用于电磁散射的矩量法择基问题的群方法
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摘要
传统的寻求基函数的方法是通过解波动方程，分离变量来实现的。这种方法目前为止，适用范围非常有限。对于对称程度较低或复杂对称边界情形，分离变量的方法是无能为力的。目前在工程电磁场领域，由于经典解析方法的这种局限性，加之计算机的速度和容量的不断发展，人们常常致力于数值方法的研究。
矩量法(MoM)，就是这些数值方法中的一种。它是通过引入基函数与权函数的方法，离散积分方程为矩阵方程的方式来进行求解的。由此可见选择基函数与权函数是MoM的重要环节。尤其是基函数，从理论上讲，有许多组函数可供选择，而实际上极少数的基函数对给定的问题是适当的；基函数选择不当，矩阵方程就不会收敛，出现病态特征。如何根据具体给定的边界问题，确定出所适应的基函数(这一类问题称为矩量法的择基问题)，目前还没有定法，有些还要靠一些先验知识和实践经验。
群论作为近代数学的一个重要分支，在物理、化学等许多领域得到了越来越广泛的应用。群论的一个特点就是对于具有复杂对称结构的物体给予描述上的简捷；而其另外一个特点就是能够根据对称性，寻找出适定基函数。
正是基于上述情况，如何按照散射体的对称结构，通过群论方法系统地研究、寻找适合给定边界情况的基函数，并将其用于矩量法，是本论文的主要工作。例如，对于球对称物体的求解问题，根据波动方程通过分离变量法所找出的基函数为球谐函数，这一球谐函数完全可以通过群论的方法得到，本文中对此有详细的阐述。此外，本文针对具有一定对称特征的二维物理边界(以四方柱为例)，给出了详尽的群方法寻基过程：先通过散射体的对称特征建立一个对称变换群，然后以群的特征标分解方法，利用类特征标的正交关系，把复合对称群分解成基群的直和，再根据所选取的对称位置的点函数，在对称变换过程中，利用置换叠加，找出点函数的变化曲线；最后根据变化曲线找出对应解析函数，再进行约化、归一，从而最终形成满足对称特征的基函数。结果表明，群论所寻基函数用于矩量法时，具有收敛速度较快的特点。本文最后还给出了群方法应用于电磁散射的展望。
西南交通大学硕士研究生学位论文第日页
把群论用于电磁应用领域，属于全新的开创性事业，本文在这方面进
行了初步而有价值的探索。为完善矩量法的核心环节和矩量法的进一步广
泛应用，提供坚实的理论依据，具有非常重要的理论意义和应用价值。
The traditional way to find basis functions of EM problems is to solve wave equations by separating variables, which is helpless for the problems with less symmetrical or complicated symmetrical boundaries. Therefore in EM engineering field, people always commit to research numerical methods along with the development of computer science.
The method of moments (MoM) is one of the powerful numerical techniques, whose basic principle is to convert the integral equation with a given boundary-value problem into a matrix equation by using basis functions and testing ones. The matrix equation can be solved by digital computer. It is obvious that how to choose basis and testing functions is a key point in MoM, especially the choice of basis functions. There are many sets of basis functions in theory, but few are suitable. And unsuitable basis functions can not lead to convergence of the matrix equation. The proper basis functions can not be obtained by rules and sometimes we need a lot of experience.
However, as an important branch of mathematics, group theory is applied broadly in many fields including physics and chemistry. It is noted by simple descriptions on complicated symmetrical boundaries and it can find classified basis functions by symmetrical transformations.
Therefore, in this thesis how to use the group method to obtain the suitable basis functions of MoM in EM scattering problems is our main job. For example, we can get the same result of spherical harmonic basis functions both by separating variables and group theory given in this paper. Besides, according to the symmetrical structure of 2D scattering boundary, first we can have a group including all the symmetric transformations, and divide the group into direct sum of representations by orthogonal decomposition of characters. And then, with the curve changes of point function, the orthonormal basis function satisfied with the symmetric boundary can be attained. Finally we put it into use in MoM. The results indicate that the harmonic basis functions from group theory. can lead to fast solution. In conclusion, we give some expectations of group theory connected with EM scattering problems.
