Fast Direct Solution Of Electromagnetic Field Integral Equation

Many aspects of modern scientific research are closely linked with the target object of the electromagnetic scattering and radiation characteristics such as microstrip antennas, high-speed PCB board, microwave circuits and millimeter-wave integrated circuits analysis design, geological exploration, target detection, environmental monitoring, and many other applications. Especially with the development of computer technology, computational electromagnetics has become the study of electromagnetic scattering and radiation characteristics of an important interdisciplinary. In computational electromagnetics, the discretization of the electromagnetic integral equations generally leads to a dense system of linear equations, and it is prohibitively expensive for both CPU time and memory to solve the resulting equation system for electrically large problems, if the conventional methods such as Gaussian elimination is utilized. In view of this, the study on the fast algorithms has become one of popular research areas in the field of computational electromagnetics. This paper did some research work on low rank compression and decomposition method, fast direct inverse matrix method. Main jobs are as follows:Firstly, based on the method of moments(MoM), we present an introduction to the adaptive cross approximation algorithm (ACA), its basic concepts and detailed implementation process. In ACA algorithm, the rank deficiency of non-diagonal blocks of matrix system is fully utilized. Base on this feature, all the the non-diagonal blocks can be represented as the products of two low-rank matrix, and thus, much less number of matrix elements are calculated and less memory is required during the matrix filling process. The numerical examples are presented to verify that ACA really reduces the space and time complexity of matrix, and also the error is controllable.Secondly, base on the representation we get from the ACA, a fast direct inverse method is proposed:1. In ACA algorithm, the products of two low-rank matrix are utilized to approximate non-diagonal matrix blocks. So the non-diagonal blocks of the coefficient matrix obtained by ACA have the form:Aη≈UηVη,(i≠j), Uη、Vη are low rank matrix. Further, split and operation with Uη、Vη will eventually convert Aη into this form:Aη1≈LiMijRj.2.we got the matrix inversion formula of A:A-1=A(I-LSD C). Because all L, S, R are low rank matrices, the computational complexity will be greatly reduced when the above formula is applied to an right hand side vector of linear system. Finally, Matlab codes are implemented and applied to several standard structures, which do verify the correctness and efficiency of our algorithm.