<p> The goal of this thesis is to build hybrid deterministic/Monte Carlo algorithms for solving the neutron transport equation and associated <i> k</i>-eigenvalue problem. We begin by introducing and deriving the transport equation before discussing a series of deterministic methods for solving the transport equation. To begin we consider moment-based acceleration techniques for both the one and two-dimensional fixed source problems. Once this machinery has been developed, we will apply similar techniques for computing the dominant eigenvalue of the neutron transport equation. We'll motivate the development of hybrid methods by describing the deficiencies of deterministic methods before describing Monte Carlo methods and their advantages. We conclude the thesis with a chapter describing the detailed implementation of hybrid methods for both the fixed-source and <i>k</i>-eigenvalue problem in both one and two space dimensions. We'll use a series of test problems to demonstrate the effectiveness of these algorithms before hinting at some possible areas of future work.</p>

This research develops an analytical method for predicting the hydrodynamic force experienced by a long slender solid body of arbitrary cross-sectional shape and body centreline configuration, subjected to an unbounded uniform fluid flow. It is assumed the slenderness parameter, K (the ratio of the body cross-sectional length scale the body length) is small ($ ll 1$), the body centreline radius of curvature is everywhere large (of order body length), the cross-sectional shape varies slowly alone the body length, and the Reynolds number $R sb{e}$, based on the body length is of order unity. / The inner flow solution for an arbitrary cross-section is illustrated by applying the complex variable method for a body with an elliptical cross-section, which is extendable to any cross-sectional shape. / The novelty of this research is the improvement of the approximation of the force per unit length in slender body theory when inertia effects are not negligibly small. (Abstract shortened by UMI.)

The main focus of this thesis is on investigating the minimum size of the Representative Volume Element (RVE) and finite-size scaling of properties of random linear and nonlinear elastic composites. The RVE is a material volume which accurately describes the overall behavior of a heterogeneous solid, and is the core assumption of continuum mechanics theory. If the composite microstructure admits the assumption of spatial homogeneity and ergodicity, the RVE can be attained within a specific accuracy on a finite length-scale. Determining this scale is the key objective of this thesis. / In order to theoretically analyze the scale-dependence of the apparent response of random microstructures, essential and natural boundary conditions which satisfy Hill's averaging theorem in finite deformation elasticity are first considered. It is shown that the application of the partitioning method and variational principles in nonlinear elasticity and thermoelasticity, under the two above-mentioned boundary conditions, leads to the hierarchy of mesoscale bounds on the effective strain- and free-energy functions, respectively. These theoretical derivations lay the ground for the quantitative estimation of the scale-dependence of nonlinear composite responses and their RVE size. / The hierarchies were computed for planar matrix-inclusion composites with the microstructure modeled by a homogeneous Poisson point field. Various nonlinear composites with Ogden-type strain-energy function are considered. The obtained results are compared with those where both matrix and inclusions are described by a neo-Hookean strain-energy function as well as with the results obtained from the linear elasticity theory. The trends toward the RVE are also computed for nonlinear elastic composites subjected to non-isothermal loading. The accuracy of the RVE size estimation is calculated in terms of the discrepancy between responses under essential and natural boundary conditions. Overall, the results show that the trends toward the RVE as well as its minimum size are functions of the deformation, deformation mode, temperature, and the mismatch between material properties of the phases. / The last part of the thesis presents an investigation of the size effect on thermoelastic damping of a micro-/nanobeam resonator. It does not follow the framework described above. The main concern here is the size and the vibration frequency, at which the classical Fourier law of heat conduction is no longer valid, and the finite speed of heat propagation has to be taken into account.

<p> The magnetohydrodynamics (MHD) model describes the flow of electrically conducting fluids in the presence of magnetic fields. A principal application of MHD is the modeling of plasma physics, ranging from plasma confinement for thermonuclear fusion to astrophysical plasma dynamics. MHD is also used to model the flow of liquid metals, for instance in magnetic pumps, liquid metal blankets in fusion reactor concepts, and aluminum electrolysis. The model consists of a non-self-adjoint, nonlinear system of partial differential equations (PDEs) that couple the Navier-Stokes equations for fluid flow to a reduced set of Maxwell's equations for electromagnetics. </p><p> In this dissertation, we consider computational issues arising for the MHD equations. We focus on developing fast computational algorithms for solving the algebraic systems that arise from finite element discretizations of the fully coupled MHD equations. Emphasis is on solvers for the linear systems arising from algorithms such as Newton's method or Picard iteration, with a main goal of developing preconditioners for use with iterative methods for the linearized systems. In particular, we first consider the linear systems arising from an exact penalty finite element formulation of the MHD equations. We then draw on this research to develop solvers for a formulation that includes a Lagrange multiplier within Maxwell's equations. We also consider a simplification of the MHD model: in the MHD kinematics model, the equations are reduced by assuming that the flow behavior of the system is known. In this simpler setting, we allow for epistemic uncertainty to be present. By mathematically modeling this uncertainty with random variables, we investigate its implications on the physical model.</p>