This paper has made a beginning and valuable probe for applying the group theory to electromagnetic scattering field for getting the basis functions,
which reinforces the kernel part of MoM, and provides a theoretic support to the appliances of MoM. So it has the most significance.
矩量法(MoM)，就是这些数值方法中的一种。它是通过引入基函数与权函数的方法，离散积分方程为矩阵方程的方式来进行求解的。由此可见选择基函数与权函数是MoM的重要环节。尤其是基函数，从理论上讲，有许多组函数可供选择，而实际上极少数的基函数对给定的问题是适当的；基函数选择不当，矩阵方程就不会收敛，出现病态特征。如何根据具体给定的边界问题，确定出所适应的基函数(这一类问题称为矩量法的择基问题)，目前还没有定法，有些还要靠一些先验知识和实践经验。
群论作为近代数学的一个重要分支，在物理、化学等许多领域得到了越来越广泛的应用。群论的一个特点就是对于具有复杂对称结构的物体给予描述上的简捷；而其另外一个特点就是能够根据对称性，寻找出适定基函数。
正是基于上述情况，如何按照散射体的对称结构，通过群论方法系统地研究、寻找适合给定边界情况的基函数，并将其用于矩量法，是本论文的主要工作。例如，对于球对称物体的求解问题，根据波动方程通过分离变量法所找出的基函数为球谐函数，这一球谐函数完全可以通过群论的方法得到，本文中对此有详细的阐述。此外，本文针对具有一定对称特征的二维物理边界(以四方柱为例)，给出了详尽的群方法寻基过程：先通过散射体的对称特征建立一个对称变换群，然后以群的特征标分解方法，利用类特征标的正交关系，把复合对称群分解成基群的直和，再根据所选取的对称位置的点函数，在对称变换过程中，利用置换叠加，找出点函数的变化曲线；最后根据变化曲线找出对应解析函数，再进行约化、归一，从而最终形成满足对称特征的基函数。结果表明，群论所寻基函数用于矩量法时，具有收敛速度较快的特点。本文最后还给出了群方法应用于电磁散射的展望。
西南交通大学硕士研究生学位论文第日页
把群论用于电磁应用领域，属于全新的开创性事业，本文在这方面进
行了初步而有价值的探索。为完善矩量法的核心环节和矩量法的进一步广
泛应用，提供坚实的理论依据，具有非常重要的理论意义和应用价值。
The traditional way to find basis functions of EM problems is to solve wave equations by separating variables, which is helpless for the problems with less symmetrical or complicated symmetrical boundaries. Therefore in EM engineering field, people always commit to research numerical methods along with the development of computer science.
The method of moments (MoM) is one of the powerful numerical techniques, whose basic principle is to convert the integral equation with a given boundary-value problem into a matrix equation by using basis functions and testing ones. The matrix equation can be solved by digital computer. It is obvious that how to choose basis and testing functions is a key point in MoM, especially the choice of basis functions. There are many sets of basis functions in theory, but few are suitable. And unsuitable basis functions can not lead to convergence of the matrix equation. The proper basis functions can not be obtained by rules and sometimes we need a lot of experience.
However, as an important branch of mathematics, group theory is applied broadly in many fields including physics and chemistry. It is noted by simple descriptions on complicated symmetrical boundaries and it can find classified basis functions by symmetrical transformations.
Therefore, in this thesis how to use the group method to obtain the suitable basis functions of MoM in EM scattering problems is our main job. For example, we can get the same result of spherical harmonic basis functions both by separating variables and group theory given in this paper. Besides, according to the symmetrical structure of 2D scattering boundary, first we can have a group including all the symmetric transformations, and divide the group into direct sum of representations by orthogonal decomposition of characters. And then, with the curve changes of point function, the orthonormal basis function satisfied with the symmetric boundary can be attained. Finally we put it into use in MoM. The results indicate that the harmonic basis functions from group theory. can lead to fast solution. In conclusion, we give some expectations of group theory connected with EM scattering problems.
