Electromagnetic wave propagation in complex dispersive media is governed by the time dependent Maxwell’s equations coupled to equations that describe the evolution of the induced macroscopic polarization. We consider “polydispersive” materials represented by distributions of dielectric parameters in a polarization model. The work focuses on a novel computational framework for...
We consider numerical methods for finding approximate solutions to Ordinary Differential Equations (ODEs) with parameters distributed with some probability by the Generalized Polynomial Chaos (GPC) approach. In particular, we consider those with forcing functions that have a random parameter in both the scalar and vector case. We then consider linear...
In this thesis, we investigate the problem of simulating Maxwell's equations in dispersive dielectric media. We begin by explaining the relevance of Maxwell's equations to
21st century problems. We also discuss the previous work on the numerical simulations of
Maxwell's equations. Introductions to Maxwell's equations and the Yee finite difference...
Accurate modeling and simulation of wave propagation in dispersive dielectrics such as water, human tissue and sand, among others, has a variety of applications. For example in medical imaging, electromagnetic waves are used to interrogate human tissue in a non-invasive manner to detect anomalies that could be cancerous. In non-destructive...
We consider wide bandwidth electromagnetic pulse interrogation problems for the determination of dielectric response parameters in complex dispersive materials. We couple Maxwell’s equations with an auxiliary ODE modeling dielectric polarization. A problem of particular interest is to identify parameters in a standard polarization model (e.g., Debye or Lorentz) using time-domain...
In this report we consider the Debye model along with Maxwell's equations (Maxwell-Debye) to model electromagnetic wave propagation in dispersive media that exhibit orientational polarization. We construct and analyze a sequential operator splitting method for the discretization of the Maxwell-Debye system. Energy analysis indicates that the operator splitting scheme is...
We study the stability properties of, and the phase error present in, a finite element scheme for Maxwell’s
equations coupled with a Debye or Lorentz polarization model. In one dimension we consider a second order
formulation for the electric field with an ordinary differential equation for the electric polarization added...
We study the stability properties of, and the phase error present in, several higher order (in space) staggered finite difference schemes for Maxwell's equations coupled with a Debye or Lorentz polarization model. We present a novel expansion of the symbol of finite difference approximations, of arbitrary (even) order, of the...
Water conservation and water quality are rapidly increasing in importance in all areas of the world. The ability to accurately measure soil water content and salinity, over a wide variety of conditions, is key to meeting this need. A set of forward prediction models and waveform interpretation algorithms to extract...
A new second order accurate nonuniform grid spacing technique which does not
depend on supraconvergence is developed for Finite Difference Time Domain (FDTD)
simulation of general three dimensional structures. The technique is useful for FDTD
simulations of systems which require finer details in small regions of the simulation space by...
In this thesis we analyze a model for Kerr optical materials consisting of Maxwell's equations along with the dispersive Duffing model. We consider Duffing models with cubic and quintic polynomial nonlinearities. We assume a traveling wave solution to this nonlinear electromagnetic system and analyze it using the theory of dynamical...
In this dissertation, we introduce a family of fully discrete finite difference time-domain (FDTD) methods for Maxwell’s equations in linear and nonlinear materials. Onecategory of methods is constructed using multiscale techniques involving operator splittings. We present the sequential splitting scheme, the Strang Marchuk splitting scheme,the weighted sequential splitting scheme including...
In this thesis we construct compatible discretizations of Maxwell's equations. We use the term compatible to describe numerical methods for Maxwell's equations which obey many properties of vector Calculus in a discrete setting. Compatible discretizations preserve the exterior Calculus ensuring that the divergence of the curl and the curl of...
The aim of this dissertation is to construct a virtual element method (VEM) for models in magneto-hydrodynamics (MHD), an area that studies the behavior and properties of electrically conducting fluids such as a plasma. MHD models are a coupling of the Maxwell’s equations for electromagnetics and models for fluid flow....
In advanced integrated circuit (IC) processes, the metal fill inserted to meet foundry imposed density requirements degrades the performance of interconnects and passive components which ultimately affects the overall circuit performance. Accounting for this degradation through electromagnetic and equivalent circuit modeling is becoming a critical aspect of IC design. However,...
We present a novel strategy for minimizing the numerical dispersion error in edge discretizations of the time-domain electric vector wave equation on square meshes based on the mimetic finite difference (MFD) method. We compare this strategy, called M-adaptation, to two other discretizations, also based on square meshes. One is the...
The electrical performance of on-chip interconnects has become a limiting factor to the performance of modern integrated circuits including RFICs, mixed-signal circuits, as well as high-speed VLSI circuits due to increasing operating frequencies, chip areas, and integration densities. It is advantageous to have fast and accurate closed-form expressions for the...
This dissertation concerns several problems in the fields of light interaction with nanostructured media, metamaterials, and plasmonics. We present a technique capable of extending operational bandwidth of hyperbolic metamaterials based on interleaved highly-doped InGaAs and undoped AlInAs multilayer stacks. The experimental results confirm theoretical predictions, exhibiting broadband negative refraction response...
One of the most significant challenges faced by hydrogeologic modelers is the disparity between the spatial and temporal scales at which fundamental flow, transport, and reaction processes can best be understood and quantified (e.g., microscopic to pore scales and seconds to days) and at which practical model predictions are needed...
As scientists, we have an affinity for order: repeating crystal structures, Euclidean space, continuous functions… Even concepts such as defects are discussed in the context of distracting from order. Things are most logical and best described in straightforward, taxonomical fashion. But what happens when the very application desired is dependent...