A osteoporose é uma degradação óssea prevalente que é caracterizada pela perda de massa óssea e deteriorização da microarquitetura do osso trabecular. A perda da massa óssea é normalmente medida por meio da Densidade Mineral Óssea (DMO), porém é fato que esta medida não é suficiente para identificar completamente a fragilidade óssea e, consequentemente, o risco de fratura de um paciente. Portanto, o estudo da estrutura trabecular tornou-se de grande importância. Neste trabalho, é feita uma análise da conectividade trabecular, utilizando-se, para isso, ajustes lineares dos valores da Característica de Euler-Poincaré (CEP), calculada para pares de imagens tomográficas. Relacionando os achados com dados clínicos e medidas da DMO, percebeu-se que a CEP não está correlacionada aos mesmos, nem diretamente com a razão área trabecular/medula. São ainda abordados aqui conceitos de Estereologia, discutidos alguns de seus métodos, bem como algumas técnicas de processamento de imagens, que são ferramentas de estudo dos parâmetros histomorfométricos utilizados na investigação da microarquitetura trabecular. / Osteoporosis is a prevalent bone disorder tbat is cbaracterized by the loss af bone mass and the deterioration of the trabecnlar bone microarchitecture. The loss of bone mass is normally measured by the Bone Mineral Density (BMD), however i1.is known tbat tbis measure is not sufficient to fully identify the bone fragility and its consequent future risk for a patient. Therefore, the study af the bone strueture has become of great importance nowadays. ln this work, we investigate the applieabillty oHhe Euler. poinearé Charaeteristic (CE P) to estiroate the trabecular bone connectivity, using, for these, pairs af tomografic images. Thc resnlt will be comparcd to cUnic data and nieasure of BMD. Was noticied that the CEP values are not related with them, even in direet way, with the ratio between trabecnlar and no trabeeular areas. Besides, an introduct.ion to Stereology concepts are provided and some image processing techniques are discussed. These are important tools to the study of histomophometric paraIlleters that are UBedto investigate the trabeenlar microarchitecture.

Nosso objetivo neste trabalho é estudar a obstrução de Euler de uma função, este conceito foi definido por J.-P. Brasselet, D. Massey, A. J. Parameswaran e J. Seade, e generaliza dois conceitos importantes, a obstrução de Euler definida por R. D. MacPherson assim como o número de Milnor de uma função. O resultado principal deste trabalho mostra a relação existente entre a obstrução de Euler e a obstrução de Euler de uma função / Our goal in this work is to study the Euler obstruction of a function, this concept was defined by J.-P. Brasselet, D. Massey, A. J. Parameswaran and J. Seade, and it generalizes two important concepts, the Euler obstruction defined by R. D. MacPherson and the Milnor number of a function. The main result of this study shows the relation between the Euler obstruction and the Euler obstruction of a function

M.Sc. / There have been an extensive study on solutions of differential equations modeling physical phenomena that blows up in finite time. The blow-up time often represents an important change in the properties of such models and hence it is very important to compute it as accurate as possible. In this work, an adaptive in time numerical method for computing blow-up solutions for nonlinear ODEs is introduced. The method is named implicit midpoint-implicit Euler method (IMIE) and is based on the implicit Euler and the implicit midpoint method. The method is used to compute blow-up time for different examples of ODEs, PDEs and VIDEs. The PDEs studied are reaction-diffusion equations whereby the method of lines is first used to discretize the equation in space to obtain a system of ODEs. Quadrature rules are used to approximate the integral in the VIDE to get a system of ODEs. The IMIE method is then used then to solve the system of ODEs. The results are compared to results obtained by the PECEIE method and Matlab solvers ode45 and ode15s. The results show that the IMIE method gives better results than the PECE-IE and ode15s and compares quite remarkably with the 4th order ode45 yet it is of order 1 with order 2 superconvergence at the mesh points.

