基于有限元方法的电磁结构拓扑优化
|
摘要
拓扑优化设计是现代结构优化设计的一个重要研究方向，该方法为工程师创新设计出低成本高品质的产品提供了保障。经过近二十年的发展，拓扑优化设计方法已由结构设计领域向其他领域发展，包括航空航天、微机电系统、电磁、流体等领域。耦合结构的拓扑优化亦是拓扑优化设计中的重要的研究课题之一及难点，电热固致动装置的优化设计属于这一类。使用拓扑优化设计方法对电磁相关结构进行设计可有效缩短产品开发周期，提高产品质量。
“基于有限元方法的电磁结构拓扑优化”的研究工作是探索一种把现代拓扑优化设计理念应用于电磁结构设计的方法，重点是建立电磁致动机构拓扑优化设计数学模型，并编程实现该设计。主要研究工作包括：
(1)为了更高效地进行电磁结构拓扑优化设计，推导了电磁结构的有限元方程，设计了适于拓扑优化的有限元实现流程，编写了相应的程序。通过计算不同边界条件、不同形式网格的电场问题和磁场问题的数值解对有限元方法的准确性进行评价。对实现拓扑优化设计而言，这些有限元计算程序使拓扑优化设计程序简化，接口工作减少，为基于有限元分析的电磁结构拓扑优化奠定了良好的基础。
(2)编程实现了以三角形为基单元的微机电系统中柔性机构拓扑优化设计方法。分析了三角形单元的优越性。详细推导了三角形单元下的有限元法实现过程及其在拓扑优化时的敏度，给出了基于准则法的柔性机构拓扑优化数学模型。使用Michell结构、微夹钳机构二个典型算例验证了设计的拓扑优化方法的有效性和正确性。应用该算法设计了三角形微夹钳和微型扭矩产生器，得到了较好的拓扑形式。给出了不同参数情况下微型扭矩产生器的拓扑形式。
(3)将拓扑优化设计方法用于存在耦合场作用的微致动器结构的设计。针对不同驱动方式的微致动器，考虑了其防热性能、承载能力以及功能，建立了拓扑优化数学模型，利用拓扑优化的思想，寻求电磁相关的多场耦合结构的最优拓扑分布。设计出特定状态下满足要求的电热致动器和压电致动器的拓扑形式，实现满足功能及约束条件下高效地使用材料、节约能源。
Topology optimization is a hot research field for its ability to make engineer get low-costand high-quality products. Applications of topology optimization have also spread to otherphysics and multi-physics disciplines including electromagnetism, fluids, MEMS, optics andacoustics. Topology optimization is now a rather well-established field that after almost twodecades of emphasis on structural design is now being applied for optimal design in such diverseareas as electro-magnetism and fluids. Topology optimization of coupled structure has becomeone of the most important topics of engineering applications. Design optimization of structureand thermal actuated and electronically actuated mechanism have been an interesting area ofresearch in the field of engineering design for its ability to shorten the design cycle andenhancing product quality.
This dissertation emphasizes the approaches and engineering computations required toeffectively design this kind of actuator. The research of this thesis focuses on an optimal designof actuator using topology optimization. Main works are included as follows.
(1) This dissertation studies computational methods of finding numerical solutions toMaxwell’s equations and gives an introduction to the finite element method with applications inelectromagnetics and its numerical implementation. The accuracy of the finite element method isevaluated by computing the numerical solution based on different meshes under differentboundary conditions. Both triangular and quadrilateral elements are employed for diffrentanalysis. Several examples are illustrated to verify the accuracy and effectiveness of the code inthis work. Additionally, the solid model is simplified and the depicting parameters are eliminatedconsiderably compared with the whole structure, through which optimization can beimplemented easily.
(2) This dissertation establishes the SIMP models of topology optimization for rotatingcompliant mechanisms. This work gives the mathematical formulation of rotating compliantmechanism, sensitivity analysis, and OC solving algorithm. The advantages of using triangleelement are presented. Numerical instabilities are eliminated in the implementation of progamming. Numrical examples of Michell structure and micro gripper show the rationality andefficiency of this algorithm. The micro triangle gripper and rotating actuator are designedthrough topology optimzation established in this study.
(3) This dissertation establishes a computational algorithm for topology optimization ofcoupled solid-thermal-electronically structure and actuator. Based on the proposed algorithm, thepaper gives the numerical implementation of topology optimization and visualization. Itintegrates the thermal force and electrical force into the mahtematical model for the designing ofactuator through topology optimization. A thermal actuator and piezoelectric actuator aredesigned using the mathematical topology optimization formulation. The validity and robustnessof the new algorithm are verified by some widely used examples in topology optimization andthen can be used to design mechanism and/or actuator under coupled field.
“基于有限元方法的电磁结构拓扑优化”的研究工作是探索一种把现代拓扑优化设计理念应用于电磁结构设计的方法，重点是建立电磁致动机构拓扑优化设计数学模型，并编程实现该设计。主要研究工作包括：
(1)为了更高效地进行电磁结构拓扑优化设计，推导了电磁结构的有限元方程，设计了适于拓扑优化的有限元实现流程，编写了相应的程序。通过计算不同边界条件、不同形式网格的电场问题和磁场问题的数值解对有限元方法的准确性进行评价。对实现拓扑优化设计而言，这些有限元计算程序使拓扑优化设计程序简化，接口工作减少，为基于有限元分析的电磁结构拓扑优化奠定了良好的基础。
(2)编程实现了以三角形为基单元的微机电系统中柔性机构拓扑优化设计方法。分析了三角形单元的优越性。详细推导了三角形单元下的有限元法实现过程及其在拓扑优化时的敏度，给出了基于准则法的柔性机构拓扑优化数学模型。使用Michell结构、微夹钳机构二个典型算例验证了设计的拓扑优化方法的有效性和正确性。应用该算法设计了三角形微夹钳和微型扭矩产生器，得到了较好的拓扑形式。给出了不同参数情况下微型扭矩产生器的拓扑形式。
(3)将拓扑优化设计方法用于存在耦合场作用的微致动器结构的设计。针对不同驱动方式的微致动器，考虑了其防热性能、承载能力以及功能，建立了拓扑优化数学模型，利用拓扑优化的思想，寻求电磁相关的多场耦合结构的最优拓扑分布。设计出特定状态下满足要求的电热致动器和压电致动器的拓扑形式，实现满足功能及约束条件下高效地使用材料、节约能源。
Topology optimization is a hot research field for its ability to make engineer get low-costand high-quality products. Applications of topology optimization have also spread to otherphysics and multi-physics disciplines including electromagnetism, fluids, MEMS, optics andacoustics. Topology optimization is now a rather well-established field that after almost twodecades of emphasis on structural design is now being applied for optimal design in such diverseareas as electro-magnetism and fluids. Topology optimization of coupled structure has becomeone of the most important topics of engineering applications. Design optimization of structureand thermal actuated and electronically actuated mechanism have been an interesting area ofresearch in the field of engineering design for its ability to shorten the design cycle andenhancing product quality.
This dissertation emphasizes the approaches and engineering computations required toeffectively design this kind of actuator. The research of this thesis focuses on an optimal designof actuator using topology optimization. Main works are included as follows.
(1) This dissertation studies computational methods of finding numerical solutions toMaxwell’s equations and gives an introduction to the finite element method with applications inelectromagnetics and its numerical implementation. The accuracy of the finite element method isevaluated by computing the numerical solution based on different meshes under differentboundary conditions. Both triangular and quadrilateral elements are employed for diffrentanalysis. Several examples are illustrated to verify the accuracy and effectiveness of the code inthis work. Additionally, the solid model is simplified and the depicting parameters are eliminatedconsiderably compared with the whole structure, through which optimization can beimplemented easily.
(2) This dissertation establishes the SIMP models of topology optimization for rotatingcompliant mechanisms. This work gives the mathematical formulation of rotating compliantmechanism, sensitivity analysis, and OC solving algorithm. The advantages of using triangleelement are presented. Numerical instabilities are eliminated in the implementation of progamming. Numrical examples of Michell structure and micro gripper show the rationality andefficiency of this algorithm. The micro triangle gripper and rotating actuator are designedthrough topology optimzation established in this study.
