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.

Significant effort has recently been directed towards the development of numerically efficient iterative techniques for the solution of boundary integral equation formulations of time harmonic scattering problems. The primary result of this effort has been the development of several advanced numerical techniques which enable the dense matrix-vector products associated with the iterative solution of boundary integral equations to be rapidly computed. However, an important aspect of this problem which has yet to be adequately addressed is the development of rapidly convergent iterative techniques to complement the relatively more mature numerical algorithms which expedite the matrix-vector product operation. To this end, a class of efficient iterative methods for boundary integral equation formulations of two-dimensional scattering problems is presented. This development is based on an attempt to approximately factor (i.e., renormalize) the boundary integral formulation of an arbitrary scattering problem into a product of one-way wave operators and a corresponding coupling operator which accounts for the interactions between oppositely propagating waves on the surface of the scatterer. The original boundary integral formulation of the scattering problem defines the coupling between individual equivalent sources on the surface of the scatterer. The renormalized version of this equation defines the coupling between the forward and backward propagating fields obtained by re-summing the individual equivalent sources present in the original boundary integral formulation of the scattering problem. An important feature of this class of rapidly convergent iterative techniques is that they are based on an attempt to incorporate the important physical aspects of the scattering problem into the iterative procedure. This leads to rapidly convergent iterative series for a number of two-dimensional scattering problems. The iterative series obtained using this renormalization procedure are much more rapidly convergent than the series obtained using Krylov subspace techniques. In fact, for several of the geometries considered the number of iterations required to achieve a specified residual error is independent of the size of the scatterer. This desirable property of the iterative methods presented here is not shared by other iterative schemes for wave scattering problems. Moreover, because the approach used to develop these iterative series depends only on the assumption that the total field can be approximately represented by a summation of independent and oppositely directed waves (and not on the presence of special geometries, etc.), the proposed iterative methods are very general and are thus applicable to a large number of complex scattering problems. / 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.

The purpose of this problem is to make a comparative study of methods in teaching Gregg shorthand. The problem is to compare the methods of approach, procedures, and techniques used, and to determine what has been accomplished in the way of experiments which have been performed by the different writers.

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This thesis presents an approximate solution of second order relative motion equations. The equations of motion for a Keplerian orbit in spherical coordinates are expanded in Taylor series form using reference conditions consistent with that of a circular orbit. Only terms that are linear or quadratic in state variables are kept in the expansion. A perturbation method is employed to obtain an approximate solution of the resulting nonlinear differential equations. This new solution is compared with the previously known solution of the linear case to show improvement, and with numerical integration of the quadratic differential equation to understand the error incurred by the approximation. In all cases, the comparison is made by computing the difference of the approximate state (analytical or numerical) from numerical integration of the full nonlinear Keplerian equations of motion. / Master of Science

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Typescript (photocopy). / Digitized by Kansas Correctional Industries

Reactive ion etching has been used to obtain anisotropic silicon trenches with small sidewall angles. This work demonstrates that the sidewall angle can be controlled by the wafer temperature and there exists an Arrhenius-type relationship among isotropic polymer deposition rate, thickness of polymer, and sidewall angle.