This paper has made a beginning and valuable probe for applying the group theory to electromagnetic scattering field for getting the basis functions,
which reinforces the kernel part of MoM, and provides a theoretic support to the appliances of MoM. So it has the most significance.
引文
[1] J.A.Stratton《Electromagnetic Theory》 McGraw-Hill Book Co., New York 1941
[2] 符果行 《经典电磁理论方法》 电子科技大学出版社 1998：108～160
[3] 王志良，任伟 《电磁散射理论》 四川科学技术出版社 1993
[4] 王秉中 《计算电磁学》科学出版社2002
[5] 汪茂光 《几何绕射理论》西北电讯工程学院出版社1987
[6] M.A.Morgen "Finite Element and Finite Difference Methods in Electromagnetic Scattering" PIER.2, Elsevier, New York 1989
[7] 李忠元 《电磁场边界元法》北京工业出版社 1988
[8] P. C. Waterman "Matrix Formulation of Electromagnetic Scattering" PIEEE 53,1965:805～812
[9] Akira Ishimaru "'Electromagnetic Wave Propagation, Radiation and Scattering" Physical Review D, Vol.3,No.4:356～361
[10] R.F.Harrington "Field Computations By Moment Methods" MaeMillan, New York 1968
[11] 李世智 《电磁辐射与散射问题的矩量法》 电子工业出版社 1985
[12] 朱峰“用于电磁散射的T-MATRIX方法及其相关的减元理论”西南交通大学博士论文 1997：9
[13] 倪光正 《电磁场计算方法》 高等教育出版社 1997
[14] 龚中麟，徐承和 《近代电磁理论》 高等教育出版社 1992
[15] 张钧《电磁场边值问题的积分方程解法》高等教育出版社1989：151～170
[16] A.J.Poggio and E.K.Miller "Integral equation solutions of three dimensional scattering problems" Computer Techniques for Electromagnetics Chap.4, Elmsford. New York 1973
[17] N.Morita N. Kumagai and J. R. Mautz "Integral Equation Methods for Electromagnetics" Norwood, MA Artech House 1990
[18] C.T.Tai "Direct integration of field equations" IEEE APS Int. Symp. Dig., Vol.2 Atlanta, Georgia 1998:884
[19] 杨儒贵 《电磁理论中的辅助函数》 高等教育出版社 1989：122～146
[20] 傅君眉，冯恩信 《高等电磁理论》 西安交通大学出版社 2000：62～103
[21] 杨儒贵 《电磁定理和原理及其应用》 西南交通大学出版社1989：23～30
[22] J.M.Jin, J.L.Volakis, and V.V.Liepa "a Moment method solution of the volume-surface integral equation using isoparametric elements and pointmatching" IEEE Trans. Microwave Theory Tech., vol.37 1989:1641～1645
[23] J.R.Mautz and R.F.Harrington "H-field, E-field, and combined-field solutions for conducting body of revolution" AEU. Vol.32 1978
[24] S.M.Tao and D.R.Wilton "E-field, H-field, and combined-field solutions for arbitrarily shaped three-dimensional dielectric bodies" Electromagn., vol.10, no.4 1990
[25] Y.Chang and R.F.Harrington "A surface formulation for characteristic modes of material bodies" IEEE Trans. Attennas Propagat., vol.251977
[26] T.K.Wu and L.L.Tsai "Scattering from arbitrarily-shaped lossy dielectric bodies of revolution" Radio Sci., vol. 12 1977
[27] J.R.Mautz and R.F. Harrington "Electromagnetic scattering from a homogeneous material body of revolution" AEU, vol.33 1979
[28] D.E.Livesay and K.M.Chen "Electromagnetic fields induced inside arbitrarily shaped biological bodies" IEEE Trans. Microwave Theory Tech., vol.22 1974
