This work investigates numerically the process of Teflon ablation using a finite-volume discretization, implicit time integration and a domain decomposition method in three-dimensions. The interest in Teflon stems from its use in Pulsed Plasma Thrusters and in thermal protection systems for reentry vehicles. The ablation of Teflon is a complex process that involves phase transition, a receding external boundary where the heat flux is applied, an interface between a crystalline and amorphous (gel) phase and a depolymerization reaction which happens on and beneath the ablating surface. The mathematical model used in this work is based on a two-phase model that accounts for the amorphous and crystalline phases as well as the depolymerization of Teflon in the form of an Arrhenius reaction equation. The model accounts also for temperature-dependent material properties, for unsteady heat inputs and boundary conditions in 3D. The model is implemented in 3D domains of arbitrary geometry with a finite volume discretization on unstructured grids. The numerical solution of the transient reaction-diffusion equation coupled with the Arrhenius-based ablation model advances in time using implicit Crank-Nicolson scheme. For each time step the implicit time advancing is decomposed into multiple sub-problems by a domain decomposition method. Each of the sub-problems is solved in parallel by Newton-Krylov non-linear solver. After each implicit time-advancing step, the rate of ablation and the fraction of depolymerized material are updated explicitly with the Arrhenius-based ablation model. After the computation, the surface of ablation front and the melting surface are recovered from the scalar field of fraction of depolymerized material and the fraction of melted material by post-processing. The code is verified against analytical solutions for the heat diffusion problem and the Stefan problem. The code is validated against experimental data of Teflon ablation. The verification and validation demonstrates the ability of the numerical method in simulating three dimensional ablation of Teflon.

This project considers the diffusion of water molecules through a cellular medium in which the cells are modeled by square compartments placed symmetrically in a square domain. We assume the diffusion process is governed by the 2D diffusion equations and the solution is provided by implementing the Crank-Nicolson scheme. These results are verified and illustrated to agree well with the finite element method using the Comsol Multiphysics package. The model is used to compute the values of the apparent diffusion coefficient, (ADC) which is a measure that is derived from diffusion weighted MRI data and can be used to identify, e.g., regions of ischemia in the brain. With our model, it is possible to examine how the value of the apparent diffusion coefficient is affected whenever the extracellular space is varied. We observe that the average distance that the water molecules travel in a definite time is highly dependent on the geometrical properties of the cellular media. / UOIT

The scope of this thesis is in the time-dependent formulation of the two dimensional model of resonant electron-diatomic molecule collisions in the range of low energies. In its time independent form the model was previously numerically solved without the Born-Oppenheimer approximation with use of modern tools such as the finite element method with discrete variable representation (FEM-DVR) or exterior complex scaling (ECS). Within the scope of this model we numerically solve the evolution problem, with use of the Crank-Nicolson method and the Padé approximation. Later we evaluate the cross section of the elastic and some inelastic processes with the correlation function approach. At last we make a comparison of the evolution and the cross sections to time dependent formulation of the local complex potential approximation of the electron-molecule collisions.

In molecules, electronic state transitions can occur via quantum coupling of the states. If the coupling is due to the kinetic energy of the molecular nuclei, then electronic transitions are best represented in the adiabatic frame. If the coupling is instead facilitated through the potential energy of the nuclei, then electronic transitions are better represented in the diabatic frame. In our study, we modeled these latter transitions, called ``nonadiabatic transitions.'' For one nuclear degree of freedom, we modeled the de-excitation of a diatomic molecule. For two nuclear degrees of freedom, we modeled the de-excitation of an ethane-like molecule undergoing cis-trans isomerization. For both cases, we studied the dependence of the de-excitation on the nuclear configuration and potential energy of the molecule. We constructed a numerical model to solve the time-dependent Schr\"{o}dinger Equation for two coupled wave functions. Our algorithm takes full advantage of the sparseness of the numerical system, leading to a final set of equations that is solved recursively using nothing more than the Tridiagonal Algorithm. We observed that the most effective de-excitation occurred when the molecule transitioned from a stable equilibrium configuration to an unstable equilibrium configuration. This same mechanism is known to drive fast electronic transitions in the adiabatic frame. We concluded that while the adiabatic and diabatic frames are strongly opposed physically, the mathematical mechanism driving electronic transitions in the two frames is in some sense the same.