In this paper we present detailed Euler-Euler Large Eddy Simulations (LES) of dispersed bubbly flow in a rectangular bubble column. The motivation of this study is to investigate potential of this approach for the prediction of bubbly flows, in terms of mean quantities. The set of physical models describing the momentum exchange between the phases was chosen according to previous experiences of the authors. Experimental data, Euler-Lagrange LES and unsteady Euler-Euler Reynolds-Averaged Navier-Stokes model are used for comparison. It was found that the presented modelling combination provides good agreement with experimental data for the mean flow and liquid velocity fluctuations. The energy spectrum made from the resolved velocity from Euler-Euler LES is presented and discussed.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22970号 / 理博第4647号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 坂上 貴之, 教授 泉 正己, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Σκοπός της εργασίας είναι η παρουσίαση των εξισώσεων κίνησης του στερεού σώματος και η μελέτη δύο σημαντικών επιλύσιμων περιπτώσεων κίνησης στρόβου (Lagrange, Euler) . Στo πρώτο κεφάλαιο περιγράφουμε την κίνηση ενός στερεού σώματος χρησιμοποιώντας την ομάδα στροφών. Αποδεικνύουμε το θεώρημα Chasles το οποίο μας δείχνει πως η μετακίνηση ενός στερεού μπορεί να αποσυντεθεί σε περιστροφή γύρω από έναν άξονα και μεταφορά πάνω σε αυτόν. Στη συνέχεια σκοπός μας είναι η κατανόηση της γωνιακής ταχύτητας ενός στερεού σώματος. Σημαντικό ρόλο σε αυτή την πορεία παίζει τόσο το αδρανειακό όσο και το ενσωματωμένο στο στερεό σύστημα αναφοράς. Έπειτα δίνονται οι ορισμοί της ενέργειας, της στροφορμής, της ροπής και οι εκφράσεις τους συναρτήσει γνωστών πλέον εννοιών από τα προηγούμενα. Το κεφάλαιο ολοκληρώνεται με την Δυναμική που έχει ως αντικείμενο μελέτης και έρευνας τη κίνηση των σωμάτων υπό την επίδραση δυνάμεων, και καταλήγει στην παρουσίαση των εξισώσεων Euler. Στο δεύτερο κεφάλαιο στρέφουμε το ενδιαφέρον μας στις εφαρμογές και παρουσιάζουμε την επίλυση δύο σημαντικών προβλημάτων της μηχανικής: η κίνηση ενός συμμετρικού στρόβου που κινείται υπό την επίδραση του βάρους του έχοντας ένα σταθερό σημείο (ο στρόβος του Lagrange) και η κίνηση ενός στερεού που κινείται χωρίς την επίδραση εξωτερικών ροπών (ο στρόβος του Euler). Οι λύσεις εκφράζονται μέσω Ελλειπτικών Συναρτήσεων. Τέλος, στο τρίτο κεφάλαιο παρατίθενται σχόλια στις εργασίες των Holmes - Marsden και των Heijden - Yagasaki που αφορούν την ύπαρξη χαοτικής συμπεριφοράς στην διαταραγμένη περίπτωση Lagrange, που αναφέρεται σε σχεδόν συμμετρικό στρόβο. / Σκοπός της εργασίας είναι η παρουσίαση των εξισώσεων κίνησης του στερεού σώματος και η μελέτη δύο σημαντικών επιλύσιμων περιπτώσεων κίνησης στρόβου (Lagrange, Euler) . Στo πρώτο κεφάλαιο περιγράφουμε την κίνηση ενός στερεού σώματος χρησιμοποιώντας την ομάδα στροφών. Αποδεικνύουμε το θεώρημα Chasles το οποίο μας δείχνει πως η μετακίνηση ενός στερεού μπορεί να αποσυντεθεί σε περιστροφή γύρω από έναν άξονα και μεταφορά πάνω σε αυτόν. Στη συνέχεια σκοπός μας είναι η κατανόηση της γωνιακής ταχύτητας ενός στερεού σώματος. Σημαντικό ρόλο σε αυτή την πορεία παίζει τόσο το αδρανειακό όσο και το ενσωματωμένο στο στερεό σύστημα αναφοράς. Έπειτα δίνονται οι ορισμοί της ενέργειας, της στροφορμής, της ροπής και οι εκφράσεις τους συναρτήσει γνωστών πλέον εννοιών από τα προηγούμενα. Το κεφάλαιο ολοκληρώνεται με την Δυναμική που έχει ως αντικείμενο μελέτης και έρευνας τη κίνηση των σωμάτων υπό την επίδραση δυνάμεων, και καταλήγει στην παρουσίαση των εξισώσεων Euler. Στο δεύτερο κεφάλαιο στρέφουμε το ενδιαφέρον μας στις εφαρμογές και παρουσιάζουμε την επίλυση δύο σημαντικών προβλημάτων της μηχανικής: η κίνηση ενός συμμετρικού στρόβου που κινείται υπό την επίδραση του βάρους του έχοντας ένα σταθερό σημείο (ο στρόβος του Lagrange) και η κίνηση ενός στερεού που κινείται χωρίς την επίδραση εξωτερικών ροπών (ο στρόβος του Euler). Οι λύσεις εκφράζονται μέσω Ελλειπτικών Συναρτήσεων. Τέλος, στο τρίτο κεφάλαιο παρατίθενται σχόλια στις εργασίες των Holmes - Marsden και των Heijden - Yagasaki που αφορούν την ύπαρξη χαοτικής συμπεριφοράς στην διαταραγμένη περίπτωση Lagrange, που αναφέρεται σε σχεδόν συμμετρικό στρόβο.

