Author ORCID Identifier

https://orcid.org/0000-0003-3103-3510

Date Available

5-2-2025

Year of Publication

2025

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Engineering

Department/School/Program

Electrical and Computer Engineering

Faculty

Dr. John C. Young

Faculty

Dr. Daniel Lau

Abstract

Methods like the Method of Moments (MoM) or the locally-corrected Nyström (LCN) method are employed to discretize and solve electromagnetic integral equations. This process results in large, dense systems of linear equations that must be solved. In many cases, the elements of the system matrix can be computed analytically or approximated with high-order numerical methods. In this thesis, various approaches are presented to improve the accuracy and efficiency of integral equation solutions.

The second chapter derives a modified form of the low-rank matrix approximation algorithm known as the adaptive cross approximation (ACA). The original ACA has been observed to lose error control for some complicated low-rank matrix structures. The modified algorithm resolves these issues in most cases by compressing a weighted-average form of the original matrix. The original matrix can easily be recovered from the averaged matrix. The modified ACA maintains controlled accuracy more consistently and accelerates electromagnetic simulations by compressing submatrices associated with sufficiently displaced source-field cell geometries. The compression algorithms may be embedded in a variety of high-order integral equation methods.

The third and fourth chapters derive analytic forms for certain integrals that occur in integral equation simulations. The third chapter covers two-dimensional integrals with natural logarithm kernels over curvilinear source domains. The fourth chapter comprises a set of integrals over linear triangular domains with singular and hyper-singular kernels. Some of the integrals were derived previously, but the literature lacks a thorough analysis of their accuracy. The remaining integrals have not been derived previously and are presented here within.

The fifth chapter presents mixed-order, divergence-conforming pyramidal basis functions for the locally-corrected Nyström method. The use of the LCN method with pyramidal elements has not been published in the literature. First- and second-order cell geometries are discussed. Zero- and first-order divergence-conforming bases are provided. A set of specialized quadrature rules optimized for the LCN method are presented. The method is thoroughly characterized for accuracy.

The sixth chapter discusses some simple approaches for solving integral equations with machine learning models. Supervised and unsupervised learning is performed with fully connected neural networks with dense layers. A training procedure is presented, and the model predicted solutions are compared with analytic reference solutions.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2025.130

Funding Information

This work was supported between 2021-2024 by the Department of Education’s Graduate Assistance in Areas of National Need Fellowship Program. It was also supported between 2024-2025 by Office of Naval Research Grant N00014-21-1-2599.

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