On the hybridization of dipole moment (DM) and finite methods for efficient solution of multiscale problems

Recently, the solution of multiscale problems that are not only large, but contain fine features as well, has emerged as one of the key areas in Computational Electromagnetics that present a considerable challenge to us. Some examples of such problems are: RFID sensors mounted on complex platforms; nanowire antennas placed close to a relatively large size object; metamaterials containing inclusions with fine features; and scatterers with finite but small thicknesses and/or with narrow cracks or slits. Despite concerted and prolonged efforts by a large group of researchers, as well as by commercial software developersboth of whom have dedicated their attention to the solution of multiscale problemsit appears that a satisfactory resolution of these problems has remained quite elusive, except when we resort to a brute force approach. The latter approach obviously requires extensive computational resources, both in terms of CPU time and memory; yet, the solution it yields can still be considerably inaccurate when the feature size is very small compared to the wavelength. One might argue that considerable progress has been made in recent years toward the handling of CEM problems characterized by a very large number of DoFs (degrees of freedom), often upward of millions for MoM (Method of Moments) problems, and even exceeding billions for some Finite Methodsfor example the FDTD. However, a close examination reveals that although the problems analyzed in the past have often been physically largesometimes tens of wavelengths in sizetheir shapes have been relatively simple and devoid of fine features; furthermore, their material characteristics have typically been assumed to be homogeneous and perfectly conducting. In real world problems, the objects of interest often have fine features that are small compared to the wavelength. We then face a challenge and even run into a bottleneck when dealing with these multiscale problems because of the heavy burden they place on both the CPU time and memory when conventional Finite Methods, such as the FEM or the FDTD, are used for their analysis. These methods also have to handle mesh generation problems when modeling objects with fine features, which exacerbate the situation even further. Furthermore, the system matrix becomes ill-conditioned if the FEM is used for multiscale problems, and there is no systematic way to find a pre-conditioner that fixes the problem. In addition, the low-frequency problem renders the ill-conditioning situation even worse than it would be with just the multiscale problem alone at higher frequencies. Attempting to circumvent these difficulties by going to the MoM does not help us escape these difficulties altogether either. This is because the MoM matrix becomes highly illconditioned when dealing with objectssuch as thin shells or nanowiresespecially when their material characteristics are lossy, as is typically the case in practice.