Short Course: Near-fields in Computational Electromagnetics: A Novel Regularization Technique

Presented by

Alireza Baghai-Wadji - University of Cape Town, Department of Electrical Engineering

Abstract

Method of Moments (MoMs), in its originally conceived form, or in any of its more sophisticated advanced variations, continues to be one of the most powerful numerical techniques in computational electromagnetics available today. The steps involved are well understood and the accuracies achievable are impressive. Since the involved system matrices are dense, several powerful strategies have been proposed to remedy this deficiency without compromising the distinguishing features of the MoMs substantially. The application of the MoMs involves choosing appropriate basis- and weighting functions, constructing dyadic Green's functions, and regularizing weak-, strong- or hyper-strong singularities. The latter step becomes considerably more challenging when simulating microwave device characteristics in multi-scale multi-physics environments.

This course focuses on the regularization of dyadic Green's functions and the near-field phenomena and features the following computationally relevant topics in great detail: (i) Basis- and weighting functions will be reviewed by introducing general notions of functional spaces and their duals. Wavelets and dual wavelets, as well as frames and dual frames will be introduced in terms of illustrative examples. A brief discourse on joint time-frequency transforms and multi-resolution analysis will be offered to further clarify modern notions of analysis- and synthesis functions, based on the overarching concept of the resolution of identity. These considerations naturally pave the way for constructing customized basis- and weighting functions. Optimally-localized smooth functions including Wannier functions will be discussed. (ii) Maxwell's equations will be transformed into an equivalent diagonalized form with its utmost favourable properties. Diagonalization is amenable to symbolic computation, and its realization follows a recipe with a few simple steps only. Following the recipe, Maxwell's equations in free space, anisotropic-, and bi-anisotropic media can be handled with comparable simplicity. Diagonalization opens the door to a myriad of powerful techniques for investigating the nature of Green's functions' singularities, and their systematic and automatic regularization. (iii) It will be demonstrated how basis- and weighting functions can be constructed from the asymptotic tails of dyadic Green's functions in spectral domain. (iv) The culmination of the course is the introduction of a recently-developed regularization technique, which will be applied to dyadic Green's functions - in closed as well as numerical forms. A wealth of carefully-chosen examples from electro- and magneto-statics, electrodynamics, photonics and plasmonics in two- and three-dimensional isotropic, anisotropic, and bi-anisotropic (dispersive) media will be presented.

A comprehensive manuscript, well over 250 pages, will be made available to course participants.