(3) This dissertation establishes a computational algorithm for topology optimization ofcoupled solid-thermal-electronically structure and actuator. Based on the proposed algorithm, thepaper gives the numerical implementation of topology optimization and visualization. Itintegrates the thermal force and electrical force into the mahtematical model for the designing ofactuator through topology optimization. A thermal actuator and piezoelectric actuator aredesigned using the mathematical topology optimization formulation. The validity and robustnessof the new algorithm are verified by some widely used examples in topology optimization andthen can be used to design mechanism and/or actuator under coupled field.
引文
[1] M.P. Bendson, O. Sigmund. Topology Optimization Theory, Methods and Applications. Springer.
[2] Hans A Eschenauer, Niels Olhoff. Topology optimization of continuum Structures: A review. ASME:Appllied Mechanics Review,2001,54(4):331-390.
[3]李芳,凌道盛.工程结构优化设计发展综述.工程设计学报,2002,9(5):229-235.
[4]郭中泽,张卫红,陈裕泽.结构拓扑优化设计综述.机械设计,2007,24(8):1-6.
[5]满宏亮.工程结构拓扑优化的理论研究及应用:[硕士学位论文].长春:吉林大学,2007.
[6]贾海朋.结构与柔性机构拓扑优化:[博士学位论文].大连:大连理工大学,2004.
[7]李震.柔性机构拓扑优化方法及其在微机电系统中的应用:[博士学位论文].大连:大连理工大学,2005
[8] Dorn WS, Gomory RE, Greenberg HJ. Automatic design of optimal structures. Journal of Mechanics,1964,3:25-52.
[9] Bendsoe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenizationmethod. Computer Methods in Applied Mechanics and Engineering,1988,71(2):197-224.
[10] Cherkaev AV, Palais R. Optimal design of three dimensional axisymmetric elastic structures. Struct.Optim.1996,12:35-45.
[11] Diaz A, Lipton R. Optimal material layout for3D elastic Structures, Struct. Optim,1997,13:60-64.
[12] Olhoff N, Ronholt E, Seheel J. Topology optimization of three-dimensional structures using optimummicrostructures. Struct.OPtim.1998,16:l-18.
[13] Jeonghoon Yoo, Noboru Kikuchi, John L. Volakis. Structural Optimization in Magnetic Devices by theHomogenization Design Method. IEEE Trans. Magn.,2000,36(3):574-580.
[14] J. Yoo. Structural Optimization in Magnetic Fields Using the Homogenization Design Method–Part II:Reduction of Vibration Caused by Magnetic Forces. Arch. Comput. Meth. Engng.,2002,9(3):257-282.
[15] Bendson MP. Optimal shape design as a material distribution problem. Struet. Optim.,1989, l:193-202.
[16] Jeonghoon Yoo, Hyeoksoo. Hong. A modified density approach for topology optimization in magneticfields. Solids and Structures,2004,41:2461-2477.
[17] Jae Seok Choi, Jeonghoon Yoo. Structural Topology Optimization of Magnetic Actuators Using GeneticAlgorithms and ON/OFF Sensitivity. IEEE Trans. Magn.,2009,45(5):2276-2279.
[18] Norio Takahashi, Shunsuke Nakazaki, Daisuke Miyagi. Examination of Optimal Design Method ofElectromagnetic Shield Using ON/OFF Method. IEEE Trans. Magn.,2009,49(10):1546-1549.
[19] N. Takahashi, T. Yamada, D. Miyagi. Examination of Optimal Design of Imp Motor Using ON/OFFMethod. IEEE Trans. Magn.,2010,46(8):3149-3152.
[20] Xie YM, Steven GP. A simple evolutionary procedure for structural optimization. Computers andStructures.1993,49:885-896.
[21] Querin OM, Young V, Steven GP. Computational efficiency and validation of bi-directional evolutionarystructural optimization. Computer methods in applied mechanics and engineering.2000,189:559-573.
[22] Sethian JA., Wiegmann A. Structural boundary design via level set and immersed interface methods.Journal of Computational Physics,2000,163(2):489-528.
[23] T. J. Barth, J.A. Sethian. Numerical Schemes for the Hamilton-Jacobi and Level Set Equations onTriangulated Domains. J. Comput. Phys.,1998,145:1-40.
[24] J. Kwack, S. Min, J. Hong. Optimal Stator Design of Interior Permanent Magnet Motor to Reduce TorqueRipple Using the Level Set Method. IEEE Trans. Magn.,2010,46(6):2108-2111.
[25] Sunghoon Lim, Takayuki Yamada, Seungjae Min, et al. Topology Optimization of a Magnetic ActuatorBased on a Level Set and Phase-Field Approach[J]. IEEE Trans. Magn.,2011,47(5):1318-1321.
[26] Hokyung Shim, Vinh Thuy Tran Ho, Semyung Wang, et al. Tortorelli. Level Set-Based TopologyOptimization for Electromagnetic Systems. IEEE Trans. Magn.,2009,45(3):1582-1585.
[27] S. Park, S. Min, S. Yamasaki, et al. Magnetic Actuator Design Using Level Set Based TopologyOptimization. IEEE Trans. Magn.,2008,44(11):4037-4040.
[28] Sang-In Park, Seungjae Min. Magnetic Actuator Design for Maximizing Force Using Level Set BasedTopology Optimization. IEEE Trans. Magn.,2009,45(5):2336-2339.
[29] Sang-in Park, Seungjae Min. Optimal Topology Design of Magnetic Devices Using Level-Set Method.IEEE Trans. Magn.,2009,45(3):1610-1613.
[30] Jin-kyu Byun, Song-yop Hahn. Topology Optimization of Electrical Devices Using Mutual Energy andSensitivity. IEEE Trans. Magn.,1999,35(5):3718-3720.
[31] Y. Okamoto, K. Akiyama, N. Takahashi.3-D Topology Optimization of Single-Pole-Type Head by UsingDesign Sensitivity Analysis. IEEE Trans. Magn.,2006,42(4):1087-1090.
[32]易龙,彭云,孙秦.基于改进的灵敏度分析的有限元优化技术研究.机械强度.2008,30(3):483-487.
[33]盛新庆.计算电磁学要论（第一版）.北京：科学出版社,2004
[34]张慧,杨仕友,倪光正.快速全局优化算法在电磁场逆问题中的应用.机电工程,2007,24(10):28-30.
[35]张慧.电磁场逆问题分析计算的快速全局优化算法研究.浙江大学,2007.
[36]缪希仁,张培铭.基于遗传算法的电磁电器优化设计.低压电器,1998,18(5):12-15.
[37]陈丽安,张培铭等.基于免疫遗传算法的智能化电磁电器全局优化设计.电工电能新技术,2003,21(1):17-20.
[38]罗隆福,黄守道,童调生等.遗传算法用于电机电器优化设计的展望.湖南大学学报,1997,24(5):55-58
[39]倪光正,杨仕友,钱秀英等.工程电磁场数值计算.北京:机械工业出版社,2004.
[40]颜威利,杨庆新,汪友华等.电气工程电磁场数值分析.北京:机械工业出版社,2005.
[41]谢德馨,杨仕友.工程电磁场数值分析与综合.北京:机械工业出版社,2009.
[42]程志光,高桥则雄,博扎德等.电气工程电磁热场模拟与应用.北京:科学出版社,2009.
[43]谢德馨,阎秀恪,张奕黄等.旋转电机绕组磁链的三维有限元分析.中国电机工程学报,2006,26(21):143-148.
[44]谢德馨.用有限元法计算矩形槽内载流导体的集肤效应.哈尔滨电工学院学报,1981,4(1):1-23.
[45]吕殿利,汪友华,韩婷彦等.特高压变压器线圈的电场计算与分析.河北大学学报(自然科学版),2011,31(2):144-149.
[46]侯小全,周光厚,廖毅刚等.电磁场有限元技术在发电机优化设计中的应用.东方电气评论,2010,24(96):44-51.
[47]汪友华,颜威利,张冠生等.自适应模拟退火法在电磁场逆问题中的应用.中国电机工程学报,1995,15(4):234-239.
[48]白凤仙,邵玉槐,孙建中.利用智能型模拟退火算法进行开关磁阻电机磁极几何形状的优化.中国电机工程学报.2003,23(1):126-141.
