In this thesis, we consider linear extremal problems in the Hp spaces. For many of these extremal problems, a unique solution can be guaranteed. We will examine some of the classical examples of extremal problems in these spaces. With this framework in place we will then consider a particular problem which does not always have a unique solution.

La simulation numérique des phénomènes physiques est devenue un élément incontournable dans la boite à outils de l'ingénieur mécanicien. Des outils robustes et modulables, basés sur les méthodes classiques d'approximation, sont désormais couramment utilisés dans l'industrie. Cependant, ces outils nécessitent des moyens de calculs importants lorsqu'ils sont utilisés pour résoudre des problèmes complexes. Même si les progrès remarquables de l'industrie informatique rendent de tels moyens de calcul toujours plus abordables, il s'avère aujourd'hui nécessaire de proposer des méthodes d'approximation innovantes permettant de mieux exploiter les ressources informatiques disponibles. Les méthodes de réduction de modèle sont présentées comme un candidat idéal pour atteindre cet objectif. Parmi celles-ci, les méthodes basées sur la construction d'une approximation à variables séparées se sont révélées être très efficaces pour approcher la solution d'une grande variété de problèmes, réduisant les coûts numériques de plusieurs ordres de grandeur. Néanmoins, l'efficacité de ces méthodes dépend considérablement du problème traité. Aussi, on se propose ici d'évaluer l'intérêt d'une approximation à variables séparées espace-temps dans le cadre de problèmes académiques de dynamique transitoire. On définit tout d'abord la meilleure approximation (au sens d'un problème de minimisation) de la solution d'un problème transitoire, sous la forme d'une représentation à variables séparées espace-temps. Le calcul de cette approximation étant basé sur l'hypothèse que la solution du problème de référence est connue (méthode a posteriori), la suite du manuscrit est dédiée à la construction d'une telle approximation sans autres connaissances a priori sur la solution de référence, que les opérateurs du problème espace-temps dont elle est solution (méthode a priori). Un formalisme générique, basé sur une représentation tensorielle des opérateurs du problème espace-temps est alors introduit dans un cadre multichamps. On développe ensuite un solveur exploitant ce format générique, pour construire une approximation à variables séparées espace-temps de la solution d'un problème transitoire. Ce solveur est basé sur la décomposition généralisée propre de la solution (Proper Generalized Decomposition - PGD). Un état de l'art des algorithmes existants permet alors d'évaluer l'efficacité des définitions classiques de la PGD pour approcher la solution de problèmes académiques de dynamique transitoire. Les résultats obtenus mettant en défaut l'optimalité de la PGD la plus robuste, une nouvelle définition, récemment introduite dans la littérature, est appliquée dans un cadre multichamps à la résolution d'un problème d'élastodynamique 2D. Cette nouvelle définition, basée sur la minimisation du résidu dans une norme idéale, permet finalement d'obtenir une très bonne approximation de la meilleure approximation de rang donné, sans avoir à calculer un grand nombre de modes espace-temps. / Numerical simulation of physical phenomena has become an indispensable part of the mechanical engineer's toolbox. Robust and flexible tools, based on classical approximation methods, are now commonly used in industry. However, these tools require lots of computational resources to solve complex problems. Even if such resources are more and more affordable thanks to the remarkable progress in computer industry, it is now necessary to propose innovative approximation methods in order to better exploit the impressive amount of computational resources that are todays available. Reduced order modeling techniques are presented as ideal candidates to address this issue. Among these, methods based on the construction of low rank separated approximations have been shown to be very efficient to approach solutions of a wide variety of problems, reducing computational costs by several orders of magnitude. Nonetheless, efficiency of these methods significantly depends on the considered problem. In this manuscript, we propose to evaluate interest of space-time separated representations to approach solutions of academical transient dynamic problems. We first define the best space-time separated approximation (with respect to a minimization problem) of a given solution of a transient problem. The construction of this approximation being based on the hypothesis that the problem's solution is known (a posteriori method), the following of the manuscript is dedicated to the construction of such an approximation without any other knowledge on the reference solution than the operators of the space-time problem from which it is solution (a priori method). We then introduce a generic formalism, based on the tensor product structure of the operators of the space-time problem, in a multi-field framework. Next, this formalism is used to develop a generic solver that builds a separated approximation of a transient problem's solution, with the help of the Proper Generalized Decomposition (PGD). A state of art of existing algorithms is done, and efficiency of classical definitions of PGD to approach solutions of several academical transient dynamic problems is evaluated. Numerical results highlight the lack of optimality of the more robust PGD. Therefore, a new PGD definition, recently introduced in literature, is applied to solution of an elastodynamic problem in a multi-field framework. This new definition is based on minimization of an ideal residual norm and allows to find a very good approximation of the best approximation of a given rank, without having to calculate more space-time modes than needed.

