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
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摘要
现代电气工程领域应用的许多设备都包含运动的导体,如电机、电磁开关、电磁线圈炮、磁悬浮列车等,对这些设备进行参数分析和优化设计都离不开对其中的电磁场进行求解。运动导体涡流电磁问题一直是计算电磁学领域的研究热点和难点,传统的电磁场数值算法如有限元法需要对求解域进行网格剖分,在处理运动导体导致的模型形变问题时往往会遇到较大的困难。而新近发展起来的无单元算法则免除了网格的剖分,在处理耦合导体运动的电磁问题方面表现出了巨大的发展潜力。本文研究了一种纯无单元算法——径向基点配置法,并将其应用于二维运动导体涡流电磁问题的求解和分析,全文的工作主要包括以下几个方面：
1、提出了一种新的叠加径向基点配置法。该方法基于场源叠加原理和径向基点配置法求解方程的线性特性,可以方便地对电磁场控制方程中包含的未知量进行解耦；通过引入运动坐标系来建立径向基节点模型,我们可以处理导体以任意形式运动的涡流磁场问题,磁场控制方程通过构造时域迭代来离散,便于考察任意时刻系统内的磁场以及涡流的分布情况。
2、验证了本文提出的叠加径向基点配置法在分析稳态及瞬态电磁问题时与常规径向基点配置法的等价性；并通过工程问题算例验证了该算法在求解运动导体涡流问题时的有效性。此外还利用该算法构造了磁场—力—运动—电路综合分析模型对一个包含运动导体的实际工程装置的动态过程进行了模拟,并考察了装置的主要模型参数对其工作效率的影响。
3、对径向基点配置法在更复杂电磁问题求解领域的应用进行了初步探索。针对非线性介质电磁问题,提出了基于牛顿迭代和拓延法初值优化的迭代型径向基点配置求解格式,并通过算例考察了其收敛性；针对多介质求解域电磁问题,将区域分解径向基点配置法从稳态扩展到瞬态问题求解领域,结合叠加径向基法推导了基于时域迭代格式的多介质运动导体涡流问题求解公式,并且指出该算法在求解涡流问题时可以有效地降低迭代过程中求解的矩阵方程的维数,从而极大地提高计算效率。
Many devices in the modern electrical engineering field contain moving conductors, such as the electric machine, electromagnetic switch, coil gun, maglev train and so on. To analyze and optimize their parameters, the electromagnetic field in these devices must be calculated. The moving conductor eddy current problem has always been an attractive and difficult problem in the computational electromagnetics. The traditional numerical method, for example, the finite element method needs to mesh the solving domain to construct the shape function. This leads to a great difficulty when the model varies because of the movement of the conductor. The newly developed meshless method, which is free from domain meshing, shows a deep potential to deal with motion coupled electromagnetic problems. The radial basis function (RBF) collocation method, which is a kind of pure meshless method, is studied and applied to solve two dimensional moving conductor eddy current problem in this paper. The structure of the paper is as follows.
1. A novel superposition RBF collocation method is presented. This method, which is based on the source superposition principle and the linear property of the governing equations of RBF collocation method, could conveniently decouple the unknowns in the electromagnetic field equations. The arbitrary movement of the conductor is expressed through introducing moving coordinate systems to build the node model. The governing equations of the field are discretized through time domain iteration, which enable us to analyze the magnetic field and eddy current in the system at any time.
2. The equality of the proposed method and the normal RBF collocation method is examined in both static and transient electromagnetic problems. According to numerical examples, we also prove that the proposed method is effective to solve moving conductor eddy current problems. Furthermore, a global model combined the magnetic field, electromagnetic force, conductor movement and the circuit equations is presented to simulate the dynamic process of a practical engineering device containing moving conductors. The model parameters are also analyzed for optimization.
3. The application of the RBF collocation method in complex media electromagnetic problems is presented. First, an iterative RBF collocation scheme on the basis of Newton iteration and continuation optimization of the initial value is presented to deal with nonlinear media electromagnetic problems. The convergence property of this method is analyzed with a numerical example. Moreover, the domain decomposition method (DDM) combined RBF collocation method, which has been used to solve electrostatic problems, is extended to compute transient electromagnetic field. Through combining the proposed superposition RBF collocation method and the DDM, a time domain iterative scheme for multi-media moving conductor eddy current problems is presented. One advantage of this method is that the dimensions of the matrix equations solved in each iteration step could be reduced and the efficiency of computation could be greatly improved.
1、提出了一种新的叠加径向基点配置法。该方法基于场源叠加原理和径向基点配置法求解方程的线性特性,可以方便地对电磁场控制方程中包含的未知量进行解耦；通过引入运动坐标系来建立径向基节点模型,我们可以处理导体以任意形式运动的涡流磁场问题,磁场控制方程通过构造时域迭代来离散,便于考察任意时刻系统内的磁场以及涡流的分布情况。
2、验证了本文提出的叠加径向基点配置法在分析稳态及瞬态电磁问题时与常规径向基点配置法的等价性；并通过工程问题算例验证了该算法在求解运动导体涡流问题时的有效性。此外还利用该算法构造了磁场—力—运动—电路综合分析模型对一个包含运动导体的实际工程装置的动态过程进行了模拟,并考察了装置的主要模型参数对其工作效率的影响。
3、对径向基点配置法在更复杂电磁问题求解领域的应用进行了初步探索。针对非线性介质电磁问题,提出了基于牛顿迭代和拓延法初值优化的迭代型径向基点配置求解格式,并通过算例考察了其收敛性；针对多介质求解域电磁问题,将区域分解径向基点配置法从稳态扩展到瞬态问题求解领域,结合叠加径向基法推导了基于时域迭代格式的多介质运动导体涡流问题求解公式,并且指出该算法在求解涡流问题时可以有效地降低迭代过程中求解的矩阵方程的维数,从而极大地提高计算效率。
Many devices in the modern electrical engineering field contain moving conductors, such as the electric machine, electromagnetic switch, coil gun, maglev train and so on. To analyze and optimize their parameters, the electromagnetic field in these devices must be calculated. The moving conductor eddy current problem has always been an attractive and difficult problem in the computational electromagnetics. The traditional numerical method, for example, the finite element method needs to mesh the solving domain to construct the shape function. This leads to a great difficulty when the model varies because of the movement of the conductor. The newly developed meshless method, which is free from domain meshing, shows a deep potential to deal with motion coupled electromagnetic problems. The radial basis function (RBF) collocation method, which is a kind of pure meshless method, is studied and applied to solve two dimensional moving conductor eddy current problem in this paper. The structure of the paper is as follows.
1. A novel superposition RBF collocation method is presented. This method, which is based on the source superposition principle and the linear property of the governing equations of RBF collocation method, could conveniently decouple the unknowns in the electromagnetic field equations. The arbitrary movement of the conductor is expressed through introducing moving coordinate systems to build the node model. The governing equations of the field are discretized through time domain iteration, which enable us to analyze the magnetic field and eddy current in the system at any time.
2. The equality of the proposed method and the normal RBF collocation method is examined in both static and transient electromagnetic problems. According to numerical examples, we also prove that the proposed method is effective to solve moving conductor eddy current problems. Furthermore, a global model combined the magnetic field, electromagnetic force, conductor movement and the circuit equations is presented to simulate the dynamic process of a practical engineering device containing moving conductors. The model parameters are also analyzed for optimization.
3. The application of the RBF collocation method in complex media electromagnetic problems is presented. First, an iterative RBF collocation scheme on the basis of Newton iteration and continuation optimization of the initial value is presented to deal with nonlinear media electromagnetic problems. The convergence property of this method is analyzed with a numerical example. Moreover, the domain decomposition method (DDM) combined RBF collocation method, which has been used to solve electrostatic problems, is extended to compute transient electromagnetic field. Through combining the proposed superposition RBF collocation method and the DDM, a time domain iterative scheme for multi-media moving conductor eddy current problems is presented. One advantage of this method is that the dimensions of the matrix equations solved in each iteration step could be reduced and the efficiency of computation could be greatly improved.
引文
[1]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[J].电工技术学报,2003,18(5)：1-4.
[2]杨庆新,刘素珍,陈海燕等.工程电磁场无单元法[M].北京：科学出版社,2008.
[3]尹华杰.无网格法及其在电磁场计算中的应用展望[J].电机与控制学报,2003,7(2)：107-111.
[4]R. Hardy. Multiquadric equations of topography and other irregular surfaces[J]. Journal of Geophysical Research,1971,76(8):1905-1915.
[5]E.J. Kansa. Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—Ⅰ surface approximations and partial derivative estimates[J]. Computers and Mathematics with Applications,1990, 19(8-9):127-145.
