<p> Geometric integrators yield high-fidelity numerical results by retaining conservation laws in the time advance. A particularly powerful class of geometric integrators is symplectic integrators, which are widely used in orbital mechanics and accelerator physics. An important application presently lacking symplectic integrators is the guiding center motion of magnetized particles represented by non-canonical coordinates. Because guiding center trajectories are foundational to many simulations of magnetically confined plasmas, geometric guiding center algorithms have high potential for impact. The motivation is compounded by the need to simulate long-pulse fusion devices, including ITER, and opportunities in high performance computing, including the use of petascale resources and beyond. </p><p> This dissertation uses a systematic procedure for constructing geometric integrators &mdash; known as variational integration &mdash; to deliver new algorithms for guiding center trajectories and other plasma-relevant dynamical systems. These variational integrators are non-trivial because the Lagrangians of interest are degenerate - the Euler-Lagrange equations are first-order differential equations and the Legendre transform is not invertible. The first contribution of this dissertation is that variational integrators for degenerate Lagrangian systems are typically <i>multistep methods.</i> Multistep methods admit parasitic mode instabilities that can ruin the numerical results. These instabilities motivate the second major contribution: degenerate variational integrators. By replicating the degeneracy of the continuous system, degenerate variational integrators avoid parasitic mode instabilities. The new methods are therefore robust geometric integrators for degenerate Lagrangian systems. </p><p> These developments in variational integration theory culminate in one-step degenerate variational integrators for non-canonical magnetic field line flow and guiding center dynamics. The guiding center integrator assumes coordinates such that one component of the magnetic field is zero; it is shown how to construct such coordinates for nested magnetic surface configurations. Additionally, collisional drag effects are incorporated in the variational guiding center algorithm for the first time, allowing simulation of energetic particle thermalization. Advantages relative to existing canonical-symplectic and non-geometric algorithms are numerically demonstrated. All algorithms have been implemented as part of a modern, parallel, ODE-solving library, suitable for use in high-performance simulations.</p>

This research work presents numerical simulations of supercontinuum (SC) generation in optical waveguides based on Ge11.5As24Se64.5 chalcogenide (ChG) material. Rigorous numerical simulations were performed using finite-element and split-step Fourier methods in order to optimize the waveguides for wideband SC generation. Through dispersion engineering and by varying dimensions of the 1.8-cm-long ChG nanowires, we have investigated dispersion curves for a number of nanowire geometries and identified a promising one which can be used for generating a SC with 1300 nm bandwidth pumped at 1550 nm with a low peak power of 25 W. It was observed through successive inclusion of higher-order dispersion coefficients during SC simulations that there is a possibility of obtaining spurious results if the adequate number of dispersion coefficients is not considered. We then investigate MIR SC in dispersion-tailored, air-clad, ChG channel waveguide employing either Ge11.5As24S64.5 or MgF2 glass and ChG rib waveguide employing MgF2 glass for their lower claddings. We study the effect of waveguide parameters on the bandwidth of the SC at the output of 1-cm-long waveguides. Our results show that output can vary over a wide range depending on their design and the pump wavelength employed. At the pump wavelength of 2 μm the SC never extended beyond 4.5 μm for any of our designs. However, SC could be extended to beyond 5 μmfor a pump wavelength of 3.1 μm. A broadband SC spanning from 2 to 6 μm and extending over 1.5 octave could be generated with a moderate peak power of 500 W at a pump wavelength of 3.1 μm using an air-clad, all-ChG, channel waveguide. We show that SC can be extended even further covering the wavelength ranges 1.8-7.7 μm and 1.8-8 μm (> 2 octaves) when MgF2 glass is used for the lower claddings of ChG channel waveguide and rib waveguide, respectively. By employing the same pump source, we show that SC spectra can cover a wavelength range of 1.8-11 μm (> 2.5 octaves) in a channel waveguide and 1.8-10 μm in a rib waveguide employing MgF2 glass for their lower claddings with a moderate peak power of 3 kW. Finally we present microstrucured fibre based design made with same glass to generate SC spectra in the MIR region. Numerical simulations show that such a 1-cm-long fibre can produce a spectrum extending from 1.3 μm to beyond 11 μm (> 3 octaves) with the same pump and peak power applied before. We consider three fibre structures with microstrucured air-holes in their cladding and find their optimum designs through dispersion engineering. Among these, equiangular-spiral microstrucured fibre is found to be the most promising candidate for generating ultrawide SC in the MIR region.