Ce travail de thèse rentre dans le cadre du développement de modèles multi‐échelles de simulation de colonne d’absorption gaz‐liquide pour des applications de captage de CO2 en vue d’optimiser leur design. Il est le fruit d’une collaboration entre IFP Énergies nouvelles et l’Institut de Mécanique des Fluides de Toulouse. Les colonnes à garnissages représentent une technologie essentielle aux applications d’absorption gaz‐liquide. Dans les procédés de captage de CO2 aux amines, le solvant liquide s’écoule sur les parois du garnissage, idéalement sous forme de film ruisselant mouillant toute la surface disponible, le gaz, en régime turbulent, venant le cisailler à contre‐courant de manière à promouvoir un transfert de CO2 de la phase gaz vers la phase liquide. Un écoulement le plus homogène possible permet d’avoir les meilleures performances de transfert. Toutefois, l’expérience montre que des maldistributions, notamment de la phase liquide, peuvent apparaître même en cas de bonnes distributions en entrée de colonne. La distribution du liquide est régie par un phénomène de « dispersion » dont l’étude et la modélisation représentent le principal objet de cette thèse. Pour ce faire, ce travail de thèse s’appuie sur des travaux expérimentaux, réalisés sur une installation d’IFPEN à Lyon, et sur des travaux numériques réalisés dans l’équipe Interface de l’IMFT. Le premier axe de l’étude abordé a ainsi consisté en l’acquisition de données originales de distribution de liquide en partant d’une configuration d’alimentation sous forme d’un jet central en tête de colonne et ce pour deux types de garnissages métalliques : un garnissage structuré, le Mellapak 250.X et un garnissage vrac, l’IMTP. 40. La méthode de tomographie gamma a été mise en œuvre afin de mesurer l’atténuation d’un flux photonique par le liquide en mouvement ce qui permet d’établir des cartes de rétention locale de liquide sur une section de colonne. Les profils de rétention résultants ont été ensuite exploités afin de caractériser la dispersion de liquide dans le système pour des régimes d’écoulement allant des plus faibles aux plus fortes interactions gaz‐liquide. Ces résultats ont permis de développer un modèle simple d’advection diffusion faisant appel à un paramètre hydrodynamique clé qui est le « coefficient de dispersion », qui reproduit bien l’étalement du jet de liquide. Dans le cas du garnissage Mellapak 250.X nous avons pu mettre en évidence une dispersion qui est régie essentiellement par la géométrie du milieu. / This work is done within the framework of collaboration between IFPEN and l’Institut de Mécanique des Fluides de Toulouse (IMFT). It takes place in the general context of optimization of industrial gas‐liquid packed columns dedicated to CO2 capture using multi‐scale simulation approach of hydrodynamics and transfer. The main objective of this study is to develop predictive model for liquid dispersion in two‐phase flows in packed beds operating in the counter‐current mode. This model will help simulate the flow at the macro‐scale of a packed column. Packed columns are widely used for gas/liquid absorption processes since they generate subsequent exchange surface between phases with limited pressure drop. In the particular case of amine process, the two‐phase flow in the packing consists in thin trickling liquid films sheared by counter‐current gas flow that circulates in communicating tortuous channels. It is widely recognized that homogeneous flow ensures good separation performances. This is far from being the case at industrial scale since maldistributions, especially for liquid phase, do occur even with optimized liquid and gas distributors in the column inlets. Liquid distribution results from “dispersion” phenomenon which modelling is not fully handled in literature. Prediction of liquid dispersion has been the main objective of this Ph.D. research. This study is based upon two main approaches: an experimental study carried out using a 40 cm in diameter column operating in IFPEN (Lyon) and a numerical study carried out in Interface team in IMFT. The first part of this research focused on measuring spatial distribution of liquid hold‐up over different sections of the packed bed using a gamma ray tomography system. In order to focus into liquid dispersion phenomenon, a point source liquid feeding configuration was considered. Hold‐up maps obtained from photonic flux attenuation measurements were then used to characterize liquid dispersion from a source point for both packings (Mellapak 250.X and IMTP. 40) using a relatively simple advection‐diffusion model.