[6]E.J. Kansa. Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—Ⅱ solutions to parabolic, hyperbolic and elliptic partial differential equations[J]. Computers and Mathematics with Applications,1990,19(8-9):147-161.
[7]X. Zhang, K.Z. Song, M.W. Lu, el al. Meshless methods based on collocation with radial basis functions[J]. Computational Mechanics,2000,26(4):333-343.
[8]周可定.工程电磁场专论[M].武汉：华中工学院出版社,1986.
[9]冯慈璋,马西奎.工程电磁场导论[M].北京：高等教育出版社,2000.
[10]倪光正,杨仕友,钱秀英等.工程电磁场数值计算[M].北京：机械工业出版社,2004.
[11]刘国强,赵凌志,蒋继娅.Ansoft工程电磁场有限元分析[M].北京：电子工业出版社,2005.
[12]葛德彪,闫玉波.电磁波时域有限差分法[M].西安：西安电子科技大学出版社,2002.
[13]李庆民,刘卫东,钱家骊.电磁推力机构的一种分析方法[J].电工技术学报,2004,19(2)：20-24.
[14]娄杰,李庆民,孙庆森等.快速电磁推力机构的动态特性仿真与优化设计[J].中国电机工程学报,2005,25(16)：23-29.
[15]张惠娟.运动电磁系统涡流场有限元研究[D].天津,河北工业大学,2000.
[16]S. Hahn. An'upwind' finite element method for electromagnetic field problems in moving media[J]. International Journal for Numerical Methods in Engineering,1987, 24(11):2071-2086.
[17]D. Rodger, T. Karaguler, P.J. Leonard. A formulation for 3D moving conductor eddy current problems[J]. IEEE Transactions on Magnetics,1989,25(5):4147-4149.
[18]K. Muramatsu, T. Nakata, N. Takahashi et al. Comparison of coordinate systems for eddy current analysis in moving conductors[J]. IEEE Transactions on Magnetics, 1992,28(2):1186-1190.
[19]K. Yamazaki.3D eddy current formulation for moving conductors with variable velocity of coordinate system using edge finite elements [J]. IEEE Transactions on Magnetics,1999,35(3):1594-1598.
[20]S. Niikura, A. Kameari. Analysis of eddy current and force in conductors with motion[J]. IEEE Transactions on Magnetics,1992,28(2):1450-1453.
[21]M. Jarnienx, D. Grenier, G. Reyne et al. FEM computation of eddy current and force in moving systems application to linear induction launcher [J]. IEEE Transactions on Magnetics,1993,29(2):1989-1992.
[22]B. Davat, Z. Ren, M. Lajoie-Mazenc. The movement in field modeling[J]. IEEE Transactions on Magnetics,1985, MAG-21(6):2296-2298.
[23]Y. Kawase, S. Tatsuoka, T. Yamaguchi.3-D finite element analysis of operation characteristics of AC electromagnetic contactors [J]. IEEE Transactions on Magnetics,1994,30(5):3244-3247.
[24]Y. Kawase, O. Miyatani, T. Yamaguchi. Numerical analysis of dynamic characteristics of electromagnets using 3-D finite element method with edge elements[J]. IEEE Transactions on Magnetics,1994,30(5):3248-3251.
[25]S. Dufour, G. Vinsard, B. Laport.2-D adaptive mesh with movement[J]. IEEE Transactions on Magnetics,2001,37(5):3482-3485.
[26]K. Yamazaki, T. Saeki. Adaptive mesh generation within nonlinear iterative calculations for analyses of electric machines[J]. IEEE Transactions on Magnetics, 2003,39(3):1654-1657.
[27]K. Yamazaki, S. Watari, A. Egawa. Adaptive finite element meshing for eddy current analysis of moving conductor[J]. IEEE Transactions on Magnetics,2004, 40(2):993-996.
[28]孙玉田,杨明,李北芳.电机动态有限元法中的运动问题[J].大电机技术,1997,(6)： 35-39.
[29]P. Lombard, G. Meunier. A general purpose method for electric and magnetic combined problems for 2D, axisymmetry and transient systems[J]. IEEE Transactions on Magnetics,1993,29(2):1737-1740.
[30]P.J. Leonard, H.C. Lai, G. Hainsworth et al. Analysis of performance of tubular pulsed coil induction launchers[J]. IEEE Transactions on Magnetics,1993,29(1): 686-690.
[31]H.C. Lai, P.J. Leonard, D. Rodger et al.3D finite element dynamic simulation of electrical machines coupled to external circuits [J]. IEEE Transactions on Magnetics, 1997,33(2):2010-2013.
[32]严登俊,刘瑞芳,胡敏强等.处理电磁场有限元运动问题的新方法[J].中国电机工程学报,2003,23(8)：163-167.
[33]Y. Zhang, B. Bai, D. Xie. Analysis of the 3D transient electromagnetic field-circuit-movement coupled problem[C]. Automation Congress,2008. WAC 2008.
[34]R. Perrin-Bit, J.L. Coulomb. A three dimensional finite element mesh connection for problems involving movement[J]. IEEE Transactions on Magnetics,1995,31(3): 920-923.
[35]C.S. Biddlecombe, J. Simkin, A.P. Jay et al. Transient electromagnetic analysis coupled to electric circuits and motion[J]. IEEE Transactions on Magnetics,1998, 34(5):3182-3185.
[36]K. Tani, T. Yamada, Y. Kawase. A new technique for 3D dynamic finite element analysis of electromagnetic problems with relative movement[J]. IEEE Transactions on Magnetics,1998,35(3):3371-3374.
[37]K. Tani, T. Yamada, Y. Kawase. Dynamic analysis of linear actuator taking into account eddy currents using finite element method and 3-D mesh coupling method[J]. IEEE Transactions on Magnetics,1999,35(3):1785-1788.
[38]张宇,阮江军,刘兵等.组合网格法在电磁-机械耦合问题中的应用[J].中国电机工程学报,2007,27(33)：42-47.
[39]甘艳,阮江军,张宇等.组合网格法及其在电磁问题中的应用[J].电工技术学报,2008,23(11)：8-14.
[40]P. Ying, J. Ruan, Y. Zhang et al. A composite grid method for moving conductor eddy-current problem[J]. IEEE Transactions on Magnetics,2007,43(7):3259-3265.
[41]Y. Zhang, J. Ruan, Y. Gan et al. Application of a composite grid method in the analysis of 3-D eddy current field involving movement[J]. IEEE Transactions on Magnetics,2008,44(6):1298-1301.
[42]M. Koizumi, T. Takahashi. An integral equation method for analysis of three-dimensional magnetodynamics with moving objects [J]. IEEE Transactions on Magnetics,1993,29(2):1516-1519.
[43]D. Kim, S. Hahn, K. Choi et al. Electromagnetic force analysis of moving conducting slab by using 3D IBIEM[J]. IEEE Transactions on Magnetics,1998, 34(5):2589-2592.
[44]D. Kim, S. Hahn, I. Park et al. Computation of three-dimensional electromagnetic field including moving media by indirect boundary integral equation methodfJ]. IEEE Transactions on Magnetics,1999,35(3):1932-1938.
[45]M. Clemens, T. Weiland, M. Wilke. Transient eddy current formulation including moving conductors using the FI method[J]. IEEE Transactions on Magnetics,2000, 36(4):804-808.
[46]N. Esposito, A. Musolino, M. Raugi. Modelling of three-dimensional nonlinear eddy current problems with conductors in motion by an integral formulation[J]. IEEE Transactions on Magnetics,2002,32(3):764-767.
[47]王洋忠,王炳革,卢斌先等.三维开域涡流场A-V位有限元与边界元耦合分析方法[J].中国电机工程学报,2000,20(5)：1-4.
[48]S. Kurz, J. Fetzer, G. Lehner. Three dimensional transient BEM-FEM coupling analysis of electrodynamic levitation problems[J]. IEEE Transactions on Magnetics, 1996,32(3):1062-1065.
[49]S. Kurz, J. Fetzer, G. Lehner. A novel formalation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM-FEM coupling[J]. IEEE Transactions on Magnetics,1998,34(5):3068-3073.
[50]A. Koski, K. Forsman, T. Tarhasaari et al. Force and torque computations with hybrid methods[J]. IEEE Transactions on Magnetics,1999,35(3):1387-1390.
[51]S. Suuriniemi, K. Forsman, L. Kettunen et al. Computation of eddy currents coupled with motion[J]. IEEE Transactions on Magnetics,2000,36(4):1341-1345.
[52]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[J].中国电机工程学报,2006,26(14)：145-151.
[53]S. Barmada, A. Musolino, M. Raugi. Hybrid FEM/MOM formulation for eddy current problems with moving conductors[J]. IEEE Transactions on Magnetics,2000, 36(4):827-830.
