方法,机械强度,2005,27(1):72-77.
[34]周克民,胡云昌,利用变厚度单元进行平面连续体的拓扑优化,天津城市建设学院学报,2001,7(1):33-35.
[35]周克民,胡云昌,结合拓扑分析进行平面连续体拓扑优化,天津大学学报,2001,34(3):340-345.
[36]左孔天,陈立平,钱勤等,求解拓扑优化问题的一种移动渐进2小波混和算法,固体力学学报,2004,25(4):471-475.
[37]左孔天,陈立平,张云清等,用拓扑优化方法进行热传导散热体的结构优化设计,机械工程学报,2005,41(4):13-16,21.
[38]左孔天,陈立平,王书亭等,用拓扑优化方法进行微型柔性机构的设计研究,机械工程学报,2004,15(21):1886-1890.
[39]王书亭,左孔天,基于均匀化理论的拓扑优化算法研究,华中科技大学学报,2004,32(10):25-27,30.
[40]王书亭,左孔天,一种基于均匀化理论的拓扑优化准则法,华中科技大学学报,2004,32(10):28-30.
[41]罗震,陈立平,钟毅芳,基于凸规划方法的计算机辅助结构拓扑优化设计,计算机辅助设计与图形学学报,2005,17(4):774-781.
[42]罗震,郭文德,蒙永立等,全柔性微型机构的拓扑优化设计技术研究,航空学报,2005,26(5):617-625.
[43]罗震,陈立平,黄玉盈等,基于RAMP密度-刚度插值格式的结构拓扑优化,计算力学学报,2005,22(5):585-591.
[44]张宪民,陈永健,多输入多输出柔顺机构拓扑优化及输出耦合的抑制,机械工程学报,2006,42(3):162-165.
[45]张宪民,陈永健,考虑输出耦合时柔顺机构拓扑与压电驱动单元的优化设计,机械工程学报,2005,41(8):69-74.
[46]郭旭 . 赵 康 . 基于拓扑 描述 函 数的 连续体 结构拓扑优化 方法 . 力学学报.2004.36(5):520-526.
[47]孟宪颐 , 用边界 元 分 域法对 机 械 构件 进行 形状优化 , 机 械 工程学报,1991,27(1):84-90.
[48]张立洲,李性厚,孙焕纯,弹性平面孔洞形状优化的虚边界元法,锦州工学院院报,1991,10(2):1-7.
[49]申秀丽,温卫东,高德平,接触问题形状优化德边界元法及其工程应用,航空动力学报,1997(12):113-116.
[50]周克民,胡云昌,用可退化有限元进行平面连续体拓扑优化,应用力学学报,2002,19(1):124-126.
[51]隋允康,杨德庆,王备,多工况应力和位移约束下连续体结构拓扑优化,力学学报,2000,32(2):171-179.
[52]杨德庆,隋允康,刘正兴,多工况应力约束下连续体结构拓扑优化映射变换解法,上海交通大学学报,2000,34(8):1061-1065.
[53]于新,隋允康,平面连续体多工况应力约束下结构拓扑优化的有无复合体方法,固体力学学报,2001,18(2):200-204.
[54]王健 , 程 耿东 , 多 工 况 应 力 约束 下 连续体 结构拓扑优化设计 , 机 械 强度,2003,25(1):55-57.
[55]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,固体力学学报,2005,26(1):29-36.
[56]Bends?e M P, Optimization of Structural Topology, Shape, and Material, New York, Springer Verlag, 1997.
[57]Bends?e M P, Sigmund O, Topology Optimization: Theory, Methods and Applications, New York, Springer Verlag, 2003.
[58]Suzuki K, Kikuchi N, A homogenization method for shape and topology optimization, Computer Method in Applied Mechanics and Engineering, 1991, 93(3):291-381.
[59]Allaire G, Bonnetier E, Francfort G, etc., Shape optimization by the homogenization method, Numerische Mathematik, 1997, 76: 27-68.
[60]Bends?e M P, Kikuchi N, Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering, 1998, 71(2): 197-224.
[61]Allaire G, Homogenization and applications to material sciences, Lecture Notes atthe Newton Institue, Cambridge, September 1999.
[62]Byun J K, Lee J H, Inverse Problem application of topology optimization method with mutual energy concept and design sensitivity, IEEE Transactions on Magnetics, 2000, 36(4): 1144-1147.
[63]Allaire G, Shape optimization by the homogenization method, New York, Springer Verlag, 2001.
[64]Ambrosio L, Buttazzo G, An optimal design Problem with Perimeter Penalization, Calculus of Variations, 1993, 1: 55-69.
[65]Bends?e M P, Sigmund O, Material interpolation schemes in topology optimization, Archive of Applied Mechanics, 1999, 69: 635-654.
[66]Rietz A, Sufficiency of a finite exponent in SIMP (Power Law) methods, Structural and Multidisciplinary Optimization, 2001, 21(2): 159-163.
[67]Martinez J M, A note on the theoretical convergence properties of SIMP method, Structural and Multidisciplinary Optimization, 2004, 29(4): 319-323.
[68]Xie Y M, Steven G P, A simple evolutionary procedure for structural optimization, Computers & Structures, 1993, 49: 885-896.
[69]Xie Y M, Steven G P, Evolutionary Structural Optimization, New York, Springer Verlag, 1997.
[70]Querin O M, Steven G P, Evolutionary structural optimization using an additive algorithm, Finite Elements in Analysis and Design, 2000, 34(3-4): 291-308.
[71]Querin O M, Steven G P, Xie Y M, Evolutionary structural optimisation (ESO) using a bidirectional algorithm, Engineering Computations, 1998, 15(8): 1031-1048.
[72]Annicchiarico W, Cerrolaza M, An Evolutionary approach for the shape optimization of general boundary elements models, Electronic Journal of Boundary Elements, 2001, 2: 251-266.
[73]Chang S, Youn S, Material clouds method for topology optimization, International Journal for Numerical Methods in Engineering, 2005, 65(10): 1585-1607.
