A diagonalized multilevel fast multipole method with spherical harmonics expansion of the k-space Integrals

Diagonalization of the fast multipole method (FMM) for the Helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves. The resulting k-space integral over the Ewald sphere is numerically evaluated. Storing the k-space quadrature samples of the method...

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Bibliographic Details
Published in:IEEE transactions on antennas and propagation 2005-02, Vol.53 (2), p.814-817
Main Author: Eibert, T.F.
Format: Article
Language:English
Subjects:
Applied sciences
Electromagnetic radiation
Electromagnetic scattering
Exact sciences and technology
Fast integral methods
fast multipole methods
Integral equations
Iterative methods
Moment methods
Radar scattering
Radiocommunications
Radiowave propagation
Redundancy
Robustness
Sampling methods
Studies
Telecommunications
Telecommunications and information theory
Testing
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Summary:Diagonalization of the fast multipole method (FMM) for the Helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves. The resulting k-space integral over the Ewald sphere is numerically evaluated. Storing the k-space quadrature samples of the method of moments (MoM) basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems. In this paper, it is proposed to expand the k-space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules. Aggregations, plane wave translations, and disaggregations in the realized multilevel fast multipole method (MLFMM) are carried out using the k-space samples of a numerical quadrature rule. However, the incoming plane waves on the finest MLFMM level are expanded in spherical harmonics again. Thus, due to the orthonormality of spherical harmonics, the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients. Overall, the resulting MLFMM can save a considerable amount of memory without compromising accuracy and numerical speed.
ISSN:0018-926X
1558-2221
DOI:10.1109/TAP.2004.841310