[54]M. Schinnerl, J. Schoberl, M. Kaltenbacher et al. Multigrid method for the three-dimensional simulation of nonlinear magnetomechal systems[J]. IEEE Transactions on Magnetics,2002,38(3):1497-1511.
[55]R.V. Sabariego, J. Gyselinck, C. Geuzaine et al. Application of the fast multipole method to the 2D finite element-boundary element analysis of electromechanical devices[J]. International Journal for Computation and Mathematics in Electrical and Electronic Engineering,2003,22(3):659-673.
[56]R.V. Sabariego, J. Gyselinck, P. Dular et al. Fast multipole acceleration of the hybrid finite-elment/boundary element analysis of 3-D eddy-current problems[J]. IEEE Transactions on Magnetics,2004,40(2):1278-1281.
[57]陶文铨,吴学红,戴艳俊.无网格数值求解方法[J].中国电机工程学报,2010,30(5)：1-10.
[58]L.B. Lucy. A numerical approach to the testing of the fission hypothesis[J]. The Astrophysical Journal,1977,8(12):1013-1024.
[59]J.J. Moraghan. An introduction to SPH[J]. Computer Physics Communications, 1988,48:89-96.
[60]R.A. Gingold, J.J. Moraghan. Smoothed particle hydrodynamics:theory and applications to nonspherical stars[J]. Mon Not Roy Astrou Soc,1977,18:375-389.
[61]G.R. Johnson, S.R. Beissel. Normalized smoothing functions for SPH impact computations [J]. International Journal for Numerical Methods in Engineering,2002, 53:825-843.
[62]J.W. Swegle, D.L. Hicks, S.W. Attaway. Smoothed particle hydrodynamics stability analysis[J]. Journal of Computational Physics,1995,116(1):123-134.
[63]J.K. Chen, J.E. Beraun, C.J. Jih. An improvement for tensile instability in smoothed particle hydrodynamics [J]. Computational Mechanics,1999,23(4):279-287.
[64]C.T. Dyka. Addressing tension instability in SPH methods. Technical Report NRL/MR/6384, NRL,1994.
[65]G. Ala, E. Francomano, A. Tortorici et al. A smoothed particle interpolation scheme for transient electromagnetic simulation[J]. IEEE Transactions on Magnetics,2006, 42(4):647-650.
[66]P. Lancaster, K. Salkauskas. Surfaces generated by moving least square methods[J]. Mathematics of Computation,1981,37(155):141-158.
[67]B. Nayroles, G. Touzot, P. Villon. Generalizing the finite element method:Diffuse approximation and diffuse elements[J]. Computational Mechanics,1992,10(5): 307-318.
[68]T. Belytschko, Y.Y. Liu, L. Gu. Element-free Galerkin methods[J]. International Journal for Numerical Methods in Engineering,1994,37:229-256.
[69]P. Krysl, T. Belytschko. The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks[J]. International Journal for Numerical Methods in Engineering,1999,44(6):767-800.
[70]S. Li, W. Hao, W.K. Liu. Numerical simulations of large deformation of thin shell structures using meshfree methods[J]. Computational Mechanics,2000,25:102-116.
[71]C. Herault, Y. Marechal. Boundary and interface conditions in meshless method[J]. IEEE Transactions on Magnetics,1999,35(3):1450-1453.
[72]V. Cingoski, N. Miyamoto, H. Yamashita. Element-free Galerkin method for electromagnetic field computations[J]. IEEE Transactions on Magnetics,1998,34(5): 3236-3239.
[73]S.L. Ho, S. Yang, J.M. Machado et al. Application of a meshless method in electromagnetics[J]. IEEE Transactions on Magnetics,2001,37(5):3198-3202.
[74]L. Xuan, Z. Zeng, B. Shanker et al. Element-free Galerkin method for static and quasi-static electromagnetic field computation[J]. IEEE Transactions on Magnetics, 2004,40(1):12-20.
[75]L. Xuan, Z. Zeng, B. Shanker et al. Meshless Method for Numerical Modeling of Pulsed Eddy Currents[J]. IEEE Transactions on Magnetics,2004,40(6):3457-3462.
[76]G.F. Parreira, E.J. Silva, A.R. Fonseca et al. The Element-Free Galerkin Method in Three-Dimensional Electromagnetic Problems[J]. IEEE Transactions on Magnetics, 2006,42(4):711-714.
[77]O. Bottauscio, M. Champi, A. Manzin. Element-Free Galerkin Method in Eddy-Current Problems With Ferromagnetic Media[J]. IEEE Transactions on Magnetics,2006,42(4):1577-1584.
[78]A. Manzin, O. Bottauscio. Element-Free Galerkin Method for the Analysis of Electromagnetic-Wave Scattering[J]. IEEE Transactions on Magnetics,2008,44(6): 1366-1369.
[79]Y. Zhang, K.R. Shao, D.X. Xie et al. Meshless Method Based on Orthogonal Basis for Computational Electromagnetics[J].IEEE Transactions on Magnetics,2005, 41(5):1432-1435.
[80]S.Y. Yang, G. Ni, J. Qian et al. Wavelet-Galerkin method for computations of electromagnetics[J]. IEEE Transactions on Magnetics,1998,34(5):2493-2496.
[81]杨仕友,倪光正.小波-伽辽金有限元法及其在电磁场数值计算中的应用[J].中国电机工程学报,1999,19(1)：56-61.
[82]S.Y. Yang, S.L. Ho, P.H. Ni et al. A Combined Wavelet-FE Method for Transient Electromagnetic-Field Computations[J]. IEEE Transactions on Magnetics,2006, 42(4):571-574.
[83]V. Cingoski, N. Miyamoto, H. Yamashita. Hybrid element-free Galerkin-finite element method for electromagnetic field computations [J]. IEEE Transactions on Magnetics,2000,36(4):1543-1547.
[84]S.L. Ho, S.Y. Yang, G.Z. Ni et al. Numerical Analysis of Thin Skin Depths of 3-D Eddy-Current Problems Using a Combination of Finite Element and Meshless Methods[J]. IEEE Transactions on Magnetics,2004,40(2):1354-1357.
[85]W.K. Liu, S. Sun, Y.F. Zhang. Reproducing kernel particle methods[J]. International Journal for Numerical methods in Fluids,1995,20:1081-1106.
[86]W.K. Liu, S. Li, T. Belytschko. Moving least-square reproducing kernel method Part Ⅰ methodology and convergence[J]. Computer Methods in Applied Mechanics and Engineering,1997,143:113-154.
[87]S. Li, W.K. Liu. Moving least-square reproducing kernel method Part Ⅱ Fourier analysis[J]. Computer Methods in Applied Mechanics and Engineering,1996,139: 159-193.
[88]W.K. Liu, Y. Chen, R.A. Uras et al. Generalized multiple scale reproducing kernel particle methods [J]. Computer Methods in Applied Mechanics and Engineering, 1996,139:91-157.
[89]S.A. Viana, R.C. Mesquita. Moving Least Square Reproducing Kernel Method for Electromagnetic Field Computation[J]. IEEE Transactions on Magnetics,1999, 35(3):1372-1375.
[90]N.R. Aluru. A point collocation method for meshless analysis of microelectronic and micro electromechanical devices[J]. Computational Electronics,1998,54-57.
[91]C.A. Durate, J.T. Oden. Hp clouds:a h-p meshless method[J]. Numerical method for partial differential equations,1996,12:673-705.
[92]J.T. Oden, C.A. Durate, O.C. Zienkiewicz. A new could-based hp finite element method[J]. International Journal for Numerical Methods in Engineering,1998,50: 160-170.
[93]T.J. Liszka, C.A. Durate, W.W. Tworzydlo. Hp-meshless cloud method[J]. Computer Methods in Applied Mechanics and Engineering,1996,139:263-288.
[94]T. Zhu, J. Zhang, S.N. Atluri. A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach[J]. Computational Mechanics,1998,21:223-235.
[95]S.N. Atluri, T. Zhu. A new meshless local Petrov-Galerkin(MLPG) approach in computational mechanics[J]. Computational Mechanics,1998,22:117-127.
[96]S.A. Viana, D. Rodger, H.C. Lai. Meshless local Petrov-Galerkin method with radial basis functions applied to electromagnetics [J]. IEE Proceedings Science, Measurement and Technology,2004,151(6):449-451.
[97]W.L. Nicomedes, R.C. Mesquita, F.J.S. Moreira. A local boundary integral equation (LBIE) method in 2D electromagnetic wave scattering, and a meshless discretization approach[C]. Microwave and Optoelectronics Conference (IMOC),2009,133-137.