[74]Sigmund O, Petersson J, Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Structural and Multidisciplinary Optimization, 1998, 16(1): 68-75.
[75]Sigmund O, On the design of compliant mechanism using topology optimization, Mechanics of Structures and Machines, 1997, 25(4): 495-526.
[76]Wang M Y, Wang S, Bilateral filtering for structural topology optimization, International Journal for Numerical Methods in Engineering, 2005, 63 (13): 1911-1938.
[77]Haber R B, Jog C S, Bends?e M P, A new approach to variable-topology shape design using a constraint on perimeter, Structural Optimization, 1996, 11: 1-12.
[78]Eschenauer H A, Schumacher A, Bubble method for topology and shape optimization of structures, Structural Optimization, 1994, 8: 42-51.
[79]Eschenauer H A, Schumacher A, Topology and shape optimization procedures using hole positioning criteria, Topology Optimization in Structural Mechanics, 1997, 135-196.
[80]Poulsen T A, Topology optimization in wavelet space, International Journal for Numerical Methods in Engineering, 2002, 53(4): 567-582.
[81]Petersson J, Sigmund O, Slope constrained topology optimization, International Journal for Numerical Methods in Engineering, 1998, 41(8): 1417-1434.
[82]Jang G W, Jeong H, Checkerboard-free topology optimization using non-conforming finite elements, International Journal for Numerical Methods in Engineering, 2003, 57(12): 1717-1735.
[83]Jang G W, Lee S, Topology optimization using non-conforming finite elements: three-dimensional case, International Journal for Numerical Methods in Engineering, 2005, 63(6): 859-975.
[84]Ruiter M J, Keulen F, Topology optimization: approaching the material distribution problem using a topological function description, Computational techniques for materials, composites and composite structures, 2000, 111-119.
[85]Ruiter M J, Keulen F, Topology Optimization Using the Topology Description Function, The Fourth World Congress of Structural and Multidisciplinary Optimization, Dalian, China, 2001, 19-20.
[86]Ruiter M J, Keulen F, The Topological Derivative in the Topology Description Function Approach, Engineering Design Optimization, Product and Process Improvement, Proceedings of the 4th ASMO-UK/ISSMO conference,Newcastle-upon-Tyne, UK, 2002, 160-167.
[87]Ruiter M J, Keulen F, Topology optimization using a topology description function, Structural and multidisciplinary optimization, 2004, 26(6): 406-416.
[88]Guo X, Zhao K, Wang M Y, A new approach for simultaneous shape and topology optimization based on implicit topology description functions, Proceedings of the ASME Design Engineering Technical Conference, 2004, 1: 503-512.
[89]Sethian J A, Wiegann A, Structural boundary design via level set and immersed interface methods, Journal of Computational Physics, 2000, 163(2): 489-528.
[90]Osher S J, Santosa F, Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum, Journal of Computational Physics, 2001, 171(1): 272-288.
[91]Allaire G, Jouve F, Toader A M, Structural optimization using sensitivity analysis and a level-set method, Journal of Computational Physics, 2004, 194(1): 363-393.
[92]Allaire G, Gournay F, Jouve F, etc., Structural optimization using topological and shape sensitivity via a level set method, Control and Cybernetics, 2005, 34(1): 59-81.
[93]Allaire G, Jouve F, A level-set method for vibration and multiple loads structural optimization, Computer Methods in Applied Mechanics and Engineering, 2005, 194(30-33): 3269-3290.
[94]Allaire G, Topology optimization with the homogenization and the level-set method, Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 2005, 170: 1-13.
[95]Wang M Y, Wang X, Guo D, A level set method for structural topology optimization, Computer Methods in Applied Mechanics and Engineering, 2003, 192(1-2): 227-246.
[96]Wang X, Wang M Y, Structural shape and topology optimization in a level-set-based framework of region representation, Structural and Multidisciplinary Optimization, 2004, 27(1-2): 1-19.
[97]Wang M Y, Wang X, “Color” level sets: a multi-phase method for structural topology optimization with multiple materials, Computer Methods in Applied Mechanics and Engineering, 2004, 193(6-8): 469-496.
[98]Wang M Y, Chen S K, Wang X, etc., Design of multi-material compliant mechanisms using level-set methods, Journal of Mechanical Design, 2005, 127(5): 941-956.
[99]Wang M Y, Wang X, PDE-driven level sets, shape sensitivity, and curvature flow for structural topology optimization, CMES-Computer Modeling in Engineering and Sciences, 2004, 6(4): 373-395.
[100]Mei Y, Wang X, A level set method for structural topology optimization and its applications, Advances in Engineering for Software, 2004, 35(7): 415-441.
[101]Mei Y, Wang X, A level set method for microstructure design of composite materials, Acta Mechanica Solida Sinica, 2004, 17(3): 239-250.
[102]梅玉林,王德伦,拓扑优化的水平集方法及其在刚性结构、柔性机构和材料设计中的应用,[学位论文],大连,大连理工大学,2003.
[103]Kwak J, Cho S, Topological shape optimization of geometrically nonlinear structures using level set method, Computers and Structures 2005, 83(27): 2257-2268.
[104]Cho S, Ha S, Yang Y, Topological Shape Optimization of Geometrically Nonlinear Structures Using Level Set and Meshfree Methods, 6th World Congress on Structural and Multidisciplinary Optimization, Brazil, 2005.
[105]Ha S, Cho S, Topological Shape Optimization of Heat Conduction Problems using Level Set Approach, Numerical Heat Transfer Part B: Fundamentals, 2005, 48(1): 67-88.
[106]Cho S, Ha S, Park C, Topological shape optimization of power flow problems at high frequencies using level set approach, International Journal of Solids and Structures, 2006, 43(1): 172-192.
[107]Design of a small satellite, http://www.topopt.dtu.dk/.
[108]Haber E, A multilevel, level set method for optimizing eigenvalues in shape design problems, Journal of Computational Physics, 2004, 198(2): 518-534.