[49]杨庆新,安金龙,马振平等.基于最小二乘支持向量机和自适应模拟退火算法的电磁场逆问题全局优化方法.电工技术学报,2008,23(11):1-7.
[50] Dong-Hun Kim, Soo Bum Lee, Byung Man Kwank, et al. Smooth Boundary Topology Optimization forElectrostatic Problems through the Combination of Shape and Topological Design Sensitivities. IEEETrans. Magn.,2008,44(6):1002-1005.
[51] Dong-Hun Kim, Jan K. Sykulski, David A. Lowther. The Implications of the Use of Composite Materialsin Electromagnetic Device Topology and Shape Optimization. IEEE Trans. Magn.,2009,45(3):1154-1157.
[52]王金铭,谢德馨,于超等.基于Tabu法的变压器铁心柱截面积优化.电工技术学报,2002,17(2):13-16.
[53]杨仕友,倪光正,钱金银.电机电磁场逆问题数值计算的改进TABU算法.中国电机工程学报,1998,18(2):83-86.
[54]马文阁,干于,杨仕友.改进的区域消除法及在电磁场逆问题中的应用.电机与控制学报,2000,4(1):10-12.
[55]安斯光,杨仕友,李桃.改进的禁忌搜索算法及其在电磁场逆问题中的应用.电力系统保护与控制,2010,38(24):30-33.
[56] Chen-Yi Liao, Wei-Ping Lee, Xianghan Chen, et al. Dynamic and adjustable particle swarm optimization.Proceedings of the8th International Conference on Evolutionary Computing, Canada,2007:19-21.
[57] J. Yoo, H.-J. Soh. An optimal design of magnetic actuators using topology optimization and the responsesurface method. Microsyst Technol,2005,11:1252-1261.
[58] Niarchos D. Magnetic MEMS: Key issues and some applications. Sensors and actuators A,2003,109:166-173
[59]谢德馨,唐任远,王尔智等.计算电磁学的现状与发展趋势—第14届COMPUMAG会议综述.电工技术学报,2003,18(5):1-4.
[60]谢德馨,唐任远.计算电磁学近年来的若干重要成果—第15届COMPUMAG会议概述.电工技术学报,2005,20(9):1-5.
[61] D.A. Lowther. New trends in the simulation of magnetic devices. Mathematics and Computers inSimulation,1995,38:257-262.
[62] A. Taflove, Computational electrodynamics: the finite-difference time-domain method. Boston,MA:Artech House,1995
[63]杨阳.电磁场时域有限差分数值方法的研究:[博士学位论文].南京:南京理工大学,2005.
[64]焦超群.细薄结构的时域有限差分法在电磁屏蔽中的应用研究:[博士学位论文].北京:华北电力大学,2006.
[65]刘士利.改进的边界元法及其在电场计算中的应用:[博士学位论文].北京:华北电力大学,2011.
[66]陶瑞民.永磁电机电磁场的边界元方法.电气技术,2011,11:20-22.
[67]姜保军,孙力,李波.电磁检测中的开域电磁场数值计算.中国电机工程学报,2005,25(8):156-160.
[68]林庆华,栗保明.有限元/边界元耦合法计算电磁轨道炮三维瞬态涡流场.南京理工大学学报(自然科学版),2010,34(2):217-221.
[69]关海爽,马文阁.电磁场数值计算新方法的研究.辽宁工学院学报,2006,26(4):230-233.
[70]国海广,范丽丽,潘晓东.解决电大尺寸电磁场问题的改进矩量法研究.现代电子技术,2009,6:130-133.
[71] Walton C. Gibson. The method of MoMents in electromagnetics. Boca Raton: Chapman&Hall/CRC,2008.
[72] Kensuke Yokoi. A numerical method fro free-surface flows and its application to droplet impact on a thinliquid layer. Scientific Computing,2008,35(6):372-396
[73] J Zou, J S Yuan, X S Ma, ec tl. Magnetic field analysis of iron-core reactor coils by the finite-volumemethod. IEEE Trans. on Magn,2004,40(2):814-817.
[74] J Zou, Y Q Xie, J S Yuan,ec tl. Analysis of the thin plate eddy-current problem by finite volume method.IEEE Trans. on Magn,2004,40(2):1370-1373.
[75] Xie Yaoqin, Yuan Jiansheng, Ma Xinshan, ec tl. Calculation of EEG problems with anisotropicconductiong media by the finite volume method. IEEE Trans. on Magn,2001,37(5):3749-3752.
[76]甘艳,阮江军,张宇等.TEAM9-1问题讨论与混合有限元法有限体积法的应用.电工技术学报,2009,24(1):1-7.
[77]甘艳,阮江军,张宇.有限元法与有限体积法相结合处理运动电磁问题.中国电机工程学报,2006,26(14):145-151.
[78]李锦彪,谢德馨.变压器电磁场问题的自适应有限元分析.变压器,2010,47(1):36-39
[79]闵越,肖静.变压器三维电场分析方法探讨.山东大学学报(工学版),2007,37(6):54-57.
[80]曲利岩.电磁场有限元分析技术的研究和实现:[硕士学位论文].杭州:浙江大学,2002.
[81]姬丽娟.交流接触器电磁机构有限元分析与多场耦合技术的研究:[硕士学位论文].天津:河北工业大学,2005.
[82]张淮清,俞集辉,郑亚利.径向基函数及其耦合方法在电磁场数值计算中的应用.计算物理,2009,26(2):299-310.
[83]唐章宏,顿月芹,袁建生.求解开域电磁场问题的区域映射有限元法.清华大学学报(自然科学版),2009,49(1):5-8.
[84]谢德馨,刘奇林.电工钢片各向异性磁场有限元分析.沈阳工业大学学报,2010,32(3):241-245.
[85]张为民,谢德馨,白保东等.强磁场永磁机构组装过程的动态有限元分析.中国电机工程学报,2007,27(18):61-66.
[86]崔芮华,贾峰,柳杰等.交流接触器电磁机构动态仿真分析.仿真技术,2010,22(7-1):286-288.
[87] K. Kadded, R.R. Saldanha, J.L. Coulomb. Mathematical Minimization of the Time Harmonics of theE.M.F. of a DC-PM Machine using a Finite Element Method. IEEE Trans. Magn.,1993,29(2):1747-1752.
[88]张勇.计算电磁学的无单元方法研究:[博士学位论文].武汉:华中科技大学,2006.
[89] Herault C, Marechal Y. Boundary and interface conditions in meshless methods. IEEE Trans. on Magn,1999,35(3):1450-1453.
[90] Gingoski V, Miyamonto N, Yanashita H. Element-free Galerkin method for electromagnetic fieldcomputations. IEEE Trans. on Magn,1998,34(5):3236-3239.
[91] Yang Qingxin, Liu Suzhen, Chen Haiyan. Three-Dimensional element free Galerkin method and itsapplications in electromagnetic field numerical analysis.电工技术学报,2006,21(9):116-121.
[92] Viana S A, Mesquita R C. Moving least square reproducing Kernel method for electromagnetic fieldcomputation. IEEE Trans. on Magn,1999,35(3):1372-1375.
[93]张淮清.电磁场计算中的径向基函数无网格法研究:[博士学位论文].重庆:重庆大学,2008.
[94]杨光源.运动导体涡流电磁问题的径向基点配置型无单元算法研究:[博士学位论文].华中科技大学,2011.
[95]刘素贞,杨庆新,陈海燕.无单元法和有限元法的比较研究.河北工业大学学报,2000,29(5):66-68.
[96] Ho S L, Yang S, Machado J M, et al. Application of a meshless method in electromagnetic. IEEE Trans.on Magn,2001,37(5):3198-3202.
[97]刘素贞,杨庆新,陈海燕等.无单元法在电磁场数值计算中的应用研究.电工技术学报,2001,16(2):30-33.
[98]康晓明.电压互感器的有限元分析与优化设计:[博士学位论文].天津:天津大学,2007.
[99]狄维娜.交流接触器电磁机构的动静态特性分析与仿真:[硕士学位论文].天津:河北工业大学,2007.
[100]高艳丰.电压互感器电磁场有限元分析与优化设计技术的研究:[硕士学位论文].天津:河北工业大学,2006.
