A wide variety of phenomena in condensed matter systems is driven primarily by interactions between electrons in the system. This work is concerned with the application of functional renormalization group (fRG) as a generalized solver for the multi-band Hamiltonians that describe these systems. We consider a decoupled formulation of the fRG equations that is optimized in the frequency and momentum domains and retains the flow of relevant modes in the system. Approximate truncations that extend the scheme to arbitrary multiband systems are addressed. This optimized decoupling is then used to derive the flow equations that describe fluctuations in model Hamiltonians for Cuprate and Pnictide superconductors. We construct a full phase diagram of the systems studied as a function of doping, temperature and coupling. Access to the frequency modes in the system allows us to explore the impact of coupling phonons to these model Hamiltonians. Alterations to the diagram due to electron-phonon coupling is derived. The results of the decoupled formulation is in agreement with results in the literature for many of the models considered. Further the fRG captures the sensitivity of susceptibilities of Cuprate Hamiltonians to band structure, the enhanced role of Hunds coupling in Pnictide systems and the impact of phonons in multiband Hamiltonians.

<p> Earthquakes and tsunamis represent two of the most devastating natural disasters faced by humankind. Earthquakes can occur in matters of seconds, with little to no warning. The governing variables of earthquakes, namely the stress profiles of vast regions of the earth's crust, cannot be measured in a comprehensive manner. Similarly, tsunami parameters are often accurately determined only minutes before waves make landfall. We are therefore left only with statistical analyses of past events to produce hazard forecasts for these disasters. Unfortunately, the events that cause the most damage also occur infrequently, and most regions have scientific records of earthquakes going back only a century, with modern instrumentation being widely distributed only in the past few decades. The 2011 <i>M=</i>9 Tohoku earthquake and tsunami, which killed close to sixteen thousand people, is the perfect case study of a country heavily invested in earthquake and tsunami risk reduction, yet being unprepared for a once-in-a-millennium event. </p><p> Physics-based simulations are some of the most promising tools for learning more about these systems. These tools can be used to study many thousands of years worth of synthetic seismicity. Additionally, scaling laws present in such complex geophysical systems can provide insights into dynamics otherwise hidden from view. This dissertation represents a collection of studies using these two tools. First, the Virtual Quake earthquake simulator is introduced, along with some of my contributions to its functionality and maintenance. A method based on Omori aftershock scaling is presented for verifying the spatial distribution of synthetic earthquakes produced by long-term simulators. The use of aftershock ground motion records to improve constraints on those same aftershock models is then explored. Finally, progress in constructing a tsunami early warning system based on the coupling of Virtual Quake and the Tsunami Squares wave simulator is presented. Taken together, these studies demonstrate the versatility and strength of complexity science and computational methods in the context of hazard analysis.</p><p>