[109]Ma Z, Kikuchi N, Cheng H, Topological design for vibrating structures, Computer Methods in Applied Mechanics and Engineering, 1995, 121(1-4): 259-280.
[110]Allaire G, Aubry S, Jouve F, Eigenfrequency optimization in optimal design,Computer Methods in Applied Mechanics and Engineering, 2001, 190(28): 3365-3579.
[111]Xie Y M, Steven G P, Evolutionary Structural optimization for dynamic problems. Computers and Structures, 1996, 58(6): 1067-1073.
[112]Bruns T E, Topology optimization of nonlinear elastic structures and compliant mechanisms, [Dissertation], University of Michigan, Annarbor, 1990.
[113]Buhl T, Pedersen C B W, Sigmund O, Stiffness design of geometrically nonlinear structures using topology optimization, Structural and Multidisciplinary Optimization, 2000, 19(2): 93-104.
[114]Jung D, Gea H C, Topology optimization of nonlinear structures, Finite Elements in Analysis and Design, 2004, 40(1): 1417-1427.
[115]Stegmann J, Lund E, Nonlinear topology optimization of laminated shells, Proceeding 15th Nordic Seminar on Computational Mechanics, Aalborg, Denmark, 2002, 215-218.
[116]Lin J, Shape optimization of nonlinear structure under fatigue loading, [Dissertation], University of Cincinnati, Mechanical Engineering, 2001.
[117]郭旭,赵康,拓扑相关载荷作用下结构形状/拓扑优化的水平集方法,中国计算力学大会,2003,627-633.
[118]Guo X, Zhao K, Gu Y X, Topology optimization with design-dependent loads by level set approach, Proceeding of 10th AIAA/ISSMO Multidisciplinary analysis and Optimization conference, 2004.
[119]Kharmanda G, Olhoff N, Reliability-based topology optimization as a new strategy to generate different structural topologies, Proceedings of 15th Nordic Seminar on Computational Mechanics, Aalborg, Denmark, 2002, 18-19.
[120]Kharmanda G, Olhoff N, Mohamed A, etc., Reliability-based topology optimization, Structural and Multidisciplinary Optimization, 2004, 26(5): 295-307.
[121]Ananthasuresh G K, Kota S, Gianchandani Y, A methodical approach to the design of compliant micromechanisms, Solid-State sensor and Actuator Workshop, 1994, 189-192.
[122]Nishiwaki S, Frecker M I, Min S, etc., Topology optimization of compliantmechanisms using the homogenization method, International Journal for Numerical Methods in Engineering, 1998, 42(3): 535-559.
[123]Sigmund O, Torquato S, Composites with external thermal expansion coefficients, Applied Physics Letters, 1996, 69(21): 3203-3205.
[124]Sigmund O, Design of multiphysics actuators using topology optimization-Part I: One-material structures, Computer Methods in Applied Mechanics and Engineering, 2001, 190(49-50): 6577-6604.
[125]Sigmund O, Design of multiphysics actuators using topology optimization-Part II: Two-material structures, Computer Methods in Applied Mechanics and Engineering, 2001, 190(49-50): 6605-6627.
[126]Sigmund O, Systematic Design and optimization of Multi-Material, Multi-Degree-of-Freedom Micro Actuators, Technical Proceedings of the 2000 International Conference on Modeling and Simulation of Microsystems, 2000, 36-39.
[127]Larsen U D, Sigmund O, Bouwstra S, etc., Design and Fabrication of Compliant Micromechanisms and Structures with Negative Poisson’s Ratio, Journal of Micro electromechanical System, 1997, 6(2): 99-106.
[128]Sigmund O, Topology optimization: a tool for the tailoring of structure and materials, Philosophical Transactions: Mathematical, Physical and Engineering Sciences (Series A), 2000, 358(1765): 211-227.
[129]Yin L, Ananthasuresh G K, Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme, Structural Multidisciplinary Optimization, 2001, 23(1): 49-62.
[130]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.
[131]夏再忠 , 过 增 元 , 用 生 命 演 化过程 模 拟 导热 优化 , 自 然 科 学 进展,2001,11(8):845-851.
[132]Cheng X, Li Z, Guo Z, Constructs of highly effective heat transport paths by bionic optimization, Science in China, 2003, 46(3): 296-302.
[133]过增元,程新广,夏再忠,最小热量传递势容耗散原理及其再导热优化中的应用,科学通报,2003,48(1):21-25.
[134]程新广,李志信,过增元,基于最小热量传递势容耗散原理的导热优化,工程热物理学报,2003,24(1):94-96.
[135]Li Q, Steven G P, Osvaldo M, etc., Shape and topology design for heat conduction by evolutionary structural optimization, International Journal of Heat and Mass Transfer, 1999, 42(17): 3361-3371.
[136]Gaunaoui K E, Allaire G, Homogenization of a conductive and radiative heat transfer problem, simulation with cast3m, Proceeding of HT2005, 2005 ASME Summer Heat Transfer Conference, San Francisco, California, USA, July 17-22, 2005.
[137]顾元宪,赵红兵,陈飚松等,热-应力耦合结构灵敏度分析方法,力学学报,2001,33(5):685-691.
[138]陈飚松,林巍,顾元宪,热传导问题灵敏度分析的伴随法,计算力学学报,2003,20(4):383-390.
[139]Kim D H, Sykulski J K, Lowther D A, A Novel Scheme for Material Updating in Source Distribution Optimization of Magnetic Devices Using Sensitivity Analysis, IEEE Transactions on Magneics, 2005, 41(5): 1752-1755.
[140]Yoo J, Kikuchi N, Volakis J L, Structural optimization in magnetic devices by the homogenization design method, IEEE Transactions on Magneics, 2000, 36(3): 574-580.
[141]Bobaru F, Rachakonda S, Optimal shape profiles for cooling fins of high and low conductivity, International Journal of Heat and Mass Transfer, 2004, 47(23): 4953-4966.
