自然单元法在电磁场数值计算中的应用
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摘要
在电磁场数值分析中,有限元法(Finite Element Method,FEM)已经得到了成功而广泛的应用,但是在遇到自适应计算或电磁场设计的逆问题时,FEM前处理或网格划分过程复杂且费时费力。自然单元法(Natural Element Method,NEM)可以有效的克服上述问题,它只需节点信息而不需单元或网格信息,可以完全的摆脱有限元法中前处理网格的麻烦,更有利于实现电磁场的自适应分析。本文主要研究NEM在处理电磁场问题方面的可行性。
首先,本文阐述并深入研究了基于有限元法的电磁场数值分析方法,建立了基于Galerkin及Petrov-Galerkin的有限元法的系统平衡方程。
其次,本文从基于有限元的电磁场数值分析入手,结合电磁场的有关理论和自然单元法的基本原理,对基于自然单元法的电磁场数值分析进行公式推导。本文构造的自然邻接插值函数具有δ特性且在电磁场凸域边界上的相邻点之间是严格线性的,这使得自然单元法能方便地施加电磁场的本质边界条件,也可方便地处理其边界面。由于基于Galerkin方法形成的系统平衡方程,需要在背景三角积分网格里采用3个以上的积分点完成系统平衡方程的数值积分,使得计算过程复杂,计算时间长,因而本文采用了基于Petrov-Galerkin的方法形成系统平衡方程。该方法的积分能直接在子域上完成,不需要额外的背景积分网格,且只在三角形积分网格的中心采用一个积分点进行数值积分,从而简化了计算过程,缩短了计算时间。
最后,本文以具有解析解的电磁场问题为例,编写相应的Fortran程序,利用自然单元法对算例进行电磁场数值计算。从算例结果可得:在相同的条件下,自然单元法比有限元法的精度高且更加的接近于解析解;自然单元法可以用较少的剖分单元达到有限元法较多剖分单元的精度。从而本文成功地将自然单元法引入到了电磁场数值分析中。
The Finite Element Method (FEM) has been widely used in the numerical calculation of electromagnetic field, but on the hand of adaptive calculation or inverse problem of electromagnetic field design, the FEM pre-processing or mesh generation cost long time and more works. The Natural Element Method (NEM) can solve these issues effectively, it needs only the points information but not the cell or grid information, totally avoiding the troubles of pre-processing grid in FEM, it is convenient to realize the adaptive analysis of electromagnetic field. This paper mainly research on the feasibility of NEM in processing electromagnetic field.
Firstly, this paper deeply researches on the electromagnetic field numerical calculation methods based on FEM, it also describes the establishment of the FEM system equilibrium equations based on the Galerkin and Petrov-Galerkin.
Secondly, starting from the electromagnetic field numerical calculation based on FEM, combined with the relative theories of electromagnetic field and NEM, this paper deduces the functions in electromagnetic field numerical calculation based on NEM. The natural neighbor interpolation function is constructed with theδcharacteristic, and it is strictly linear on the border of the points in the adjacent, so the NEM can easily impose the essential boundary conditions, also can easily deal with its boundary surface. The system equilibrium equation based on the Galerkin needs three or more integral pitch winding points in background triangle integral grid to accomplish numerical integration, so this makes the calculation process complicated and long time consumed, so this paper uses the system equilibrium equation based on Petrov-Galerkin. The integral of this method can be finished directly in sub-domain, no additional background integral grid, and the central of triangle integral grid only needs one point to accomplish numerical integral, simplifying the calculation process and saving the time.
Finally, this paper takes the example of the electromagnetic field issues with linear analytical solution, compiles corresponding Fortran programs, uses NEM to accomplish electromagnetic field numerical calculation. According to the example result, the conclusion is available: compared with the FEM, the NEM has high precision and more close to analytical solution, the NEM can reach high precision with fewer units. Therefore, this paper successfully brings the NEM to electromagnetic field numerical calculation.