Distributed optimal control problems with transient convection dominated diffusion convection reaction equations are considered. The problem is discretized in space by using three types of discontinuous Galerkin (DG) method: symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), incomplete interior penalty Galerkin (IIPG). For time discretization, Crank-Nicolson and backward Euler methods are used. The discretize-then-optimize approach is used to obtain the finite dimensional problem. For one-dimensional unconstrained problem, Newton-Conjugate Gradient method with Armijo line-search. For two-dimensional control constrained problem, active-set method is applied. A priori error estimates are derived for full discretized optimal control problem. Numerical results for one and two-dimensional distributed optimal control problems for diffusion convection equations with boundary layers confirm the predicted orders derived by a priori error estimates.

A numerical model for describing the kinetics of intracellular water transport during cryopreservation was developed. As ice is formed outside the cell, depleting the extracellular liquid of water, the cell will experience an osmotic pressure difference across its membrane, which causes cell dehydration and concomitant shrinkage. Although Mazur (1963) has previously modeled this phenomenon as a two-compartment system with membrane limited transport, the assumption of well-mixed compartments breaks down at large Biot numbers. Therefore, we have developed a numerical solution to this moving-boundary problem, including diffusive transport in the intracellular liquid, in addition to the osmotically driven membrane flux. Our model uses a modified Crank-Nicolson scheme with a non-uniform Eulerian-Lagrangian grid, and is able to reproduce predictions from Mazurs model at low Biot numbers, while generating novel predictions at high Biot numbers. Given that cell damage may result from excessive water loss, our model can be used to predict freezing methods that minimize the probability of cell injury during the cryopreservation process.

This thesis interrogates the ways in which the body-centred practices of women film artists embrace the materiality of direct animation in order to foreground gendered, subjective positions. Through the researcher's own creative practice, it investigates how this mode of film-making, in which the artist works through physical engagement with the film materials and the material processes of film-making, might be understood as feminine and/or feminist. Direct animation foregrounds touch as the primary sense. Its practices are process-based and highly experimental, because images are made through the agency of the body operating within restrictive parameters, making results difficult to predict or control with precision. For these reasons, direct animation has not been embraced by mainstream, narrative-focused, studio-based models of production, unlike other forms of two and three dimensional animation. It has remained a specialist area for the individual artist and auteur, and, to date, there is a paucity of commentary about direct animation practices, and what exists has been dominated by male voices. In order to develop ideas about the ways in which women represent themselves in an expanded film-making praxis that is focused on the body and materiality of process, this PhD inquiry, encompassing a body of films with written contextualisation, is situated in the context of the direct animation practices of three artists (Caroline Leaf, Annabel Nicolson, and Margaret Tait); and informed by conceptual frameworks provided by Luce Irigaray, Julia Kristeva and Hélène Cixous. This thesis proposes, via interaction between these three axes of research, that women film artists, operating independently, are able to create a female imaginary that represents women and is recognised by them, by constructing positions of practice outside the dominant symbolic modes of patriarchy, which evolve through the maternal body and the materialities of the feminine.