[29] D.H.Schaubert, D.R.Wilton and A.W.Glisson "A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies" IEEE Trans.Attennas Propagat., vol.32 1984
[30] M.Rao, D.R.Wilton and A.W.Glisson "Electromagnetic scattering by surfaces of arbitrary shape" IEEE Trans. Attennas Propagat., vol.301982
[31] G.Kang, J.M.Song, W.C.Chew, K.Donepudi and J.M.Jin "A novel gridrobust higher-order vector basis function for the method of moments" IEEE Trans.Attennas Propagat., vol.30 1982
[32] A.F.Peterson, S.L.Ray and R.Mittra "Computational Methods for
Electromagnetics" New York: IEEE Press, 1997
[33] 洪伟，章文勋 《矩量法的误差分析》GTD与MM讨论会论文 1984
[34] E.Wigner "Group Theory and Its Application to the Quantum Mechamcs of Atomic Spectra" Academic Press, New York 1959
[35] A.I.Solomon "Group Theory of Superfluidity", J. Math. Phy., 1971
[36] C.J.Bradley and A.P.Crachnell "The Mathematical theory of symmetry in Solids" Oxford: Clarendon Press 1972
[37] G. Burns and A.M. Glazer "Space groups for solid state scientists" Academic Press, New York 1978
[38] B.G. Wybourne "Classical groups for physicists" Wiley, New York1974
[39] 陶瑞宝《物理学中的群论》上海科学技术出版社 1986
[40] 高崇寿《群论及其在粒子物理学中的应用》高等教育出版社1992
[43] 马中骐，戴安英 《群论及其在物理中的应用》 北京理工大学出版社1988
[44] A．W．约什 《物理学中的群论基础》 科学出版社 1986：74～80
[45] 马中骐 《物理学中的群论》 科学出版社 1998：99～106
[46] 陈金全 《群表示论的新途径》上海科学技术出版社 1984：309～327
[47] 梁昆淼 《数学物理方程》 高等教育出版社，1998：273
[48] 金建铭 《电磁场有限元方法》 西安电子科技大学出版社 1998
[49] 张玉胜，汪文秉 《改进的FDTD法分析研究无限长导体柱棱边上TM散射之间的相互影响》电波科学学报 第9卷第4期 1994：17～22
[50] 杨儒贵《高等电磁理论讲义》2001：58
[2] 符果行 《经典电磁理论方法》 电子科技大学出版社 1998：108～160
[3] 王志良，任伟 《电磁散射理论》 四川科学技术出版社 1993
[4] 王秉中 《计算电磁学》科学出版社2002
[5] 汪茂光 《几何绕射理论》西北电讯工程学院出版社1987
[6] M.A.Morgen "Finite Element and Finite Difference Methods in Electromagnetic Scattering" PIER.2, Elsevier, New York 1989
[7] 李忠元 《电磁场边界元法》北京工业出版社 1988
[8] P. C. Waterman "Matrix Formulation of Electromagnetic Scattering" PIEEE 53,1965:805～812
[9] Akira Ishimaru "'Electromagnetic Wave Propagation, Radiation and Scattering" Physical Review D, Vol.3,No.4:356～361
[10] R.F.Harrington "Field Computations By Moment Methods" MaeMillan, New York 1968
[11] 李世智 《电磁辐射与散射问题的矩量法》 电子工业出版社 1985
[12] 朱峰“用于电磁散射的T-MATRIX方法及其相关的减元理论”西南交通大学博士论文 1997：9
[13] 倪光正 《电磁场计算方法》 高等教育出版社 1997
[14] 龚中麟，徐承和 《近代电磁理论》 高等教育出版社 1992
[15] 张钧《电磁场边值问题的积分方程解法》高等教育出版社1989：151～170
[16] A.J.Poggio and E.K.Miller "Integral equation solutions of three dimensional scattering problems" Computer Techniques for Electromagnetics Chap.4, Elmsford. New York 1973
[17] N.Morita N. Kumagai and J. R. Mautz "Integral Equation Methods for Electromagnetics" Norwood, MA Artech House 1990
[18] C.T.Tai "Direct integration of field equations" IEEE APS Int. Symp. Dig., Vol.2 Atlanta, Georgia 1998:884
