A new method to perform direct numerical simulations of wall-bounded flows has been developed and implemented. The method uses high-order compact finite differences in wall-normal (for channel flow) or radial direction (for pipe flow) on a collocated grid, which gives high-accuracy results without the effectfof filtering caused by frequent interpolation as required on a staggered grid. The use of compact finite differences means that extreme clustering near the wall leading to small time steps in high-Reynolds number simulations is avoided. The influence matrix method is used to ensure a completely divergence-freesolution and all systems of equations are solved in banded form, which ensures an effcient solution procedure with low requirements for data storage. The method is unique in the sense that exactly divergence-free solutions on collocated meshes are calculated using arbitrary dffierence matrices. The code is validated for two flow cases, i.e. turbulent channel and turbulent pipe flow at relatively low Reynolds number. All tests show excellent agreement with analytical and existing results, confirming the accuracy and robustness ofthe method. The next step is to eciently parallelise the code so that high-Reynolds number simulations at high resolution can be performed. We furthermore investigated rare events occurring in the near-wall region of turbulent wall-bounded flows. We find that negative streamwise velocities and extreme wall-normal velocity uctuations are found rarely (on the order of 0:01%), and that they occur more frequently at higher Reynolds number. These events are caused by strong vortices lying further away from the wall and it appears that these events are universal for wall-bounded flows. / <p>QC 20150303</p>

In this thesis, we develop a feedback-invariant theory of local controllability for affine distributions. We begin by developing an unexplored notion in control theory that we call proper small-time local controllability (PSTLC). The notion of PSTLC is developed for an abstraction of the well-known notion of a control-affine system, which we call an affine system. Associated to every affine system is an affine distribution, an adaptation of the notion of a distribution. Roughly speaking, an affine distribution is PSTLC if the local behaviour of every affine system that locally approximates the affine distribution is locally controllable in the standard sense. We prove that, under a regularity condition, the PSTLC property can be characterized by studying control-affine systems. The main object that we use to study PSTLC is a cone of high-order tangent vectors, or variations, and these are defined using the vector fields of the affine system. To better understand these variations, we study how they depend on the jets of the vector fields by studying the Taylor expansion of a composition of flows. Some connections are made between labeled rooted trees and the coefficients appearing in the Taylor expansion of a composition of flows. Also, a relation between variations and the formal Campbell-Baker-Hausdorff formula is established. After deriving some algebraic properties of variations, we define a variational cone for an affine system and relate it to the local controllability problem. We then study the notion of neutralizable variations and give a method for constructing subspaces of variations. Finally, using the tools developed to study variations, we consider two important classes of systems: driftless and homogeneous systems. For both classes, we are able to characterize the PSTLC property. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2010-01-11 20:11:45.466

In this dissertation, Richardson extrapolation and other computational techniques are used to develop a series of high accuracy high efficiency solution techniques for solving partial differential equations (PDEs). A Richardson extrapolation-based sixth-order method with multiple coarse grid (MCG) updating strategy is developed for 2D and 3D steady-state equations on uniform grids. Richardson extrapolation is applied to explicitly obtain a sixth-order solution on the coarse grid from two fourth-order solutions with different related scale grids. The MCG updating strategy directly computes a sixth-order solution on the fine grid by using various combinations of multiple coarse grids. A multiscale multigrid (MSMG) method is used to solve the linear systems resulting from fourth-order compact (FOC) discretizations. Numerical investigations show that the proposed methods compute high accuracy solutions and have better computational efficiency and scalability than the existing Richardson extrapolation-based sixth order method with iterative operator based interpolation. Completed Richardson extrapolation is explored to compute sixth-order solutions on the entire fine grid. The correction between the fourth-order solution and the extrapolated sixth-order solution rather than the extrapolated sixth-order solution is involved in the interpolation process to compute sixth-order solutions for all fine grid points. The completed Richardson extrapolation does not involve significant computational cost, thus it can reach high accuracy and high efficiency goals at the same time. There are three different techniques worked with Richardson extrapolation for computing fine grid sixth-order solutions, which are the iterative operator based interpolation, the MCG updating strategy and the completed Richardson extrapolation. In order to compare the accuracy of these Richardson extrapolation-based sixth-order methods, truncation error analysis is conducted on solving a 2D Poisson equation. Numerical comparisons are also carried out to verify the theoretical analysis. Richardson extrapolation-based high accuracy high efficiency computation is extended to solve unsteady-state equations. A higher-order alternating direction implicit (ADI) method with completed Richardson extrapolation is developed for solving unsteady 2D convection-diffusion equations. The completed Richardson extrapolation is used to improve the accuracy of the solution obtained from a high-order ADI method in spatial and temporal domains simultaneously. Stability analysis is given to show the effects of Richardson extrapolation on stable numerical solutions from the underlying ADI method.