[98]G.R. Liu, Y.T. Gu. A point interpolation method for two-dimensional solids[J]. International Journal for Numerical Methods in Engineering,2001,50:937-951.
[99]J.G. Wang, G.R. Liu. A point interpolation meshless method based on radial basis functions[J]. International Journal for Numerical Methods in Engineering,2002,54: 1623-1648.
[100]Y.L. Wu, G.R. Liu. A meshJ.G. Wang, G.R. Liu. A meshfree formulation for local radial point interpolation method (LRPIM) for incompressible flow simulation [J]. Computational Mechanics,2003,30:355-365.
[101]S.A. Viana, D. Rodger, H.C. Lai. Application of the Local Radial Point Interpolation Method to Solve Eddy-Current Problems[J]. IEEE Transactions on Magnetics,2006, 42(4):591-594.
[102]S.J. Lai, B.Z. Wang, Y. Duan. Meshless radial basis function method for transient electromagnetic computations[J]. IEEE Transactions on Magnetics,2008,44(10): 2288-2295.
[103]Y. Yu, Z. Chen. A 3-D radial point interpolation method for meshless time-domain modeling[J]. IEEE Transactions on Microwave Theory and Techniques,2009,57(8): 2015-2020.
[104]Y.X. Mukherjee, S. Mukherjee. The boundary node method for potential problems[J]. International Journal for Numerical Methods in Engineering,1997, 40(5):797-815.
[105]G.R. Liu, Y.T. Gu. A point interpolation method for two-dimensional solids[J]. International Journal for Numerical Methods in Engineering,2001,50:973-951.
[106]Y. Zhang, K.R. Shao, Y.G. Guang et al. A Boundary Meshless Method for Transient Eddy Current Problems[J]. IEEE Transactions on Magnetics,2005,41(10): 4090-4092.
[107]Y. Zhang, K.R. Shao, Y.G. Guang et al. A Comparison of Point Interpolative Boundary Meshless Method Based on PBF and RBF for Transient Eddy Current Analysis[J]. IEEE Transactions on Magnetics,2007,43(4):1497-1500.
[108]H.K. Kim, K.Y. Park, D.W. Kim et al. Electromagnetic field analysis using the point collocation method based on the FMLSRK approximation[C]. Electrical Machines and Systems,2003(ICEMS 2003),2003,2:751-753.
[109]N.R. Aluru. A point collocation method for meshless analysis of microelectronic and micro electromechanical devices[C]. Extended Abstracts of 1998 Sixth International Workshop on Computational Electronics,1998:54-57.
[110]M.J. Mitchell, R. Qiao, N.R. Aluru. Meshless analysis of steady-state electro-osmotic transport[J]. Journal of Microelectromechanical Systems,2000,9(4): 435-449.
[111]M. Sharan, E.J. Kansa, S. Gupta. Applications of the mutiquadric method for solution of elliptic partial differential equations[J]. Computers & Mathematics with Applications,1997,84:275-302.
[112]Z.M. Wu, R. Schaback. Local error estimates for radial basis function interpolation of scattered data. M. Sharan, E.J. Kansa, S. Gupta. Applications of the mutiquadric method for solution of elliptic partial differential equations [J]. IMA Journal of Numerical Analysis,1993,13(1):13-27.
[113]C. Franke, R. Schaback. Soving partial differential equations by collocation using radial basis functions[J]. Computers & Mathematics with Applications,1998, 93:73-82.
[114]Z.M. Wu. Compactly supported positive definite radial functions[J]. Advances in Computational Mathematics,1995,4:283-292.
[115]G.E. Fasshauer. Solving partial differential equations by collocation with radial basis functions[C]. Proceedings of Chamonix,1996,26(4):1-8.
[116]X. Zhang, K.Z. Song, M.W. Lu, el al. Meshless methods based on collocation with radial basis functions [J]. Computational Mechanics,2000,26(4):333-343.
[117]A.I. Fedoseyev, M.J. Friedman, E.J. Kansa. Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary [J]. Computers & Mathematics with Applications,2002,43:439-455.
[118]R.K. Beatson, J.B. Cherrie, C.T. Mouat. Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration[J]. Advances in Computational Mathematics,1999,11:253-270.
[119]L. Ling, E.J. Kansa. A least-squares preconditioner for radial basis functions collocation methods [J]. Advances in Computational Mathematics,2005,23:31-54.
[120]D. Brown, L. Ling, E.J. Kansa et al. On approximate cardinal preconditioning methods for solving PDEs with radial basis functions[J]. Engineering Analysis with Boundary Elements,2005,29:343-343.
[121]M.R. Dubal. Domain decomposition and local refinement for multiquadric approximations[J]. Appl. Sci. Comput.,1994,1:146-171.
[122]A.S.M. Wong, Y.C. Hon, T.S. Li et al. Multizone decomposition for simulation of time-dependent problems using the multiquadric scheme[J]. Computers and Mathematics with Applications,1999,37:23-43.
[123]R.K. Beatson, W.A. Light, S. Billings. Fast solution of the radial basis function interpolation methods:Domain decomposition methods[J]. SIAM Journal of Scientific Computing,2000,22(5):1717-1740.
[124]E.J. Kansa, Y.C. Hon. Circumventing the ill-conditioning problem with multiquadric radial basis functions:Applications to elliptic partial differential equations[J]. Computers and Mathematics with Applications,2000,39:123-137.
[125]Y.C. Hon, Z.M. Wu. Additive Schwarz domain decomposition with radial basis approximation[J]. Internat. J. Appl. Math.,2002,4:81-98.
[126]X. Zhou, Y.C. Hon, J. Li. Overlapping domain decomposition method by radial basis functions[J]. Applied Numerical Mathematics,2003,44:241-255.
[127]M.S. Ingber, C.S. Chen, J.A. Tanski. A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations [J]. International Journal for Numerical Methods in Engineering,2004,60: 2183-2201.
[128]J. Li, Y.C. Hon. Domain decomposition for radial basis meshless methods[J]. Numerical Methods for Partial Differential Equations,2004,20(3):450-462.
[129]L. Ling, E.J. Kansa. Preconditioning for radial basis functions with domain decomposition methods[J]. Mathematical and Computer Modelling,2004,40: 1413-1427.
[130]Y. Duan, P.F. Tang, T.Z. Huang et al. Coupling projection domain decomposition method and Kansa's method in electrostatic problems [J]. Computer Physics Communications,2009,180:209-214.
[131]Y.C. Hon, Z.M. Wu. A quasi-interpolation method for solving ordinary differential equations[J]. International Journal for Numerical Methods in Engineering,2000,48: 1187-1197.
[132]Z.M. Wu. Dynamically knots setting in meshless method for solving time dependent propagations equation[J]. Computer Methods in Applied Mechanics and Engineering,2004,193:1221-1229.
[133]R. Chen, Z.M. Wu. Solving partial differential equation by using multiquadric quasi-interpolation[J]. Applied Mathematics and Computation,2007,186: 1502-1510.
[134]E.J. Kansa, R.E. Carlson. Improved accuracy of multiquadric interpolation using variable shape parameters [J]. Computers and Mathematics with Applications,1992, 24(12):99-120.
[135]J. Wertz, E.J. Kansa, L. Ling. The role of the multiquadric shape parameters in solving elliptic partial differential equations[J]. Computers and Mathematics with Applications,2006,51:1335-1348.
[136]张勇.计算电磁学的无单元方法研究[D].武汉,华中科技大学,2006.
[137]张淮清.电磁场计算中的径向基函数无网格法研究[D].重庆,重庆大学,2008.
[138]S.A. Viana, D. Rodger, H.C. Lai. Meshless Multiquadric RBF Method Applied to Two-dimensional Boundary Value Problems[C].12th Biennial IEEE Conference on Electromagnetic Field Computation,2006:359.
[139]Y. Zhang, K.R. Shao, Y.G. Guo et al. An improved multiquadric collocation method for 3-D electromagnetic problems[J]. IEEE Transactions on Magnetics,2007,43(4): 1509-1512.
[140]张淮清,俞集辉.电磁场边值问题求解的径向基函数方法[J].高电压技术,2010,36(2)：531-536.
[141]Y. Zhang, K.R. Shao, Y.G. Guo et al. Multiquadrics collocation method for transient eddy current problems[J]. IEEE Transactions on Magnetics,2006,42(10): 3183-3185.
[142]张淮清,俞集辉.瞬态涡流问题求解的时域径向基函数方法[J].电机与控制学报,2008,12(6)：685-690.
[143]P.L. Jiang, S.Q. Li, C.H. Chan. Analysis of Elliptical Waveguides by a Meshless collocation method with, the Wendland Radial Basis Functions[J]. Microwave and Optical Technology Letters,2002,32(2):162-165.
