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Author ORCID Identifier

https://orcid.org/0009-0003-9329-1459

AccessType

Open Access Dissertation

Document Type

dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Electrical and Computer Engineering

Year Degree Awarded

2023

Month Degree Awarded

May

First Advisor

Marinos N. Vouvakis

Subject Categories

Computational Engineering | Other Electrical and Computer Engineering

Abstract

Maxwell’s equations and the laws of Electromagnetics (EM) govern a plethora of electrical, optical phenomena with applications on wireless, cellular, communications, medical and computer hardware technologies to name a few. A major contributor to the technological progress in these areas has been due to the development of simulation and design tools that enable engineers and scientists to model, analyze and predict the EM interactions in their systems of interest. At the core of such tools is the field of Computational Electromagnetics (CEM), which studies the solution of Maxwell’s equations with the aid of computers. The advances in these applications technologies, in return, demand increasingly more efficient and accurate CEM methods. Among the many CEM methodologies that are currently in broad use, the Boundary Element Method (BEM) or surface Method of Moments (MoM), is perhaps the most popular in solving electrically large or electrically small multi-layered structures. In BEM, the surfaces of conductors and dielectrics are discretized to triangular or quadrilateral elements and the equivalent currents on them are convolved with the appropriate Green’s function at all observations on the mesh to produce a fully populated impedance matrix to be solved with an appropriate excitation. The reliability, accuracy and speed of BEM, among others, critically depends on the method used to perform the singular four-dimensional convolution integrals between source and observation surface currents through a Green’s function, that exhibits a singularity when observation and source elements touch or overlap. Large literature has been devoted in addressing this important issue, and methods involving using singularity subtraction, cancellation or even full 4D integral evaluations. Each of these approaches offer certain advantages, but they tend to require thousands of (often complicated) function evaluations for a single impedance matrix singular integration, it is noted that a typical problem may involve tens or hundreds of millions of such singular integrations. In this dissertation, an unconventional approach of calculating all weakly singular and near weakly singular integrals, encountered in the BEM solution of the Electric Field Integral Equation (EFIE), as well as near singular integrals encountered in the BEM solution of the Magnetic Field Integral Equation (MFIE) in flat triangular meshes, is presented. Instead of specialized integration rules such as singularity subtraction or cancellation, universal look-up-tables and multi-dimensional interpolation are used. Firstly, frequency independent integral expressions, equivalent to the original EFIE-BEM, MFIE-BEM element matrix expressions are derived, in order to facilitate the construction of said universal look-up-tables of integrals. The domain of these functions is discretized by hp refinement, i.e., the size, h and approximation order, p, of the interpolation elements of the entire interpolation domain can be varied independently. Because of the high-dimensional nature of the interpolation domain, from three dimensional to six dimensional, the interpolation over each element is performed with either sparse grids or low-rank tensor train approximations. The integrals are pre-computed into the tables using a state-of-the-art singularity subtraction method at maximum accuracy. Consequently, during run-time, these tables are loaded and any arbitrary singular integral is recovered by multi-dimensional interpolation. The method is compared to a state-of-the-art singularity subtraction technique for the lowest order Rao-Wilton-Glisson (RWG) basis functions in various PEC flat triangular meshes. For EFIE common triangle, weakly singular, in accuracy, while offering over 150× speed-ups. Similarly for EFIE common edge, near weakly singular, interactions it shows about 50× speed-ups but at a somewhat lower, yet acceptable, accuracy. The tensor decomposition approach improves the accuracy to the level of the state-of-the-art and offers about 20× speed-ups, while it also has a controllable accuracy and speed. Lastly, for MFIE common edge, near hyper singular, interactions accuracy is improved by 1 − 2 decimal digits, while offering 20× speed-ups. For a typical BEM run using the single level fast multiple method (FMM) accelerator, the end-to-end set-up time speed improvement with the proposed approach is 15 − 20%.

DOI

https://doi.org/10.7275/34114903

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