[101]姚缨英,高昌燮,柳在燮等.基于有限元分析计算设计灵敏度和网格更新的三维形状优化算法.中国电机工程学报,2004,24(5):130-136.
[102] Yingying Yao, Jae Seop Ryu, Chang Seop Koh, et al. A Novel Mesh Regeneration Using StructuralDeformation Analysis for3-D Shape Optimization of Electromagnetic Devices. IEEE Trans. Magn.,2004,40(2):1009-1012.
[103]顾元宪,魏鸿磊,赵国忠.基于灵敏度的电磁结构形状优化方法研究.机械强度,2003,25(6):651-656
[104] JAlfredo Canelas, Jean R. Roche, José Herskovits. The inverse electromagnetic shaping problem. StructMultidisc Optim,2009,38:389-403.
[105] Wei Li, Zi Yan Ren, Young Woo Jeong, et al. Optimal Shape Design of a Thomson-Coil ActuatorUtilizing Generalized Topology Optimization Based on Equivalent Circuit Method. IEEE Trans. Magn.,2011,47(5):1246-1249.
[106] Y. Kim, I. Park. Topology Optimization of Rotor in Synchronous Reluctance Motor Using Level SetMethod and Shape Design Sensitivity. IEEE Trans. Magn.,2010,46(3):1093-1096.
[107] H. Shim, T.H.Vinh Thuy, S. Wang et al. Topological Shape Optimization of Electromagnetic ProblemsUsing Level Set Method and Radial basis Function. Comput. Model. Eng. Sci.,2008,37(2):175-202.
[108] Jangwon Lee, Semyung Wang. Topological Shape Optimization of Permanent Magnet in Voice CoilMotor Using Level Set Method. IEEE Trans. Magn.,2012,48(2):931-934.
[109] J. Lee, S. Wang. Topological shape Optimization of Permanent Magnet in Voice Coil Motor Using LevelSet Method. IEEE Trans. Magn.,2012,48(2):931-934
[110] M. Li. D. A. Lowther. The Application of Topological Gradients to Defect Identification in MagneticFlux Leakage-Type NDT. IEEE Trans. Magn.,2010,46(8):3221-3224.
[111] Jeonghoon Yoo. Modified Method of Topology Optimizationin Magnetic Fields. IEEE Trans. Magn.,2004,40(4):1796-1801.
[112] Chang-Hwan Im, Hyun-Kyo Jung, Yong-Joo Kim. Hybrid Genetic Algorithm for ElectromagneticTopology Optimization. IEEE Trans. Magn.,2003,39(5):2163-2169.
[113] Chao Wang, Qiuliang Wang, Hui Huang, et al. Electromagnetic optimization design of a HTS magnetusing the improved hybrid genetic algorithm. Cryogenics,2006,46:349-353.
[114] Kota Watanabe, Felipe Campelo, Yosuke Iijima, et al. Optimization of Inductors Using EvolutionaryAlgorithms and Its Experimental Validation. IEEE Trans. Magn.,2010,46(8):3393-3396.
[115] J. Lee, J. Yoo. Topology Optimization of the Permanent Magnet Type MRI Considering the MagneticField Homogeneity. J. Magn. Mater,2010,322(9-12):1651-1654.
[116]任自艳,谢德馨,李会香.开放式MRI永磁型主磁体的匀场方法.电工技术学报,2010,25(3):1-5.
[117] Youguang Guo, Jianguo Zhu, D. G. Dorrell. Design and Analysis of a Claw Pole Permanent MagnetMotor with Molded Soft Magnetic Composite Core. IEEE Trans. Magn.,2009,45(10):4582-4585.
[118] Franz Zürcher, Thomas Nussbaumer, Wolfgang Gruber, et al. Design and Development of a26-Pole and24-Slot Bearingless Motor. IEEE Trans. Magn.,2009,45(10):4594-4597
[119] Thibaut Labbe, Bruno Dehez. Topology Optimization Method Based on the Maxwell Stress Tensor forthe Design of Ferromagnetic Parts in Electromagnetic Actuators. IEEE Trans. Magn.2011,47(9):2188-2193.
[120] T.Labbe, B.Dehez. Convexity-Oriented Method for the Topology Optimization of FerromagneticMoving Parts in Electromagnetic Actuators Using the Magnetic Energy. IEEE Trans. Magn.,2010,46(12):4016-4022.
[121] Sigmund, J. Petersson. Numerical Instabilities in Topology Optimization: A Survey on ProceduresDealing with Checkerboards, Mesh-dependencies and Local Minima. Struct. Optim,1998,16(1):68-75.
[122] Sunghoon Lim, Seungjae Min. Design Optimization of Permanent Magnet Actuator Using Multi-PhaseLevel-Set Model. IEEE Trans. Magn.,2012,48(4):1641-1644.
[123] Feustel, O. Krusemark, J. Muller. Numerical simulation and optimization of planar electromagneticactuators. Sensors and Actuators,1998,70:276-282.
[124] Smaranda Nitu, Constantim Nitu, Gheorghe Tuluca, et al. Electromagnetic actuator with magnetic storedenergy. Materials Processing Technology,2007,181:153-158.
[125] Ingo Hahn. Heuristic Structural Optimization of the Permanent Magnets Used in a Surface MountedPermanent-Magnet Synchronous Machine. IEEE Trans. Magn.,2012,48(1):118-127.
[126] Jaewook Lee, Noboru Kikuchi. Structural Topology Optimization of Electrical Machinery to MaximizeStiffness with Body Force Distribution. IEEE Trans. Magn.,2010,46(10):3790-3794.
[127] Semyung Wang, Jenam Kang, Jeongchae Noh. Topology Optimization of a Single-Phase InductionMotor for Rotary Compressor. IEEE Trans. Magn.,2004,40(3):1591-1596.
[128] Arnold J. Rix, Maarten J. Kamper. Radial-Flux Permanent-Magnet Hub Drives: A Comparison Based onStator and Rotor Topologies. IEEE Trans. Industrial Electronics,2012,59(6):2475-2483.
[129] Neil Hodgins, Ozan Keysan, Alasdair S. McDonald, et al. Design and Testing of a Linear Generator forWave-Energy Applications. IEEE Trans. Industrial Electronics,2012,59(6):2094-2103.
[130] Derek N. Dyck, David A. Lowther. Automated Design of Magnetic Devices by Optimizing MaterialDistribution. IEEE Trans. Magn.,1996,32(3):1188-1193.
[131] Dong-Hun Kim, Jan K. Sykulski, David A. Lowther. A Novel Scheme for Material Updating in SourceDistribution Optimization of Magnetic Devices Using Sensitivity Analysis. IEEE Trans. Magn.,2005,41(5):1752-1755.
[132] Takeo Ishikawa, Kouki Yonetake, Nobuyuki Kurita. An Optimal Material Distribution Design ofBrushless DC Motor by Genetic Algorithm Considering a Cluster of Material. IEEE Trans. Magn.,2011,47(5):1310-1313.
[133] Sang-in Park, Seungjae Min. Design of Magnetic Actuator With Nonlinear Ferromagnetic MaterialsUsing Level-Set Based Topology Optimization. IEEE Trans. Magn.,2010,46(2):618-621.
[134] Jin-Kyu Byun, Joon-Ho Lee, Il-Han Park. Node-Based Distribution of Material Properties for TopologyOptimization of Electromagnetic Devices. IEEE Trans. Magn.,2004,40(2):1212-1215.
[135] Jaewook Lee, Ercan M. Dede, Tsuyoshi Nomura.Simultaneous Design Optimization of PermanentMagnet, Coils, and Ferromagnetic Material in Actuators. IEEE Trans. Magn.,2011,47(12):4712-4716.
[136] Jae Seok Choi, Jeonghoon Yoo. Simultaneous Structural Topology Optimization of ElectromagneticSources and Ferromagnetic Materials. Comput. Methods Appl. Mech. Engrg.,2009,198:2111-2121.
[137] Tsuyoshi Nomura．Topology optimization of electromagnetic materials. Met materials and AutomotiveApplications,2006,41(4):9-15.
[138] Tsuyoshi Nomura, Shinji Nishiwaki, Kazuo Sato, et al. Topology optimization for the design of periodicmicrostructures composed of electromagnetic materials. Finit Elements in Analysis and Design,2009,45:210-226.
