Pouvoir déterminer la structure et la composition de l’intérieur de la Terre est un enjeu scientifique fondamental, pour la compréhension de l’organisation de la Terre profonde, des mécanismes des séismes et leur localisation en lien avec la prévention du risque sismique, pour la détection et l’exploitation des ressources naturelles telles que l’eau ou les hydrocarbures, ou encore pour toutes les activités de construction et de prévention associées au génie civil. Pour cela, les ondes sismiques sont un outil de choix. L’utilisation d’une approximation haute fréquence pour la modélisation de la propagation des ondes est avantageuse en termes de coût de calcul dès lors que plusieurs centaines, voire milliers, ou plus de longueurs d’ondes doivent être propagées. À la place de l’équation des ondes linéaire, l’approximation haute fréquence fournit trois équations aux dérivées partielles fondamentales. L’équation Eikonal, non linéaire, permet d’obtenir le temps de trajet. Une deuxième équation fournit l’angle d’émergence. L’équation Eikonal et l’équation des angles appartiennent toutes deux à la grande famille des équations de Hamilton-Jacobi. Enfin, l’équation de transport permet de calculer l’amplitude.Le tracé des rais sismiques est une technique lagrangienne qui utilise la méthode des caractéristiques pour obtenir un ensemble d’équations différentielles ordinaires à partir de ces équations aux dérivées partielles. Ces équations peuvent être intégrées facilement, donnant ainsi accès au temps de trajet et à l’amplitude le long des rais. Très largement utilisés dans la communauté géophysique du fait de leur simplicité, les outils de tracé de rais ne sont pas pour autant les plus efficaces et les plus robustes en pratique pour des applications d’imagerie et d’inversion haute résolution. En lieu et place, il peut être utile de résoudre directement les équations aux dérivées partielles par une méthode eulérienne. Durant les trois dernières décennies, une multitude de solveurs ont été développés pour l’équation Eikonal, la plupart utilisant la méthode des différences finies. Ces différents travaux visent à obtenir le meilleur compromis entre précision, coût de calcul, robustesse, facilité d’implémentation et souplesse d’utilisation.Dans cette thèse, je développe une approche différente, se basant principalement sur une méthode Galerkine discontinue. Dans le champ des mathématiques, cette méthode a été largement utilisée pour résoudre les lois de conservation et les équations de Hamilton-Jacobi. Très peu de travaux ont porté sur l'utilisation de cette méthode pour la résolution de l’équation Eikonal statique dans un contexte géophysique, et ce malgré le haut niveau de précision qu'elle apporte. C’est pourquoi, en me basant sur des travaux mathématiques, je propose un nouveau solveur Eikonal adapté au contexte géophysique. Les milieux hétérogènes complexes, anisotropes, et incluant des variations topographiques sont correctement pris en compte, avec une précision sans précédent. En y intégrant de manière robuste une stratégie de balayage rapide, je montre que ce solveur présente une très grande efficacité en deux comme en trois dimensions.J'utilise également ce solveur pour calculer l’angle d’émergence. Je développe par ailleurs un solveur voisin en volumes finis pour la résolution de l’équation de transport, permettant ainsi le calcul de l’amplitude. La variable d’état adjoint pour la tomographie sismique des temps et des pentes vérifiant une équation de transport semblable, je montre qu'on peut également la calculer à l'aide de ce solveur en volumes finis. En conséquence, je propose et analyse un ensemble consistant de solveurs pour la communauté géophysique. Ces outils devraient s’avérer utiles pour une large palette d’applications. Finalement, en guise d’illustration, je les utilise dans des schémas d’imagerie sismique, dans le but de démontrer le bénéfice apporté par une approximation haute fréquence dans ce type de schémas. / Recovering information on the structure and the composition of the Earth's interior is a fundamental issue for a large range ofapplications, from planetology to seismology, natural resources assessment, and civil engineering. Seismic waves are a very powerful tool for that purpose. Using a high-frequency approximation for the numerical modeling of seismic wave propagation is computationally advantageous when hundreds, thousands, or more of wavelengths have to be propagated. Instead of the linear wave equation, the high-frequency approximation yields three fundamental partial differential equations. The nonlinear Eikonal equation leads to traveltime. A second equation is derived for the take-off angle. Both Eikonal and angle equations belong to the wide Hamilton-Jacobi family of equations. In addition, the transport equation leads to the amplitude.As a Lagrangian approach, seismic ray tracing employs the method of characteristics to derive a set of ordinary differential equations from these partial differential equations. They can be easily integrated, thus yielding traveltime and amplitude along rays. Widely used in the geophysical community for their simplicity, the ray-tracing tools might not be the most efficient and robust