The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint.

In a challenging and constantly changing world, students are required to develop advanced thinking skills such as critical systematic thinking, decision making and problem solving. This challenge requires developing critical thinking abilities which are essential in unfamiliar situations. A central component in current reforms in mathematics and science studies worldwide is the transition from the traditional dominant instruction which focuses on algorithmic cognitive skills towards higher order cognitive skills. The transition includes, a component of scientific inquiry, learning science from the student's personal, environmental and social contexts and the integration of critical thinking. The planning and implementation of learning strategies that encourage first order thinking among students is not a simple task. In an attempt to put the importance of this transition in mathematical education to a test, we propose a new method for mathematical instruction based on the infusion approach put forward by Swartz in 1992. In fact, the model is derived from two additional theories., that of Ennis (1989) and of Libermann and Tversky (2001). Union of the two latter is suggested by the infusion theory. The model consists of a learning unit (30h hours) that focuses primarily on statistics every day life situations, and implemented in an interactive and supportive environment. It was applied to mathematically gifted youth of the Kidumatica project at Ben Gurion University. Among the instructed subjects were bidimensional charts, Bayes law and conditional probability; Critical thinking skills such as raising questions, seeking for alternatives and doubting were evaluated. We used Cornell tests (Ennis 1985) to confirm that our students developed critical thinking skills.

Les travaux de cette thèse, réalisés au sein de l’équipe CLEF/DMFN de l’ONERA (Office National d’ Etudes et de Recherches Aérospatiales) en partenariat avec le laboratoire DynFluid et le CIRT (Consortium Industrie-Recherche en Turbomachines) s’inscrivent dans une demarche d’amélioration des outils de simulations pour les turbomachines. Compte tenu de ce contexte, l’objectif de cette étude est de développer de nouvelles méthodes pour le traitement des raccords non coincidents dans les turbomachines qui soit à la fois d’ordre elevé et conservatifs. Les développements proposés sont validés et composés de configurations de difficulté croissante. / A high-order and conservative method is developed for the numerical treatment of interface conditions in patched grids, based on the use of a ctitious grid methodology. The proposed approach is compared with a non-conservative interpolation of the state variables from the neighbouring domain for selected internal fow problems.

Nous présentons dans cette thèse une approche nouvelle de la segmentation d’images, avec des descripteurs a priori utilisant des champs de Markov d’ordre supérieur. Nous représentons le modèle de forme par un graphe de distribution de points qui décrit les informations a priori des invariants de pose grâce à des cliques L1 discrètes d’ordre supérieur. Chaque clique de triplet décrit les variations statistiques locales de forme par des mesures d’angle,ce qui assure l’invariance aux transformations globales (translation, rotation et échelle). L’apprentissage d’une structure de graphe discret d’ordre supérieur est réalisé grâce à l’apprentissage d’un champ de Markov aléatoire utilisant une décomposition duale, ce qui renforce son efficacité tout en préservant sa capacité à rendre compte des variations.Nous introduisons la connaissance a priori d’une manière innovante pour la segmentation basée sur un modèle. Le problème de la segmentation est ici traité par estimation statistique d’un maximum a posteriori (MAP). L’optimisation des paramètres de la modélisation- c’est à dire de la position des points de contrôle - est réalisée par le calcul d’une fonction d’énergie globale de champs de Markov (MRF). On combine ainsi les calculs statistiques régionaux et le suivi des frontières avec la connaissance a priori de la forme.Les descripteurs invariants sont estimés par des potentiels de Markov d’ordre 2, tandis que les caractéristiques régionales sont transposées dans un espace de caractéristiques et calculées grâce au théorème de la Divergence.De plus, nous proposons une nouvelle approche pour la segmentation conjointe de l’image et de sa modélisation ; cette méthode permet d’obtenir une segmentation plus fine lorsque la délimitation précise d’un objet est recherchée. Un modèle graphique combinant l’information a priori et les informations de pixel est développé pour réaliser l’unité des modules "top-down" et "bottom-up". La cohérence entre l’image et sa modélisation est assurée par une décomposition qui associe les parties du modèle avec la labellisation de chaque pixel.Les deux champs de Markov d’ordre supérieur considérés sont optimisés par les algorithmes de l’état de l’art. Les résultats prometteurs dans les domaines de la vision par ordinateur et de l’imagerie médicale montrent le potentiel de cette méthode appliquée à la segmentation. / In this thesis, we propose a novel framework for knowledge-based segmentation using high-order Markov Random Fields (MRFs). We represent the shape model as a point distribution graphical model which encodes pose invariant shape priors through L1 sparse higher order cliques. Each triplet clique encodes the local shape variation statistics on the angle measurements which inherit invariance to global transformations (i.e. translation,rotation and scale). A sparse higher-order graph structure is learned through MRF training using dual decomposition, producing boosting efficiency while preserving its ability to represent the shape variation.We incorporate the prior knowledge in a novel framework for model-based segmentation.We address the segmentation problem as a maximum a posteriori (MAP) estimation in a probabilistic framework. A global MRF energy function is defined to jointly combine regional statistics, boundary support as well as shape prior knowledge for estimating the optimal model parameters (i.e. the positions of the control points). The pose-invariant priors are encoded in second-order MRF potentials, while regional statistics acting on a derived image feature space can be exactly factorized using Divergence theorem. Furthermore, we propose a novel framework for joint model-pixel segmentation towardsa more refined segmentation when exact boundary delineation is of interest. Aunified model-based and pixel-driven integrated graphical model is developed to combine both top-down and bottom-up modules simultaneously. The consistency between the model and the image space is introduced by a model decomposition which associates the model parts with pixels labeling. Both of the considered higher-order MRFs are optimized efficiently using state-of the-art MRF optimization algorithms. Promising results on computer vision and medical image applications demonstrate the potential of the proposed segmentation methods.