[139] M. Y. Wang, X. Wang. Color’ Level Sets: A Multi-Phase Method for Structural Topology Optimizationwith Multiple Materials. Comput. Methods Appl. Mech. eng.,2004,193:469-496.
[140] Takeo Ishikawa, Kyoichi Nakayama. Topology Optimization of Rotor Structure in Brushless DC MotorWith Concentrated Windings Using Genetic Algorithm Combined With Cluster of Material. IEEE Trans.Magn.,2012,48(2):899-902.
[141] Frederico G, Guimar es, Jaime A. Ramírez. A Pruning Method for Neural Networks and Its Applicationfor Optimization in Electromagnetics. IEEE Trans. Magn.,2004,40(2):1160-1163.
[142](美)金建铭著,王建国译.电磁场有限元方法.西安:西安电子科技大学出版社,1998.
[143]何红雨.电磁场数值计算法与MATLAB实现.武汉:华中科技大学出版社,2003.
[144] Herbert De Gersema, Kay Hameyerb, Thomas Weilanda. Field-circuit coupled models inelectromagnetic simulation. Computational and Applied Mathematics,2004,168:125-133.
[145]张洋,白保东,谢德馨.三维瞬态涡流–电路–运动系统耦合问题的新解法.中国电机工程学报,2008,28(9):139-144.
[146]张倩,胡仁喜,康仕廷等编著.ANSYS12.0电磁学有限元分析从入门到精通.北京:机械工业出版社,2010.
[147] Hyeoksoo Hong, Jeonghoon Yoo. Shape Design of the Surface Mounted Permanent Magnet in aSynchronous Machine. IEEE Trans. Magn.,2011,47(8):2109-2117.
[148] Shinji Nishiwaki, Tsuyoshi Nomura, Shinya Kinoshita, et al. Topology optimization for cross-sectiondesigns of electromagnetic waveguides targeting guiding characteristics. Finite Elements in Analysis andDesign,2009,45:944-957.
[149] T.Labbe, B.Dehez. Design of Magnetic Thrusts Using Topology Optimization Methods. J.Comput. Math.Elect. Electron. Eng.,2011,30(2):641-655.
[150] Semyung Wang, D. Youn, H. Moon1, et al. Topology Optimization of Electromagnetic SystemsConsidering Magnetization Direction. IEEE Trans. Magn.2005,41(5):1808-1811.
[151] Jae Seok Choi, Jeonghoon Yoo, Shinji Nishiwaki, et al. Optimization of Magnetization Directions in a3-D Magnetic Structure. IEEE Trans. Magn.,2010,46(6):1603-1606.
[152] Jeonghoon Yoo, Seunjin Yang, Jae Seok Choi. Optimal Design of an electromagnetic Coupler toMaximize Force to a Specific Direction. IEEE Trans. Magn.,2008,44(7):1737-1742.
[153] X. Huang, Y.M. Xie. Topology optimization of nonlinear structures under displacement loading.Engineering Structures,2008,30:2057-2068.
[154] Volker Schulz. Simultaneous solution approaches for large optimization problems. Computational andApplied Mathematics,2004,64-165:629-641.
[155] Jin-kyu Byun, Ju-hyun Lee, Il-han Park, et al. Inverse Problem Application of Topology OptimizationMethod with Mutual Energy Concept and Design Sensitivity. IEEE Trans. Magn.,2000,6(4):1144-1147.
[156] O. de la Barrière, S. Hlioui, H. Ben Ahmed, et al.3-D Formal Resolution of Maxwell Equations for theComputation of the No-Load Flux in an Axial Flux Permanent-Magnet Synchronous Machine. IEEETrans. Magn.,2012,48(2):128-136.
[157] Sung-Roc Jang, Hong-Je Ryoo, Suk-Ho Ahn, et al. Development and Optimization of High-VoltagePower Supply System for Industrial Magnetron. IEEE Trans. Industrial Electronics,2012,59(6):1453-1461.
[158] Osama A. Mohammed, Riaz Merchant, Fuat G. Uler. An Intelligent System for Design Optimization ofElectromagnetic Devices. IEEE Trans. Magn.,1994,30(5):3633-3636.
[159] T.Labbe, B.Dehez. Convexity-Oriented Mapping Method for the Topology Optimization ofElectromagnetic Devices Composed of Iron and Coils. IEEE Trans. Magn.,2010,46(5):1177-1185.
[2] Hans A Eschenauer, Niels Olhoff. Topology optimization of continuum Structures: A review. ASME:Appllied Mechanics Review,2001,54(4):331-390.
[3]李芳,凌道盛.工程结构优化设计发展综述.工程设计学报,2002,9(5):229-235.
[4]郭中泽,张卫红,陈裕泽.结构拓扑优化设计综述.机械设计,2007,24(8):1-6.
[5]满宏亮.工程结构拓扑优化的理论研究及应用:[硕士学位论文].长春:吉林大学,2007.
[6]贾海朋.结构与柔性机构拓扑优化:[博士学位论文].大连:大连理工大学,2004.
[7]李震.柔性机构拓扑优化方法及其在微机电系统中的应用:[博士学位论文].大连:大连理工大学,2005
[8] Dorn WS, Gomory RE, Greenberg HJ. Automatic design of optimal structures. Journal of Mechanics,1964,3:25-52.
[9] Bendsoe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenizationmethod. Computer Methods in Applied Mechanics and Engineering,1988,71(2):197-224.
[10] Cherkaev AV, Palais R. Optimal design of three dimensional axisymmetric elastic structures. Struct.Optim.1996,12:35-45.
[11] Diaz A, Lipton R. Optimal material layout for3D elastic Structures, Struct. Optim,1997,13:60-64.
[12] Olhoff N, Ronholt E, Seheel J. Topology optimization of three-dimensional structures using optimummicrostructures. Struct.OPtim.1998,16:l-18.
[13] Jeonghoon Yoo, Noboru Kikuchi, John L. Volakis. Structural Optimization in Magnetic Devices by theHomogenization Design Method. IEEE Trans. Magn.,2000,36(3):574-580.
[14] J. Yoo. Structural Optimization in Magnetic Fields Using the Homogenization Design Method–Part II:Reduction of Vibration Caused by Magnetic Forces. Arch. Comput. Meth. Engng.,2002,9(3):257-282.
[15] Bendson MP. Optimal shape design as a material distribution problem. Struet. Optim.,1989, l:193-202.
[16] Jeonghoon Yoo, Hyeoksoo. Hong. A modified density approach for topology optimization in magneticfields. Solids and Structures,2004,41:2461-2477.
[17] Jae Seok Choi, Jeonghoon Yoo. Structural Topology Optimization of Magnetic Actuators Using GeneticAlgorithms and ON/OFF Sensitivity. IEEE Trans. Magn.,2009,45(5):2276-2279.
[18] Norio Takahashi, Shunsuke Nakazaki, Daisuke Miyagi. Examination of Optimal Design Method ofElectromagnetic Shield Using ON/OFF Method. IEEE Trans. Magn.,2009,49(10):1546-1549.
[19] N. Takahashi, T. Yamada, D. Miyagi. Examination of Optimal Design of Imp Motor Using ON/OFFMethod. IEEE Trans. Magn.,2010,46(8):3149-3152.
[20] Xie YM, Steven GP. A simple evolutionary procedure for structural optimization. Computers andStructures.1993,49:885-896.
[21] Querin OM, Young V, Steven GP. Computational efficiency and validation of bi-directional evolutionarystructural optimization. Computer methods in applied mechanics and engineering.2000,189:559-573.
[22] Sethian JA., Wiegmann A. Structural boundary design via level set and immersed interface methods.Journal of Computational Physics,2000,163(2):489-528.
[23] T. J. Barth, J.A. Sethian. Numerical Schemes for the Hamilton-Jacobi and Level Set Equations onTriangulated Domains. J. Comput. Phys.,1998,145:1-40.
[24] J. Kwack, S. Min, J. Hong. Optimal Stator Design of Interior Permanent Magnet Motor to Reduce TorqueRipple Using the Level Set Method. IEEE Trans. Magn.,2010,46(6):2108-2111.