Neste trabalho determinamos relações entre a obstrução de Euler de uma função analítica com singularidade isolada f e o número de Milnor de f definido por Bruce e Roberts para funções definidas em espaços singulares. Apresentamos também uma generalização da obstrução de Euler de uma função analítica com singularidade isolada para o caso de uma aplicação \'f : (V, 0) seta (\'C POT. k\', 0) onde (V, 0) é o germe de uma variedade analítica complexa, equidimensional de dimensão \' n > OU = k\' , e uma fórmula para calcular a obstrução de Euler de k-referenciais, em termos da obstrução de Euler de f / In this work we determine relations between the local Euler obstruction of an analytic function singular at the origin to the case of a analytic map \'f : (V, 0) seta (\'C POT. k, 0\'), where (V, 0) is the germ of a complex analytic variety, equidimensional of dimension \' n > OU = k\', and a formula which computes the local Euler obstruction, defined for k-frames, in the local Euler obstruction of f

O desenvolvimento de uma ferramenta de Dinâmica de Fluidos Computacional que utiliza Método de Elementos Finitos baseada na discretização de Galerkin descontínuo é apresentado neste trabalho com objetivo de resolver a equação de Euler linearizada para escoamento compressível em duas dimensões usando malhas estruturadas e não estruturadas. Procuramos utilizar esta ferramenta como um propagador de ondas sonoras para estudar fenômenos aeroacústicos. O problema de Riemann presente no fluxo convectivo da equação de Euler é tratado com um método upwind HLL e para o avanço da solução no tempo é usado o método de Runge-Kutta explícito de 4 estágios com segunda ordem de precisão. A eficiência computacional, a convergência do método e a precisão são testadas através de simulações de escoamentos já apresentadas na literatura. A taxa de convergência para altas ordens de aproximação é assintótica que é um resultado compatível com a formulação Galerkin descontínuo. / The development of a Computation Fluid Dynamic Tool based on the Finite Element Method with discontinuous Galerkin discretization is presented in this work. The aim of this study is to solve the compressible linearized Euler\'s equation in two dimensions on structured and non structured meshes. This tool has been used as a means to study aeroacoustics phenomena. The Riemann\'s problem presented on a convective flow in Euler\'s equation is tackled by a HLL\'s method and the time integration being used is the four-stage Runge-Kutta explicit method with second order of accuracy. The computational efficiency, the convergence of the method and the accuracy are tested by comparing our results for flow simulations with those that are available in the literature. The convergence rate to high approximation order is asymptotic and it shows a result which is compatible with a discontinuous Galerkin formulation.

Neste trabalho, definimos a obstrução local de Euler de um espaço analítico complexo singular (X, \'x IND.0\'), denotada por Eu(X, \'x IND.0\'), e a obstrução local de Euler de uma função holomorfa f definida neste espaço, com uma singularidade isolada em \'x IND. 0\', denotada por \'Eu IND. f\' (X, \'x IND.0\'); e apresentamos duas fórmulas para seus respectivos cálculos. Em seguida, através de uma abordagem geométrica, determinamos as relações entre \'Eu IND. f\' (X,\'x IND.0\') e algumas generalizações do número de Milnor para funções em espaços singulares / In this work we define the local Euler obstruction of a complex analytic singularity (X, \'x IND.0\'), denoted Eu(X, \'x IND.0\'), and the local Euler obstruction of a holomorphic function f defined on this space, with an isolated singularity at \'x IND. 0\', denoted \'Eu IND. f\' (X, \'x IND.0\'); and we present two formulas for their respective calculations. Next, using a geometric approach, we determine the relations between \'Eu IND.f\' (X, \'x IND.0\') and several generalizations of the Milnor number for functions on singular spaces