Implementation of a broadband multilevel fast multipole algorithm for multiscale electromagnetics problems

Abstract

Fast multipole method (FMM) in computational physics and its multilevel version, i.e., multilevel fast multipole algorithm (MLFMA) in computational electromagnetics are among the best known methods to solve integral equations (IEs) in the frequency-domain. MLFMA is well-accepted in the computational electromagnetic (CEM) society since it provides a full-wave solution regarding Helmholtz-type electromagnetics problems. This is done by discretizing proper integral equations based on a predetermined formulation and solving them numerically with O(N logN) complexity, where N is the number of unknowns. In this dissertation, we present two broadband and efficient methods in the context of MLFMA, one for surface integral equations (SIEs) and another for volume integral equations (VIEs), both of which are capable of handling large multiscale electromagnetics problems with a wide dynamic range of mesh sizes. By invoking a novel concept of incomplete-leaf tree structures, where only the overcrowded boxes are divided into smaller ones for a given population threshold, a versatile method for both types of IEs has been achieved. Regarding SIEs, for geometries containing highly overmeshed local regions, the proposed method is always more efficient than the conventional MLFMA for the same accuracy, while it is always more accurate if the efficiency is comparable. Regarding VIEs, for inhomogeneous dielectric objects possessing variable mesh sizes due to different electrical properties, in addition to obtaining similar results from the proposed method, a reduction in the storage is also achieved. Several canonical and also real-life examples are provided to demonstrate the superior efficiency and accuracy of the proposed algorithm in comparison to the conventional MLFMA.