ones for practical high-resolution imaging and inversion applications. Instead, it might be desirable to directly solve the partial differential equations in an Eulerian way. In the three last decades, plenty of Eikonal solvers have been designed, mostly based on finite-difference methods. Successive works try to find the best compromise between accuracy, computational efficiency, robustness, ease of implementation, and versatility.In this thesis, I develop a different approach, mainly based on the discontinuous Galerkin method. This method has been intensively used in the mathematical field for solving conservation laws and time-dependent Hamilton-Jacobi equations. Only few investigations have been done regarding its use for solving the static Eikonal equation in a geophysical context, despite the high level of accuracy allowed by this method. Therefore, improving upon mathematical studies, I propose a new Eikonal solver suitable for the geophysical context. Complex heterogeneous anisotropic media with non-flat topographies are correctly handled, with an unprecedented accuracy. Combined with a fast-sweeping strategy in a robust way, I show that this new solver exhibits a high computational efficiency, in two dimensions as well as in three dimensions.I also employ this solver for the computation of the take-off angle. I design an additional finite-volume solver for solving the transport equation, leading to the computation of amplitude. With this solver, I also consider the computation of the adjoint-state variable for seismic tomography, since it satisfies a similar transport equation. Eventually, I propose a whole set of consistent solvers to the geophysical community. These tools should be useful in a wide range of applications. As an illustration, I finally use them in advanced seismic imaging schemes, in order to demonstrate the benefit brought by the high-frequency approximation in this kind of schemes.

Orientador: José Carlos Miguel / Banca: Nelson Antonio Pirola / Banca: Dagoberto Buim Arena / Resumo: Esta pesquisa tem como objeto de análise o Ensino da Geometria no Ciclo II do Ensino Fundamental e algumas implicações políticas pedagógicas que cercam este tema. Trabalho realizado inicialmente em duas salas de 5ª séries de uma Escola Pública Estadual, de um pequeno município às margens do Rio Tietê, na qual um grupo de 20 alunos e seus conhecimentos geométricos, foram o foco inicial desta investigação. Os alunos com idades entre dez e doze anos, pertencentes cada dez, respectivamente, a uma das duas classes de 5ª séries do período da manhã, tendo dois respectivos professores de Matemática, de metodologia e crenças pedagógicas diferentes: a construtivista e a tradicional, que nos levaram a realizar um total de 200 atividades - cada aluno foi avaliado com dez atividades que contemplam fazeres geométricos, pertinentes à Grade Curricular de Matemática. Estas foram analisadas de maneira qualitativa, e independente da crença metodológica do professor, na sua maioria os alunos apresentaram frágil e preocupante desempenho quanto aos conhecimentos geométricos. Tais resultados nos levam a ampliar os questionamentos, assim como os grupos pesquisados, que se completa com 20 professores de Matemática, que respondem a 140 questões sobre o objeto de estudo e seu questionamento principal, o pensamentos dos 2 Professores responsáveis pelas duas 5ª séries envolvidas na pesquisa, e depoimentos de 4 PCOPs - Professores Coordenadores de Matemática de 4 Oficinas Pedagógicas de Diretorias de Ensino do interior paulista. No universo de respostas analisadas, à luz de uma metodologia qualitativa, surgem apontamentos para a situação caótica do Ensino da Geometria. Quanto à categoria docente, as conseqüências de grande carência de conteúdos geométricos na sua Formação Acadêmica e outros que implicam diretamente na produção de conhecimentos matemáticos... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The object of this study is the teaching of Geometry in junior high school and the political and pedagogical implications connected to the theme. I conducted this research in two fifth grade classrooms from a public school of a small town located on the bank of the Tietê River. I initially surveyed a group of 20 students and their knowledge of Geometry. I chose 10 students from each class, ages ranging from 10 to 12 years old, who studied in the morning and had two different teachers whose approaches to the teaching of Geometry was diverse: one followed the traditional model whereas the other adopts Constructivism. With the students, we did 200 activities involving the knowledge of Geometry. Each student performed ten tasks about the syllabus of Geometry. Such tasks were qualitatively analyzed. Throughout the research, both my questionings and the groups enlarged and 20 Math teachers answered 140 questions about my object of study and its main questionings. Also answering the questions, the two teachers responsible for the classes as well as four teachers from different regional Boards of Education. When their answers were analyzed I noticed a lack of geometrical knowledge in their academic formation. Besides this scarce knowledge of the VAN HIELE theoretical bases of the subject they teach, other factors influence performance in class, such as the low salaries they get. These will have damaging effects on the students' learning process and led me to conclude that we need more solid educational policies, capable of transforming this scenario. The study also shows a documental analysis of more than three decades of syllabuses elaborated by the State Board of Education of the state of São Paulo and ho the teaching of Geometry was carelessly handled. / Mestre