<p> Many problems in image and video processing may be formulated in the language of constrained optimization. Algorithms for solving general constrained optimization problems may not guarantee solutions or be computationally efficient, particularly if the problem is nonlinear or non-convex. Oftentimes these constrained optimization problems may be relaxed into the form of a convex problem. This allows for the use of convex solvers such as the Augmented Lagrangian method and the Split Bregman iteration. In this thesis, we will study the advantages of incorporating convexity into constrained optimization problems. These problems will be motivated from the standpoint of hyperspectral image processing, particularly the detection and identification of airborne chemicals in gas cloud releases. </p>

This dissertation is concerned with experimental and analytical studies of the mechanics of interaction where a pneumatic tire is loaded vertically on a supporting ground, i.e., rigid base, clay and sand. / The numerous experiments were conducted under various conditions to characterized the interactions in terms of the experimental results, e.g. axle displacement, contact area, contact pressure, etc. The results of pressure distribution indicate that recognizing a tire as a pneumatic body is crucial in establishing a rational theory for tire-supporting ground interaction problems. The pressure distribution and contact area obtained in the experiments are also utilized in validating an analytical approach (i.e. First Analytical Approach) established in the dissertation. / A hypothetical description of the progress of tire deformation is discussed based on the experimental results. The discussion helps in providing a better understanding of the mechanics of the interaction, and for selecting basic analytical and/or numerical tools in establishing the present analytical methods. / In the analytical work, the two distinct analytical approaches (i.e. First and Second Analytical Approaches) are established under the plane strain condition in predicting contact length and pressure. However, the first analytical approach is emphasized in this dissertation, while the second one is rather a complementary work. / In the first analytical approach, the real contact profile is taken into account, while the existing contact theories (by Hertz, Muskhelishvili, etc.) essentially ignore the real kinematics of contact surfaces on which the pattern of pressure distribution greatly depends. In this first analytical approach, the following steps are taken: (1) transform a tire-supporting ground interaction problem into an equivalent free boundary (-value) problem of the deformed supporting ground; this is done so that the complex factors inherent to pneumatic tires are not directly taken into the analytical formulation; (2) determine the modulus of elasticity of the deformed supporting ground by taking into account the contact profile; (3) find the contact length and pressure by means of the complex variable method. / The contact length and pressure analytically obtained are in close agreement with those obtained through experimentation. An attempt has also been made to solve the sliding interaction problems. / The second analytical approach, which is an iterative technique combining the incremental finite element method and the complex variable method, is established fundamentally to solve an interaction problem between an elastic solid and a nonlinear elastic half-plane. Two different types of interaction problems are solved, i.e. tire-clay and rigid wheel-snowpack interactions. Numerical results on contact length for both problems showed acceptable agreement with the experimental results, while those on sinkage obtained for the rigid wheel did not.

This study considers a number of problems which remain in nonlinear consolidation modelling and prediction, despite the considerable research effort which has already been devoted to the subject. Nonlinear consolidation models refer to those models capable of accounting for material nonlinearity either explicitly, in the formulation of the governing equation, or implicitly in the numerical-solution technique. A nonlinear model can be characterised by its generality, appropriateness for modelling consolidation and prediction-capability. These three characteristics are not consistent; a model's superiority with respect to one characteristic does not imply its overall superiority. This inconsistency between the characteristics is resolved by a proposed model. This model is seen as an improvement over two available and widely used models; the Gibson et al. (1967) and Yong et al. (1983) models. / Nonlinear multiple-layer analysis requires the satisfaction of continuity conditions at the inter-layer boundaries. Existing continuity procedures seek to reduce the problem to a tractable single-layer problem, ignore the interaction between layers and are unsuitable for use with nonlinear models. Procedures, based on the trial-function technique, are proposed which satisfy the continuity conditions and facilitate these analyses. Various procedures are necessary to define the initial consolidation status of a soil depending on whether the field data is complete or incomplete. Procedures which can acknowledge incomplete data by calling for bounded analyses and yet can take full advantage of available data are proposed. / A finite-difference numerical-solution algorithm is developed for use with the proposed non-linear model. This algorithm is efficient, versatile and more suitable for multiple-layer analysis than the Yong et al. (1983) algorithm, on which it is based. / The findings of this study are successfully field validated on the basis of three case histories; the consolidation of highly compressible organic soils underlying two embankments in Poland and subsidence due to groundwater withdrawal in Bangkok, Thailand.