[144]P. Kowalczyk, M. Mrozowski. Mesh-free approach to Hemholtz equation based on radial basis functions[C],15th International Conference on Microwaves, Radar and Wireless Communications,2004,2(17-19):702-705.
[145]K. Li, Q.B. Huang, L.G. Lin. A meshless method for 3-D wave propagation problems[C].2010 Xvth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED),2010:196-200.
[146]J.H. Yu, H.Q. Zhang. Solving waveguide eigenvalue problem by using radial basis function method[C]. Automation Congress,2008, WAC 2008:1-5.
[147]T. Ishikawa, Y. Tsukui, M. Matsunami. A combined method for the global optimization using radial basis function and deterministic approach[J]. IEEE Transactions on Magnetics,1999,35(3):1730-1733.
[148]S.L. Ho, S.Y. Yang, G.Z. Ni et al. A response surface methodology based on improved compactly supported radial basis function and its application to rapid optimizations of electromagnetic devices[J]. IEEE Transactions on Magnetics,2005, 41(6):2111-2117.
[149]S.L. Ho, S.Y. Yang, G.Z. Ni et al. A fast global optimizer based on improved CS-RBF and stochastic optimal algorithm[J]. IEEE Transactions on Magnetics,2006, 42(4):1175-1178.
[150]G. Lei, G.Y. Yang, K.R. Shao et al. Electromagnetic device design based on RBF models and two new sequential optimization strategies[J]. IEEE Transactions on Magnetics,2010,46(8):3181-3184.
[151]S.L. Ho, S.Y. Yang, H.C. Wong et al. A Meshless Collocation Method Based on Radial Basis Functions and Wavelets[J]. IEEE Transactions on Magnetics,2004, 40(2):1021-1024.
[152]F.G. Guimaraes, R.R. Saldanha, R.C. Mesquita et al. A meshless method for electromagnetic field computation based on the multiquadric technique[J]. IEEE Transactions on Magnetics,2007,43(4):1281-1284.
[153]R.K. Gordon, W.E. Hutchcraft. Using elliptical basis functions in a meshless method to determine electromagnetic fields near material interfaces[C]. IEEE Antennas and Propagation Society International Symposium,2007:3592-3595.
[154]Y. Zhang, G.Y. Yang, K.R. Shao et al. Multiscale combined radial basis function collocation method for eddy currents analysis in high-speed moving conductors[J]. IEEE Transactions on Magnetics,2009,45(10):3973-3976.
[155]W. Zhao, Y. Duan, F. Rong et al. A novel algorithm for transient eddy current problems[C]. International Conference on Microwave and Millimeter Wave Technology (ICMMT),2010:592-594.
[156]S.J. Lai, B.Z. Wang, Y. Duan. Static-field problems solved by RBF-MLM with natural conformal node distribution[C]. International Conference on Microwave and Millimeter Wave Technology (ICMMT),2010:827-830.
[157]谢德馨,宋志明,徐子宏.三维涡流计算中的矢量磁位与标量电位[J].沈阳工业大学学报,1989,11(2)：1-14.
[158]Karl H, Fetzer J, Kurz S, et al. Description of TEAM workshop problem 28: An electrodynamic levitation device [C]. Proc. TEAM Workshop,6th Round, Rio de Janeiro, Brazil,1997.
[159]H.C. Lai, D.Rodger, P.C. Coles. A 3-D overlapping finite-element scheme for modeling movement[J]. IEEE Transactions on Magnetics,2004,40(2):533-536.
[160]Y. Wang, Z.X. Ren. EML technology research in China[J]. IEEE Transactions on Magnetics,1999,35(1):44-46.
[161]李梅武,崔英,薛飞.航母飞机起飞的最佳选择——电磁弹射系统[J].舰船科学技术,2008,30(2)：34-37.
[162]S. Barmada, A. Musolino, M. Raugi et al. Analysis of the performance of a multi-stage pulsed linear induction launcher[J]. IEEE Transactions on Magnetics, 2001,37(1):111-115.
[163]李庆阳,莫孜中,祁力群.非线性方程组的数值解法[M].北京：科学出版社,1987.
[2]杨庆新,刘素珍,陈海燕等.工程电磁场无单元法[M].北京：科学出版社,2008.
[3]尹华杰.无网格法及其在电磁场计算中的应用展望[J].电机与控制学报,2003,7(2)：107-111.
[4]R. Hardy. Multiquadric equations of topography and other irregular surfaces[J]. Journal of Geophysical Research,1971,76(8):1905-1915.
[5]E.J. Kansa. Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—Ⅰ surface approximations and partial derivative estimates[J]. Computers and Mathematics with Applications,1990, 19(8-9):127-145.
[6]E.J. Kansa. Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—Ⅱ solutions to parabolic, hyperbolic and elliptic partial differential equations[J]. Computers and Mathematics with Applications,1990,19(8-9):147-161.
[7]X. Zhang, K.Z. Song, M.W. Lu, el al. Meshless methods based on collocation with radial basis functions[J]. Computational Mechanics,2000,26(4):333-343.
[8]周可定.工程电磁场专论[M].武汉：华中工学院出版社,1986.
[9]冯慈璋,马西奎.工程电磁场导论[M].北京：高等教育出版社,2000.
[10]倪光正,杨仕友,钱秀英等.工程电磁场数值计算[M].北京：机械工业出版社,2004.
[11]刘国强,赵凌志,蒋继娅.Ansoft工程电磁场有限元分析[M].北京：电子工业出版社,2005.
[12]葛德彪,闫玉波.电磁波时域有限差分法[M].西安：西安电子科技大学出版社,2002.
[13]李庆民,刘卫东,钱家骊.电磁推力机构的一种分析方法[J].电工技术学报,2004,19(2)：20-24.
[14]娄杰,李庆民,孙庆森等.快速电磁推力机构的动态特性仿真与优化设计[J].中国电机工程学报,2005,25(16)：23-29.
[15]张惠娟.运动电磁系统涡流场有限元研究[D].天津,河北工业大学,2000.
[16]S. Hahn. An'upwind' finite element method for electromagnetic field problems in moving media[J]. International Journal for Numerical Methods in Engineering,1987, 24(11):2071-2086.
[17]D. Rodger, T. Karaguler, P.J. Leonard. A formulation for 3D moving conductor eddy current problems[J]. IEEE Transactions on Magnetics,1989,25(5):4147-4149.
[18]K. Muramatsu, T. Nakata, N. Takahashi et al. Comparison of coordinate systems for eddy current analysis in moving conductors[J]. IEEE Transactions on Magnetics, 1992,28(2):1186-1190.
[19]K. Yamazaki.3D eddy current formulation for moving conductors with variable velocity of coordinate system using edge finite elements [J]. IEEE Transactions on Magnetics,1999,35(3):1594-1598.
[20]S. Niikura, A. Kameari. Analysis of eddy current and force in conductors with motion[J]. IEEE Transactions on Magnetics,1992,28(2):1450-1453.
[21]M. Jarnienx, D. Grenier, G. Reyne et al. FEM computation of eddy current and force in moving systems application to linear induction launcher [J]. IEEE Transactions on Magnetics,1993,29(2):1989-1992.
[22]B. Davat, Z. Ren, M. Lajoie-Mazenc. The movement in field modeling[J]. IEEE Transactions on Magnetics,1985, MAG-21(6):2296-2298.
[23]Y. Kawase, S. Tatsuoka, T. Yamaguchi.3-D finite element analysis of operation characteristics of AC electromagnetic contactors [J]. IEEE Transactions on Magnetics,1994,30(5):3244-3247.
[24]Y. Kawase, O. Miyatani, T. Yamaguchi. Numerical analysis of dynamic characteristics of electromagnets using 3-D finite element method with edge elements[J]. IEEE Transactions on Magnetics,1994,30(5):3248-3251.
[25]S. Dufour, G. Vinsard, B. Laport.2-D adaptive mesh with movement[J]. IEEE Transactions on Magnetics,2001,37(5):3482-3485.
[26]K. Yamazaki, T. Saeki. Adaptive mesh generation within nonlinear iterative calculations for analyses of electric machines[J]. IEEE Transactions on Magnetics, 2003,39(3):1654-1657.
[27]K. Yamazaki, S. Watari, A. Egawa. Adaptive finite element meshing for eddy current analysis of moving conductor[J]. IEEE Transactions on Magnetics,2004, 40(2):993-996.
[28]孙玉田,杨明,李北芳.电机动态有限元法中的运动问题[J].大电机技术,1997,(6)： 35-39.
[29]P. Lombard, G. Meunier. A general purpose method for electric and magnetic combined problems for 2D, axisymmetry and transient systems[J]. IEEE Transactions on Magnetics,1993,29(2):1737-1740.
