Philosophiae Doctor - PhD / Fitted operator finite difference methods (FOFDMs) for singularly perturbed problems have been explored for the last three decades. The construction of these numerical schemes is based on introducing a fitting factor along with the diffusion coefficient or by using principles of the non-standard finite difference methods. The FOFDMs based on the latter idea, are easy to construct and they are extendible to solve partial differential equations (PDEs) and their systems. Noting this flexible feature of the FOFDMs, this thesis deals with extension of these methods to solve interior layer problems, something that was still outstanding. The idea is then extended to solve singularly perturbed time-dependent PDEs whose solutions possess interior layers. The second aspect of this work is to improve accuracy of these approximation methods via methods like Richardson extrapolation. Having met these three objectives, we then extended our approach to solve singularly perturbed two-point boundary value problems with variable diffusion coefficients and analogous time-dependent PDEs. Careful analyses followed by extensive numerical simulations supporting theoretical findings are presented where necessary.

Surface chemistry plays a key role in science and technology because materials interact with their environments through their surfaces. Understanding surface chemistry can help alter/improve the properties of materials. However, surface characterization and modification often require multiple characterization and synthesis techniques. Silicon/silica-based materials are technologically important, so studying their surface properties can enable future advancements. In this dissertation, I explore surface modification and characterization of different types of Si/SiO2 thin films, including silicon wafers, fused silica capillary columns, and oblique angle sputtered Si/SiO2 thin films. In Chapters 2-5, I first present a method to rapidly silanize silica surfaces using a gas-phase synthesis that employs a small aminosilane that passivates/deactivates silicon wafers and the inner surfaces of capillary columns. This deposition takes place in a flow-through, atmospheric pressure, gas-phase reactor. This surface modification results in a significant decrease in the number of free surface silanols, which was confirmed by high-sensitivity low energy ion scattering (HS-LEIS), X-ray photoelectron spectroscopy (XPS), and spectroscopic ellipsometry (SE). I then show that this silanization inhibits atomic layer deposition (ALD) of zinc oxide (ZnO), which is an important optical thin film material. Finally, I performed in-depth characterization of thin films of oblique angle deposited porous Si/SiO2. These films have been used as the active coatings in solid phase microextraction (SPME) devices. The characterization and analysis in this study were mainly by scanning transmission electron microscopy (STEM) and various computational microstructural characterization techniques, e.g., two-point statistics. The rest of my dissertation focuses on XPS data analysis and interpretation. I first show box plots as a simple graphical tool for determining overfitting in XPS peak fitting. I next present a series of chemometrics/informatics analyses of an XPS image dataset from a patterned silicon surface with different oxide thicknesses. This dataset was probed via an initial, graphical analysis of the data, summary statistics with a focus on pattern recognition entropy (PRE), principal component analysis (PCA), multivariate curve resolution (MCR), and cluster analysis (CA).

Sensory adaptation is an important aspect of perception. A seemingly non-beneficial consequence of adaptation is that it produces perceptual illusions. For instance, following focal adaptation, the perceived separation between stimuli straddling the adapted attribute or region is often exaggerated. This type of illusion, known as perceptual repulsion, is both a consequence of and a clue to the brain’s coding strategies and how they are influenced by recent sensory events. Adaptation-induced perceptual repulsion has been well documented in vision (e.g. the tilt aftereffect) and to a lesser extent in audition, but rarely studied in touch. The present thesis investigated the effects of adaptation on tactile spatial perception using a combination of human psychophysics and computational modeling. In a two-interval forced choice task, participants compared the perceived separation between two point-stimuli applied on the forearms successively. The point of subjective equality was extracted as a measure of perceived two-point distance. We showed that tactile spatial perception is subject to an adaptation-induced repulsion illusion: vibrotactile adaptation focally reduced tactile sensitivity and significantly increased the perceived distance between points straddling the adapted skin site (Chapter 2). This repulsion illusion, however, was not observed when the intervening skin was desensitized with topical anesthesia instead of vibrotactile adaptation, suggesting that peripheral desensitization alone is insufficient to induce the illusion (Chapter 3). With Bayesian perceptual modeling, we showed that the illusion was consistent with the hypothesis that the brain decodes tactile spatial input without awareness of the adaptation state in the nervous system (Chapter 4). Together, the empirical and theoretical work furthers the understanding of dynamic tactile spatial coding as the somatosensory system adapts to the sensory environment. Its main findings are consistent with the adaptation- induced repulsion illusions reported in vision and audition, suggesting that perception in different sensory modalities shares common processing features and computational principles. / Thesis / Doctor of Philosophy (PhD) / Sensory adaptation can shape how we perceive the world. In this thesis, we showed that the perception of space in touch is pliable and subject to the influence of adaptation. Psychophysical testing in human participants showed that vibratory adaptation induced an illusion that expanded the perceived distance between stimuli on the skin. This illusion provides clues into how information about space in touch is normally processed and interpreted by the brain. In addition, we developed a computational model that used a powerful statistical framework – Bayesian inference – to probe touch on a theoretical basis. To the best of our knowledge, the present thesis provides the first combined psychophysical and computational study on the effects of adaptation on tactile spatial perception. Our findings suggest that touch shares some common information processing principles with vision and hearing, and adaptation plays a functionally similar role in mediating this process across the senses.