The present thesis studies the stability of a rotating cylindrical shell containing a co-rotating axial viscous flow. The system can be thought of as a long thin-walled pipe carrying an internal axial flow while the whole is in a frame of reference rotating at a prescribed rate. The equations of the previously solved inviscid model are rederived and the problem is studied further. The results obtained for purely axial flow are reproduced, but as expected from literature, it is impossible to obtain satisfactory results for the system subjected to rotation due to the presence of singularities in the flow pressure solution. A hypothetical physical explanation for these singularities is put forward and has similarities with the phenomenon of atmospheric flow blocking. / Considering the unsuccessful results obtained with the inviscid theory, it is believed that the added realism brought in by the introduction of viscosity in the theory can lead to a successful model. Assuming a travelling-wave perturbation scheme, the linear Donnell-Mushtari thin shell equations are coupled with the fluid stresses obtained by solving numerically the incompressible Navier-Stokes equation for a laminar or turbulent flow. A novel triple-perturbation approach is established to consider the interaction between the fluid and the structure. This triple-perturbation approach is in essence a superposition of three fluid fields caused by the three components of the shell deformation for a given oscillation mode. It is found that the usual technique for linear aeroelasticity studies consisting of applying the fluid boundary conditions at the undeformed position of the wall instead of the instantaneous deformed position greatly alters the stability of the system. To remedy to this problem, three different corrections are applied and tested on the carefully derived model. The dynamics of the system subjected to purely axial flow with no rotation is successfully studied with the viscous model for both laminar and turbulent flow conditions. Because no experimental or previous theoretical data is available, it is impossible to validate the results obtained in the laminar regime. For the turbulent regime, as the Reynolds number is increased, the results tend more and more towards those obtained with the inviscid theory. / The results obtained for small rates of rotation show that both in the laminar and in the turbulent regime, the system tends to be stabilised when subjected to a small rate of rotation. On the other hand, this tendency should be reversed for higher rates of rotation, but it is impossible to show this due to the limitations of the root-finding method employed.

Under certain conditions, flow down an inclined plane destabilizes and a persistent series of interfacial waves develop. An interest in determining under what conditions a flow becomes unstable and how the interface develops has motivated researchers to derive several models for analyzing this problem. The first part of this thesis compares three models for flow down a wavy, inclined plane with the goal of determining which best predicts features of the flow. These models are the shallow-water model (SWM), the integral-boundary-layer (IBL) model, and the weighted residual model (WRM). The model predictions for the critical Reynolds number for flow over an even bottom are compared to the theoretical value, and the WRM is found to match the theoretical value exactly. The neutral stability curves predicted by the three models are compared to two sets of experimental data, and again the WRM most closely matches the experimental data. Numerical solutions of the IBL model and the WRM are compared to numerical solutions of the full Navier-Stokes equations; both models compare well, although the WRM matches slightly better. Finally, the critical Reynolds numbers for the IBL model and the WRM for flow over a wavy incline are compared to experimental data. Both models give results close to the data and perform equally well. These comparisons indicate that the WRM most accurately models the flow. In the second part of the thesis, the WRM is extended to include the effects of bottom heating and permeability. The model is used to predict the effect of heating and permeability on the stability of the flow, and the results are compared to theoretical predictions from the Benney equation and to a perturbation solution of the Orr-Sommerfeld equation from the literature. The results indicate that the model does faithfully predict the theoretical critical Reynolds number with heating and permeability, and both effects destabilize the flow. Finally, numerical simulations of the model equations are compared to full numerical solutions of the Navier-Stokes equations for the case with bottom permeability. The results are found to agree, which indicates that the WRM remains appropriate when permeability is included.

This work seeks, through numerical simulations as well as analysis, to derive from relatively simple models an intuitive understanding of the dynamics and behaviour of flow in lakes near the bottom boundary. The main body is divided into two equally important sections, the analysis and simulation of the effects of nonlinear (quadratic) bottom drag on the flow, and the simulation of the effects of topography on lake dynamics as it relates to the redistribution of sediment from the lakebed. The simulations all follow a structured scheme, beginning with relatively simple one-dimensional models to build intuition and proceeding to full two-dimensional simulations using the weakly nonhydrostatic shallow water equations. Thus this work seeks to build an understanding of the behaviour of the modified shallow water equations (a good representation of lake behaviour) and to analyze the effects of nonlinear drag and bottom topography on these systems. The nonlinear drag chapters demonstrate that the addition of a nonlinear friction term, while very efficient at removing energy from the system, also causes interesting new behaviour. In the pendulum (a good one dimensional analogy to the shallow water equations) the presence of nonlinear drag alters the parameter space enough to induce or destroy chaotic behaviour. A phenomenon worth considering in relation the shallow water equations. Additionally, the presence of drag causes as a cascade in spectral space, similar to the classical turbulent cascade. This work considers this effect and seeks to differentiate it from the turbulent cascade wherever and whenever possible. The final section of the thesis deals with the presence and effects of bottom topography (namely protrusions from the lake bed) on wave velocities due to a basin-scale seiche. This section examines both the dynamics of the system, through deflection about topography and the modification of the wave due to nonlinearity and bathymetry, as well as the relationship between lake dynamics and sediment redistribution. Finally, possible future directions are suggested as natural extensions to the work already done, as well as more sophisticated numerical models which could provide further insight into the problems discussed herein.