[30]P.J. Leonard, H.C. Lai, G. Hainsworth et al. Analysis of performance of tubular pulsed coil induction launchers[J]. IEEE Transactions on Magnetics,1993,29(1): 686-690.
[31]H.C. Lai, P.J. Leonard, D. Rodger et al.3D finite element dynamic simulation of electrical machines coupled to external circuits [J]. IEEE Transactions on Magnetics, 1997,33(2):2010-2013.
[32]严登俊,刘瑞芳,胡敏强等.处理电磁场有限元运动问题的新方法[J].中国电机工程学报,2003,23(8)：163-167.
[33]Y. Zhang, B. Bai, D. Xie. Analysis of the 3D transient electromagnetic field-circuit-movement coupled problem[C]. Automation Congress,2008. WAC 2008.
[34]R. Perrin-Bit, J.L. Coulomb. A three dimensional finite element mesh connection for problems involving movement[J]. IEEE Transactions on Magnetics,1995,31(3): 920-923.
[35]C.S. Biddlecombe, J. Simkin, A.P. Jay et al. Transient electromagnetic analysis coupled to electric circuits and motion[J]. IEEE Transactions on Magnetics,1998, 34(5):3182-3185.
[36]K. Tani, T. Yamada, Y. Kawase. A new technique for 3D dynamic finite element analysis of electromagnetic problems with relative movement[J]. IEEE Transactions on Magnetics,1998,35(3):3371-3374.
[37]K. Tani, T. Yamada, Y. Kawase. Dynamic analysis of linear actuator taking into account eddy currents using finite element method and 3-D mesh coupling method[J]. IEEE Transactions on Magnetics,1999,35(3):1785-1788.
[38]张宇,阮江军,刘兵等.组合网格法在电磁-机械耦合问题中的应用[J].中国电机工程学报,2007,27(33)：42-47.
[39]甘艳,阮江军,张宇等.组合网格法及其在电磁问题中的应用[J].电工技术学报,2008,23(11)：8-14.
[40]P. Ying, J. Ruan, Y. Zhang et al. A composite grid method for moving conductor eddy-current problem[J]. IEEE Transactions on Magnetics,2007,43(7):3259-3265.
[41]Y. Zhang, J. Ruan, Y. Gan et al. Application of a composite grid method in the analysis of 3-D eddy current field involving movement[J]. IEEE Transactions on Magnetics,2008,44(6):1298-1301.
[42]M. Koizumi, T. Takahashi. An integral equation method for analysis of three-dimensional magnetodynamics with moving objects [J]. IEEE Transactions on Magnetics,1993,29(2):1516-1519.
[43]D. Kim, S. Hahn, K. Choi et al. Electromagnetic force analysis of moving conducting slab by using 3D IBIEM[J]. IEEE Transactions on Magnetics,1998, 34(5):2589-2592.
[44]D. Kim, S. Hahn, I. Park et al. Computation of three-dimensional electromagnetic field including moving media by indirect boundary integral equation methodfJ]. IEEE Transactions on Magnetics,1999,35(3):1932-1938.
[45]M. Clemens, T. Weiland, M. Wilke. Transient eddy current formulation including moving conductors using the FI method[J]. IEEE Transactions on Magnetics,2000, 36(4):804-808.
[46]N. Esposito, A. Musolino, M. Raugi. Modelling of three-dimensional nonlinear eddy current problems with conductors in motion by an integral formulation[J]. IEEE Transactions on Magnetics,2002,32(3):764-767.
[47]王洋忠,王炳革,卢斌先等.三维开域涡流场A-V位有限元与边界元耦合分析方法[J].中国电机工程学报,2000,20(5)：1-4.
[48]S. Kurz, J. Fetzer, G. Lehner. Three dimensional transient BEM-FEM coupling analysis of electrodynamic levitation problems[J]. IEEE Transactions on Magnetics, 1996,32(3):1062-1065.
[49]S. Kurz, J. Fetzer, G. Lehner. A novel formalation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM-FEM coupling[J]. IEEE Transactions on Magnetics,1998,34(5):3068-3073.
[50]A. Koski, K. Forsman, T. Tarhasaari et al. Force and torque computations with hybrid methods[J]. IEEE Transactions on Magnetics,1999,35(3):1387-1390.
[51]S. Suuriniemi, K. Forsman, L. Kettunen et al. Computation of eddy currents coupled with motion[J]. IEEE Transactions on Magnetics,2000,36(4):1341-1345.
[52]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[J].中国电机工程学报,2006,26(14)：145-151.
[53]S. Barmada, A. Musolino, M. Raugi. Hybrid FEM/MOM formulation for eddy current problems with moving conductors[J]. IEEE Transactions on Magnetics,2000, 36(4):827-830.
[54]M. Schinnerl, J. Schoberl, M. Kaltenbacher et al. Multigrid method for the three-dimensional simulation of nonlinear magnetomechal systems[J]. IEEE Transactions on Magnetics,2002,38(3):1497-1511.
[55]R.V. Sabariego, J. Gyselinck, C. Geuzaine et al. Application of the fast multipole method to the 2D finite element-boundary element analysis of electromechanical devices[J]. International Journal for Computation and Mathematics in Electrical and Electronic Engineering,2003,22(3):659-673.
[56]R.V. Sabariego, J. Gyselinck, P. Dular et al. Fast multipole acceleration of the hybrid finite-elment/boundary element analysis of 3-D eddy-current problems[J]. IEEE Transactions on Magnetics,2004,40(2):1278-1281.
[57]陶文铨,吴学红,戴艳俊.无网格数值求解方法[J].中国电机工程学报,2010,30(5)：1-10.
[58]L.B. Lucy. A numerical approach to the testing of the fission hypothesis[J]. The Astrophysical Journal,1977,8(12):1013-1024.
[59]J.J. Moraghan. An introduction to SPH[J]. Computer Physics Communications, 1988,48:89-96.
[60]R.A. Gingold, J.J. Moraghan. Smoothed particle hydrodynamics:theory and applications to nonspherical stars[J]. Mon Not Roy Astrou Soc,1977,18:375-389.
[61]G.R. Johnson, S.R. Beissel. Normalized smoothing functions for SPH impact computations [J]. International Journal for Numerical Methods in Engineering,2002, 53:825-843.
[62]J.W. Swegle, D.L. Hicks, S.W. Attaway. Smoothed particle hydrodynamics stability analysis[J]. Journal of Computational Physics,1995,116(1):123-134.
[63]J.K. Chen, J.E. Beraun, C.J. Jih. An improvement for tensile instability in smoothed particle hydrodynamics [J]. Computational Mechanics,1999,23(4):279-287.
[64]C.T. Dyka. Addressing tension instability in SPH methods. Technical Report NRL/MR/6384, NRL,1994.
[65]G. Ala, E. Francomano, A. Tortorici et al. A smoothed particle interpolation scheme for transient electromagnetic simulation[J]. IEEE Transactions on Magnetics,2006, 42(4):647-650.
[66]P. Lancaster, K. Salkauskas. Surfaces generated by moving least square methods[J]. Mathematics of Computation,1981,37(155):141-158.
[67]B. Nayroles, G. Touzot, P. Villon. Generalizing the finite element method:Diffuse approximation and diffuse elements[J]. Computational Mechanics,1992,10(5): 307-318.
[68]T. Belytschko, Y.Y. Liu, L. Gu. Element-free Galerkin methods[J]. International Journal for Numerical Methods in Engineering,1994,37:229-256.
[69]P. Krysl, T. Belytschko. The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks[J]. International Journal for Numerical Methods in Engineering,1999,44(6):767-800.
[70]S. Li, W. Hao, W.K. Liu. Numerical simulations of large deformation of thin shell structures using meshfree methods[J]. Computational Mechanics,2000,25:102-116.
[71]C. Herault, Y. Marechal. Boundary and interface conditions in meshless method[J]. IEEE Transactions on Magnetics,1999,35(3):1450-1453.
[72]V. Cingoski, N. Miyamoto, H. Yamashita. Element-free Galerkin method for electromagnetic field computations[J]. IEEE Transactions on Magnetics,1998,34(5): 3236-3239.
[73]S.L. Ho, S. Yang, J.M. Machado et al. Application of a meshless method in electromagnetics[J]. IEEE Transactions on Magnetics,2001,37(5):3198-3202.
[74]L. Xuan, Z. Zeng, B. Shanker et al. Element-free Galerkin method for static and quasi-static electromagnetic field computation[J]. IEEE Transactions on Magnetics, 2004,40(1):12-20.
[75]L. Xuan, Z. Zeng, B. Shanker et al. Meshless Method for Numerical Modeling of Pulsed Eddy Currents[J]. IEEE Transactions on Magnetics,2004,40(6):3457-3462.