Neste trabalho apresentamos uma análise de possíveis processos de tunelamento em sistemas unidimensionais através do estudo do potencial de barreira dupla com região intermediária confinante, conseguimos verificar a existência de um processo de tunelamento que ocorre através da ocupação virtual da região intermediária. Uma modelagem deste fenômeno é proposta baseando-se em uma teoria perturbativa realizada em termos de estados \"quase-localizados\" da partícula. Além da descrição qualitativa do fenômeno, determinamos as condições físicas para que este processo de tunelamento, também chamado de cotunelamento ou tunelamento de ordem superior, ocorra. Como resultado, recuperamos com boa aproximação o coeficiente de transmissão exato do sistema escolhido. Por fim, um outro resultado obtido durante o desenvolvimento deste trabalho foi uma prova conclusiva da convergência do método de diferenças finitas FDTD aplicada à equação de Schrödinger. Esta prova, diferente das demais encontradas na literatura, conseguiu demonstrar de maneira conclusiva a inclusão do limite superior para o passo temporal, de maneira a assegurar a convergência das soluções numéricas, algo até então testado numericamente, mas sem uma prova rigorosa da sua validade. / In this work, we present a study about tunnelling processes occurring in one-dimensional systems. Choosing a double well potential with a confining region as a case study, we verified that there is a tunnelling process which is due to virtual occupation of the confining region. Using perturbative theory of quasi-localized states to describe the particles dynamics, we provide a qualitative description of the phenomenon of tunnelling through virtual occupation, and we are capable of determining the conditions for which such a tunneling (also known as cotunnelling or high-order tunneling) should be present. With this analysis we could calculate with good approximation the particle transmission coefficient through the barrier. Finally, we also provide a rigorous proof of the convergence conditions for the numerical calculation of the Schrödinger equation using the finite difference method.