This thesis aims to acquire, deepen and promote understanding of computing techniques for high dimensional scattered data approximation with radial basis functions. The main contributions of this thesis include sufficient conditions for the sovability of compactly supported radial basis functions with different shapes, near points preconditioning techniques for high dimensional interpolation systems with compactly supported radial basis functions, a heterogeneous hierarchical radial basis function interpolation scheme, which allows compactly supported radial basis functions of different shapes at the same level, an O(N) algorithm for constructing hierarchical scattered data set andan O(N) algorithm for sparse kernel summation on Cartesian grids. Besides the main contributions, we also investigate the eigenvalue distribution of interpolation matrices related to radial basis functions, and propose a concept of smoothness matching. We look at the problem from different perspectives, giving a systematic and concise description of other relevant theoretical results and numerical techniques. These results are interesting in themselves and become more interesting when placed in the context of the bigger picture. Finally, we solve several real-world problems. Presented applications include 3D implicit surface reconstruction, terrain modelling, high dimensional meteorological data approximation on the earth and scattered spatial environmental data approximation.

No description available.

This thesis reports a rock slope hazard investigation located in the Central Andes of Chile, where two significant rock mass wasting events were recognized. Dating using cosmogenic nuclide for ³⁶Cl showed that the deposits were post-glacial in age, corresponding to the Upper Pleistocene Portillo Rock Avalanche (PRA) and a Holocene rock slump and rockslide. The pre-historic landslide deposits underlie both a key transportation route between Chile and Argentina and an important ski resort. The purpose of this research was to investigate the likely failure mechanism and characterise the runout path and volume of the PRA. The insights gained on the back analysis of the slope were used in later stages to assess the hazard potential of a recurring major rockslide. The distinct element code UDEC was used to evaluate the failure mechanism. Elasto-plastic modelling results showed that sliding and shearing along the bedding planes together with brittle fracturing and shearing through the toe of the slope likely had occurred. Runout simulations were carried out using DAN3D. Combinations of rheologies were tested and ranked based on their ability to represent the current distribution of the debris by means of pre-failure topography reconstruction and volume estimates of the deposits. Results showed that the best basal rheological combination for the PRA was frictional during the rockslide and Voellmy when entrainment became important. In contrast, a constant frictional basal rheology best represented the Holocene rock slump. The performance of the present-day state of the slope was tested under different scenarios. Under static condition the slope proved to be stable indicating a stabilized geometrical profile with time. Also, the slope proved to be stable under increased pore water pressures at its toe. Finally the modelled slope was subjected to a seismic load (M=7.8) and its crest failed due to an outward rotation of blocks, probably aided by topographic amplification. The runout simulations showed that the leading edge of the flow could override part of the International Santiago-Mendoza Corridor with no direct impact to the Portillo Ski Resort. Overall, though, under this highly unlikely dynamic condition for the site, the hazard level is very low. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

The purpose of this research study is to propose guidelines for providing a positive user experience for pupils using Dr Math®. User experience was found to have a positive impact on the acceptance and adoption of a product. Thus the proposed guidelines contribute in maximizing the adoption and acceptance of Dr Math® among pupils. This study begins with an introductory chapter that describes the problem that forms the basis for this research. The chapter defines the objectives that this study is intended to achieve in order to accomplish its ultimate goal. The methodology followed to conduct this research study as well as its scope are also defined here. The results from a preliminary survey revealed that despite its potential accessibility, Dr Math® has a low adoption rate. However, when compared to other mobile learning (m-learning) applications for mathematics learning, Dr Math® is more popular. Thus Dr Math® was selected as a case for study. Chapter 2 of this study provides a detailed description of Dr Math® as a local mobile application for mathematics learning. It was found that the affordability and accessibility of Dr Math® did not necessarily imply a high adoption rate. There are various possible barriers to its low adoption. User experience (UX), which is the focus of this study, is one of them. Thus, a subsequent chapter deals with UX. Chapter 3 discusses UX, its scope, components and definition and places particular emphasis on its significance in the success of any product. The chapter also highlights the characteristics of a positive UX and the importance of designing for this outcome. In Chapter 4, a discussion and justification of the methodology used to conduct this research is discussed. This study primarily employs a qualitative inductive approach within an interpretivism paradigm. An exploratory single case study was used to obtain an in-depth analysis of the case. Data was collected using Dr Math® log files as a documentary source. Gathered data was then analysed and organized into themes and categories using qualitative content analysis as outlined in Chapter 5. Also the findings obtained from the results, which are mainly the factors that were found to have an impact on the user interaction with Dr Math®, are presented here. The identified factors served as a basis from which the guidelines presented in Chapter 6 were developed. Chapter 7 presents the conclusions and recommendations of the research. From both theoretical and empirical work, it was concluded that Dr Math® has the potential to improve mathematics learning in South Africa. Its adoption rate, however, is not satisfying: hence, the investigation of the factors impacting on the user interaction with Dr Math®, from which the proposed guidelines are based.

No description available.