首先,本文阐述并深入研究了基于有限元法的电磁场数值分析方法,建立了基于Galerkin及Petrov-Galerkin的有限元法的系统平衡方程。
其次,本文从基于有限元的电磁场数值分析入手,结合电磁场的有关理论和自然单元法的基本原理,对基于自然单元法的电磁场数值分析进行公式推导。本文构造的自然邻接插值函数具有δ特性且在电磁场凸域边界上的相邻点之间是严格线性的,这使得自然单元法能方便地施加电磁场的本质边界条件,也可方便地处理其边界面。由于基于Galerkin方法形成的系统平衡方程,需要在背景三角积分网格里采用3个以上的积分点完成系统平衡方程的数值积分,使得计算过程复杂,计算时间长,因而本文采用了基于Petrov-Galerkin的方法形成系统平衡方程。该方法的积分能直接在子域上完成,不需要额外的背景积分网格,且只在三角形积分网格的中心采用一个积分点进行数值积分,从而简化了计算过程,缩短了计算时间。
最后,本文以具有解析解的电磁场问题为例,编写相应的Fortran程序,利用自然单元法对算例进行电磁场数值计算。从算例结果可得:在相同的条件下,自然单元法比有限元法的精度高且更加的接近于解析解;自然单元法可以用较少的剖分单元达到有限元法较多剖分单元的精度。从而本文成功地将自然单元法引入到了电磁场数值分析中。
The Finite Element Method (FEM) has been widely used in the numerical calculation of electromagnetic field, but on the hand of adaptive calculation or inverse problem of electromagnetic field design, the FEM pre-processing or mesh generation cost long time and more works. The Natural Element Method (NEM) can solve these issues effectively, it needs only the points information but not the cell or grid information, totally avoiding the troubles of pre-processing grid in FEM, it is convenient to realize the adaptive analysis of electromagnetic field. This paper mainly research on the feasibility of NEM in processing electromagnetic field.
Firstly, this paper deeply researches on the electromagnetic field numerical calculation methods based on FEM, it also describes the establishment of the FEM system equilibrium equations based on the Galerkin and Petrov-Galerkin.
Secondly, starting from the electromagnetic field numerical calculation based on FEM, combined with the relative theories of electromagnetic field and NEM, this paper deduces the functions in electromagnetic field numerical calculation based on NEM. The natural neighbor interpolation function is constructed with theδcharacteristic, and it is strictly linear on the border of the points in the adjacent, so the NEM can easily impose the essential boundary conditions, also can easily deal with its boundary surface. The system equilibrium equation based on the Galerkin needs three or more integral pitch winding points in background triangle integral grid to accomplish numerical integration, so this makes the calculation process complicated and long time consumed, so this paper uses the system equilibrium equation based on Petrov-Galerkin. The integral of this method can be finished directly in sub-domain, no additional background integral grid, and the central of triangle integral grid only needs one point to accomplish numerical integral, simplifying the calculation process and saving the time.
Finally, this paper takes the example of the electromagnetic field issues with linear analytical solution, compiles corresponding Fortran programs, uses NEM to accomplish electromagnetic field numerical calculation. According to the example result, the conclusion is available: compared with the FEM, the NEM has high precision and more close to analytical solution, the NEM can reach high precision with fewer units. Therefore, this paper successfully brings the NEM to electromagnetic field numerical calculation.
引文
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[2] X. Zhang, K. Z. Song, M. W Lu, X. Liu, Least-square collocation meshless method[J]. Int. J. Numer Meth Engrg., 2001, 51: 1089-1100
[3] V. Cingoski, N. Miyamoto, H. Yamashita, Element-free Galerkin method for electromagnetic field co- mputations[J]. IEEE Trans. Mag., 1998, 34: 3236-3239
[4] A. V. Simone, C. M. Renato, Moving least square reproducing kernel method for electromagnetic field computation[J]. IEEE Tans. Mag., 1999, 35(3): 1372-1375
[5] V. Cingoski, N. Miyamoto, H. Yamashita, Hybrid element-free Galerkin-finite element method for electromagnetic field computations[J]. IEEE. Trans. Mag., 2000, 36: 1543-1547