<div>Optimal control theory (OCT) exists since the 1950s. However, with the advent of modern computers, the design community delegated the task of solving the optimal control problems (OCPs) largely to computationally intensive direct methods instead of methods that use OCT. Some recent work showed that solvers using OCT could leverage parallel computing resources for faster execution. The need for near real-time, high quality solutions for OCPs has therefore renewed interest in OCT in the design community. However, certain challenges still exist that prohibits its use for solving complex practical aerospace problems, such as landing human-class payloads safely on Mars.</div><div><br></div><div>In order to advance OCT, this thesis introduces Epsilon-Trig regularization method to simply and efficiently solve bang-bang and singular control problems. The Epsilon-Trig method resolves the issues pertaining to the traditional smoothing regularization method. Some benchmark problems from the literature including the Van Der Pol oscillator, the boat problem, and the Goddard rocket problem verified and validated the Epsilon-Trig regularization method using GPOPS-II.</div><div><br></div><div>This study also presents and develops the usage of trigonometry for incorporating control bounds and mixed state-control constraints into OCPs and terms it as Trigonometrization. Results from literature and GPOPS-II verified and validated the Trigonometrization technique using certain benchmark OCPs. Unlike traditional OCT, Trigonometrization converts the constrained OCP into a two-point boundary value problem rather than a multi-point boundary value problem, significantly reducing the computational effort required to formulate and solve it. This work uses Trigonometrization to solve some complex aerospace problems including prompt global strike, noise-minimization for general aviation, shuttle re-entry problem, and the g-load constraint problem for an impactor. Future work for this thesis includes the development of the Trigonometrization technique for OCPs with pure state constraints.</div>

Un mascaret est une vague remontant contre le courant d’un fleuve lorsque la marée se propage dans un estuaire. À son passage, le mascaret induit une forte turbulence et un fort mélange dont les effets sur la vie de l’estuaire sont encore mal quantifiés. Ici, le phénomène est étudié expérimentalement et numériquement en utilisant un modèle d’onde positive se propageant contre un courant permanent.L’étude en laboratoire a permis de mesurer les variations de la surface libre, de la vitesse de l’écoulement ainsi que des échelles de turbulence. Lors de son passage, des fluctuations importantes de la surface libre et de la vitesse de l’écoulement sont observées, ainsi que des variations des échelles de turbulences. Des structures turbulentes semblent se former près du fond sous le front de l’onde et montent dans la colonne d’eau après le passage du front.La simulation numérique fut réalisée à partir de données expérimentales d’onde positive ondulée sur fond lisse. Une validation des méthodes numériques a été réalisée pour différente configuration. Les résultats des simulations d’onde positives donnent une cartographie détaillée de l’écoulement dans tout le canal. De plus, la simulation a permis d’identifier une inversion de la vitesse près des parois lors du passage des crêtes des ondes générant dans certaines configurations des structures turbulentes. / Tidal bores are surge waves propagating upstream rivers as the tide rushes into estuaries. They induce large turbulences and mixing of the river and estuary flow of which effects remain scarcely studied. Herein, tidal bores are investigated experimentally and numerically with an idealised model of positive surges propagating upstream an initially steady flow. The experimental work estimated flow changes and typical turbulent length scale evolution induced by undular bores with and without breaking roller. The bore passage was associated with large free surface and flow velocity fluctuations, together with some variations of the integral turbulent scales. Coherent turbulent structures appeared in the wake of leading wave near the bed and moved upward into the water column during the bore propagation. The numerical simulations were based on previous experimental work on undular bores. Some test cases were realised to verify the accuracy of the numerical methods. The results gave access to the detailed flow evolution during the bore propagation. Large velocity reversals were observed close to the no-slip boundaries. In some configurations, coherent turbulent structures appeared against the walls in the wake of the bore front.