Missing page 91 / <p> This report presents a numerical method which can be used to solve the advection equation </p> <p> (∂ɸ/∂t) + (∂[u(x,t)ɸ]/∂x) = S(x,t) </p> where: </p> <p> ɸ ≣ concentration field </p> <p> u(x,t) ≣ velocity field </p> <p> S(x,t) ≣ source term </p> <p> Central to this method are the concept of particle path and the Eulerian interpretation of the time rate of change of the concentration field ɸ. </p> In actual comparison tests for particular cases with known solutions this method proved to be at least two orders of magnitude more accurate than the usual one sided upwind finite difference method. </p> / Thesis / Master of Engineering (ME)

Many complexities arise when writing software for computational physics. The choice of underlying data structures, physics model representation, and numerical methods used for the solver all add to the overall complexity of a code and significantly affect the simulation speed and accuracy of the solution. This work has integrated multiple recently developed software tools into a unified framework called ELEMENTS. ELEMENTS contains tools to address the complexities of data representation and numerical methods implementation for computational physics applications. ELEMENTS consists of multiple software packages: Elements, MATAR, Swage, Geometry, and SLAM. MATAR is a performance portability and productivity implementation of data-oriented design that leverages KOKKOS for multi-architecture portability. MATAR's data-oriented design allows for highly efficient memory use through the use of contiguous memory allocation and access for optimal performance. The elements library contains the requisite mathematical functions for a wide range of numerical methods and high order field representation, including the Serendipity basis set that allows for a higher-order solution with fewer degrees of freedom than the more standard tensor product elements. Swage is a novel mesh class capable of representing all of the geometric entities required to implement low and high-order continuous and discontinuous Galerkin methods on unstructured hexahedral meshes as well as connectivity structures between the disparate index spaces. SLAM is a library for linear algebra solvers and tools for linking to external solver packages. Combining these tools allows for the research and development of novel methods for solving problems in computational physics. This work discusses the ELEMENTS package and reviews multiple numerical methods built using ELEMENTS.

Thesis (M. A. (English Studies)) --University of Limpopo, 2019 / There is a universal consensus among educationalists and cognitive development theorists that integration of higher order thinking (HOT) in language teaching has farreaching positive implications in learners‘ future. Their extensive body of research clearly indicates the interrelationship between language and thinking. It shows that to develop well-rounded learners who can later deal capably with varying demands of the 21st century, teaching them linguistic and cognitive skills concurrently is a prerequisite. However, there is still a dearth of language teaching classroom-based data to be collected to ascertain which language pedagogic practices promote thinking or not. Hence, a qualitative exploratory case study was conducted to address this gap. The study was undertaken in five intermediate English FAL classes in Mankweng circuit. The aim was to establish whether HOT is encouraged in the intermediate English FAL classes. The study used two data analysis techniques: firstly, Tesch‘s inductive coding technique was used to analyse semi-structured interview results sourced from five English FAL teachers. They were sampled for the study to assess their conceptualisation of HOT and its application in their language classes. Contrastingly, Anderson and Krathwohl‘s (2001) framework was used to analyse one Grade 4 English workbook. To determine if its exercises‘ instructional verbs were promoting HOT or not; to check if the questions in its exercises were equally distributed over all the six levels of Bloom's revised Taxonomy of the cognitive domain; and to evaluate if there was an incremental introduction of HOTs in its exercises through the year. The results revealed the following: the five teachers could not conceptualise HOT and showed poor knowledge of how to teach it in their classes. The instructional verbs did not comprehensively encourage HOT; those which did were only pitched at the third level of thinking i.e. apply; most of the questions were in favour of low order thinking and there was little incremental introduction of the three top levels of Bloom‘s revised taxonomy in Grade 4 English FAL workbook specifically analyse, evaluate and create/design. Key words: High order thinking skills, cognitive domain, high order thinking and Bloom‘s revised taxonomy.

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Dans cette thèse, on s’intéresse à de nouveaux schémas de discrétisation afin de résoudre les équations couplées de la poroélasticité et nous présentons des résultats analytiques et numériques concernant des problèmes issus de la poromécanique. Nous proposons de résoudre ces problèmes en utilisant les méthodes Hybrid High-Order (HHO), une nouvelle classe de méthodes de discrétisation polyédriques d’ordre arbitraire. Cette thèse a été conjointement financée par le Bureau de Recherches Géologiques et Minières (BRGM) et le LabEx NUMEV. Le couplage entre l’écoulement souterrain et la déformation géomécanique est un sujet de recherche crucial pour les deux institutions de cofinancement. / In this manuscript we focus on novel discretization schemes for solving the coupled equations of poroelasticity and we present analytical and numerical results for poromechanics problems relevant to geoscience applications. We propose to solve these problems using Hybrid High-Order (HHO) methods, a new class of nonconforming high-order methods supporting general polyhedral meshes. This Ph.D. thesis was conjointly founded by the Bureau de recherches géologiques et minières (BRGM) and LabEx NUMEV. The coupling between subsurface flow and geomechanical deformation is a crucial research topic for both cofunding institutions.