<p> Block copolymers are composed of chemically distinct polymer chains that can be covalently linked in a variety of sequences and architectures. They are ubiquitous as ingredients of consumer products and also have applications in advanced plastics, drug delivery, advanced membranes, and next generation nano-lithographic patterning. The wide spectrum of possible block copolymer applications is a consequence of block copolymer self-assembly into periodic, meso-scale morphologies as a function of varying block composition and architecture in both melt and solution states, and the broad spectrum of physical properties that such mesophases afford. </p><p> Materials exploration and discovery has traditionally been pursued through an iterative process between experimental and theoretical/computational collaborations. This process is often implemented in a trial-and-error fashion, and from the computational perspective of generating phase diagrams, usually requires some existing knowledge about the competitive phases for a given system. Self-Consistent Field Theory (SCFT) simulations have proven to be both qualitatively and quantitatively accurate in the determination, or forward mapping, of block copolymer phases of a given system. However, it is possible to miss candidates. This is because SCFT simulations are highly dependent on their initial configurations, and the ability to map phase diagrams requires a priori knowledge of what the competing candidate morphologies are. The unguided search for the stable phase of a block copolymer of a given composition and architecture is a problem of global optimization. SCFT by itself is a local optimization method, so we can combine it with population-based heuristic algorithms geared at global optimization to facilitate forward mapping. In this dissertation, we discuss the development of two such methods: Genetic Algorithm + SCFT (GA-SCFT) and Particle Swarm Optimization + SCFT (PSO-SCFT). Both methods allow a population of configurations to explore the space associated with the numerous states accessible to a block copolymer of a given composition and architecture. </p><p> GA-SCFT is a real-space method in which a population of SCFT field configurations &ldquo;evolves&rdquo; over time. This is achieved by initializing the population randomly, allowing the configurations to relax to local basins of attraction using SCFT simulations, then selecting fit members (lower free energy structures) to recombine their fields and undergo mutations to generate a new &ldquo;generation&rdquo; of structures that iterate through this process. We present results from benchmark testing of this GA-SCFT technique on the canonical AB diblock copolymer melt, for which the theoretical phase diagram has long been established. The GA-SCFT algorithm successfully predicts many of the conventional mesophases from random initial conditions in large, 3-dimensional simulation cells, including hexagonally-packed cylinders, BCC-packed spheres, and lamellae, over a broad composition range and weak to moderate segregation strength. However, the GA-SCFT method is currently not effective at discovery of network phases, such as the Double-Gyroid (GYR) structure. </p><p> PSO-SCFT is a reciprocal space approach in which Fourier components of SCFT fields near the principal shell are manipulated. Effectively, PSO-SCFT facilitates the search through a space of reciprocal-space SCFT seeds which yield a variety of morphologies. Using intensive free energy as a fitness metric by which to compare these morphologies, the PSO-SCFT methodology allows us to agnostically identify low-lying competitive and stable morphologies. We present results for applying PSO-SCFT to conformationally symmetric diblock copolymers and a miktoarm star polymer, AB₄, which offers a rich variety of competing sphere structures. Unlike the GA-SCFT method we previously presented, PSO-SCFT successfully predicts the double gyroid morphology in the AB-diblock. Furthermore, PSO-SCFT successfully recovers the A₁₅ morphology at a composition where it is expected to be stable in the miktoarm system, as well as several competitive metastable candidates, and a new sphere morphology belonging to the hexagonal space group 191, which has not been seen before in polymer systems. Thus, we believe the PSO-SCFT method provides a promising platform for screening for competitive structures in a given block copolymer system.</p><p>

A theoretical analysis of the orientation of biofunctionalized thiolate has been made by changing the surface configuration. The results show that it is possible to match the experimental data by changing the molecular density and also that it is possible to match the experimental data using only hollow sites in the gold surface as placements for the molecules. Some configurations that match available data using only hollow sites positions have been suggested. Moving away from the (sqrt(3) x sqrt(3)) R30° configuration result in a large energy gain for Bor Capped.

<p> This thesis presents a space-time extended finite element method (space-time XFEM) based on the Heaviside enrichment for transient problems with moving interfaces, and its applications to the fluid-structure interaction (FSI) analysis. The Heaviside-enriched XFEM is a promising method to discretize partial differential equations with discontinuities in space. However, significant approximation errors are introduced by time stepping schemes when the interface geometry changes in time. The proposed space-time XFEM applies the finite element discretization and the Heaviside enrichment in both space and time with elements forming a space-time slab. A simple space-time scheme is introduced to integrate the weak form of the governing equations. This scheme considers spatial intersection configuration at multiple temporal integration points. Standard spatial integration techniques can be applied for each spatial configuration. Nitsche's method and the face-oriented ghost-penalty method are extended to the proposed space-time XFEM formulation. The stability, accuracy and flexibility of the space-time XFEM for various interface conditions including moving interfaces are demonstrated with structural and fluid problems. Moreover, the space-time XFEM enables analyzing complex FSI problems using moving interfaces, such as FSI with contact. Two FSI methods using moving interfaces (full-Eulerian FSI and Lagrangian-immersed FSI) are studied. The Lagrangian-immersed FSI method is a mixed formulation of Lagrangian and Eulerian descriptions. As solid and fluid meshes are independently defined, the FSI is computed between non-matching interfaces based on Nitsche's method and projection techniques adopted from computational contact mechanics. The stabilized Lagrange multiplier method is used for contact. Numerical examples of FSI and FSI-contact problems provide insight into the characteristics of the combination of the space-time XFEM and the Lagrangian-immersed FSI method. The proposed combination is a promising method which has the versatility for various multi-physics simulations and the applicability such as optimization.</p><p>