[25] Sunghoon Lim, Takayuki Yamada, Seungjae Min, et al. Topology Optimization of a Magnetic ActuatorBased on a Level Set and Phase-Field Approach[J]. IEEE Trans. Magn.,2011,47(5):1318-1321.
[26] Hokyung Shim, Vinh Thuy Tran Ho, Semyung Wang, et al. Tortorelli. Level Set-Based TopologyOptimization for Electromagnetic Systems. IEEE Trans. Magn.,2009,45(3):1582-1585.
[27] S. Park, S. Min, S. Yamasaki, et al. Magnetic Actuator Design Using Level Set Based TopologyOptimization. IEEE Trans. Magn.,2008,44(11):4037-4040.
[28] Sang-In Park, Seungjae Min. Magnetic Actuator Design for Maximizing Force Using Level Set BasedTopology Optimization. IEEE Trans. Magn.,2009,45(5):2336-2339.
[29] Sang-in Park, Seungjae Min. Optimal Topology Design of Magnetic Devices Using Level-Set Method.IEEE Trans. Magn.,2009,45(3):1610-1613.
[30] Jin-kyu Byun, Song-yop Hahn. Topology Optimization of Electrical Devices Using Mutual Energy andSensitivity. IEEE Trans. Magn.,1999,35(5):3718-3720.
[31] Y. Okamoto, K. Akiyama, N. Takahashi.3-D Topology Optimization of Single-Pole-Type Head by UsingDesign Sensitivity Analysis. IEEE Trans. Magn.,2006,42(4):1087-1090.
[32]易龙,彭云,孙秦.基于改进的灵敏度分析的有限元优化技术研究.机械强度.2008,30(3):483-487.
[33]盛新庆.计算电磁学要论（第一版）.北京：科学出版社,2004
[34]张慧,杨仕友,倪光正.快速全局优化算法在电磁场逆问题中的应用.机电工程,2007,24(10):28-30.
[35]张慧.电磁场逆问题分析计算的快速全局优化算法研究.浙江大学,2007.
[36]缪希仁,张培铭.基于遗传算法的电磁电器优化设计.低压电器,1998,18(5):12-15.
[37]陈丽安,张培铭等.基于免疫遗传算法的智能化电磁电器全局优化设计.电工电能新技术,2003,21(1):17-20.
[38]罗隆福,黄守道,童调生等.遗传算法用于电机电器优化设计的展望.湖南大学学报,1997,24(5):55-58
[39]倪光正,杨仕友,钱秀英等.工程电磁场数值计算.北京:机械工业出版社,2004.
[40]颜威利,杨庆新,汪友华等.电气工程电磁场数值分析.北京:机械工业出版社,2005.
[41]谢德馨,杨仕友.工程电磁场数值分析与综合.北京:机械工业出版社,2009.
[42]程志光,高桥则雄,博扎德等.电气工程电磁热场模拟与应用.北京:科学出版社,2009.
[43]谢德馨,阎秀恪,张奕黄等.旋转电机绕组磁链的三维有限元分析.中国电机工程学报,2006,26(21):143-148.
[44]谢德馨.用有限元法计算矩形槽内载流导体的集肤效应.哈尔滨电工学院学报,1981,4(1):1-23.
[45]吕殿利,汪友华,韩婷彦等.特高压变压器线圈的电场计算与分析.河北大学学报(自然科学版),2011,31(2):144-149.
[46]侯小全,周光厚,廖毅刚等.电磁场有限元技术在发电机优化设计中的应用.东方电气评论,2010,24(96):44-51.
[47]汪友华,颜威利,张冠生等.自适应模拟退火法在电磁场逆问题中的应用.中国电机工程学报,1995,15(4):234-239.
[48]白凤仙,邵玉槐,孙建中.利用智能型模拟退火算法进行开关磁阻电机磁极几何形状的优化.中国电机工程学报.2003,23(1):126-141.
[49]杨庆新,安金龙,马振平等.基于最小二乘支持向量机和自适应模拟退火算法的电磁场逆问题全局优化方法.电工技术学报,2008,23(11):1-7.
[50] Dong-Hun Kim, Soo Bum Lee, Byung Man Kwank, et al. Smooth Boundary Topology Optimization forElectrostatic Problems through the Combination of Shape and Topological Design Sensitivities. IEEETrans. Magn.,2008,44(6):1002-1005.
[51] Dong-Hun Kim, Jan K. Sykulski, David A. Lowther. The Implications of the Use of Composite Materialsin Electromagnetic Device Topology and Shape Optimization. IEEE Trans. Magn.,2009,45(3):1154-1157.
[52]王金铭,谢德馨,于超等.基于Tabu法的变压器铁心柱截面积优化.电工技术学报,2002,17(2):13-16.
[53]杨仕友,倪光正,钱金银.电机电磁场逆问题数值计算的改进TABU算法.中国电机工程学报,1998,18(2):83-86.
[54]马文阁,干于,杨仕友.改进的区域消除法及在电磁场逆问题中的应用.电机与控制学报,2000,4(1):10-12.
[55]安斯光,杨仕友,李桃.改进的禁忌搜索算法及其在电磁场逆问题中的应用.电力系统保护与控制,2010,38(24):30-33.
[56] Chen-Yi Liao, Wei-Ping Lee, Xianghan Chen, et al. Dynamic and adjustable particle swarm optimization.Proceedings of the8th International Conference on Evolutionary Computing, Canada,2007:19-21.
[57] J. Yoo, H.-J. Soh. An optimal design of magnetic actuators using topology optimization and the responsesurface method. Microsyst Technol,2005,11:1252-1261.
[58] Niarchos D. Magnetic MEMS: Key issues and some applications. Sensors and actuators A,2003,109:166-173
[59]谢德馨,唐任远,王尔智等.计算电磁学的现状与发展趋势—第14届COMPUMAG会议综述.电工技术学报,2003,18(5):1-4.
[60]谢德馨,唐任远.计算电磁学近年来的若干重要成果—第15届COMPUMAG会议概述.电工技术学报,2005,20(9):1-5.
[61] D.A. Lowther. New trends in the simulation of magnetic devices. Mathematics and Computers inSimulation,1995,38:257-262.
[62] A. Taflove, Computational electrodynamics: the finite-difference time-domain method. Boston,MA:Artech House,1995
[63]杨阳.电磁场时域有限差分数值方法的研究:[博士学位论文].南京:南京理工大学,2005.
[64]焦超群.细薄结构的时域有限差分法在电磁屏蔽中的应用研究:[博士学位论文].北京:华北电力大学,2006.
[65]刘士利.改进的边界元法及其在电场计算中的应用:[博士学位论文].北京:华北电力大学,2011.
[66]陶瑞民.永磁电机电磁场的边界元方法.电气技术,2011,11:20-22.
[67]姜保军,孙力,李波.电磁检测中的开域电磁场数值计算.中国电机工程学报,2005,25(8):156-160.
[68]林庆华,栗保明.有限元/边界元耦合法计算电磁轨道炮三维瞬态涡流场.南京理工大学学报(自然科学版),2010,34(2):217-221.
[69]关海爽,马文阁.电磁场数值计算新方法的研究.辽宁工学院学报,2006,26(4):230-233.
[70]国海广,范丽丽,潘晓东.解决电大尺寸电磁场问题的改进矩量法研究.现代电子技术,2009,6:130-133.
[71] Walton C. Gibson. The method of MoMents in electromagnetics. Boca Raton: Chapman&Hall/CRC,2008.
[72] Kensuke Yokoi. A numerical method fro free-surface flows and its application to droplet impact on a thinliquid layer. Scientific Computing,2008,35(6):372-396
[73] J Zou, J S Yuan, X S Ma, ec tl. Magnetic field analysis of iron-core reactor coils by the finite-volumemethod. IEEE Trans. on Magn,2004,40(2):814-817.
[74] J Zou, Y Q Xie, J S Yuan,ec tl. Analysis of the thin plate eddy-current problem by finite volume method.IEEE Trans. on Magn,2004,40(2):1370-1373.
[75] Xie Yaoqin, Yuan Jiansheng, Ma Xinshan, ec tl. Calculation of EEG problems with anisotropicconductiong media by the finite volume method. IEEE Trans. on Magn,2001,37(5):3749-3752.