[76]G.F. Parreira, E.J. Silva, A.R. Fonseca et al. The Element-Free Galerkin Method in Three-Dimensional Electromagnetic Problems[J]. IEEE Transactions on Magnetics, 2006,42(4):711-714.
[77]O. Bottauscio, M. Champi, A. Manzin. Element-Free Galerkin Method in Eddy-Current Problems With Ferromagnetic Media[J]. IEEE Transactions on Magnetics,2006,42(4):1577-1584.
[78]A. Manzin, O. Bottauscio. Element-Free Galerkin Method for the Analysis of Electromagnetic-Wave Scattering[J]. IEEE Transactions on Magnetics,2008,44(6): 1366-1369.
[79]Y. Zhang, K.R. Shao, D.X. Xie et al. Meshless Method Based on Orthogonal Basis for Computational Electromagnetics[J].IEEE Transactions on Magnetics,2005, 41(5):1432-1435.
[80]S.Y. Yang, G. Ni, J. Qian et al. Wavelet-Galerkin method for computations of electromagnetics[J]. IEEE Transactions on Magnetics,1998,34(5):2493-2496.
[81]杨仕友,倪光正.小波-伽辽金有限元法及其在电磁场数值计算中的应用[J].中国电机工程学报,1999,19(1)：56-61.
[82]S.Y. Yang, S.L. Ho, P.H. Ni et al. A Combined Wavelet-FE Method for Transient Electromagnetic-Field Computations[J]. IEEE Transactions on Magnetics,2006, 42(4):571-574.
[83]V. Cingoski, N. Miyamoto, H. Yamashita. Hybrid element-free Galerkin-finite element method for electromagnetic field computations [J]. IEEE Transactions on Magnetics,2000,36(4):1543-1547.
[84]S.L. Ho, S.Y. Yang, G.Z. Ni et al. Numerical Analysis of Thin Skin Depths of 3-D Eddy-Current Problems Using a Combination of Finite Element and Meshless Methods[J]. IEEE Transactions on Magnetics,2004,40(2):1354-1357.
[85]W.K. Liu, S. Sun, Y.F. Zhang. Reproducing kernel particle methods[J]. International Journal for Numerical methods in Fluids,1995,20:1081-1106.
[86]W.K. Liu, S. Li, T. Belytschko. Moving least-square reproducing kernel method Part Ⅰ methodology and convergence[J]. Computer Methods in Applied Mechanics and Engineering,1997,143:113-154.
[87]S. Li, W.K. Liu. Moving least-square reproducing kernel method Part Ⅱ Fourier analysis[J]. Computer Methods in Applied Mechanics and Engineering,1996,139: 159-193.
[88]W.K. Liu, Y. Chen, R.A. Uras et al. Generalized multiple scale reproducing kernel particle methods [J]. Computer Methods in Applied Mechanics and Engineering, 1996,139:91-157.
[89]S.A. Viana, R.C. Mesquita. Moving Least Square Reproducing Kernel Method for Electromagnetic Field Computation[J]. IEEE Transactions on Magnetics,1999, 35(3):1372-1375.
[90]N.R. Aluru. A point collocation method for meshless analysis of microelectronic and micro electromechanical devices[J]. Computational Electronics,1998,54-57.
[91]C.A. Durate, J.T. Oden. Hp clouds:a h-p meshless method[J]. Numerical method for partial differential equations,1996,12:673-705.
[92]J.T. Oden, C.A. Durate, O.C. Zienkiewicz. A new could-based hp finite element method[J]. International Journal for Numerical Methods in Engineering,1998,50: 160-170.
[93]T.J. Liszka, C.A. Durate, W.W. Tworzydlo. Hp-meshless cloud method[J]. Computer Methods in Applied Mechanics and Engineering,1996,139:263-288.
[94]T. Zhu, J. Zhang, S.N. Atluri. A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach[J]. Computational Mechanics,1998,21:223-235.
[95]S.N. Atluri, T. Zhu. A new meshless local Petrov-Galerkin(MLPG) approach in computational mechanics[J]. Computational Mechanics,1998,22:117-127.
[96]S.A. Viana, D. Rodger, H.C. Lai. Meshless local Petrov-Galerkin method with radial basis functions applied to electromagnetics [J]. IEE Proceedings Science, Measurement and Technology,2004,151(6):449-451.
[97]W.L. Nicomedes, R.C. Mesquita, F.J.S. Moreira. A local boundary integral equation (LBIE) method in 2D electromagnetic wave scattering, and a meshless discretization approach[C]. Microwave and Optoelectronics Conference (IMOC),2009,133-137.
[98]G.R. Liu, Y.T. Gu. A point interpolation method for two-dimensional solids[J]. International Journal for Numerical Methods in Engineering,2001,50:937-951.
[99]J.G. Wang, G.R. Liu. A point interpolation meshless method based on radial basis functions[J]. International Journal for Numerical Methods in Engineering,2002,54: 1623-1648.
[100]Y.L. Wu, G.R. Liu. A meshJ.G. Wang, G.R. Liu. A meshfree formulation for local radial point interpolation method (LRPIM) for incompressible flow simulation [J]. Computational Mechanics,2003,30:355-365.
[101]S.A. Viana, D. Rodger, H.C. Lai. Application of the Local Radial Point Interpolation Method to Solve Eddy-Current Problems[J]. IEEE Transactions on Magnetics,2006, 42(4):591-594.
[102]S.J. Lai, B.Z. Wang, Y. Duan. Meshless radial basis function method for transient electromagnetic computations[J]. IEEE Transactions on Magnetics,2008,44(10): 2288-2295.
[103]Y. Yu, Z. Chen. A 3-D radial point interpolation method for meshless time-domain modeling[J]. IEEE Transactions on Microwave Theory and Techniques,2009,57(8): 2015-2020.
[104]Y.X. Mukherjee, S. Mukherjee. The boundary node method for potential problems[J]. International Journal for Numerical Methods in Engineering,1997, 40(5):797-815.
[105]G.R. Liu, Y.T. Gu. A point interpolation method for two-dimensional solids[J]. International Journal for Numerical Methods in Engineering,2001,50:973-951.
[106]Y. Zhang, K.R. Shao, Y.G. Guang et al. A Boundary Meshless Method for Transient Eddy Current Problems[J]. IEEE Transactions on Magnetics,2005,41(10): 4090-4092.
[107]Y. Zhang, K.R. Shao, Y.G. Guang et al. A Comparison of Point Interpolative Boundary Meshless Method Based on PBF and RBF for Transient Eddy Current Analysis[J]. IEEE Transactions on Magnetics,2007,43(4):1497-1500.
[108]H.K. Kim, K.Y. Park, D.W. Kim et al. Electromagnetic field analysis using the point collocation method based on the FMLSRK approximation[C]. Electrical Machines and Systems,2003(ICEMS 2003),2003,2:751-753.
[109]N.R. Aluru. A point collocation method for meshless analysis of microelectronic and micro electromechanical devices[C]. Extended Abstracts of 1998 Sixth International Workshop on Computational Electronics,1998:54-57.
[110]M.J. Mitchell, R. Qiao, N.R. Aluru. Meshless analysis of steady-state electro-osmotic transport[J]. Journal of Microelectromechanical Systems,2000,9(4): 435-449.
[111]M. Sharan, E.J. Kansa, S. Gupta. Applications of the mutiquadric method for solution of elliptic partial differential equations[J]. Computers & Mathematics with Applications,1997,84:275-302.
[112]Z.M. Wu, R. Schaback. Local error estimates for radial basis function interpolation of scattered data. M. Sharan, E.J. Kansa, S. Gupta. Applications of the mutiquadric method for solution of elliptic partial differential equations [J]. IMA Journal of Numerical Analysis,1993,13(1):13-27.
[113]C. Franke, R. Schaback. Soving partial differential equations by collocation using radial basis functions[J]. Computers & Mathematics with Applications,1998, 93:73-82.
[114]Z.M. Wu. Compactly supported positive definite radial functions[J]. Advances in Computational Mathematics,1995,4:283-292.
[115]G.E. Fasshauer. Solving partial differential equations by collocation with radial basis functions[C]. Proceedings of Chamonix,1996,26(4):1-8.
[116]X. Zhang, K.Z. Song, M.W. Lu, el al. Meshless methods based on collocation with radial basis functions [J]. Computational Mechanics,2000,26(4):333-343.
[117]A.I. Fedoseyev, M.J. Friedman, E.J. Kansa. Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary [J]. Computers & Mathematics with Applications,2002,43:439-455.
[118]R.K. Beatson, J.B. Cherrie, C.T. Mouat. Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration[J]. Advances in Computational Mathematics,1999,11:253-270.