[19] 杨儒贵 《电磁理论中的辅助函数》 高等教育出版社 1989：122～146
[20] 傅君眉，冯恩信 《高等电磁理论》 西安交通大学出版社 2000：62～103
[21] 杨儒贵 《电磁定理和原理及其应用》 西南交通大学出版社1989：23～30
[22] J.M.Jin, J.L.Volakis, and V.V.Liepa "a Moment method solution of the volume-surface integral equation using isoparametric elements and pointmatching" IEEE Trans. Microwave Theory Tech., vol.37 1989:1641～1645
[23] J.R.Mautz and R.F.Harrington "H-field, E-field, and combined-field solutions for conducting body of revolution" AEU. Vol.32 1978
[24] S.M.Tao and D.R.Wilton "E-field, H-field, and combined-field solutions for arbitrarily shaped three-dimensional dielectric bodies" Electromagn., vol.10, no.4 1990
[25] Y.Chang and R.F.Harrington "A surface formulation for characteristic modes of material bodies" IEEE Trans. Attennas Propagat., vol.251977
[26] T.K.Wu and L.L.Tsai "Scattering from arbitrarily-shaped lossy dielectric bodies of revolution" Radio Sci., vol. 12 1977
[27] J.R.Mautz and R.F. Harrington "Electromagnetic scattering from a homogeneous material body of revolution" AEU, vol.33 1979
[28] D.E.Livesay and K.M.Chen "Electromagnetic fields induced inside arbitrarily shaped biological bodies" IEEE Trans. Microwave Theory Tech., vol.22 1974
[29] D.H.Schaubert, D.R.Wilton and A.W.Glisson "A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies" IEEE Trans.Attennas Propagat., vol.32 1984
[30] M.Rao, D.R.Wilton and A.W.Glisson "Electromagnetic scattering by surfaces of arbitrary shape" IEEE Trans. Attennas Propagat., vol.301982
[31] G.Kang, J.M.Song, W.C.Chew, K.Donepudi and J.M.Jin "A novel gridrobust higher-order vector basis function for the method of moments" IEEE Trans.Attennas Propagat., vol.30 1982
[32] A.F.Peterson, S.L.Ray and R.Mittra "Computational Methods for
Electromagnetics" New York: IEEE Press, 1997
[33] 洪伟，章文勋 《矩量法的误差分析》GTD与MM讨论会论文 1984
[34] E.Wigner "Group Theory and Its Application to the Quantum Mechamcs of Atomic Spectra" Academic Press, New York 1959
[35] A.I.Solomon "Group Theory of Superfluidity", J. Math. Phy., 1971
[36] C.J.Bradley and A.P.Crachnell "The Mathematical theory of symmetry in Solids" Oxford: Clarendon Press 1972
[37] G. Burns and A.M. Glazer "Space groups for solid state scientists" Academic Press, New York 1978
[38] B.G. Wybourne "Classical groups for physicists" Wiley, New York1974
[39] 陶瑞宝《物理学中的群论》上海科学技术出版社 1986
[40] 高崇寿《群论及其在粒子物理学中的应用》高等教育出版社1992
[43] 马中骐，戴安英 《群论及其在物理中的应用》 北京理工大学出版社1988
[44] A．W．约什 《物理学中的群论基础》 科学出版社 1986：74～80
[45] 马中骐 《物理学中的群论》 科学出版社 1998：99～106
[46] 陈金全 《群表示论的新途径》上海科学技术出版社 1984：309～327
[47] 梁昆淼 《数学物理方程》 高等教育出版社，1998：273
[48] 金建铭 《电磁场有限元方法》 西安电子科技大学出版社 1998
[49] 张玉胜，汪文秉 《改进的FDTD法分析研究无限长导体柱棱边上TM散射之间的相互影响》电波科学学报 第9卷第4期 1994：17～22
[50] 杨儒贵《高等电磁理论讲义》2001：58