A Study On Fast Algorithms With Degenerate Kernels For Electromagnetic Scattering Problems

It has been an important way to obtain electromagnetic scattering characteristics of targets with numerical methods, which can be used to model and simulate the targets of arbitrary shape. Among these methods, the method of moments (MoM) is rigorius, and widely employed. However, the generalized impedance matrix given by the moment-method is dense and full populated, which result in high time and memory consumption in an iterative solver. In order to alleviate this problem, quite a few fast algorithms have been developed, in which the key point is to efficiently represent the far-field submatrices of well-separated blocks. The degenerate kernel technique is versatile technique to compress the submatrices.Based on the hierarchical matrix (H matrix) framework, the degenerate kernel functions used to approximate the Green's function is studied in this dissertation, which are adopted to form fast solvers of electromagnetic scattering problems. The main contents and contributions are as follows:1. A fast algorithm to find the common edges for RWG basis function with linear complexities is proposed. The basic idea is to obtain the common edges by comparing vertex indices of the adjacent triangles. The computational complexity is reduced from O(N2) to O(N), where N is the number of the triangular elements. The proposed algorithm can be easily integrated into universal simulation tools and solvers. Moreover, it is extended to find the shared facets of tetrahedral elements successfully.2. The degenerate kernel functions constructed with Lagrange polynomials and radial basis function (RBF) are derived to approximate the Green's function, which is a oscillatory kernel. The accuracy and effiency are compared, and less interpolation points are needed for radial basis function to achieve a given accuracy. For the PEC scattering problems, a near-surface interpolation scheme is proposed, in which only the points near the surface of the target is employed. Compared with the conventional 3-D uniformly grid, the number of interpolation points is reduced from O(k3) to O(k2), where k is the number of interpolation points on one dimension. Hence the computational efficiency is improved, while good accuracy can be maintained.3. Based on hierarchical bases hierarchical matrix framework, a fast algorithm which combines uniform and near-surface interpolation grids is proposed for solving PEC electromagnetic scattering problems. Interpolation degenerate kernel with Gauss radial basis function (RBF) is used to approximate far-field interaction matrices. In order to take advantage of both grids, uniform grid is employed for smaller blocks, while near-surface grid is used for bigger blocks. Compared with the algorithm where only uniform grid is employed, both the computational time and memory storage of the proposed method are reduced.4. A novel exponential degenerate kernel function is proposed, which can separate the interaction between the source and field points and provide a diagonal form of the translation matrix. Compared with the multipole expansion, less expansion terms are needed for the given accuracy for simulations. Hence, the exponential degenerate kernel can be used to accelerate the matrix-vector products in an iteration solver. Furthermore, a new scheme of direction vector on the unit sphere is presented, in which the expansion terms are further decreased to half of the multipole expansion, and increase slower with the increase of the cube sizes than the multipole expansion. Numerical results verify the effectiveness of the proposed method.