Various models are currently in existence for determining the deflection of lithospheric plates under an applied transverse load. The most popular models treat lithospheric plates as thin elastic or thin viscoelastic plates. The equations governing the deflection of such plates have been solved successfully in two dimensions using integral transform techniques. Three dimensional models have been solved using Fourier Series expansions assuming a sinusoidal variation for the load and deflection. In the engineering context, the finite element technique has also been employed. The current aim, however, is to develop an efficient solver for the three dimensional elastic and viscoelastic problems using finite difference techniques. A variety of loading functions may therefore be considered with minimum work involved in obtaining a solution for different forcing functions once the main program has been developed. The proposed method would therefore provide a valuable technique for assessing new models for the loading of lithospheric plates as well as a useful educational tool for use in geophysics laboratories. The multigrid method, which has proved to be a fast, efficient solver for elliptic partial differential equations, is examined as the basis for a solver of both the elastic and viscoelastic problems. The viscoelastic problem, being explicitly time-dependent, is the more challenging of the two and will receive particular attention. Multigrid proves to be a very effective method applicable to the solution of both the elastic and viscoelastic problems. / Science, Faculty of / Mathematics, Department of / Graduate

D.Ing. / This study is a numerical investigation into the behaviour of cracks in uPVC pipes under pressure. This study is a continuation of a Masters dissertation which showed that leakage exponents vary significantly from the theoretical orifice exponent of 0.5 for cracks in pipes for different materials. This study looks at the behaviour of cracks in more detail and specifically with regard to the parameters of the pipe and crack. Using Finite Element Analysis the relationship between the pressure head and the leak area in pipes with longitudinal, spiral and circumferential cracks was investigated. It was found that the longitudinal, spiral and circumferential crack areas increase linearly with pressure. The slope of this linear relationship depends on various parameters, including loading state, pipe dimensions and pipe material properties. The effect that the individual pipe parameters had on the pressure-area slope was investigated. These parameters included the material properties of the pipe (Young’s modulus, Poisson’s ratio and longitudinal stress), the geometry of the pipe (internal diameter and wall thickness) as well as the geometry of the crack (length of the crack and the width of the crack). Once the effect of the pressure-area slope m is known, the link between the conventional leakage exponent N1 and the pressure-area slope m was further investigated and the effect of each parameter on the leakage exponent N1 was found. Using various data techniques the above data was combined and processed to find mathematical relationships that give reasonable descriptions of the pressure-area slopes of longitudinal, spiral and circumferential cracks. Once these equations for the pressure-area slopes were determined it was possible to obtain three new relationships for leakage from longitudinal, spiral and circumferential cracks.

Orientador: Amauri Garcia / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-08-07T13:54:54Z (GMT). No. of bitstreams: 1 Boeira_AlexandrePitol_D.pdf: 5616217 bytes, checksum: 09ba745cbcd2adadf17f39376225a70d (MD5) Previous issue date: 2006 / Resumo: A porosidade é um dos mais importantes defeitos encontrados em componentes fundidos. Neste trabalho, a macrossegregação e a formação de porosidade são investigadas por meio de modelagem numérica e experimentos de solidificação unidirecional vertical ascendente. As composições locais, ao longo dos lingotes, para ligas binárias de alumínio e cobre (AI-4,5%Cu, AI-6,2%Cu e AI-8,1 %Cu) são previstas pelo modelo de macrossegregação e servem de parâmetro de entrada para as simulações da correspondente microporosidade. Os efeitos da gravidade na redistribuição do soluto e na formação da microporosidade também são considerados pelo modelo. Um espectrômetro de tluorescência de raios X foi usado para determinar os perfis de segregação ao longo dos lingotes. As medidas de microporosidade foram realizadas através da análise picnométrica. Os perfis experimentais de segregação e de porosidade nos lingotes são comparados com as previsões teóricas fornecidas pelo modelo numérico, considerando um perfil do coeficiente de transferência de calor metal molde transitório, determinado experimentalmente. Observa-se uma excelente concordância entre os perfis simulados e experimentais de segregação inversa do cobre. A simulação da formação de porosidade, para canais anisotrópicos, mostra-se bastante alinhada com os resultados da análise experimental. Neste contexto, a fração volumétrica de poros apresenta uma tendência ascendente à medida que se distancia da base refrigerada / Abstract: Porosity is one of the most important defects in metaIs casting. In this research macrosegregation and porosity formation are investigated by a numerical modeling technique and by upward vertical unidirectional solidification experiments. The local composition, along the castings length of aluminum cooper binary alloys (AI-4.5 wt % Cu, AI-6.2 wt % Cu and AI-8.l wt % Cu) are predicted by the macrosegregation model and are used as an input parameter for simulation of the corresponding microporosity. The effects exerted by gravity upon the solute redistribution and microporosity formation are also encompassed by this model. X-ray fluorescence spectrometer was used to determine the segregation profiles along the castings. The measurement of microporosity was performed using picnometric analysis. The experimental segregation profiles and porosity evolution along the castings are compared with theoretical predictions furnished by the numerical mo deI, where the transient metal/mold heat transfer coefficient was experimentally determined. An excellent agreement between the simulated and experimental inverse copper profile has been observed. The simulation of porosity formation for anisotropic channels has fitted better the results of the experimental analysis. In this context, the volumetric fraction of pores has shown an ascending trend from the chill to the top of the ingot / Doutorado / Materiais e Processos de Fabricação / Doutor em Engenharia Mecânica