Filamentous fungi are important in an enormous variety of ways to our life, with examples ranging from bioremediation, through the food and drinks industry to human health. These organisms can form huge networks stretching metres and even kilometres. However, their mode of growth is by the extension of individual hyphal tips only a few microns in diameter. Tip growth is mediated by the incorporation of new wall building materials at the soft apex. Just how this process is controlled (in fungi and in cell elongation in other organisms) has been the subject of intense study over many years and has attracted considerable attention from mathematical modellers. In this thesis, we consider mathematical models of fungal tip growth that can be classified as either geometrical or biomechanical. In every model we examine, a 2-D axisymmetric semihemisphere-like curve represents half the medial section of fungal tip geometry. A geometrical model for the role of the Spitzenkorper in the tip growth was proposed by Bartnicki-Garcia et al (1989), where a number of problems with the mathematical derivation were pointed out by Koch (2001). A suggestion is given as an attempt to revise the derivation by introducing a relationship between arc length of a growing tip, deposition of wall-building materials and tip curvature. We also consider two types of geometrical models as proposed by Goriely et al (2005). The first type considers a relationship between the longitudinal curvature and the function used to model deposition of wall-building materials. For these types of models, a generalized formulae for the tip shape is introduced, which allows localization of deposition of wall-building materials to be examined. The second type considers a relationship between longitudinal and latitudinal curvatures and the function used to model deposition of wall-building materials. For these types of models, a new formulation of the function used to model deposition of wall-building materials is introduced. Finally, a biomechanical model as proposed by Goriely et al (2010). Varying arc length of the stretchable region on the tip suggests differences in geometry of tip shape and the effective pressure profile. The hypothesis of orthogonal growth is done by focusing only on the apex of a "germ tube". Following that, it suggests that material points on the tip appear to move in a direction perpendicular to the tip either when surface friction is increased or decreased.

Thesis (M.A. (Mathematics Education)) --University of Limpopo, 2006 / Traditionally, mathematics has been perceived as objective, abstract, absolute and universal subject that is devoid of social and cultural influences. However, the new perspective has led to the perceptions that mathematics is a human endeavour, and therefore it is culture-bound and context-bound. Mathematics is viewed as a human activity and therefore fallible. This research was set out to explore geometrical concepts involved in the traditional circular buildings in Mopani district of Limpopo Province and relate them to the classroom learning in grade 11 classes. The study was conducted in a very remote place and a sample of two traditional circular houses from Xitsonga and Sepedi cultures was chosen for comparison purposes because of their cultural diversity. The questions that guided my exploration were: • Which geometrical concepts are involved in the design of the traditional circular buildings and mural decorations in Mopani district of the Limpopo Province?. How do the geometrical concepts in the traditional circular buildings relate to the learning of circle geometry in grade 11 class?. The data were gathered through my observations and the learners’ observations, my interviews with the builders and with the learners, and the grade 11 learners’ interaction with their parents or builders about the construction and decorations of the traditional circular houses. I used narrative configurations to analyse the collected data. Inductive analysis, discovery and interim analysis in the field were employed during data analysis. From my own analysis and interpretations, I found that there are many geometrical concepts such as circle, diameter, semi-circle, radius, centre of the circle etc. that are involved in the design of the traditional circular buildings. In the construction of these houses, these concepts are involved from the foundation of the building to the roof level. All these geometrical concepts can be used by both educators and learners to enhance the teaching and learning of circle geometry. Further evidence emerged that teaching with meaning and by relating abstract world to the real world makes mathematics more relevant and more useful.