[6] S. Y Yang, G Z. Ni, J. R. Cardoso, S. L. Ho, J. M. Machado, A combined wavelet-element free Galerkin method for numerical calculation of electromagnetic fields[J]. IEEE Trans. Mag., 2003, 39: 1413-1416
[7]刘素贞,杨庆新,陈海燕,寇晓东.无单元法在电磁场数值计算中的应用研究[J] .电工技术学报,2001, 16:3 0-33
[8]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: 107-111
[9] Sibson R.A vector identity for the Dirichlet tessellations in the plane[J]. The computer journal. 1978, 21(2): 168-173
[10] Jean B,Malcolm S.A numerical method for solving partial differential equation on highly irregular evolving grids[J] . Narure, 1995, 376: 655-660
[11] Sukumar N,Moran B,Semenov A Y et al. Natural neighbour Galerkin methods[J] . Int J Numer Meth Engrg, 2001, 50: 1-27
[12] Sukumar N,Moran B, Belytschko T. The natural element method in solid mechanics[J] , Int J Numer Methods Engrg. 1998, 43: 839-887
[13] Belikov VV,Ivanov VD, Kontorovich VK et al. The non-Sibsonian interpolation:a new method of interpolation of the values of a function on an arbitrary set of points[J] . Comput Mathematics and Mathematical Physics, 1997, 37(1): 9-15
[14]蔡永昌,朱合华,王建华.基于Voronoi结构的无网格局部Petrov-Galerkin方法[J] .力学学报, 2003(35),2 1:87-193
[15]朱怀球,吴启航.一种基于Vorono cells的C∞插值基函数及其在流体力学中的若干应用[J] .北京大学学报(自然科学版), 2001, 37(5): 669-678
[16] Yoo Jeong Wahn, Moran Brian,Chen Jiun Shyan.Stabilized conforming nodal integration in the natural-element method[J] .Int J Numer Methods Engrg, 2004, 60: 861-890
[17] Zhou J X,Wen J B,Zhang H Y et al.A nodal interpolation and post-processing technique based on Voronoi diagram for Galerkin meshless methods[J].Comput Methods Appl Mech Engrg,2003, 192: 3831-3843
[18] Sukumar N,Moran B,Belytschko T. The natural element method in solid mechanics[J]. International Journal for Numerical Methods in Engineering,1998,43: 839–887
[19] N.Sukumar.Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids[J]. Department ofcivil and environmental engineering University of California,Davis, CA95616, U.S.A
[20] Armando Duarte C.A review of some meshless methods to solve partial differential equations[J].TIAM Report 95-106
[21]秦伶俐,黄文彬.无网格方法在结构计算上的改进[D].中国农业大学,2005
[22]曹国金,姜弘道.无单元法研究和应用现状及动态[J] .力学进展,2002,32(4) 5:26-534
[23] Braun J,Sambridge M.A numerical method for sovling partial differential equations on highly irregular evolveing grids[J].Nature 1995,376:655-660
[24]刘金义,刘爽.Voronoi图应用综述[J] .工程图学学报,2004,2 1:25-132
[25] Sukumar N.Sibson and non. Sibsonian interpolants for elliptic partial differential equations [A] In: Proceeding of First MIT Conference on Computational Fluid and Solid Mechanics[C].[s.l.]: [s.n.],2001,1665-1667
[26] AtluriSN,Zhu T.New Concepts in Meshless Methods[J].International Journal for Numerical Methods in Engineering,2000,47(1.3):5 37-556
[27]戴斌.自然单元法算法与应用[M] .上海:同济大学,2003
[28] N.SukumarandB.Moran. Natural Neighbor Interpolant for Partial Differential Equations[J],Department Of Civil Engineering Northwestern University Evanston,February,1999
[29]于国辉,李永鑫,董万杰.Voronoi图和它的偶图Delaunay镶嵌在数值分析中的应用[J] .中国科技信息,2005,15(8):1 4-15
[30] Sukumar N, Moran B, Belytschko T. The nature element method in solid mechanics [J].Int.J.Num. meth. Eng.,1998, (43):8 39-887
[31]朱合华,杨宝红,蔡永昌等.无网格自然单元法在弹塑性分析中的应用[J].岩土力学,2004,25(4): 671-674
[32] O. C.监凯维奇.有限元法[M].科学出版社,1985
[33]骆少明,蔡永昌,张湘伟.面向对象的无网格伽辽金法.机械工程学报,2000, 36 (10):2 3-26
[34]龙述尧,许敬晓.弹性力学问题的局部边界积分方程方法[J].力学学报,2000,32(5):5 66-578
[35]王凯,周慎杰,单国骏.基于局部Petrov-Galerkin离散方案的无网格法[J].计算力学学报,2006,23(5): 518-523
[36]龙述尧.弹性力学问题的局部Petrov-Galerkin方法[J].力学学报,2001,23(4):5 08-518
[37]熊渊博,崔洪雪,龙述尧.大变形问题分析的局部Petrov-Galerkin法[J].计算力学学报,2009,26(3):353-357
[38]盛剑霓.电磁场数值分析[M].科学出版社,1984
[39] [美]海特编.工程电磁学[M] .徐世安,周乐柱译.科学出版社,2004
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