| 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. |