[142]Hughes T J R, The Finite Element Method linear static and dynamic finite element analysis, Mineola New York, Dover Publications, 2000.
[143]Donoso A, Pedregal P, Optimal design of 2D conducting graded materials by minimizing quadratic functionals in the field, Structural and Multidisciplinary Optimization, 2005, 30(5): 360-367.
[144]Donoso A, Numerical simulations in 3D heat conduction: minimizing the quadratic mean temperature gradient, Preprint.
[145]Grabovsky Y, Optimal design problems for Two-Phase conducting composites with weakly discontinuous objective functionals, Advances in AppliedMathematics, 2001, 27(4): 683-704.
[146]Osher S, Sethian J A, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 1988, 79: 12-49.
[147]Sethian J A, Level Set Methods and Fast Marching Methods, Cambridge United Kingdom, Cambridge University Press, 1999.
[148]Osher S, Fedkiw R, Level Set Method and Dynamic Implicit Surfaces, New York, Springer Verlag, 2003.
[149]Osher S, Fedkiw R, Level set methods: an overview and some recent results, Journal of Computational Physics, 2001, 169(2): 463-502.
[150]Osher S, Paragios N, Geometric Level Set Methods in Imaging, Vision, and Graphics, New York, Springer Verlag, 2003.
[151]Malladi R, Sethian J A, Vemuri B C, Shape modeling with front propagation: a level set approach, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1995, 17(2): 158-175.
[152]Chan T, Vese L, Active contours without edges, IEEE Transactions on Image Processing, 2001, 10(2): 266-277.
[153]Sapiro G, Geometric Partial Differential Equations and Image Analysis, Cambridge United Kingdom, Cambridge University Press, 2001.
[154]Kass M, Witkin A, Terzopoulos D, Snakes: active contour models, International Journal of Computer Vision, 1987, 1(4): 321-331.
[155]Caselles V, Kimmel R, Sapiro G, Geodesic active contours, International Journal of Computer Vision, 1997, 22(1): 61-79.
[156]Novotny A A, Feijoo R A, Taroco E, etc., Topological sensitivity analysis, Computer Methods in Applied Mechanics and Engineering, 2003, 192(9-10): 803-829.
[157]Sokolowski J, Zolesio J P, Introduction to Shape Optimization: Shape Sensitivity Analysis, New York, Springer Verlag, 1992.
[158]Pironneau O, Optimal Shape Design for Elliptic Systems, New York, Springer Verlag, 1983.
[159]Liu Z, Korvink J G, Huang R, Structural topology optimization: fully coupledlevel set method via FEMLAB. Structural and Multidisciplinary Optimization, 2005, 29(6): 407-417.
[160]Sokolowski J, Zochowski A, On the topological derivative in shape optimization, SIAM Journal on Control Optimization, 1999, 37(4): 1251-1272.
[161]Burger M, Hackl B, Ring W, Incorporating topological derivatives into level set method, Journal of Computational Physics, 2004, 194(1), 344-362.
[162]Shi Y, Karl W C, A fast level set method without solving PDEs, ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings II, 2005, 97-100.
[163]Sigmund O, A 99 line topology optimization code written in Matlab, Structural and Multidisciplinary Optimization, 2001, 21(2): 120-127.
[164]Vese L, Chan T, A multiphase level set frame work for image segmentation using the Mumford and Shah Model, International Journal of Computer Vision, 2002, 50(3): 271-293.
[165]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(2): 90-108.
[166]王春会,张光,连续体结构拓扑优化设计,[学位论文],西安,西北工业大学,2005.
[167]跨克工作室 ,有限 元分 析基 础篇Ansys与Matlab, 北京 ,清华 大学 出版社,2002.
[168]李家春,叶邦彦,汤勇等,基于密度法的热传导结构拓扑优化准则算法,华南理工大学学报,2006,34(2):27-32.
[169]汪雷,张卫红,材料与结构的传热性能优化设计,[学位论文],西安,西北工业大学,2006.
[2] 程耿东,工程结构优化设计基础,北京,水利电力出版社,1984.
[3] 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,工程力学,2005,22:107-118.
[4] Optimization in automotive engineering, http://www.fe-design.de.
[5] 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):3-149.
[6] Haber R B, Bends?e M P, Problem formulation, solution procedure and geometric modeling: Key issues in variable-topology optimization, Proceedings 7th AIAA /USAF /NASA /ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, MI, 1998, 1864-1873.
[7] Eschenauer H A,Olhoff N, Topology optimization of continuum structures: A review. Applied Mechanics Reviews, 2001, 54(4): 331-390.
[8] Michell A G M, The limits economy in frame structure, Philosophical Magazine, Series, 1904, 6(8): 589-597.
[9] 程耿东,连续体结构优化设计,大连,大连理工大学出版社,1989.
[10]程耿东,张东旭,受应力约束的平面弹性体的拓扑优化,大连理工 大学学报,1995,35:1-9.
[11]王 健 , 程 耿 东 , 应 力 约束 下 薄 板 结 构的 拓 扑 优化 , 固 体 力 学 学报,1997,18(4):317-322.
[12]叶红玲,隋允康,连续体结构静力拓扑优化方法与软件开发,[学位论文],北京,北京工业大学,2005.
[13]杨德庆,刘正兴,隋允康,连续体结构拓扑优化设计的ICM方法,上海交通大学学报,1999,33(6):734-736.
[14]杨德庆,隋允康,刘正兴等,应力和位移约束下连续体结构拓扑优化,应用数学和力学,2000,21(1):17-24.
[15]隋允康,杨德庆,孙焕纯,统一骨架与连续体的结构拓扑优化的ICM理论与方法,计算力学学报,2000,17(1):28-33.
[16]隋允康,任旭春,龙连春等,基于ICM 方法的刚架拓扑优化,计算力学学报,2003,20(3):286-289.
[17]隋 允 康 , 彭 细 荣 , 结 构 拓 扑 优 化 ICM 方 法 的 改 善 , 力 学 学报,2005,37(2):190-198.