The efficiency of the conventional, explicit finite difference time domain (FDTD)method is constrained by the upper limit on the temporal discretization, imposed by the Courant–Friedrich–Lewy (CFL) stability condition. Therefore, there is a growing interest in overcoming this limitation by employing unconditionally stable FDTD methods for which time-step and space-step can be independently chosen. Unconditionally stable Crank Nicolson method has not been widely used in time domain electromagnetics despite its high accuracy and low anisotropy. There has been no work on the Crank Nicolson FDTD (CN–FDTD) method for frequency dependent medium. In this thesis a new three-dimensional frequency dependent CN–FDTD (FD–CN–FDTD) method is proposed. Frequency dependency of single–pole Debye materials is incorporated into the CN–FDTD method by means of an auxiliary differential formulation. In order to provide a convenient and straightforward algorithm, Mur’s first-order absorbing boundary conditions are used in the FD–CN–FDTD method. Numerical tests validate and confirm that the FD–CN–FDTD method is unconditionally stable beyond the CFL limit. The proposed method yields a sparse system of linear equations which can be solved by direct or iterative methods, but numerical experiments demonstrate that for large problems of practical importance iterative solvers are to be used. The FD–CN–FDTD sparse matrix is diagonally dominant when the time-stepis near the CFL limit but the diagonal dominance of the matrix deteriorates with the increase of the time-step, making the solution time longer. Selection of the matrix solver to handle the FD–CN–FDTD sparse system is crucial to fully harness the advantages of using larger time-step, because the computational costs associated with the solver must be kept as low as possible. Two best–known iterative solvers, Bi-Conjugate Gradient Stabilised (BiCGStab) and Generalised Minimal Residual (GMRES), are extensively studied in terms of the number of iteration requirements for convergence, CPU time and memory requirements. BiCGStab outperforms GMRES in every aspect. Many of these findings do not match with the existing literature on frequency–independent CN–FDTD method and the possible reasons for this are pointed out. The proposed method is coded in Fortran and major implementation techniques of the serial code as well as its parallel implementation in Open Multi-Processing (OpenMP) are presented. As an application, a simulation model of the human body is developed in the FD–CN–FDTD method and numerical simulation of the electromagnetic wave propagation inside the human head is shown. Finally, this thesis presents a new method modifying the frequency dependent alternating direction implicit FDTD (FD–ADI–FDTD) method. Although the ADI–FDTD method provides a computationally affordable approximation of the CN–FDTD method, it exhibits a loss of accuracy with respect to the CN-FDTD method which may become severe for some practical applications. The modified FD–ADI–FDTD method can improve the accuracy of the normal FD–ADI–FDTD method without significantly increasing the computational costs.

Most of PDE-based restoration models and their numerical realizations show a common drawback: loss of fine structures. In particular, they often introduce an unnecessary numerical dissipation on regions where the image content changes rapidly such as on edges and textures. This thesis studies the magnitude data/imagery of magnetic resonance imaging (MRI) which follows Rician distribution. It analyzes statistically that the noise in the magnitude MRI data is approximately Gaussian of mean zero and of the same variance as in the frequency-domain measurements. Based on the analysis, we introduce a novel partial differential equation (PDE)-based denoising model which can restore fine structures satisfactorily and simultaneously sharpen edges as needed. For an efficient simulation we adopt an incomplete Crank-Nicolson (CN) time-stepping procedure along with the alternating direction implicit (ADI) method. The algorithm is analyzed for stability. It has been numerically verified that the new model can reduce the noise satisfactorily, outperforming the conventional PDE-based restoration models in 3-4 alternating direction iterations, with the residual (the difference between the original image and the restored image) being nearly edgeree. It has also been verified that the model can perform edge-enhancement effectively during the denoising of the magnitude MRI imagery. Numerical examples are provided to support the claim.

Portfolio selection has always been a fundamental challenge in the field of finance and captured the attention of researchers in the financial area. Merton's portfolio problem is an optimization problem in finance and aims to maximize an investor's portfolio. This thesis studies Merton's Optimal Investment-Consumption Problem under the Jourdain--Sbai stochastic volatility model and seeks to maximize the expected discounted utility of consumption and terminal wealth. The results of our study can be split into three main parts. First, we derived the Hamilton--Jacobi--Bellman equation related to our stochastic optimal control problem.  Second, we simulated the optimal controls, which are the weight of the risky asset and consumption. This has been done for all the three models within the scope of the Jourdain--Sbai model: Quadratic Gaussian, Stein & Stein, and Scott's model. Finally, we developed the system of equations after applying the Crank-Nicolson numerical scheme when solving our HJB partial differential equation.