H-matrix Methods And Its Application For Solving Electromagnetic Problems By Integral Equation

As a main method in computational electromagnetic, integral equation method(IEM) has very important value in the actual electromagnetic engineering. It’sunnecessary to set the truncation boundary and absorbing boundary conditions becauseit satisfies the radiation condition automatically. Compared with the differential method,it has less number of unknowns. But the IEM also faces the challenges including greatCPU and memory consumption, the convergence of the iteration for solving electricallylarge objects, multi-scale targets and complex targets problems. In order to overcomethese challenges, some fast algorithms based on iterative and direct methods have beendeveloped rapidly, among these the domain decomposition method based on IE and theH-matrix method are typical.This paper mainly focus on H-matrix methods and its application for solvingelectromagnetic problems by IE. The impedance matrix can be compressed efficientlyby H-matrix method because of its low rank characteristic, and then the memoryconsumption can be reduced. As an algebra algorithm, H-matrix method does notdepend on detailed integral kernel and basis/testing function, it’s applicable to the IEM.At first, the basic principle of IE is introduced in detail, including the numericalmethods for solving the integral equation, respectively describes the iterative solutionand the direct solution. About the iterative solution, mainly Generalized MinimalResidual (GMRES) and Conjugate Gradient (CG) are introduced. About the directsolution, the LU decomposition and Adaptive Cross Approximation (ACA)compression method are introduced to solve some special problems. Finally, thenumerical example about the composite conducting and dielectric objects solved byvolume and surface integral equation (VSIE) method is given.Secondly, the theory about the H-matrix method and its process are introduced indetail, applied it with the Method of Moment (MoM) for solving the electromagneticproblems. Respectively aim at the perfectly electrical conductor, dielectric body,composite conductor and dielectric object, make a comparison on the memorycompression ratio of H-matrix combined with surface integral equation(SIE), volume integral equation(VIE) and volume and surface integral equation(VSIE), concluding thatthe H-matrix can reflect its advantage more better when combine with VIE.Thirdly, make a research on the high effective method of integrating H-matrixmethod with Multilevel Fast Multipole Algorithm (MLFMA). H-matrix method is usedto replace the traditional method of moments to compute the impedance matrix and thematrix-vector multiplication of the near field, further to reduce the memoryconsumption and improve the computational efficiency. Then make an investigation onthe accuracy and efficiency of the hybrid method. Based on the hybrid method,combined it with the VSIE to solve some composite conducting and dielectric objects,such as random, coated missile etc.In addition, the sparse approximate inverse (SAI) pre-conditioner technique isapplied in order to improve the convergence of matrix equation. Through H-matrixmethod to realize the multi-layer compression of the matrix, the multilevel sparseapproximate inverse (MLSAI) pre-conditioner technique is applied here to highlyimprove the matrix condition, accelerate the iteration solution. Some examples aregiven to demonstrate its efficiency.