Living organs in human bodies continuously interact with the in vivo bio-environment, while reshaping and rearranging their constituents, responding to external or internal stimuli through life cycles. For instance, living tissues adjust the growth (or turnover) rates of their constituents to develop (volumetric and mass) changes as the tissues adapt to the pathological or physiological changes in bio-environment. From the perspective of biomechanics, changes in the bio-environment will induce the growth and remodelling (G\&R) process and reset the mechanical environment. Consequently, the mechanical cues will feed back to G\&R processes. In the long run, the interaction between G\&R and the mechanical response of living organs plays an important role in regulating the organ formulation or pathological growth. To understand the interaction between the mechanical response and the G\&R process, an important ingredient in evaluating the involved mechanics is knowledge of the solid mechanical properties of the soft tissues. Residual stress, resulting from G\&R of soft tissues, is important in modelling the mechanics of soft tissues, which still presents a modelling challenge for including residual stress in cardiovascular applications. For G\&R of living organs, changes of tissue structure and volume are also important determinants for organ development. This raises academic challenges for the understanding of the evolution of material properties and mechanical response of living tissues within a dynamic environment. To investigate the stress states (residual strain or residual stress) of living organs, the experimental results showed that the arterial slices would spring open after cutting along the radial directions, which indicates the residual strain in organs estimated by the opening angle. The residual strain, which is the elastic strain between zero-stress and no-load states, indicates the existence of residual stress after removal of the external loads. The residual stress is considered to modulate the growth and remodelling process in living organs. The evolution of residual stress could relieve the information about the history of growth, which could help to better the understanding of the formation of organs and the development of diseases. Besides the residual stress, G\&R processes are regulated by other factors, while the principles governing those mechanism are still not fully understood. Obviously, improving knowledge in this particular field will give huge potential for the design and optimization of clinical treatments to efficiently save more lives. From a general mechanics perspective to investigate the G\&R process in living tissues, the questions are: How does the residual stress influence the fibre remodelling and the material properties of entire organs? How to determine the combined effects of growth (in the stressed configuration) and remodelling on the fibre structure? How to develop a framework for investigating G\&R processes occurring in the stressed configuration? For arteries, multiple layer models are developed to analytically study residual stress in living organs. For the heart, due to its complex structure and geometry, most previous studies used the unloaded configuration or one-cut configuration as the stress-free configuration to estimate the stress state. However, both experimental and theoretical studies have suggested that: 1) residual stress will significantly influence the stress distribution in the heart. 2) a simple (or single) cut does not release all the residual stress in the heart. We build a multi-cut model and show that multiple cuts are required to release the residual stresses in the left ventricle. Our results show that with the 2-cut and 4-cut models (one radial cut followed by circumferential cuts), agreement with the measured opening angles and radii can be greatly improved. This suggests that a multi-cut model should be used to predict the residual stresses in the left ventricle, at least in the middle wall region. We further show that tissue heterogeneity plays a significant role in the model results, and that an inhomogeneous model with combined radial and circumferential cuts should be used to estimate the correct order of magnitude of the residual stress in the heart. Understanding the healing and remodelling processes induced by myocardial infarction (MI) of the heart is important and the mechanical properties of the myocardium post-MI can be indicative for effective treatments aimed at avoiding eventual heart failure. MI remodelling is a multiscale feedback process between the mechanical loading and cellular adaptation. In this thesis, we use an agent-based model to describe collagen remodelling by fibroblasts regulated by chemical and mechanical cues after acute MI, and upscale into a finite element (FE) 3D left ventricular model. This enables us to study the scar healing (collagen deposition, degradation and reorientation) of a rat heart post-MI. Our results, in terms of collagen accumulation and alignment, compare well to published experimental data. In addition, we show that different shapes of the MI region can affect the collagen remodelling, and in particular, the mechanical cue plays an important role in the healing process. For volumetric growth, recently, when the idea of growth is applied to study the evolution of organ formations, it's usually assumed that growth always occurs in the natural (reference) configuration. In some researches, it is assumed that the growth could release all the residual stress, and that further growth will start from the updated but stress-free configuration. However, living organs are actually exposed to external loading all the time, while the growth should occur from the residually-stressed current configuration. In this thesis, A theoretical framework is developed to calculate the mechanical behaviour of soft tissue after introducing inhomogeneous growth in a residually-stressed current configuration, which avoids assuming that the growth occurs in a `virtual' reference configuration. Moreover, the theoretical framework is introduced to couple the growth and fibre remodelling process to describe the mechanical behaviour of living tissues.

