A Study On Degenerated Kernel Iterative Method For Solving Electromagnetic Field Integral Equations

For fast solving scattering and radiation problems of high frequency or electrically large target,a lot of fast numerical algorithms are developed. The fast multipole algorithm for the momentmethod involves the idea of degenerated kernel approach of the original integral kernel function, forlower orders of computational and storage complexities. In addition, the hierarchical matrixprovides new techniques for degenerated kernel with the separation of the field and sourcecoordinates, with which the far zone blocks matrices can be reexpressed by low-rank approximation,for the fast solving. Since the total amount of calculations related to the number of iterations, theconvergence should be accelerated for the computational efficiency when the H-matrix technique isemployed for solving the specific type of integral equation.In this thesis, the H-matrix is employed for the degenerated integral kernel function of theelectric filed integral equations. The equations can be iteratively solved with fast matrix-vectorproduct by use of the separation of the field and source coordinates. Firstly, the target of rectangularbounding box is reordered for the H-matrix structure by bintree theory. Secondly, Lagrangeinterpolation technique is used for the degenerated the kernel function of the integral equation. Thefar blocks are approximated with low-rank matrices. Finally, some scattering problems are solvedwith the electric integral equations combined with the conjugate gradient method and generalizedminimum residual method, which are typical solving algorithms, with the preconditioning technique.The diagonal, the block diagonal and nested preconditioning are considered and tested. Thenumerical results show that the single iteration of computational and storage complexities areproportional toO N logN by using the H-matrix. It is concluded that the conjugate gradientmethod and generalized minimum residual method combining with the corresponding (block)diagonal and nested preconditioning, respectively, are the better choices for the test problems.