[76]甘艳,阮江军,张宇等.TEAM9-1问题讨论与混合有限元法有限体积法的应用.电工技术学报,2009,24(1):1-7.
[77]甘艳,阮江军,张宇.有限元法与有限体积法相结合处理运动电磁问题.中国电机工程学报,2006,26(14):145-151.
[78]李锦彪,谢德馨.变压器电磁场问题的自适应有限元分析.变压器,2010,47(1):36-39
[79]闵越,肖静.变压器三维电场分析方法探讨.山东大学学报(工学版),2007,37(6):54-57.
[80]曲利岩.电磁场有限元分析技术的研究和实现:[硕士学位论文].杭州:浙江大学,2002.
[81]姬丽娟.交流接触器电磁机构有限元分析与多场耦合技术的研究:[硕士学位论文].天津:河北工业大学,2005.
[82]张淮清,俞集辉,郑亚利.径向基函数及其耦合方法在电磁场数值计算中的应用.计算物理,2009,26(2):299-310.
[83]唐章宏,顿月芹,袁建生.求解开域电磁场问题的区域映射有限元法.清华大学学报(自然科学版),2009,49(1):5-8.
[84]谢德馨,刘奇林.电工钢片各向异性磁场有限元分析.沈阳工业大学学报,2010,32(3):241-245.
[85]张为民,谢德馨,白保东等.强磁场永磁机构组装过程的动态有限元分析.中国电机工程学报,2007,27(18):61-66.
[86]崔芮华,贾峰,柳杰等.交流接触器电磁机构动态仿真分析.仿真技术,2010,22(7-1):286-288.
[87] K. Kadded, R.R. Saldanha, J.L. Coulomb. Mathematical Minimization of the Time Harmonics of theE.M.F. of a DC-PM Machine using a Finite Element Method. IEEE Trans. Magn.,1993,29(2):1747-1752.
[88]张勇.计算电磁学的无单元方法研究:[博士学位论文].武汉:华中科技大学,2006.
[89] Herault C, Marechal Y. Boundary and interface conditions in meshless methods. IEEE Trans. on Magn,1999,35(3):1450-1453.
[90] Gingoski V, Miyamonto N, Yanashita H. Element-free Galerkin method for electromagnetic fieldcomputations. IEEE Trans. on Magn,1998,34(5):3236-3239.
[91] Yang Qingxin, Liu Suzhen, Chen Haiyan. Three-Dimensional element free Galerkin method and itsapplications in electromagnetic field numerical analysis.电工技术学报,2006,21(9):116-121.
[92] Viana S A, Mesquita R C. Moving least square reproducing Kernel method for electromagnetic fieldcomputation. IEEE Trans. on Magn,1999,35(3):1372-1375.
[93]张淮清.电磁场计算中的径向基函数无网格法研究:[博士学位论文].重庆:重庆大学,2008.
[94]杨光源.运动导体涡流电磁问题的径向基点配置型无单元算法研究:[博士学位论文].华中科技大学,2011.
[95]刘素贞,杨庆新,陈海燕.无单元法和有限元法的比较研究.河北工业大学学报,2000,29(5):66-68.
[96] Ho S L, Yang S, Machado J M, et al. Application of a meshless method in electromagnetic. IEEE Trans.on Magn,2001,37(5):3198-3202.
[97]刘素贞,杨庆新,陈海燕等.无单元法在电磁场数值计算中的应用研究.电工技术学报,2001,16(2):30-33.
[98]康晓明.电压互感器的有限元分析与优化设计:[博士学位论文].天津:天津大学,2007.
[99]狄维娜.交流接触器电磁机构的动静态特性分析与仿真:[硕士学位论文].天津:河北工业大学,2007.
[100]高艳丰.电压互感器电磁场有限元分析与优化设计技术的研究:[硕士学位论文].天津:河北工业大学,2006.
[101]姚缨英,高昌燮,柳在燮等.基于有限元分析计算设计灵敏度和网格更新的三维形状优化算法.中国电机工程学报,2004,24(5):130-136.
[102] Yingying Yao, Jae Seop Ryu, Chang Seop Koh, et al. A Novel Mesh Regeneration Using StructuralDeformation Analysis for3-D Shape Optimization of Electromagnetic Devices. IEEE Trans. Magn.,2004,40(2):1009-1012.
[103]顾元宪,魏鸿磊,赵国忠.基于灵敏度的电磁结构形状优化方法研究.机械强度,2003,25(6):651-656
[104] JAlfredo Canelas, Jean R. Roche, José Herskovits. The inverse electromagnetic shaping problem. StructMultidisc Optim,2009,38:389-403.
[105] Wei Li, Zi Yan Ren, Young Woo Jeong, et al. Optimal Shape Design of a Thomson-Coil ActuatorUtilizing Generalized Topology Optimization Based on Equivalent Circuit Method. IEEE Trans. Magn.,2011,47(5):1246-1249.
[106] Y. Kim, I. Park. Topology Optimization of Rotor in Synchronous Reluctance Motor Using Level SetMethod and Shape Design Sensitivity. IEEE Trans. Magn.,2010,46(3):1093-1096.
[107] H. Shim, T.H.Vinh Thuy, S. Wang et al. Topological Shape Optimization of Electromagnetic ProblemsUsing Level Set Method and Radial basis Function. Comput. Model. Eng. Sci.,2008,37(2):175-202.
[108] Jangwon Lee, Semyung Wang. Topological Shape Optimization of Permanent Magnet in Voice CoilMotor Using Level Set Method. IEEE Trans. Magn.,2012,48(2):931-934.
[109] J. Lee, S. Wang. Topological shape Optimization of Permanent Magnet in Voice Coil Motor Using LevelSet Method. IEEE Trans. Magn.,2012,48(2):931-934
[110] M. Li. D. A. Lowther. The Application of Topological Gradients to Defect Identification in MagneticFlux Leakage-Type NDT. IEEE Trans. Magn.,2010,46(8):3221-3224.
[111] Jeonghoon Yoo. Modified Method of Topology Optimizationin Magnetic Fields. IEEE Trans. Magn.,2004,40(4):1796-1801.
[112] Chang-Hwan Im, Hyun-Kyo Jung, Yong-Joo Kim. Hybrid Genetic Algorithm for ElectromagneticTopology Optimization. IEEE Trans. Magn.,2003,39(5):2163-2169.
[113] Chao Wang, Qiuliang Wang, Hui Huang, et al. Electromagnetic optimization design of a HTS magnetusing the improved hybrid genetic algorithm. Cryogenics,2006,46:349-353.
[114] Kota Watanabe, Felipe Campelo, Yosuke Iijima, et al. Optimization of Inductors Using EvolutionaryAlgorithms and Its Experimental Validation. IEEE Trans. Magn.,2010,46(8):3393-3396.
[115] J. Lee, J. Yoo. Topology Optimization of the Permanent Magnet Type MRI Considering the MagneticField Homogeneity. J. Magn. Mater,2010,322(9-12):1651-1654.
[116]任自艳,谢德馨,李会香.开放式MRI永磁型主磁体的匀场方法.电工技术学报,2010,25(3):1-5.
[117] Youguang Guo, Jianguo Zhu, D. G. Dorrell. Design and Analysis of a Claw Pole Permanent MagnetMotor with Molded Soft Magnetic Composite Core. IEEE Trans. Magn.,2009,45(10):4582-4585.
[118] Franz Zürcher, Thomas Nussbaumer, Wolfgang Gruber, et al. Design and Development of a26-Pole and24-Slot Bearingless Motor. IEEE Trans. Magn.,2009,45(10):4594-4597
[119] Thibaut Labbe, Bruno Dehez. Topology Optimization Method Based on the Maxwell Stress Tensor forthe Design of Ferromagnetic Parts in Electromagnetic Actuators. IEEE Trans. Magn.2011,47(9):2188-2193.
[120] T.Labbe, B.Dehez. Convexity-Oriented Method for the Topology Optimization of FerromagneticMoving Parts in Electromagnetic Actuators Using the Magnetic Energy. IEEE Trans. Magn.,2010,46(12):4016-4022.