[18]隋允康,叶红玲,彭细荣,应力约束全局化策略下的连续体结构拓扑优化,力学学报,2006,38(3):364-370.
[19]隋允康,彭细荣,连续体结构考虑离散性目标的ICM方法,计算力学学报,2006,23(2):163-168.
[20]叶红玲,隋允康,应力约束下连续体结构的拓扑优化,北京工业大学学报,2006,32(4):301-305.
[21]隋允康,张学胜,龙连春等,位移约束集成化处理的连续体结构拓扑优化,固体力学学报,2006,27(1):102-107.
[22]隋允康,彭细荣,叶红玲,应力约束全局化处理的连续体结构ICM拓扑优化方法,工程力学,2006,23(7):1-7.
[23]袁 振 , 吴 长 春 , 采 用 非 协 调 元 的 连 续体 拓 扑 优 化 设计 , 力 学 学报,2003,35(2):176-180.
[24]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,中国科学技术大学学报,2001,31(6):694-699.
[25]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,固体力学学报,2003,24(1):40-45.
[26]袁振 , 吴长春 , 复 合材料 扭 转 轴截 面 微 结构拓扑优化设计 , 力学学报,2003,35(1):39-42.
[27]荣见华,谢忆民,姜节胜,渐进结构优化设计的现状与进展,长沙交通学院学报,2001,17(3):16-23.
[28]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,2003,35(5):584-591.
[29]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,工程力学,2004,21(5):64-71.
[30]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,计算力学学报,2004,21(3):322-329.
[31]罗志凡,荣见华,杜海珍,一种基于主应力的双向渐进结构拓扑优化方法,应用基础与工程科学学报,2003,11(1):98-105.
[32]罗志凡,卢耀祖,荣见华等,基于一种新的应力准则的渐进结构优化方法,同济大学学报,2005,33(3):372-375.
[33]杜海珍,荣见华,傅建林等,基于应变能的双向结构渐进优化方法,机械强度,2005,27(1):72-77.
[34]周克民,胡云昌,利用变厚度单元进行平面连续体的拓扑优化,天津城市建设学院学报,2001,7(1):33-35.
[35]周克民,胡云昌,结合拓扑分析进行平面连续体拓扑优化,天津大学学报,2001,34(3):340-345.
[36]左孔天,陈立平,钱勤等,求解拓扑优化问题的一种移动渐进2小波混和算法,固体力学学报,2004,25(4):471-475.
[37]左孔天,陈立平,张云清等,用拓扑优化方法进行热传导散热体的结构优化设计,机械工程学报,2005,41(4):13-16,21.
[38]左孔天,陈立平,王书亭等,用拓扑优化方法进行微型柔性机构的设计研究,机械工程学报,2004,15(21):1886-1890.
[39]王书亭,左孔天,基于均匀化理论的拓扑优化算法研究,华中科技大学学报,2004,32(10):25-27,30.
[40]王书亭,左孔天,一种基于均匀化理论的拓扑优化准则法,华中科技大学学报,2004,32(10):28-30.
[41]罗震,陈立平,钟毅芳,基于凸规划方法的计算机辅助结构拓扑优化设计,计算机辅助设计与图形学学报,2005,17(4):774-781.
[42]罗震,郭文德,蒙永立等,全柔性微型机构的拓扑优化设计技术研究,航空学报,2005,26(5):617-625.
[43]罗震,陈立平,黄玉盈等,基于RAMP密度-刚度插值格式的结构拓扑优化,计算力学学报,2005,22(5):585-591.
[44]张宪民,陈永健,多输入多输出柔顺机构拓扑优化及输出耦合的抑制,机械工程学报,2006,42(3):162-165.
[45]张宪民,陈永健,考虑输出耦合时柔顺机构拓扑与压电驱动单元的优化设计,机械工程学报,2005,41(8):69-74.
[46]郭旭 . 赵 康 . 基于拓扑 描述 函 数的 连续体 结构拓扑优化 方法 . 力学学报.2004.36(5):520-526.
[47]孟宪颐 , 用边界 元 分 域法对 机 械 构件 进行 形状优化 , 机 械 工程学报,1991,27(1):84-90.
[48]张立洲,李性厚,孙焕纯,弹性平面孔洞形状优化的虚边界元法,锦州工学院院报,1991,10(2):1-7.
[49]申秀丽,温卫东,高德平,接触问题形状优化德边界元法及其工程应用,航空动力学报,1997(12):113-116.
[50]周克民,胡云昌,用可退化有限元进行平面连续体拓扑优化,应用力学学报,2002,19(1):124-126.
[51]隋允康,杨德庆,王备,多工况应力和位移约束下连续体结构拓扑优化,力学学报,2000,32(2):171-179.
[52]杨德庆,隋允康,刘正兴,多工况应力约束下连续体结构拓扑优化映射变换解法,上海交通大学学报,2000,34(8):1061-1065.
[53]于新,隋允康,平面连续体多工况应力约束下结构拓扑优化的有无复合体方法,固体力学学报,2001,18(2):200-204.
[54]王健 , 程 耿东 , 多 工 况 应 力 约束 下 连续体 结构拓扑优化设计 , 机 械 强度,2003,25(1):55-57.
[55]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,固体力学学报,2005,26(1):29-36.
[56]Bends?e M P, Optimization of Structural Topology, Shape, and Material, New York, Springer Verlag, 1997.
[57]Bends?e M P, Sigmund O, Topology Optimization: Theory, Methods and Applications, New York, Springer Verlag, 2003.
[58]Suzuki K, Kikuchi N, A homogenization method for shape and topology optimization, Computer Method in Applied Mechanics and Engineering, 1991, 93(3):291-381.
[59]Allaire G, Bonnetier E, Francfort G, etc., Shape optimization by the homogenization method, Numerische Mathematik, 1997, 76: 27-68.
[60]Bends?e M P, Kikuchi N, Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering, 1998, 71(2): 197-224.
