【作者】 赵柳；

【导师】 朱峰；

【作者基本信息】 西南交通大学 ， 电工理论与新技术， 2004， 硕士

【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。更多还原

【Abstract】 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.更多还原