Ph.D. Theses (Mathematics)
http://hdl.handle.net/1957/15739
2016-09-27T05:21:14ZNew Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant
http://hdl.handle.net/1957/59294
New Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant
Do, Hieu Trung
Translation surfaces can be viewed as polygons with parallel and equal sides
identified. An affine homeomorphism φ from a translation surface to itself is called
pseudo-Anosov when its derivative is a constant matrix in SL₂(R) whose trace is larger
than 2 in absolute value. In this setting, the eigendirections of this matrix defines the
stable and unstable flow on the translation surface. Taking a transversal to the stable
flows, the first return map of the flow induces an interval exchange transformation T.
The Sah-Arnoux-Fathi invariant of φ is the sum of the wedge product between the
lengths of the subintervals of T and their translations. This wedge product does not
depend on the choice of transversal. We apply Veech’s construction of pseudo-Anosov
homeomorphisms to produce infinite families of pseudo-Anosov maps in the stratum
H(2, 2) with vanishing Sah-Arnoux-Fathi invariant, as well as sporadic examples in
other strata.
Graduation date: 2016
2016-05-27T00:00:00ZCompatible Discretizations for Maxwell's Equations with General Constitutive Laws
http://hdl.handle.net/1957/59160
Compatible Discretizations for Maxwell's Equations with General Constitutive Laws
McGregor, D. A. (Duncan A. O.)
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 a gradient are zero in a discrete setting. This compatibility of discretizations with the continuum Maxwell's equations guarantees that the numerical solutions are physically meaningful.
We focus on the construction of a class of discretizations called Mimetic Finite Differences (MFD). The MFD method is a generalization of both staggered finite differences and mixed finite elements. We construct a parameterized family of MFD methods with equivalent formal order of accuracy. For time-dependent problems, we exploit this non-uniqueness by finding parameters which are optimal with respect to a certain criteria, for example, minimizing dispersion error. Dispersion error is a numerical artifact in which individual frequencies in a wave propagate at incorrect speeds; dominating the error in wave problems over long time propagation.
The novelty of this work is the construction of an MFD discretization for Maxwell's equations which reduces dispersion error for transient wave propagation in materials that are modeled by a general class of linear constitutive laws. We provide theoretical analysis of these new discretizations including an analysis of stability and discrete divergence. We also provide numerical demonstrations to illustrate the theory.
In addition to applications in the time domain we consider equilibrium Magnetohydrodynamic (MHD) generators. MHD generators extract power directly from a plasma by passing it through a strong magnetic field. Used as a topping cycle for traditional steam turbine generator, MHD offers a theoretical thermal efficiency of 60% compared to 40% of traditional systems. However, this technology has high life cycle costs due to equipment failure. One source of failure is arcing: the formation of high density currents which damage the generator. In this work we develop, analyze, and simulate a model of these generators. We use these simulations to show the viability of detecting electrical arcs by measurements of their magnetic fields outside of the generator.
Graduation date: 2016
2016-05-26T00:00:00ZHybrid Multiscale Methods with Applications to Semiconductors, Porous Media, and Materials Science
http://hdl.handle.net/1957/59138
Hybrid Multiscale Methods with Applications to Semiconductors, Porous Media, and Materials Science
Costa, Timothy B.
In this work we consider two multiscale applications with tremendous computational
complexity at the lower scale. First, we examine a model for charge transport in semicon-
ductor structures with heterojunction interfaces. Due to the complex physical phenomena
at the interface, the model at the design scale is unable to adequately capture the behavior
of the structure in the interface region. Simultaneously it is computationally intractable to
simulate the full heterostructure on the scale required near the interface. Second, we con-
sider the problem of the simulation of fluid flow in a dynamically evolving porous medium.
The evolution of the medium strongly couples the porescale flow solutions and the macro
scale model, requiring a novel approach to communicate the porescale evolution to the
macroscale without resorting to the intractable simulation of the fluid flow problem di-
rectly on the porescale geometry. We formulate novel methods for these two applications
in the multiscale framework. For the semiconductor problem we present iterative sub-
structuring domain decomposition methods that decouple the interface computation from
the macroscale model. For the fluid flow problem we develop a reduced order three-scale
fluid flow model based on a spatial decomposition of the porescale geometry and the offline
approximation of a stochastic process describing macroscale permeability paramaterized
by the volume fraction of the evolved geometry.
Graduation date: 2016
2016-05-31T00:00:00ZSome Results in Single-Scattering Tomography
http://hdl.handle.net/1957/58015
Some Results in Single-Scattering Tomography
Sherson, Brian
Single-scattering tomography describes a model of photon transfer through a object in which photons are assumed to scatter at most once. The Broken Ray transform arises from this model, and was first investigated by Lucia Florescu, Vadim A. Markel, and John C. Schotland, [2], in 2010, followed by an inversion, [1], in the case of fixed initial and terminal directions. Later, in 2013, Katsevich and Krylov, [6], investigated settings where terminal rays were permitted to vary, either heading towards or away from a focal point, providing inversion formulas in two- and three-detector settings.
In this thesis, we will explore these transforms, give them distributional meaning, and analyze how these transforms propagate singularities. This requires analysis of the relationships between a function and its antiderivative obtained from integration over a ray.
This thesis also introduces the Polar Broken Ray transform, in which the source position is fixed, initial direction varies, and the scattering angle is held constant. We will discover that the Polar Broken Ray transform is injective on spaces of functions supported in an annulus that is bounded away from the origin, and also derive distributional meaning to the Polar Broken Ray transform, and derive a relationship between a wavefront set of a function and that of its Polar Broken Ray transform.
Provided in the appendix are implementations of numerical inversions of the Broken Ray transform with fixed initial and terminal directions, as well as the Polar Broken Ray transform.
Graduation date: 2016
2015-12-10T00:00:00Z