A MULTILEVEL FAST MULTIPOLE METHOD WITH SPHERICAL HARMONICS EXPANSION OF THE k-SPACE INTEGRALS

A novel diagonalized multilevel fast multipole method (MLFMM) for time harmonic Maxwell’s equations is proposed which realizes considerable memory savings without compromising accuracy and computational speed. The improvement is achieved by expanding the k-space integrals over the unit sphere in spherical harmonics. Since the spherical harmonics expansion is carried out for cartesian vector components of the expansion functions, only very few expansion coefficients are required. Aggregations, plane wave translations, and disaggregations are performed 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. INTRODUCTION Method of moments (MoM) solutions of integral equations are among the most successful numerical methods for the solution of electromagnetics radiation and scattering problems. Most important is its robustness and insensitivity against dispersion errors. The drawback of large computation complexity due to the fully-populated system matrices has been overcome since the introduction of fast integral methods such as the MLFMM [1][2][3]. In this paper, we concentrate on diagonalized FMM algorithms using propagating plane waves. These are typically based on numerical integration of the corresponding k-space integrals over the Ewald sphere. The quadrature sampling rate is dictated by the spectral content of the diagonalized translation operator and this results in oversampling of the k-space representations of the basis/testing functions of the method of moments (MoM) procedure employed to solve the integral equation [4]. To obtain a computationally efficient algorithm, the k-space samples of the basis/testing functions should be pre-computed and stored in core memory of the computer prior to starting the iteration procedure. Due to the aforementioned oversampling, a large amount of core memory is wasted and considerable memory savings can be achieved by using a more adequate k-space representation of the basis/testing functions. In this paper, the concept of spherical harmonics expansions of the k-space integrals motivated by an efficient k-space representation of the basis/testing functions is worked out. It is shown how a computationally efficient MLFMM algorithm can be realized leading to considerable memory reductions as compared to previous implementations. FORMULATION We consider a time harmonic (time dependence e ) surface integral equation formulation using the electric field integral equation (EFIE) for metallic objects. However, the presented ideas can be applied to MFIE and combined field integral equation (CFIE) formulations for metallic and dielectric objects as well. According to [2][5], a Galerkin-type MoM equation system can be derived as