Nesta tese foram desenvolvidas metodologias sintéticas para a abertura de N-tosilaziridinas, provenientes de aminoácidos, via arilcianocupratos de ordem superior, os quais foram gerados pela reação de troca telúrio/cobre entre teluretos arílicos e cianocupratos; tais aberturas levam à formação de estruturas análogas a fenetilaminas em bons rendimentos, as quais possuem atividades farmacológicas interessantes. Adicionalmente, um conjunto de amino teluretos contendo diferentes grupos protetores foram preparados a partir dos respectivos aminoálcoois, em bons rendimentos (72 a 83%), e avaliados como potenciais espécies nucleofílicas. Os teluretos, que contêm como grupo protetor benzoíla, foram utilizados em reações de troca telúrio/lítio, gerando as espécies dilitiadas, as quais foram capturadas com diversos eletrófilos, levando à formação de aminoálcoois de cadeia maior e também à formação de análogos de fenetilaminas em bons rendimentos. Nesta tese também foram estudadas reações que envolvem química de arinos, os quais podem ser gerados de várias maneiras. Dentre essas formas para produzir os arinos, destacamos o uso de 2-(trimetilsilil)fenil triflato, o qual, na presença de fontes de íons fluoreto em solvente polar aprótico, gera a espécie reativa. Empregamos essa metodologia para geração de benzinos na reação de inserção em ligações do tipo &#963; de disselenetos de diarila. Tendo visto o sucesso da metodologia desenvolvida, decidimos sintetizar outros arinos derivados do 2-(trimetilsilil)fenil triflato e empregá-los em outro tipo de reação de inserção em ligações do tipo &#963;, desta vez em ligações entre selênio/estanho de tributil(fenilselanil)estanana, levando a uma difuncionalização nas posições 1 e 2 do anel aromático. / In this PhD thesis we developed a synthetical methodology for the ring opening reaction of N-tosyl aziridines by high order arylcyanocuprates that have been generated by tellurium/copper exchange reaction between aryl tellurides and cyanocuprates. These ring opening reactions lead to an interesting class of phenethylamines in good yields. Additionally, a set of tellurium amines containing different protecting groups were conveniently prepared in good yields (72 - 83%) from aminoalcohols and evaluated as potencial nucleophilic species. The corresponding tellurium amines have been evaluated in the tellurium/lithium exchange reaction. The nitrogen-containing organolithium compounds were efficiently prepared by using N-Bz tellurium amines and a mixture of n-butyllithium and lithium naphthalenide (LiNp) to performe the exchange reaction. The dianion intermediates were trapped with a wide range of electrophiles, resulting in this corresponding products in good to excellent yields. The reaction was also employed in the synthesis of phenethylamines. In this work we also studied reactions involving aryne chemistry, that are reactive species. A number of methods for benzyne formation have been reported in the literature. Among them we pay particular attention to the use of 2-(trimethylsilyl)aryl triflates in the presence of fluoride ion sources and aprotic polar solvents that generate this reactive species. We employed this methodology for generation of benzyne in the insertion reaction of diaryl diselenide &#963; bond. In view of the success of the developed methodology, we decided to synthesize other ayne derivatives from 2-(trimethylsilyl)phenyl triflate and use them on another type of &#963; bond insertion reaction. The insertion reaction of arynes into Se-Sn sigma bond of tributyl(phenylselanyl)stannane, leading a double functionality in the aromatic system.

No contexto de Dinâmica de Fluidos Computacional, métodos de simulação de objetos imersos em Malhas Cartesianas têm se mostrado vantajosos tanto em termos de Custo Computacional quanto em termos de precisão numérica. Entretanto, a representação física de objetos imersos nesses domínios computacionais impõe a perda de validade dos esquemas de Diferenças Finitas empregados, na região das superfícies introduzidas. Este trabalho analisa um Método de Interfaces Imersas quanto ao desempenho em aplicações a esquemas de solução numérica de Alta Ordem de precisão. Através de Testes de Refinamento de Malha, é feita a apreciação da ordem de decaimento dos erros das soluções numéricas em comparação com as soluções analíticas para 2 problemas unidimensionais. O primeiro envolve a solução da Equação de Calor unidimensional sujeita a uma Condição Inicial Unitária, e o segundo relaciona-se ao cálculo das duas primeiras derivadas espaciais das funções analíticas Seno e Tangente Hiperbólica. Também é promovida uma análise de forma fragmentária do método, a fim de individualizar a contribuição dos elementos envolvidos no comportamento das soluções geradas. Os resultados obtidos indicam eventuais alterações na ordem de precisão dos esquemas de Diferenças Finitas originalmente aplicados. Esse comportamento e visto como uma dependência que o método escolhido apresenta em relação a função discretizada. Por fim, são elaboradas considerações sobre restrições de aplicabilidade do método escolhido. / In the Computational Fluid Dynamics context, methods for simulating immersed objects in Cartesian Grids have shown advantages regarding both Computational Cost and numerical precision. Nevertheless, the physical representation of immersed objects within these computational domains leads to the loss of validity of the emplyed Finite Dierence Schemes near the immersed surfaces. This work analizes a Immersed Interface Method regarding its performance in High Order Schemes applications. The error decay order for numerical solutions of two 1D problems is observed. The rst problem relates to the solution of the Heat Equation subjected to the unitary initial condition. The second relates to the computation of the rst two derivatives of analytical functions Sin and Hyperbolic Tangent. It\'s also conducted a fragmentary analysis, which is intended to identify the contribution of each element of this method to the character of the generated solution. The results indicate some eventual changes in the Order of the Finite Dierences Schemes employed. This behaviour is regarded as a dependency of this method to the nature of the discretized function. Finaly, some remarks regarding restrictions to this method\'s applicability are made.