Low Pressure Die Casting (LPDC) process is one the most common casting process to produce structural automotive components, such as alloy wheels and suspension components. It has been identified that cavity filling and solidification process are two most critical aspects to produce premium quality casting components.During the solidification process of casting alloy, it is a well known phenomenon that metal experiences volumetric shrinkage due to its density difference between liquid and solid phase. When this volumetric shrinkage is not properly compensated, then a casting defect commonly known as solidification shrinkage occurs. The solidification shrinkage has very detrimental effects on structural integrity required for premium quality casting such as aluminium alloy wheels.Literature and practical experiences of foundry men show that it is critical to achieve unidirectional solidification pattern by avoiding an isolated hot spot in order to minimise the solidification shrinkage. However, it is found that the geometry of industrial casting applications is often constrained by other design factors that would not naturally avoid these isolated hot spots. The subject of this research, aluminium alloy wheels, is not excluded from this constraint.In aluminium alloy wheels, an isolated hot spot is commonly observed in an area known as rim and spoke junction due to its geometry constraints. Consequently, the solidification shrinkage is commonly experienced, which is undesirable due to its detrimental effects for the structural integrity of alloy wheels. In order to minimise the solidification shrinkage, forced cooling method is applied to avoid an isolated hot spot. The control of this forced cooling is achieved by cooling media, flow rate of cooling media and duration cooling circuit. Foundry experiments in industrial environment were conducted producing aluminium alloy wheels using commercially treated A356 (Al-Si) alloy, where different durations of cooling circuit were used to understand the sensitivity of solidification shrinkage formation to the duration of cooling circuit. This was followed by metallurgical structure analysis and numerical modelling to suggest the sensitivity of cooling circuit duration in controlling solidification shrinkage.The major finding conclusion of this research is that control of the shrinkage formation is not very sensitive to the duration cooling circuit. It is suggested that as casting solidifies initially from the mould wall, it retracts away from the cast-mould interface due to thermal contraction. Consequently, air gap is formed between casting and mould interface, creating an effective thermal resistance layer. Thereafter, heat transfer across the cast-mould interface is not sensitive to the change in the cooling channel which is a distance to the cast-mould interface.Some limitations of numerical modelling and metallurgical analysis were also identified during this research and recommendations were made to improve the accuracy of local hot spot prediction in production of aluminium alloy wheels. More specifically, numerical modelling of the effect of grain refinement and use of non homogeneous material property (particularly fraction of solid) for rapidly chilled area. Fraction of eutectic rather than secondary dendrites arm spacing is a proper microstructure parameter that can be used to locate the hot spot.