Die Relaxation in der Variationsrechnung führt zu Minimierungsaufgaben mit einer quasi-konvexen Energiedichte. In der nichtlinearen Elastizität, Topologieoptimierung, oder bei Mehrphasenmodellen sind solche Energiedichten konvex mit einer zusätzlichen Kontrolle in der dualen Variablen und einem beidseitigem Wachstum der Ordnung $p$. Diese Minimierungsprobleme haben im Allgemeinen mehrere Lösungen, welche dennoch eine eindeutige Spannung $\sigma$ definieren. Die Approximation mit der „hybrid high-order“ (HHO) Methode benutzt eine Rekonstruktion des Gradienten in dem Raum der stückweisen Raviart-Thomas Finiten Elemente ohne Stabilisierung auf einer Triangulierung in Simplexen. Die Anwendung dieser Methode auf die Klasse der degenerierten, konvexen Minimierungsprobleme liefert eine eindeutig bestimmte, $H(\div)$ konforme Approximation $\sigma_h$ der Spannung. Die a priori Abschätzungen in dieser Arbeit gelten für gemischten Randbedingungen ohne weitere Voraussetzung an der primalen Variablen und erlauben es, Konvergenzraten bei glatten Lösungen vorherzusagen. Die a posteriori Analysis führt auf garantierte obere Fehlerschranken, eine berechenbare untere Energieschranke, sowie einen konvergenten adaptiven Algorithmus. Die numerischen Beispiele zeigen höhere Konvergenzraten mit zunehmenden Polynomgrad und bestätigen empirisch die superlineare Konvergenz der unteren Energieschranke. Obwohl der Fokus dieser Arbeit auf die nicht stabilisierte HHO Methode liegt, wird eine detaillierte Fehleranalysis für die stabilisierte Version mit einer Gradientenrekonstruktion im Raum der stückweisen Polynome präsentiert. / The relaxation procedure in the calculus of variations leads to minimization problems with a quasi-convex energy density. In some problems of nonlinear elasticity, topology optimization, and multiphase models, the energy density is convex with some convexity control plus two-sided $p$-growth. The minimizers may be non-unique in the primal variable, but define a unique stress variable $\sigma$. The approximation by hybrid high-order (HHO) methods utilizes a reconstruction of the gradients in the space of piecewise Raviart-Thomas finite element functions without stabilization on a regular triangulation into simplices. The application of the HHO methodology to this class of degenerate convex minimization problems allows for a unique $H(\div)$ conform stress approximation $\sigma_h$. The a priori estimates for the stress error $\sigma - \sigma_h$ in the Lebesgue norm are established for mixed boundary conditions without additional assumptions on the primal variable and lead to convergence rates for smooth solutions. The a posteriori analysis provides guaranteed error control, including a computable lower energy bound, and a convergent adaptive scheme. Numerical benchmarks display higher convergence rates for higher polynomial degrees and provide empirical evidence for the superlinear convergence of the lower energy bound. Although the focus is on the unstabilized HHO method, a detailed error analysis is provided for the stabilized version with a gradient reconstruction in the space of piecewise polynomials.

