Carried Out Based On Spectral Domain, Spherical Harmonic Multilevel Fast Multipole Algorithm Research

Research on electromagnetic scattering of complicated target is a challenged problem, which has been paid close attention to by scientists and engineers on electromagnetic field. And the research has come into a new stage with the progress of computer technique and the appearance of many efficient numerical methods. In recent years, more and more researchers plough into this field according to the appearance of method of moment, which is an numerical method on solving integral equation, and the appearance of fast multipole method (FMM) along with its expansion. How to reduce the numerical calculating cost and storage has become the main objective that the researchers pursuit during their research process. This paper is also on this purpose.Firstly, this paper pays attention on fast multipole method (FMM) and its expansion, with the name of multilevel fast multipole method (MLFMM), which are both used on solving the problem of electromagnetic scattering of complicated target. FMM accelerates the iteration on matrix and reduces the storage and computation cost to O( N^1.5) by converting the coupling impedance elements in far area into aggregation factor, translation factor and deploying factor, and the factors are stored respectively. What's more, MLFMM reduces the storage and computation cost to O( N logN) with the multilevel structures. Nevertheless, the storage and computation cost are too big to solve the electromagnetic scattering of large electrical size targets.This paper focuses on multilevel fast multipole method with spherical harmonics expansion of the k-space integral, which divided the three factors (i.e. aggregation factor, translation factor and deploying factor) in traditional fast multipole method into three parts (i.e. deploying, aggregation translation and aggregation). This improvement leads to the conversion of the storage with aggregation factor and deploying factor of the k-space integral sample into the storage of the eigenvalue extracted. Because the number of eigenvalue is far smaller than that of the sample, the amount of memory is saved considerably. This paper also proposes the analysis of the algorithm, and draws the conclusion that this method, with a appropriate level, can improve the iteration speed of the matrix in a great level without compromising accuracy. The numerical example proves the conclusion.In order to improve the iteration speed of multilevel fast multipole method with spherical harmonics expansion of the k-space integral according to the numerical performance of the translation factor, this paper proposed a new algorithm on fast far field approximation multilevel fast multipole with spherical harmonics expansion of the k-space integral (SE-FAFFA-MLFMM). The numerical result and the algorithm analysis indicate the high performance on accelerating matrix iteration.