[121] Sigmund, J. Petersson. Numerical Instabilities in Topology Optimization: A Survey on ProceduresDealing with Checkerboards, Mesh-dependencies and Local Minima. Struct. Optim,1998,16(1):68-75.
[122] Sunghoon Lim, Seungjae Min. Design Optimization of Permanent Magnet Actuator Using Multi-PhaseLevel-Set Model. IEEE Trans. Magn.,2012,48(4):1641-1644.
[123] Feustel, O. Krusemark, J. Muller. Numerical simulation and optimization of planar electromagneticactuators. Sensors and Actuators,1998,70:276-282.
[124] Smaranda Nitu, Constantim Nitu, Gheorghe Tuluca, et al. Electromagnetic actuator with magnetic storedenergy. Materials Processing Technology,2007,181:153-158.
[125] Ingo Hahn. Heuristic Structural Optimization of the Permanent Magnets Used in a Surface MountedPermanent-Magnet Synchronous Machine. IEEE Trans. Magn.,2012,48(1):118-127.
[126] Jaewook Lee, Noboru Kikuchi. Structural Topology Optimization of Electrical Machinery to MaximizeStiffness with Body Force Distribution. IEEE Trans. Magn.,2010,46(10):3790-3794.
[127] Semyung Wang, Jenam Kang, Jeongchae Noh. Topology Optimization of a Single-Phase InductionMotor for Rotary Compressor. IEEE Trans. Magn.,2004,40(3):1591-1596.
[128] Arnold J. Rix, Maarten J. Kamper. Radial-Flux Permanent-Magnet Hub Drives: A Comparison Based onStator and Rotor Topologies. IEEE Trans. Industrial Electronics,2012,59(6):2475-2483.
[129] Neil Hodgins, Ozan Keysan, Alasdair S. McDonald, et al. Design and Testing of a Linear Generator forWave-Energy Applications. IEEE Trans. Industrial Electronics,2012,59(6):2094-2103.
[130] Derek N. Dyck, David A. Lowther. Automated Design of Magnetic Devices by Optimizing MaterialDistribution. IEEE Trans. Magn.,1996,32(3):1188-1193.
[131] Dong-Hun Kim, Jan K. Sykulski, David A. Lowther. A Novel Scheme for Material Updating in SourceDistribution Optimization of Magnetic Devices Using Sensitivity Analysis. IEEE Trans. Magn.,2005,41(5):1752-1755.
[132] Takeo Ishikawa, Kouki Yonetake, Nobuyuki Kurita. An Optimal Material Distribution Design ofBrushless DC Motor by Genetic Algorithm Considering a Cluster of Material. IEEE Trans. Magn.,2011,47(5):1310-1313.
[133] Sang-in Park, Seungjae Min. Design of Magnetic Actuator With Nonlinear Ferromagnetic MaterialsUsing Level-Set Based Topology Optimization. IEEE Trans. Magn.,2010,46(2):618-621.
[134] Jin-Kyu Byun, Joon-Ho Lee, Il-Han Park. Node-Based Distribution of Material Properties for TopologyOptimization of Electromagnetic Devices. IEEE Trans. Magn.,2004,40(2):1212-1215.
[135] Jaewook Lee, Ercan M. Dede, Tsuyoshi Nomura.Simultaneous Design Optimization of PermanentMagnet, Coils, and Ferromagnetic Material in Actuators. IEEE Trans. Magn.,2011,47(12):4712-4716.
[136] Jae Seok Choi, Jeonghoon Yoo. Simultaneous Structural Topology Optimization of ElectromagneticSources and Ferromagnetic Materials. Comput. Methods Appl. Mech. Engrg.,2009,198:2111-2121.
[137] Tsuyoshi Nomura．Topology optimization of electromagnetic materials. Met materials and AutomotiveApplications,2006,41(4):9-15.
[138] Tsuyoshi Nomura, Shinji Nishiwaki, Kazuo Sato, et al. Topology optimization for the design of periodicmicrostructures composed of electromagnetic materials. Finit Elements in Analysis and Design,2009,45:210-226.
[139] M. Y. Wang, X. Wang. Color’ Level Sets: A Multi-Phase Method for Structural Topology Optimizationwith Multiple Materials. Comput. Methods Appl. Mech. eng.,2004,193:469-496.
[140] Takeo Ishikawa, Kyoichi Nakayama. Topology Optimization of Rotor Structure in Brushless DC MotorWith Concentrated Windings Using Genetic Algorithm Combined With Cluster of Material. IEEE Trans.Magn.,2012,48(2):899-902.
[141] Frederico G, Guimar es, Jaime A. Ramírez. A Pruning Method for Neural Networks and Its Applicationfor Optimization in Electromagnetics. IEEE Trans. Magn.,2004,40(2):1160-1163.
[142](美)金建铭著,王建国译.电磁场有限元方法.西安:西安电子科技大学出版社,1998.
[143]何红雨.电磁场数值计算法与MATLAB实现.武汉:华中科技大学出版社,2003.
[144] Herbert De Gersema, Kay Hameyerb, Thomas Weilanda. Field-circuit coupled models inelectromagnetic simulation. Computational and Applied Mathematics,2004,168:125-133.
[145]张洋,白保东,谢德馨.三维瞬态涡流–电路–运动系统耦合问题的新解法.中国电机工程学报,2008,28(9):139-144.
[146]张倩,胡仁喜,康仕廷等编著.ANSYS12.0电磁学有限元分析从入门到精通.北京:机械工业出版社,2010.
[147] Hyeoksoo Hong, Jeonghoon Yoo. Shape Design of the Surface Mounted Permanent Magnet in aSynchronous Machine. IEEE Trans. Magn.,2011,47(8):2109-2117.
[148] Shinji Nishiwaki, Tsuyoshi Nomura, Shinya Kinoshita, et al. Topology optimization for cross-sectiondesigns of electromagnetic waveguides targeting guiding characteristics. Finite Elements in Analysis andDesign,2009,45:944-957.
[149] T.Labbe, B.Dehez. Design of Magnetic Thrusts Using Topology Optimization Methods. J.Comput. Math.Elect. Electron. Eng.,2011,30(2):641-655.
[150] Semyung Wang, D. Youn, H. Moon1, et al. Topology Optimization of Electromagnetic SystemsConsidering Magnetization Direction. IEEE Trans. Magn.2005,41(5):1808-1811.
[151] Jae Seok Choi, Jeonghoon Yoo, Shinji Nishiwaki, et al. Optimization of Magnetization Directions in a3-D Magnetic Structure. IEEE Trans. Magn.,2010,46(6):1603-1606.
[152] Jeonghoon Yoo, Seunjin Yang, Jae Seok Choi. Optimal Design of an electromagnetic Coupler toMaximize Force to a Specific Direction. IEEE Trans. Magn.,2008,44(7):1737-1742.
[153] X. Huang, Y.M. Xie. Topology optimization of nonlinear structures under displacement loading.Engineering Structures,2008,30:2057-2068.
[154] Volker Schulz. Simultaneous solution approaches for large optimization problems. Computational andApplied Mathematics,2004,64-165:629-641.
[155] Jin-kyu Byun, Ju-hyun Lee, Il-han Park, et al. Inverse Problem Application of Topology OptimizationMethod with Mutual Energy Concept and Design Sensitivity. IEEE Trans. Magn.,2000,6(4):1144-1147.
[156] O. de la Barrière, S. Hlioui, H. Ben Ahmed, et al.3-D Formal Resolution of Maxwell Equations for theComputation of the No-Load Flux in an Axial Flux Permanent-Magnet Synchronous Machine. IEEETrans. Magn.,2012,48(2):128-136.
[157] Sung-Roc Jang, Hong-Je Ryoo, Suk-Ho Ahn, et al. Development and Optimization of High-VoltagePower Supply System for Industrial Magnetron. IEEE Trans. Industrial Electronics,2012,59(6):1453-1461.
[158] Osama A. Mohammed, Riaz Merchant, Fuat G. Uler. An Intelligent System for Design Optimization ofElectromagnetic Devices. IEEE Trans. Magn.,1994,30(5):3633-3636.
[159] T.Labbe, B.Dehez. Convexity-Oriented Mapping Method for the Topology Optimization ofElectromagnetic Devices Composed of Iron and Coils. IEEE Trans. Magn.,2010,46(5):1177-1185.