[119]L. Ling, E.J. Kansa. A least-squares preconditioner for radial basis functions collocation methods [J]. Advances in Computational Mathematics,2005,23:31-54.
[120]D. Brown, L. Ling, E.J. Kansa et al. On approximate cardinal preconditioning methods for solving PDEs with radial basis functions[J]. Engineering Analysis with Boundary Elements,2005,29:343-343.
[121]M.R. Dubal. Domain decomposition and local refinement for multiquadric approximations[J]. Appl. Sci. Comput.,1994,1:146-171.
[122]A.S.M. Wong, Y.C. Hon, T.S. Li et al. Multizone decomposition for simulation of time-dependent problems using the multiquadric scheme[J]. Computers and Mathematics with Applications,1999,37:23-43.
[123]R.K. Beatson, W.A. Light, S. Billings. Fast solution of the radial basis function interpolation methods:Domain decomposition methods[J]. SIAM Journal of Scientific Computing,2000,22(5):1717-1740.
[124]E.J. Kansa, Y.C. Hon. Circumventing the ill-conditioning problem with multiquadric radial basis functions:Applications to elliptic partial differential equations[J]. Computers and Mathematics with Applications,2000,39:123-137.
[125]Y.C. Hon, Z.M. Wu. Additive Schwarz domain decomposition with radial basis approximation[J]. Internat. J. Appl. Math.,2002,4:81-98.
[126]X. Zhou, Y.C. Hon, J. Li. Overlapping domain decomposition method by radial basis functions[J]. Applied Numerical Mathematics,2003,44:241-255.
[127]M.S. Ingber, C.S. Chen, J.A. Tanski. A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations [J]. International Journal for Numerical Methods in Engineering,2004,60: 2183-2201.
[128]J. Li, Y.C. Hon. Domain decomposition for radial basis meshless methods[J]. Numerical Methods for Partial Differential Equations,2004,20(3):450-462.
[129]L. Ling, E.J. Kansa. Preconditioning for radial basis functions with domain decomposition methods[J]. Mathematical and Computer Modelling,2004,40: 1413-1427.
[130]Y. Duan, P.F. Tang, T.Z. Huang et al. Coupling projection domain decomposition method and Kansa's method in electrostatic problems [J]. Computer Physics Communications,2009,180:209-214.
[131]Y.C. Hon, Z.M. Wu. A quasi-interpolation method for solving ordinary differential equations[J]. International Journal for Numerical Methods in Engineering,2000,48: 1187-1197.
[132]Z.M. Wu. Dynamically knots setting in meshless method for solving time dependent propagations equation[J]. Computer Methods in Applied Mechanics and Engineering,2004,193:1221-1229.
[133]R. Chen, Z.M. Wu. Solving partial differential equation by using multiquadric quasi-interpolation[J]. Applied Mathematics and Computation,2007,186: 1502-1510.
[134]E.J. Kansa, R.E. Carlson. Improved accuracy of multiquadric interpolation using variable shape parameters [J]. Computers and Mathematics with Applications,1992, 24(12):99-120.
[135]J. Wertz, E.J. Kansa, L. Ling. The role of the multiquadric shape parameters in solving elliptic partial differential equations[J]. Computers and Mathematics with Applications,2006,51:1335-1348.
[136]张勇.计算电磁学的无单元方法研究[D].武汉,华中科技大学,2006.
[137]张淮清.电磁场计算中的径向基函数无网格法研究[D].重庆,重庆大学,2008.
[138]S.A. Viana, D. Rodger, H.C. Lai. Meshless Multiquadric RBF Method Applied to Two-dimensional Boundary Value Problems[C].12th Biennial IEEE Conference on Electromagnetic Field Computation,2006:359.
[139]Y. Zhang, K.R. Shao, Y.G. Guo et al. An improved multiquadric collocation method for 3-D electromagnetic problems[J]. IEEE Transactions on Magnetics,2007,43(4): 1509-1512.
[140]张淮清,俞集辉.电磁场边值问题求解的径向基函数方法[J].高电压技术,2010,36(2)：531-536.
[141]Y. Zhang, K.R. Shao, Y.G. Guo et al. Multiquadrics collocation method for transient eddy current problems[J]. IEEE Transactions on Magnetics,2006,42(10): 3183-3185.
[142]张淮清,俞集辉.瞬态涡流问题求解的时域径向基函数方法[J].电机与控制学报,2008,12(6)：685-690.
[143]P.L. Jiang, S.Q. Li, C.H. Chan. Analysis of Elliptical Waveguides by a Meshless collocation method with, the Wendland Radial Basis Functions[J]. Microwave and Optical Technology Letters,2002,32(2):162-165.
[144]P. Kowalczyk, M. Mrozowski. Mesh-free approach to Hemholtz equation based on radial basis functions[C],15th International Conference on Microwaves, Radar and Wireless Communications,2004,2(17-19):702-705.
[145]K. Li, Q.B. Huang, L.G. Lin. A meshless method for 3-D wave propagation problems[C].2010 Xvth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED),2010:196-200.
[146]J.H. Yu, H.Q. Zhang. Solving waveguide eigenvalue problem by using radial basis function method[C]. Automation Congress,2008, WAC 2008:1-5.
[147]T. Ishikawa, Y. Tsukui, M. Matsunami. A combined method for the global optimization using radial basis function and deterministic approach[J]. IEEE Transactions on Magnetics,1999,35(3):1730-1733.
[148]S.L. Ho, S.Y. Yang, G.Z. Ni et al. A response surface methodology based on improved compactly supported radial basis function and its application to rapid optimizations of electromagnetic devices[J]. IEEE Transactions on Magnetics,2005, 41(6):2111-2117.
[149]S.L. Ho, S.Y. Yang, G.Z. Ni et al. A fast global optimizer based on improved CS-RBF and stochastic optimal algorithm[J]. IEEE Transactions on Magnetics,2006, 42(4):1175-1178.
[150]G. Lei, G.Y. Yang, K.R. Shao et al. Electromagnetic device design based on RBF models and two new sequential optimization strategies[J]. IEEE Transactions on Magnetics,2010,46(8):3181-3184.
[151]S.L. Ho, S.Y. Yang, H.C. Wong et al. A Meshless Collocation Method Based on Radial Basis Functions and Wavelets[J]. IEEE Transactions on Magnetics,2004, 40(2):1021-1024.
[152]F.G. Guimaraes, R.R. Saldanha, R.C. Mesquita et al. A meshless method for electromagnetic field computation based on the multiquadric technique[J]. IEEE Transactions on Magnetics,2007,43(4):1281-1284.
[153]R.K. Gordon, W.E. Hutchcraft. Using elliptical basis functions in a meshless method to determine electromagnetic fields near material interfaces[C]. IEEE Antennas and Propagation Society International Symposium,2007:3592-3595.
[154]Y. Zhang, G.Y. Yang, K.R. Shao et al. Multiscale combined radial basis function collocation method for eddy currents analysis in high-speed moving conductors[J]. IEEE Transactions on Magnetics,2009,45(10):3973-3976.
[155]W. Zhao, Y. Duan, F. Rong et al. A novel algorithm for transient eddy current problems[C]. International Conference on Microwave and Millimeter Wave Technology (ICMMT),2010:592-594.
[156]S.J. Lai, B.Z. Wang, Y. Duan. Static-field problems solved by RBF-MLM with natural conformal node distribution[C]. International Conference on Microwave and Millimeter Wave Technology (ICMMT),2010:827-830.
[157]谢德馨,宋志明,徐子宏.三维涡流计算中的矢量磁位与标量电位[J].沈阳工业大学学报,1989,11(2)：1-14.
[158]Karl H, Fetzer J, Kurz S, et al. Description of TEAM workshop problem 28: An electrodynamic levitation device [C]. Proc. TEAM Workshop,6th Round, Rio de Janeiro, Brazil,1997.
[159]H.C. Lai, D.Rodger, P.C. Coles. A 3-D overlapping finite-element scheme for modeling movement[J]. IEEE Transactions on Magnetics,2004,40(2):533-536.
[160]Y. Wang, Z.X. Ren. EML technology research in China[J]. IEEE Transactions on Magnetics,1999,35(1):44-46.
[161]李梅武,崔英,薛飞.航母飞机起飞的最佳选择——电磁弹射系统[J].舰船科学技术,2008,30(2)：34-37.
[162]S. Barmada, A. Musolino, M. Raugi et al. Analysis of the performance of a multi-stage pulsed linear induction launcher[J]. IEEE Transactions on Magnetics, 2001,37(1):111-115.
[163]李庆阳,莫孜中,祁力群.非线性方程组的数值解法[M].北京：科学出版社,1987.