<p> The Next Generation Science Standards (NGSS) have established computational thinking as one of the science and engineering practices that should be developed in high school classrooms. Much of the work done by scientists is accomplished through the use of computation, but many students leave high school with little to no exposure to coding of any kind. This study outlines an attempt to integrate computational physics lessons into a high school algebra-based physics course which utilizes Modeling Instruction. Specifically, it aims to determine if students who complete computational physics assignments demonstrate any difference in understanding force concepts as measured by the Force Concept Inventory (FCI) versus students who do not. Additionally, it investigates students&rsquo; attitudes about learning computation alongside physics. Students were introduced to Vpython programs during the course of a semester. The FCI was administered pre and post instruction, and the gains were measured against a control group. The Computational Modeling in Physics Attitudinal Student Survey (COMPASS) was administered post instruction and the responses were analyzed. While the FCI gains were slightly larger on average than the control group, the difference was not statistically significant. This at least suggests that incorporating computational physics assignments does not adversely affect students&rsquo; conceptual learning.</p><p>

<p> Though fundamental to our understanding of stellar, galactic, and cosmic evolution, the stellar explosions known as supernovae (SNe) remain mysterious. We know that mass loss and mass transfer are central processes in the evolution of a star to the supernova event, particularly for thermonuclear Type Ia supernovae (SNe Ia), which are in a close binary system. The circumstellar environment (CSE) contains a record of the mass lost from the progenitor system in the centuries prior to explosion and is therefore a key diagnostic of SN progenitors. Unfortunately, tools for studying the CSE are specialized to stellar winds rather than the more complicated and violent mass-loss processes hypothesized for SN Ia progenitors.</p><p> This thesis presents models for constraining the properties of a CSE detached from the stellar surface. In such cases, the circumstellar material (CSM) may not be observed until interaction occurs and dominates the SN light weeks or even months after maximum light. I suggest we call SNe with delayed interaction SNe X;n (i.e. SNe Ia;n, SNe Ib;n). I per- formed numerical hydrodynamic simulations and radiation transport calculations to study the evolution of shocks in these systems. I distilled these results into simple equations that translate radio luminosity into a physical description of the CSE. I applied my straightforward procedure to derive upper limits on the CSM for three SNe Ia: SN 2011fe, SN 2014J, and SN 2015cp. I modeled interaction to late times for the SN Ia;n PTF11kx; this led to my participation in the program that discovered interaction in SN 2015cp. Finally, I expanded my simulations to study the Type Ib;n SN 2014C, the first optically-confirmed SN X;n with a radio detection. My SN 2014C models represent the first time an SN X;n has been simultaneous modeled in the x-ray and radio wavelengths.</p><p>

<p> The light collecting apertures of space telescopes are currently limited in part by the size and weight restrictions of launch vehicles, ultimately limiting the spatial resolution that can be achieved by the observatory. A technique that can overcome these limitations and provide superior spatial resolution is interferometric imaging, whereby multiple small telescopes can be combined to produce a spatial resolution comparable to a much larger monolithic telescope. In astronomy, the spectrum of the sources in the scene are crucial to understanding the material composition of the sources. So, the ultimate goal is to have high-spatial-resolution imagery and obtain sufficient spectral resolution for all points in the scene. This goal can be accomplished through spatio-spectral interferometric imaging, which combines the aperture synthesis aspects of a Michelson stellar interferometer with the spectral capabilities of Fourier transform spectroscopy. </p><p> Spatio-spectral interferometric imaging can be extended to a wide-field imaging modality, which increases the collecting efficiency of the technique. This is the basis for NASA&rsquo;s Wide-field Imaging Interferometry Testbed (WIIT). For such an interferometer, there are two light collecting apertures separated by a variable distance known as the baseline length. The optical path in one of the arms of the interferometer is variable, while the other path delay is fixed. The beams from both apertures are subsequently combined and imaged onto a detector. For a fixed baseline length, the result is many low-spatial-resolution images at a slew of optical path differences, and the process is repeated for many different baseline lengths and orientations. Image processing and synthesis techniques are required to reduce the large dataset into a single high-spatial-resolution hyperspectral image. </p><p> Our contributions to spatio-spectral interferometry include various aspects of theory, simulation, image synthesis, and processing of experimental data, with the end goal of better understanding the nature of the technique. We present the theory behind the measurement model for spatio-spectral interferometry, as well as the direct approach to image synthesis. We have developed a pipeline to preprocess experimental data to remove unwanted signatures in the data and register all image measurements to a single orientation, which leverages information about the optical system&rsquo;s point spread function. In an experimental setup, such as WIIT, the reference frame for the path difference measured for each baseline is unknown and must be accounted for. To overcome this obstacle, we created a phase referencing technique that leverages point sources within the scene of known separation in order to recover unknown information regarding the measurements in a laboratory setting. We also provide a method that allows for the measurement of spatially and spectrally complicated scenes with WIIT by decomposing them prior to scene projection.</p><p>