<p>In this thesis we solve the Feshbach-Villars equations for spin-zero particles through use of matrix continued fractions. The Feshbach-Villars equations are derived from the Klein-Gordon equation and admit, for the Coulomb potential on an appropriate basis, a Hamiltonian form that has infinite symmetric band-matrix structure. The corresponding representation of the Green's operator of such a matrix can be given as a matrix continued fraction. Furthermore, we propose a finite dimensional representation for the potential operator such that it retains some information about the whole Hilbert space. Combining these two techniques, we are able to solve relativistic quantum mechanical problems of a spin-zero particle in a Coulomb-like potential with a high level of accuracy.

In this thesis, we explore connections between solitons, black holes, and harmonic maps in two-dimensional gravitation. Euclidean sine-Gordon theory, naturally admitting soliton solutions, and Schwarzschild-type black hole metrics, of physical interest, are studied in detail for the case of JT gravity. Establishing an explicit soliton-black hole correspondence in this setting, new solutions to the associated JT field equations are given. Consequences and concrete applications of the constructed gauge transformations are also discussed, including characterisation of the Killing vector fields and solutions to a nontrivial Eigenvalue Problem using the theory of hypergeometric equations. We next consider a generalised two-dimensional action and establish a correspondence between nonconstant curvature soliton metrics and black hole metrics. The theory is applied to completely solve the static case, as well as study other classical dilaton models, including Spherically Symmetric Gravity and String Inspired Gravity. Finally, a connection between harmonicity and generalised solitons is given through construction of harmonic maps of the plane to the 2-sphere, suggesting new solutions to field equations admitting black hole metrics. Other directions for studying the integrable systems structure of generalised two-dimensional dilaton theories are indicated.

We show that Einstein’s gravitational field equations for the Friedmann-Robertson-Lemaître-Walker (FRLW) and for two conformal versions of the Bianchi I and Bianchi V perfect fluid scalar field cosmological models, can be equivalently reformulated in terms of a single equation of either generalized Ermakov-Milne-Pinney (EMP) or (non)linear Schrödinger (NLS) type. This work generalizes or presents an alternative to similar reformulations published by the authors who inspired this thesis: R. Hawkins, J. Lidsey, T. Christodoulakis, T. Grammenos, C. Helias, P. Kevrekidis, G. Papadopoulos and F.Williams. In particular we cast much of these authors’ works into a single framework via straightforward derivations of the EMP and NLS equations from a simple linear combination of the relevant Einstein equations. By rewriting the resulting expression in terms of the volume expansion factor and performing a change of variables, we obtain an uncoupled EMP or NLS equation that is independent of the imposition of additional conservation equations. Since the correspondences shown here present an alternative route for obtaining exact solutions to Einstein’s equations, we reconstruct many known exact solutions via their EMP or NLS counterparts and show by numerical analysis the stability properties of many solutions.

In this dissertation four results are presented on the fluctuations of the spin per site around the thermodynamic magnetization in the mean-field Blume-Capel model, a basic model in statistical mechanics. The first two results refine the main theorem in a 2010 paper by R. S. Ellis, J. Machta, and P. T. Otto published in Annals of Applied Probability 20 (2010) 2118-2161. This paper provides the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model. The first main result studies the asymptotics of the centered, finite-size magnetization, giving its precise rate of convergence to 0 along parameter sequences lying in the phase-coexistence region and converging sufficiently slowly to either a second-order point or the tricritical point of the model. A simple inequality yields our second main result, which generalizes the main theorem in the Ellis-Machta-Otto paper by giving an upper bound on the rate of convergence to 0 of the absolute value of the difference between the finite-size magnetization and the thermodynamic magnetization. These first two results have direct relevance to the theory of finite-size scaling. They are consequences of the third main result. This is a new conditional limit theorem for the spin per site, where the conditioning allows us to focus on a neighborhood of the pure states having positive thermodynamic magnetization. The fourth main result is a conditional central limit theorem showing that the fluctuations of the spin per site are Gaussian in a neighborhood of the pure states having positive thermodynamic magnetization.

No description available.

No description available.

No description available.