The heat dissipation and thermal performance of ventilated brake discs strongly depends on the aerodynamic characteristics of the air flow through the rotor passages. In this thesis, the thermal convection is analyzed using an analytical method, and the velocity distribution, temperature contours and Nusselt number are determined. Then numerical models for different rotors, pillar post rotors and vane rotors are generated and numerical simulations are conducted to determine the desired parameters. To analyze more realistic vane and pillar post rotor models, commercial CFD software packages, Fluent and Gambit, are used to simulate the heat flux rate, air flow rate, velocity distributions, temperature contours, and pressure distributions inside the rotors. Furthermore, sensitivity studies have been performed, to determine the effects of a different number of vanes or pillar posts, inner and outer radii and various angles of vanes. To automate the tedious and repetitive design process of the disc rotor, a design synthesis framework, iSIGHT, is used to integrate the geometrical modeling using GAMBIT and numerical simulations based on FLUENT. Through this integrated design synthesis process, the disc rotor geometrical optimization is performed using design of experiment studies. / UOIT

Buoyant flows are characterized with unsteady large-scale structures and thus time-dependent large eddy simulation (LES) is generally favored. In this dissertation, to further explore LES for buoyant flow, an LES code based on a collocated grid system is first developed. A multigrid solver using a control strategy is developed for the pressure Poisson equations. The control strategy significantly accelerated the convergence rate. A temperature solver using a fourth-order Runge-Kutta approach is also developed. The LES code is extensively tested before it is applied. Although the collocated grid system will introduce conservation errors, in tests of a steady lid-driven cavity flow and transient start-up flow, the effect of the non-conservation of the collocated grid system was not significant. <p>In LES, the effect of SGS scales is represented by SGS models. A novel dynamic nonlinear model (DNM) for SGS stress is tested using isothermal channel flow at Reynolds number 395. The kinetic energy dissipation and geometrical characteristics of the resolved scale and SGS scale with respect to the DNM are investigated. In general, the DNM is reliable and has relatively realistic geometrical properties in comparison with the conventional dynamic model in the present study. In contrast to a pure advecting velocity field, a scalar (temperature) field displays very different characteristics. The modelling of SGS heat flux has not been as extensively studied as that of SGS stress partly due to the complexity of the scalar transport. In this dissertation, LES for a turbulent combined forced and natural convection is studied. The DNM model and a nonlinear dynamic tensor diffusivity model (DTDM-HF) are applied for the SGS stress and heat flux, respectively. The combined effect of the nonlinear models is compared to that of linear models. Notable differences between the nonlinear and linear SGS models are observed at the subgrid-scale level. At the resolved scale, the difference is smaller but relatively more distinguishable in terms of quantities related to the temperature field. <p>Finally, the geometrical properties of the resolved velocity and temperature fields of the thermal flow are investigated based on the LES prediction. Some universal geometrical patterns have been reproduced, e.g. the positively skewed resolved enstrophy generation and the alignment between the vorticity and vortex stretching vectors. The present research demonstrates that LES is an effective tool for the study of the geometrical properties of a turbulent flow at the resolved-scales. The wall imposed anisotropy on the flow structures and orientation of the SGS heat flux vector are also specifically examined. In contrast to the dynamic eddy diffusivity model, the DTDM-HF successfully predicts the near-wall physics and demonstrates a non-alignment pattern between the SGS heat flux and temperature gradient vector.

This dissertation consists of two parts, i.e. dynamic approaches for subgrid-scale (SGS) stress modelling for large eddy simulation and advanced assessment of the resolved scale motions related to turbulence geometrical statistics and topologies. The numerical simulations are based on turbulent Couette flow. The first part of the dissertation presents four contributions to the development of dynamic SGS models. The conventional integral type dynamic localization SGS model is in the form of a Fredholm integral equation of the second kind. This model is mathematically consistent, but demanding in computational cost. An efficient solution scheme has been developed to solve the integral system for turbulence with homogeneous dimensions. Current approaches to the dynamic two-parameter mixed model (DMM2) are mathematically inconsistent. As a second contribution, the DMM2 has been optimized and a modelling system of two integral equations has been rigorously obtained. The third contribution relates to the development of a novel dynamic localization procedure for the Smagorinsky model using the functional variational method. A sufficient and necessary condition for localization is obtained and a Picard's integral equation for the model coefficient is deduced. Finally, a new dynamic nonlinear SGS stress model (DNM) based on Speziale's quadratic constitutive relation [J. Fluid Mech., 178, p.459, 1987] is proposed. The DNM allows for a nonlinear anisotropic representation of the SGS stress, and exhibits a significant local stability and flexibility in self-calibration. In the second part, the invariant properties of the resolved velocity gradient tensor are studied using recently developed methodologies, i.e. turbulence geometrical statistics and topology. The study is a posteriori based on the proposed DNM, which is different than most of the current a priori approaches based on experimental or DNS databases. The performance of the DNM is further validated in terms of its capability of simulating advanced geometrical and topological features of resolved scale motions. Phenomenological results include, e.g. the positively skewed resolved enstrophy generation, the alignment between the vorticity and vortex stretching vectors, and the pear-shape joint probability function contour in the tensorial invariant phase plane. The wall anisotropic effect on these results is also examined.