[61]Allaire G, Homogenization and applications to material sciences, Lecture Notes atthe Newton Institue, Cambridge, September 1999.
[62]Byun J K, Lee J H, Inverse Problem application of topology optimization method with mutual energy concept and design sensitivity, IEEE Transactions on Magnetics, 2000, 36(4): 1144-1147.
[63]Allaire G, Shape optimization by the homogenization method, New York, Springer Verlag, 2001.
[64]Ambrosio L, Buttazzo G, An optimal design Problem with Perimeter Penalization, Calculus of Variations, 1993, 1: 55-69.
[65]Bends?e M P, Sigmund O, Material interpolation schemes in topology optimization, Archive of Applied Mechanics, 1999, 69: 635-654.
[66]Rietz A, Sufficiency of a finite exponent in SIMP (Power Law) methods, Structural and Multidisciplinary Optimization, 2001, 21(2): 159-163.
[67]Martinez J M, A note on the theoretical convergence properties of SIMP method, Structural and Multidisciplinary Optimization, 2004, 29(4): 319-323.
[68]Xie Y M, Steven G P, A simple evolutionary procedure for structural optimization, Computers & Structures, 1993, 49: 885-896.
[69]Xie Y M, Steven G P, Evolutionary Structural Optimization, New York, Springer Verlag, 1997.
[70]Querin O M, Steven G P, Evolutionary structural optimization using an additive algorithm, Finite Elements in Analysis and Design, 2000, 34(3-4): 291-308.
[71]Querin O M, Steven G P, Xie Y M, Evolutionary structural optimisation (ESO) using a bidirectional algorithm, Engineering Computations, 1998, 15(8): 1031-1048.
[72]Annicchiarico W, Cerrolaza M, An Evolutionary approach for the shape optimization of general boundary elements models, Electronic Journal of Boundary Elements, 2001, 2: 251-266.
[73]Chang S, Youn S, Material clouds method for topology optimization, International Journal for Numerical Methods in Engineering, 2005, 65(10): 1585-1607.
[74]Sigmund O, Petersson J, Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Structural and Multidisciplinary Optimization, 1998, 16(1): 68-75.
[75]Sigmund O, On the design of compliant mechanism using topology optimization, Mechanics of Structures and Machines, 1997, 25(4): 495-526.
[76]Wang M Y, Wang S, Bilateral filtering for structural topology optimization, International Journal for Numerical Methods in Engineering, 2005, 63 (13): 1911-1938.
[77]Haber R B, Jog C S, Bends?e M P, A new approach to variable-topology shape design using a constraint on perimeter, Structural Optimization, 1996, 11: 1-12.
[78]Eschenauer H A, Schumacher A, Bubble method for topology and shape optimization of structures, Structural Optimization, 1994, 8: 42-51.
[79]Eschenauer H A, Schumacher A, Topology and shape optimization procedures using hole positioning criteria, Topology Optimization in Structural Mechanics, 1997, 135-196.
[80]Poulsen T A, Topology optimization in wavelet space, International Journal for Numerical Methods in Engineering, 2002, 53(4): 567-582.
[81]Petersson J, Sigmund O, Slope constrained topology optimization, International Journal for Numerical Methods in Engineering, 1998, 41(8): 1417-1434.
[82]Jang G W, Jeong H, Checkerboard-free topology optimization using non-conforming finite elements, International Journal for Numerical Methods in Engineering, 2003, 57(12): 1717-1735.
[83]Jang G W, Lee S, Topology optimization using non-conforming finite elements: three-dimensional case, International Journal for Numerical Methods in Engineering, 2005, 63(6): 859-975.
[84]Ruiter M J, Keulen F, Topology optimization: approaching the material distribution problem using a topological function description, Computational techniques for materials, composites and composite structures, 2000, 111-119.
[85]Ruiter M J, Keulen F, Topology Optimization Using the Topology Description Function, The Fourth World Congress of Structural and Multidisciplinary Optimization, Dalian, China, 2001, 19-20.
[86]Ruiter M J, Keulen F, The Topological Derivative in the Topology Description Function Approach, Engineering Design Optimization, Product and Process Improvement, Proceedings of the 4th ASMO-UK/ISSMO conference,Newcastle-upon-Tyne, UK, 2002, 160-167.
[87]Ruiter M J, Keulen F, Topology optimization using a topology description function, Structural and multidisciplinary optimization, 2004, 26(6): 406-416.
[88]Guo X, Zhao K, Wang M Y, A new approach for simultaneous shape and topology optimization based on implicit topology description functions, Proceedings of the ASME Design Engineering Technical Conference, 2004, 1: 503-512.
[89]Sethian J A, Wiegann A, Structural boundary design via level set and immersed interface methods, Journal of Computational Physics, 2000, 163(2): 489-528.
[90]Osher S J, Santosa F, Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum, Journal of Computational Physics, 2001, 171(1): 272-288.
[91]Allaire G, Jouve F, Toader A M, Structural optimization using sensitivity analysis and a level-set method, Journal of Computational Physics, 2004, 194(1): 363-393.
[92]Allaire G, Gournay F, Jouve F, etc., Structural optimization using topological and shape sensitivity via a level set method, Control and Cybernetics, 2005, 34(1): 59-81.
[93]Allaire G, Jouve F, A level-set method for vibration and multiple loads structural optimization, Computer Methods in Applied Mechanics and Engineering, 2005, 194(30-33): 3269-3290.
[94]Allaire G, Topology optimization with the homogenization and the level-set method, Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, 2005, 170: 1-13.
[95]Wang M Y, Wang X, Guo D, A level set method for structural topology optimization, Computer Methods in Applied Mechanics and Engineering, 2003, 192(1-2): 227-246.
[96]Wang X, Wang M Y, Structural shape and topology optimization in a level-set-based framework of region representation, Structural and Multidisciplinary Optimization, 2004, 27(1-2): 1-19.