Computational fluid dynamics has progressed to the point where it is now possible to simulate flows with large eddy turbulence, free surfaces and other complex features. However, the success of these models often depends on the accuracy of the advection scheme supporting them. Two such schemes are the constrained interpolation profile method (CIP) and the interpolated differential operator method (IDO). They share the same space discretisation but differ in their respectively semi-Lagrangian and Eulerian formulations. They both belong to a family of high-order, compact methods referred to as the multi-moment methods. In the absence of sufficient information in the literature, this thesis begins by taxonomising various multi-moment space discretisations and appraising their linear advective properties. In one dimension it is found that the CIP/IDO with order (2N -1) has an identical spectrum and memory cost to the Nth order discontinuous Galerkin method. Tests confirm that convergence rates are consistent with nominal orders of accuracy, suggesting that CIP/IDO is a better choice for smooth propagation problems. In two dimensions, six Cartesian multi-moment schemes of third order are compared using both spectral analysis and time-domain testing. Three of these schemes economise on the number of moments that need to be stored, with one CIP/IDO variant showing improved isotropy, another failing to maintain its nominal order of accuracy, and one of the conservative variants having eigenvalues with positive real parts: it is stable only in a semi-Lagrangian formulation. These findings should help researchers who are interested in using multi-moment schemes in their solvers but are unsure as to which are suitable. The thesis then addresses the question as to whether a multi-moment method could be implemented on a Cartesian cut cell grid. Such grids are attractive for supporting arbitrary, possibly moving boundaries with minimal grid regeneration. A pair of novel conservative fourth order schemes is proposed. The first scheme, occupying the Cartesian interior, has unprecedented low memory cost and is proven to be conditionally stable. The second, occupying the cut cells, involves a profile reconstruction that is guaranteed to be well-behaved for any shape of cell. However, analysis of the second scheme in a simple grid arrangement reveals positive real parts, so it is not stable in an Eulerian formulation. Stability in a hybrid formulation remains open to question.

Doctor of Philosophy / Department of Physics / Chii-Dong Lin / In this thesis, we establish the theoretical tools to investigate high-order harmonic generation (HHG) by intense infrared lasers in a gaseous medium. The macroscopic propagation of both the fundamental and the harmonic fields is taken into account by solving Maxwell’s wave equations, while the single-atom (or single-molecule) response is obtained by quantitative rescattering theory. The initial spatial mode of the fundamental laser is assumed either a Gaussian or a truncated Bessel beam. On the examples of Ar, N[subscript]2 and CO[subscript]2, we demonstrate that the available experimental HHG spectra with isotropic and aligned target media can be accurately reproduced theoretically even though the HHG spectra are sensitive to the experimental conditions. We show that the macroscopic HHG can be expressed as a product of a macroscopic wave packet and a photorecombination cross section, where the former depends on laser and experimental conditions while the latter is the property of the target only. The factorization makes it possible to retrieve the single-atom or single-molecule structure information from experimental HHG spectra. As for the multiple molecular orbital contribution in HHG, it causes the disappearance of the minimum in the HHG spectrum of aligned N[subscript]2 with the increase of laser intensity, and the position of minimum in HHG spectrum of aligned CO[subscript]2 depending on many factors is also attributed to it, which could explain why the minima observed in different laboratories may differ. For an important application of HHG as ultrashort light source, we show that measured continuous harmonic spectrum of Xe due to the reshaping of the fundamental laser field can be used to produce an isolated attosecond pulse by spectral and spatial filtering in the far field. For on-going application of using HHG to ionize aligned molecules, we present the photoelectron angular distribution from aligned N[subscript]2 and CO[subscript]2 in the laboratory frame, which can be compared directly with future experiments.

