No description available.

No description available.

The development of efficient mathematical models and numerical methods for the study of haemodynamics is becoming increasingly prominent in the analysis of pathological states of the cardiovascular system. Computational models contribute to medical diagnosis processes, reducing the need of classical invasive medical techniques, which are not risk-free for patients and generate high healthcare costs. The first part of this thesis focuses on the modeling and simulation of coronary blood flow, with emphasis on stable Coronary Artery Disease (CAD), a pathological condition that occurs when an abnormal narrowing builds inside coronary vessel walls. Our goal is to develop a CCTA-based Fractional Flow Reserve (FFR) model which incorporates clinical imaging and patient-specific characteristics to predict the haemodynamic behavior and properties of individuals, reducing the need for invasive measurements. A novel aspect of the proposed methodology is the inclusion of the pressure guidewire, used in clinical settings, and the assessment of its impact on local fluid dynamics and FFR predictions. Thereafter, the second part of this dissertation is devoted to the development of numerical methods for the simulation of incompressible flows with particular emphasis on the simulation of cardiovascular haemodynamics. A novel implicit hybrid finite volume/finite element methodology for the efficient simulation of blood flow is proposed and validated. The implicit discretization of the transport-diffusion equations making use of an inexact Newton-Krylov method with an SGS preconditioner yields to an efficient scheme avoiding the severe CFL condition arising in explicit or semi-implicit methods for blood flow dynamics. Besides, the Ducros flux function employed for the nonlinear convective terms leads to a provably kinetic energy stable scheme of the advection terms. In addition, a staggered semi-implicit method for the simulation of incompressible flows in one-dimensional elastic and viscoelastic vessels is proposed. The convective stage is treated explicitly in time, while the diffusive and pressure stage are handled implicitly to avoid strict bounds on the time steps. The one-dimensional methodology is then extended to networks of vessels by introducing a local three-dimensional representation of the junction.

The chemical oxygen-iodine laser (COIL) has several important military and industrial applications. The concern of this work is do develop a partial differential equation model describing optical behavior in the COIL. Optical behavior of the COIL has traditionally been investigated via a ray tracing method. Photons are represented as discrete particles, and their behavior is described by the geometry of the system. We develop an optical model wherein photons have a wave description. In order to construct the mathematical model, we utilize the theory of paraxial wave optics and Gaussian beams. Doing so allows us to incorporate physical effects such as diffusion/diffraction and refraction into the model. After describing the optical model, we present numerical methods for obtaining approximate solutions to the model in the cases of one and two transverse directions. Results are presented illustrating the efficacy of the numerical methods. / Ph. D.

Neurostimulation continues to demonstrate tremendous success as an intervention for neurodegenerative diseases, including Parkinson's disease, in addition to a range of other neurological and psychiatric disorders. In an effort to enhance the medical efficacy and comprehension of this form of brain therapy, modeling and computational simulation are regarded as valuable tools that enable in silico experiments for a range of neurostimulation research endeavours. To fully realize the capacities of neurostimulation simulations, several areas within computation and numerics need to be considered and addressed. Specifically, simulations of neurostimulation that incorporate (i) computational efficiency, (ii) application versatility, and (iii) characterizations of cellular-level electrophysiology would be highly propitious in supporting advancements in this medical treatment. The focus of this dissertation is on these specific areas. First, preconditioners and iterative methods for solving the linear system of equations resulting from finite element discretizations of partial differential equation based transcranial electrical stimulation models are compared. Second, a software framework designed to efficiently support the range of clinical, biomedical, and numerical simulations utilized within the neurostimulation community is presented. Third, a multiscale model that couples transcranial direct current stimulation administrations to neuronal transmembrane voltage depolarization is presented. Fourth, numerical solvers for solving ordinary differential equation based ligand-gated neurotransmitter receptor models are analyzed. A fundamental objective of this research has been to accurately emulate the unique medical characteristics of neurostimulation treatments, with minimal simplification, thereby providing optimal utility to the scientific research and medical communities. To accomplish this, numerical simulations incorporate high-resolution, MRI-derived three-dimensional head models, real-world electrode configurations and stimulation parameters, physiologically-based inhomogeneous and anisotropic tissue conductivities, and mathematical models accepted by the brain modeling community. It is my hope that this work facilitates advancements in neurostimulation simulation capabilities, and ultimately helps improve the understanding and treatment of brain disease. / Ph. D.

A mathematical analysis seeking an accurate measure of the worth of numerical integration techniques used for real-time digital flight simulation problems is presented. This investigation allows the subject of best integration methods to be pursued making emphasis on the choice of practical steps and the use of available mathematical techniques to illustrate and evaluate a potential root matching approach involving a selected first-order differential system. This study allows certain evaluation techniques to be developed. Notable among these are the schemes for comparing roots of sampled ideal integrators to roots of sampled approximated integrators, the development of an integration and of an iteration formula, and the creation of a computer program.

Call number: LD2668 .T4 1978 G92 / Master of Science

In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein. / Mathematical Sciences / M.Sc. (Applied Mathematics)

published_or_final_version / abstract / Physics / Doctoral / Doctor of Philosophy

An adaptive refinement procedure combining the ideas of solving a modified difference equation and of adaptive mesh refinement is introduced. The numerical solution on a fixed grid is improved by inclusion of approximated truncation error computed from local subgrid refinement. Following this procedure, a reliable scheme has been developed for refined computations of the flow past a wedge at transonic speeds. Effects of the truncation error on the pressure, wave drag, sonic line, and shock position are investigated. By comparing the pressure drag on the wedge and the wave drag due to the shocks, the existence of a supersonic-to-supersonic shock originating from the wedge shoulder is confirmed.