<p> A fourth-order accurate finite volume method for curvilinear grids based on Hermitian interpolation and splines is here presented in one and two dimensions. The finite volume method is derived in detail starting in one dimension and then extended to two dimensions using isoparametric mapping. The method is applied to the quasi-one-dimensional Euler equations through a converging-diverging nozzle as well as the heat conduction equation through a body-fitted non-orthogonal grid. Comparisons are made between the methods presented here and similar techniques in the literature. Lastly, possible ways to improve the method&rsquo;s computational efficiency are discussed. </p><p>

<p> This dissertation contains investigations based on Molecular Dynamics (MD) of a variety of systems, from small atomic clusters to polymers in solution and in their condensed phases. The overall research is divided in three parts. First, I tested a new thermostat in the literature on the thermal equilibration of a small cluster of Lennard-Jones (LJ) atoms. The proposed thermostat is a Hamiltonian thermostat based on a logarithmic oscillator with the outstanding property that the mean value of its kinetic energy is constant independent of the mass and energy. I inspected several weak-coupling interaction models between the LJ cluster and the logarithmic oscillator in 3D. In all cases I show that this coupling gives rise to a kinetic motion of the cluster center of mass without transferring kinetic energy to the interatomic vibrations. This is a failure of the published thermostat because the temperature of the cluster is mainly due to vibrations in small atomic clusters This logarithmic oscillator cannot be used to thermostat any atomic or molecular system, small or large. </p><p> The second part of the dissertation is the investigation of the inherent structure of the polymer polyethylene glycol (PEG) solvated in three different solvents: water, water with 4% ethanol, and ethyl acetate. PEG with molecular weight of 2000 Da (PEG₂₀₀₀) is a polymer with many applications from industrial manufacturing to medicine that in bulk is a paste. However, its structure in very dilute solutions deserved a thorough study, important for the onset of aggregation with other polymer chains. I introduced a modification to the GROMOS 54A7 force field parameters for modeling PEG₂₀₀₀ and ethyl acetate. Both force fields are new and have now been incorporated into the database of known residues in the molecular dynamics package Gromacs. This research required numerous high performance computing MD simulations in the ARGO cluster of GMU for systems with about 100,000 solvent molecules. My findings show that PEG₂₀₀₀ in water acquires a ball-like structure without encapsulating solvent molecules. In addition, no hydrogen bonds were formed. In water with 4% ethanol, PEG₂₀₀₀ acquires also a ball-like structure but the polymer ends fluctuate folding outward and onward, although the general shape is still a compact ball-like structure. </p><p> In contrast, PEG₂₀₀₀ in ethyl acetate is quite elongated, as a very flexible spaghetti that forms kinks that unfold to give rise to folds and kinks in other positions along the polymer length. The behavior resembles an ideal polymer in a &thetas; solvent. A Principal Component Analysis (PCA) of the minima composing the inherent structure evidences the presence of two distinct groups of ball-like structures of PEG₂₀₀₀ in water and water with 4% ethanol. These groups give a definite signature to the solvated structure of PEG₂₀₀₀ in these two solvents. In contrast, PCA reveals several groups of avoided states for PEG₂₀₀₀ in ethyl acetate that disqualify the possibility of being an ideal polymer in a &thetas; solvent. </p><p> The third part of the dissertation is a work in progress, where I investigate the condensed phase of PEG₂₀₀₀ and study the interface between the condensed phase and the three different solvents under study. With a strategy of combining NPT MD simulations at different temperatures and pressures, PEG₂₀₀₀ condensed phase displays the experimental density within a 1% discrepancy at 300 K and 1 atm. This is a very encouraging result on this ongoing project. </p><p>