[97]Wang M Y, Wang X, “Color” level sets: a multi-phase method for structural topology optimization with multiple materials, Computer Methods in Applied Mechanics and Engineering, 2004, 193(6-8): 469-496.
[98]Wang M Y, Chen S K, Wang X, etc., Design of multi-material compliant mechanisms using level-set methods, Journal of Mechanical Design, 2005, 127(5): 941-956.
[99]Wang M Y, Wang X, PDE-driven level sets, shape sensitivity, and curvature flow for structural topology optimization, CMES-Computer Modeling in Engineering and Sciences, 2004, 6(4): 373-395.
[100]Mei Y, Wang X, A level set method for structural topology optimization and its applications, Advances in Engineering for Software, 2004, 35(7): 415-441.
[101]Mei Y, Wang X, A level set method for microstructure design of composite materials, Acta Mechanica Solida Sinica, 2004, 17(3): 239-250.
[102]梅玉林,王德伦,拓扑优化的水平集方法及其在刚性结构、柔性机构和材料设计中的应用,[学位论文],大连,大连理工大学,2003.
[103]Kwak J, Cho S, Topological shape optimization of geometrically nonlinear structures using level set method, Computers and Structures 2005, 83(27): 2257-2268.
[104]Cho S, Ha S, Yang Y, Topological Shape Optimization of Geometrically Nonlinear Structures Using Level Set and Meshfree Methods, 6th World Congress on Structural and Multidisciplinary Optimization, Brazil, 2005.
[105]Ha S, Cho S, Topological Shape Optimization of Heat Conduction Problems using Level Set Approach, Numerical Heat Transfer Part B: Fundamentals, 2005, 48(1): 67-88.
[106]Cho S, Ha S, Park C, Topological shape optimization of power flow problems at high frequencies using level set approach, International Journal of Solids and Structures, 2006, 43(1): 172-192.
[107]Design of a small satellite, http://www.topopt.dtu.dk/.
[108]Haber E, A multilevel, level set method for optimizing eigenvalues in shape design problems, Journal of Computational Physics, 2004, 198(2): 518-534.
[109]Ma Z, Kikuchi N, Cheng H, Topological design for vibrating structures, Computer Methods in Applied Mechanics and Engineering, 1995, 121(1-4): 259-280.
[110]Allaire G, Aubry S, Jouve F, Eigenfrequency optimization in optimal design,Computer Methods in Applied Mechanics and Engineering, 2001, 190(28): 3365-3579.
[111]Xie Y M, Steven G P, Evolutionary Structural optimization for dynamic problems. Computers and Structures, 1996, 58(6): 1067-1073.
[112]Bruns T E, Topology optimization of nonlinear elastic structures and compliant mechanisms, [Dissertation], University of Michigan, Annarbor, 1990.
[113]Buhl T, Pedersen C B W, Sigmund O, Stiffness design of geometrically nonlinear structures using topology optimization, Structural and Multidisciplinary Optimization, 2000, 19(2): 93-104.
[114]Jung D, Gea H C, Topology optimization of nonlinear structures, Finite Elements in Analysis and Design, 2004, 40(1): 1417-1427.
[115]Stegmann J, Lund E, Nonlinear topology optimization of laminated shells, Proceeding 15th Nordic Seminar on Computational Mechanics, Aalborg, Denmark, 2002, 215-218.
[116]Lin J, Shape optimization of nonlinear structure under fatigue loading, [Dissertation], University of Cincinnati, Mechanical Engineering, 2001.
[117]郭旭,赵康,拓扑相关载荷作用下结构形状/拓扑优化的水平集方法,中国计算力学大会,2003,627-633.
[118]Guo X, Zhao K, Gu Y X, Topology optimization with design-dependent loads by level set approach, Proceeding of 10th AIAA/ISSMO Multidisciplinary analysis and Optimization conference, 2004.
[119]Kharmanda G, Olhoff N, Reliability-based topology optimization as a new strategy to generate different structural topologies, Proceedings of 15th Nordic Seminar on Computational Mechanics, Aalborg, Denmark, 2002, 18-19.
[120]Kharmanda G, Olhoff N, Mohamed A, etc., Reliability-based topology optimization, Structural and Multidisciplinary Optimization, 2004, 26(5): 295-307.
[121]Ananthasuresh G K, Kota S, Gianchandani Y, A methodical approach to the design of compliant micromechanisms, Solid-State sensor and Actuator Workshop, 1994, 189-192.
[122]Nishiwaki S, Frecker M I, Min S, etc., Topology optimization of compliantmechanisms using the homogenization method, International Journal for Numerical Methods in Engineering, 1998, 42(3): 535-559.
[123]Sigmund O, Torquato S, Composites with external thermal expansion coefficients, Applied Physics Letters, 1996, 69(21): 3203-3205.
[124]Sigmund O, Design of multiphysics actuators using topology optimization-Part I: One-material structures, Computer Methods in Applied Mechanics and Engineering, 2001, 190(49-50): 6577-6604.
[125]Sigmund O, Design of multiphysics actuators using topology optimization-Part II: Two-material structures, Computer Methods in Applied Mechanics and Engineering, 2001, 190(49-50): 6605-6627.
[126]Sigmund O, Systematic Design and optimization of Multi-Material, Multi-Degree-of-Freedom Micro Actuators, Technical Proceedings of the 2000 International Conference on Modeling and Simulation of Microsystems, 2000, 36-39.
[127]Larsen U D, Sigmund O, Bouwstra S, etc., Design and Fabrication of Compliant Micromechanisms and Structures with Negative Poisson’s Ratio, Journal of Micro electromechanical System, 1997, 6(2): 99-106.
[128]Sigmund O, Topology optimization: a tool for the tailoring of structure and materials, Philosophical Transactions: Mathematical, Physical and Engineering Sciences (Series A), 2000, 358(1765): 211-227.
[129]Yin L, Ananthasuresh G K, Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme, Structural Multidisciplinary Optimization, 2001, 23(1): 49-62.
