Abstract: The Finite Element Method (FEM) is a versatile numerical tool for simulating the behavior of Partial Differential Equations (PDEs) over geometric models with arbitrary shapes and material parameters. Its applications are widespread as PDEs are used to model the behavior of almost all complex physical systems. Common PDEs include Schrödinger’s equation which governs the evolution of quantum systems; the Navier-Stokes equations, which describe the behavior of fluids; and Maxwells Equations, which are a macroscopic description of all Electric and Magnetic Phenomena. When leveraged against Maxwells Equations, FEM allows engineers and scientists to rapidly design a wide range of Radio-Frequency (RF) devices such as antennas, RF filters, waveguides, and many others, all of which are important to the development of communications networks, sensing networks, and computing infrastructure.

One open question within FEM research is how to maximize simulation efficiency (i.e., what is the best strategy to maximize the ratio of accuracy to computational resource usage). A typical approach is to define a coarse (or low resolution) discretization of the geometric model in question, which uses a small number of computational resources. This model is then iteratively and intelligently refined, only introducing more entropy where it is most needed to improve solution accuracy. After several iterations, this approach will have achieved a desirable balance between resource usage and solution accuracy. The focus of this work is a simple and ergonomic implementation of h- and p-refinements for FEM. When used in tandem, these two refinement strategies are amenable to the above procedure and efficiently produce accurate results on challenging Computational Electromagnetics problems.

Adviser: Banislav M. Notaros
Co-Adviser: N/A
Non-ECE Member: Michael Kirby
Member 3: Milan Ilic
Addional Members: N/A

Publications:
Corrado, Jeremiah; Harmon, Jake; Notaros, Branislav; Ilic, Milan M. (2022): FEM_2D: A Rust Package for 2D Finite Element Method Computations with Extensive Support for hp-refinement. TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.19166339.v1
Corrado, Jeremiah; Harmon, Jake; Notaros, Branislav (2021): A Refinement-by-Superposition Approach to Fully Anisotropic hp-Refinement for Improved Efficiency in CEM. TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.16695163.v1

Harmon, Jake; Corrado, Jeremiah; Notaros, Branislav (2021): A Refinement-by-Superposition hp-Method for H(curl)- and H(div)-Conforming Discretizations. TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.14807895.v1

Program of Study:
ECE541
ECE444
ECE513
CS458
GRAD510
ECE502
MATH580
CS458