This dissertation consists of two parts, i.e. dynamic approaches for subgrid-scale (SGS) stress modelling for large eddy simulation and advanced assessment of the resolved scale motions related to turbulence geometrical statistics and topologies. The numerical simulations are based on turbulent Couette flow. The first part of the dissertation presents four contributions to the development of dynamic SGS models. The conventional integral type dynamic localization SGS model is in the form of a Fredholm integral equation of the second kind. This model is mathematically consistent, but demanding in computational cost. An efficient solution scheme has been developed to solve the integral system for turbulence with homogeneous dimensions. Current approaches to the dynamic two-parameter mixed model (DMM2) are mathematically inconsistent. As a second contribution, the DMM2 has been optimized and a modelling system of two integral equations has been rigorously obtained. The third contribution relates to the development of a novel dynamic localization procedure for the Smagorinsky model using the functional variational method. A sufficient and necessary condition for localization is obtained and a Picard's integral equation for the model coefficient is deduced. Finally, a new dynamic nonlinear SGS stress model (DNM) based on Speziale's quadratic constitutive relation [J. Fluid Mech., 178, p.459, 1987] is proposed. The DNM allows for a nonlinear anisotropic representation of the SGS stress, and exhibits a significant local stability and flexibility in self-calibration. In the second part, the invariant properties of the resolved velocity gradient tensor are studied using recently developed methodologies, i.e. turbulence geometrical statistics and topology. The study is a posteriori based on the proposed DNM, which is different than most of the current a priori approaches based on experimental or DNS databases. The performance of the DNM is further validated in terms of its capability of simulating advanced geometrical and topological features of resolved scale motions. Phenomenological results include, e.g. the positively skewed resolved enstrophy generation, the alignment between the vorticity and vortex stretching vectors, and the pear-shape joint probability function contour in the tensorial invariant phase plane. The wall anisotropic effect on these results is also examined.

Buoyant flows are characterized with unsteady large-scale structures and thus time-dependent large eddy simulation (LES) is generally favored. In this dissertation, to further explore LES for buoyant flow, an LES code based on a collocated grid system is first developed. A multigrid solver using a control strategy is developed for the pressure Poisson equations. The control strategy significantly accelerated the convergence rate. A temperature solver using a fourth-order Runge-Kutta approach is also developed. The LES code is extensively tested before it is applied. Although the collocated grid system will introduce conservation errors, in tests of a steady lid-driven cavity flow and transient start-up flow, the effect of the non-conservation of the collocated grid system was not significant. <p>In LES, the effect of SGS scales is represented by SGS models. A novel dynamic nonlinear model (DNM) for SGS stress is tested using isothermal channel flow at Reynolds number 395. The kinetic energy dissipation and geometrical characteristics of the resolved scale and SGS scale with respect to the DNM are investigated. In general, the DNM is reliable and has relatively realistic geometrical properties in comparison with the conventional dynamic model in the present study. In contrast to a pure advecting velocity field, a scalar (temperature) field displays very different characteristics. The modelling of SGS heat flux has not been as extensively studied as that of SGS stress partly due to the complexity of the scalar transport. In this dissertation, LES for a turbulent combined forced and natural convection is studied. The DNM model and a nonlinear dynamic tensor diffusivity model (DTDM-HF) are applied for the SGS stress and heat flux, respectively. The combined effect of the nonlinear models is compared to that of linear models. Notable differences between the nonlinear and linear SGS models are observed at the subgrid-scale level. At the resolved scale, the difference is smaller but relatively more distinguishable in terms of quantities related to the temperature field. <p>Finally, the geometrical properties of the resolved velocity and temperature fields of the thermal flow are investigated based on the LES prediction. Some universal geometrical patterns have been reproduced, e.g. the positively skewed resolved enstrophy generation and the alignment between the vorticity and vortex stretching vectors. The present research demonstrates that LES is an effective tool for the study of the geometrical properties of a turbulent flow at the resolved-scales. The wall imposed anisotropy on the flow structures and orientation of the SGS heat flux vector are also specifically examined. In contrast to the dynamic eddy diffusivity model, the DTDM-HF successfully predicts the near-wall physics and demonstrates a non-alignment pattern between the SGS heat flux and temperature gradient vector.