[130]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.
[131]夏再忠 , 过 增 元 , 用 生 命 演 化过程 模 拟 导热 优化 , 自 然 科 学 进展,2001,11(8):845-851.
[132]Cheng X, Li Z, Guo Z, Constructs of highly effective heat transport paths by bionic optimization, Science in China, 2003, 46(3): 296-302.
[133]过增元,程新广,夏再忠,最小热量传递势容耗散原理及其再导热优化中的应用,科学通报,2003,48(1):21-25.
[134]程新广,李志信,过增元,基于最小热量传递势容耗散原理的导热优化,工程热物理学报,2003,24(1):94-96.
[135]Li Q, Steven G P, Osvaldo M, etc., Shape and topology design for heat conduction by evolutionary structural optimization, International Journal of Heat and Mass Transfer, 1999, 42(17): 3361-3371.
[136]Gaunaoui K E, Allaire G, Homogenization of a conductive and radiative heat transfer problem, simulation with cast3m, Proceeding of HT2005, 2005 ASME Summer Heat Transfer Conference, San Francisco, California, USA, July 17-22, 2005.
[137]顾元宪,赵红兵,陈飚松等,热-应力耦合结构灵敏度分析方法,力学学报,2001,33(5):685-691.
[138]陈飚松,林巍,顾元宪,热传导问题灵敏度分析的伴随法,计算力学学报,2003,20(4):383-390.
[139]Kim D H, Sykulski J K, Lowther D A, A Novel Scheme for Material Updating in Source Distribution Optimization of Magnetic Devices Using Sensitivity Analysis, IEEE Transactions on Magneics, 2005, 41(5): 1752-1755.
[140]Yoo J, Kikuchi N, Volakis J L, Structural optimization in magnetic devices by the homogenization design method, IEEE Transactions on Magneics, 2000, 36(3): 574-580.
[141]Bobaru F, Rachakonda S, Optimal shape profiles for cooling fins of high and low conductivity, International Journal of Heat and Mass Transfer, 2004, 47(23): 4953-4966.
[142]Hughes T J R, The Finite Element Method linear static and dynamic finite element analysis, Mineola New York, Dover Publications, 2000.
[143]Donoso A, Pedregal P, Optimal design of 2D conducting graded materials by minimizing quadratic functionals in the field, Structural and Multidisciplinary Optimization, 2005, 30(5): 360-367.
[144]Donoso A, Numerical simulations in 3D heat conduction: minimizing the quadratic mean temperature gradient, Preprint.
[145]Grabovsky Y, Optimal design problems for Two-Phase conducting composites with weakly discontinuous objective functionals, Advances in AppliedMathematics, 2001, 27(4): 683-704.
[146]Osher S, Sethian J A, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 1988, 79: 12-49.
[147]Sethian J A, Level Set Methods and Fast Marching Methods, Cambridge United Kingdom, Cambridge University Press, 1999.
[148]Osher S, Fedkiw R, Level Set Method and Dynamic Implicit Surfaces, New York, Springer Verlag, 2003.
[149]Osher S, Fedkiw R, Level set methods: an overview and some recent results, Journal of Computational Physics, 2001, 169(2): 463-502.
[150]Osher S, Paragios N, Geometric Level Set Methods in Imaging, Vision, and Graphics, New York, Springer Verlag, 2003.
[151]Malladi R, Sethian J A, Vemuri B C, Shape modeling with front propagation: a level set approach, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1995, 17(2): 158-175.
[152]Chan T, Vese L, Active contours without edges, IEEE Transactions on Image Processing, 2001, 10(2): 266-277.
[153]Sapiro G, Geometric Partial Differential Equations and Image Analysis, Cambridge United Kingdom, Cambridge University Press, 2001.
[154]Kass M, Witkin A, Terzopoulos D, Snakes: active contour models, International Journal of Computer Vision, 1987, 1(4): 321-331.
[155]Caselles V, Kimmel R, Sapiro G, Geodesic active contours, International Journal of Computer Vision, 1997, 22(1): 61-79.
[156]Novotny A A, Feijoo R A, Taroco E, etc., Topological sensitivity analysis, Computer Methods in Applied Mechanics and Engineering, 2003, 192(9-10): 803-829.
[157]Sokolowski J, Zolesio J P, Introduction to Shape Optimization: Shape Sensitivity Analysis, New York, Springer Verlag, 1992.
[158]Pironneau O, Optimal Shape Design for Elliptic Systems, New York, Springer Verlag, 1983.
[159]Liu Z, Korvink J G, Huang R, Structural topology optimization: fully coupledlevel set method via FEMLAB. Structural and Multidisciplinary Optimization, 2005, 29(6): 407-417.
[160]Sokolowski J, Zochowski A, On the topological derivative in shape optimization, SIAM Journal on Control Optimization, 1999, 37(4): 1251-1272.
[161]Burger M, Hackl B, Ring W, Incorporating topological derivatives into level set method, Journal of Computational Physics, 2004, 194(1), 344-362.
[162]Shi Y, Karl W C, A fast level set method without solving PDEs, ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings II, 2005, 97-100.
[163]Sigmund O, A 99 line topology optimization code written in Matlab, Structural and Multidisciplinary Optimization, 2001, 21(2): 120-127.
[164]Vese L, Chan T, A multiphase level set frame work for image segmentation using the Mumford and Shah Model, International Journal of Computer Vision, 2002, 50(3): 271-293.
[165]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(2): 90-108.
[166]王春会,张光,连续体结构拓扑优化设计,[学位论文],西安,西北工业大学,2005.
[167]跨克工作室 ,有限 元分 析基 础篇Ansys与Matlab, 北京 ,清华 大学 出版社,2002.
[168]李家春,叶邦彦,汤勇等,基于密度法的热传导结构拓扑优化准则算法,华南理工大学学报,2006,34(2):27-32.
[169]汪雷,张卫红,材料与结构的传热性能优化设计,[学位论文],西安,西北工业大学,2006.