A representação de malhas por meio de estrutura de dados e operadores topológicos e um dos focos principais da modelagem geométrica, onde permite uma implementação robusta e eficiente de mecanismos de refinamento adaptativo, alinhamento de células e acesso as relações de incidência e adjacência entre os elementos da malha, o que é de grande importância na maioria das aplicações em mecânica dos fluidos. No caso de malhas não estruturadas, a não uniformidade da decomposição celular e melhor representada por uma estrategia mais sofisticada, que são as estruturas de dados topológicas. As estruturas de dados topológicas indexam elementos de uma malha representando relações de incidência e adjacência entre elementos, garantindo acesso rápido às informações. Um dos aspectos mais comuns aos problemas tratados pela mecânica dos fluidos computacional é a complexidade da geometria do domínio onde ocorre o escoamento. O uso de estruturas de dados para manipular malhas computacionais e de grande importância pois realiza de modo eficiente as consultas às informações da malha e centraliza todas as operações sobre a malha em um único módulo, possibilitando sua extensão e adaptação em diversas situações. Este trabalho visou explorar o acoplamento de uma estrutura de dados topológica, a Mate Face, em um módulo simulador existente, de modo a gerenciar todos os acessos à malha e dispor operações e iteradores para pesquisas complexas nas vizinhanças de cada elemento na malha. O módulo simulador resolve as equações governantes da mecânica dos fluidos através da técnica de volumes finitos. Foi utilizada uma formulação que atribui os valores das propriedades aos centroides dos volumes de controle, utiliza métodos de alta ordem, os esquemas ENO e WENO, que tem a finalidade de capturar com eficiência descontinuidades presentes em problemas governados por equações diferenciais parciais hiperbólicas. As equações de Euler em duas dimensões representam os escoamentos de interesse no presente trabalho. O acoplamento da estrutura de dados Mate Face ao simulador foi realizada através da criação de uma biblioteca desenvolvida que atua como uma interface de comunicação entre os dois módulos, a estrutura de dados e o simulador, que foram implementados em diferentes linguagens de programação. Deste modo, todas as funcionalidades existentes na Mate Face tornaram-se acessíveis ao simulador na forma de procedimentos. Um estudo sobre malhas dinâmicas foi realizado envolvendo o método das molas para movimentação de malhas simulando-se operações de arfagem. A idéia foi verificar a aplicabilidade deste método para auxiliar simulações de escoamentos não estacionarios. Uma outra vertente do trabalho foi estender a estrutura Mate Face de forma a representar elementos não suportados a priori, de modo a flexibilizar o seu uso em simulações de escoamentos baseados no método de volumes finitos espectrais. O método dos volumes espectrais e utilizado para se obter alta resolução espacial do domínio computacional, que também atribui valores das propriedades aos centroides dos volumes de controle, porém, os volumes de controle são particionados em volumes menores de variadas topologias. Assim, uma extensão da Mate Face foi desenvolvida para representar a nova malha para a aplicação do método, representando-se cada particionamento localmente em cada volume espectral. Para todas as etapas deste trabalho, realizaram-se experimentos que validaram a utilizaação da estrutura de dados Mate Face junto a métodos numéricos. Desta forma, a estrutura pode auxiliar as ferramentas de simulações de escoamentos de fluidos no gerenciamento e acesso à malha computacional / The storage and access of grid files by data structures and topologic operators is one of the most important goals of geometric modeling research field, which allows an efficient and stable implementation of adaptive refinement mechanisms, cells alignment and access to incidence and adjacency properties from grid elements, representing great concernment in the majority of applications from fluid mechanics. In the case of non-structured grids, the cellular decomposition if non-uniform and is better suited by a more sophisticated strategy - the topologic data structs. The topologic data structs index grid elements representing incidence and adjacency properties from grid elements, ensuring quick access to information. One of most common aspects from problems solved by computational fluid mechanic is the complexity of the domain geometry where the fluid ows. The usage of data structures to manipulate computational grids is of great importance because it performs efficiently queries on grid information and centers all operations to the grid on a unique module, allowing its extension and flexible usage on many problems. This work aims at exploring the coupling of a topologic data structure, the Mate Face, on a solver module, by controlling all grid access providing operators and iterators that perform complex neighbor queries at each grid element. The solver module solves the governing equations from fluid mechanics though the finite volume technique with a formulation that sets the property values to the control volume centroids, using high order methods - the ENO and WENO schemes, which have the purpose of efficiently capture the discontinuities appearing in problems governed by hyperbolic conservation laws. The two dimensional Euler equations are considered to represent the flows of interest. The coupling of the Mate Face data structure to the solver module was achieved by a creation of a library that acts as an interface layer between both modules, the Mate Face and the solver, which had been implemented using different programming languages. Therefore, all Mate Face class methods are available to the solver module though the interface library in the form of procedures. A study of dynamic grids was made by using spring methods for the moving grid under pitch movement case. The goal was to analyze the applicability of such method to aid non stationary simulations. Another contribution of this work was to show how the Mate Face can be extended in order to deal with non-supported types of elements, allowing it to aid numeric simulations using the spectral finite volume method. The spectral nite volume method is used to obtain high spatial resolution, also by setting the property values to the control volume centroids, but here the control volumes are partitioned into smaller volumes of different types, from triangles to hexagons. Then, an extension of the Mate Face was developed in order to hold the new generated grid by the partitioning specfied by the spectral finite volume method. The extension of Mate Face represents all partitioned elements locally for each original control volume. For all implementations and proposals from this work, experiments were performed to validate the usage of the Mate Face along with numeric methods. Finally, the data structure can aid the fluid flow simulation tools by managing the grid file and providing efficient query operators

Atrial fibrillation (AF) is a troublesome disease often overlooked by more serious myocardial infarctions. Up until now, there has been very little or no use of high order spectral techniques in order to evaluate the organization of the atrium during AF. Cross-bicoherence algorithm can be used alongside a surrogate data threshold in order to determine significant phase coupling interactions, giving rise to an organizational metric. This proposed algorithm is used to show rotigaptide, a gap junction coupling drug, significantly increases the organization of the atria during episodes of AF due to improvement of cell-to-cell coupling.