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

• Published in: IEEE transactions on antennas and propagation Vol. 53; no. 2; pp. 814 - 817
• Main Author: Eibert, T.F.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-02-2005; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• 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; Diagonalization; Radiation scattering; iterative methods; Redundancy; Moment method; Helmholtz equation; Iterative method; Algorithm; Harmonic function; Integral method; Index Terms-Fast integral methods; integral equations; Accuracy; Electromagnetic wave propagation; Plane wave; Spherical harmonic; Integral equation; fast multipole methods; Sampling; Multilevel system
• ISSN: 0018-926X; 1558-2221
• DOI: 10.1109/TAP.2004.841310

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.