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Cette thèse porte sur différents modèles de propagation en épidémiologie évolutive. L'objectif est d'en faire une analyse mathématique rigoureuse puis d'en tirer des enseignements biologiques. Dans un premier temps nous envisageons le cas d'une population d'hôtes répartis de manière homogène dans un espace linéaire, dans laquelle se propage un pathogène pouvant muter entre deux phénotypes plus ou moins virulents. Ce phénomène de mutation est à l'origine d'une interaction entre les dynamiques évolutive et épidémiologique du pathogène. Nous étudions la vitesse de propagation de l'épidémie et l'existence de fronts progressifs, ainsi que l'influence sur la vitesse de différents facteurs biologiques, comme des effets stochastiques liés à la taille de la population d'hôtes (explorations numériques). Dans un deuxième temps nous envisageons une hétérogénéité spatiale périodique dans la population d'hôtes, et l'existence de fronts pulsatoires pour le système de réaction-diffusion (non-coopératif) associé. Enfin nous considérons un pathogène pouvant muter vers un grand nombre de phénotypes différents et étudions l'existence de fronts potentiellement singuliers, modélisant ainsi une concentration sur un trait optimal. / In this thesis we consider several models of propagation arising in evolutionary epidemiology. We aim at performing a rigorous mathematical analysis leading to new biological insights. At first we investigate the spread of an epidemic in a population of homogeneously distributed hosts on a straight line. An underlying mutation process can shift the virulence of the pathogen between two values, causing an interaction between epidemiology and evolution. We study the propagation speed of the epidemic and the influence of some biologically relevant quantities, like the effects of stochasticity caused by the hosts' finite population size (numerical explorations), on this speed. In a second part we take into account a periodic heterogeneity in the hosts' population and study the propagation speed and the existence of pulsating fronts for the associated (non-cooperative) reaction-diffusion system. Finally, we consider a model in which the pathogen is allowed to shift between a large number of different phenotypes, and construct possibly singular traveling waves for the associated nonlocal equation, thus modelling concentration on an optimal trait.

Le moteur downsizé à allumage commandé constitue l’une des voies principales explorées par les constructeurs automobiles pour améliorer le rendement et réduire les émissions de dioxyde de carbone des motorisations essence. Il s’agit de combiner une réduction de la cylindrée avec une forte suralimentation afin d’améliorer le rendement du moteur, en particulier à faibles et moyennes charges. Leur mise au point est limitée par l’augmentation des combustions anormales, dont le contrôle par forte dilution peut également entraîner l'apparition de variabilités cycliques importantes. Actuellement, la compréhension des nombreux paramètres intervenant dans l'apparition de ces phénomènes et de leurs interactions, reste encore imparfaite. Dans ce contexte, l’objectif de ce travail est de contribuer à la compréhension des mécanismes impliqués dans les processus de propagation des flammes turbulentes. Cette étude est réalisée dans une enceinte de combustion sphérique haute pression haute température, équipée de ventilateurs générant une turbulence homogène et isotrope. La première partie de ce travail est consacrée à l’étude de la combustion prémélangée laminaire isooctane/air. Dans un deuxième temps, l’aérodynamique de l’écoulement dans l’enceinte est finement caractérisée par Vélocimétrie Laser Doppler et Vélocimétrie par Images de Particules. Enfin, la propagation des flammes turbulentes est étudiée en termes de vitesse à partir de visualisations par ombroscopie. Une loi unifiée, permettant de décrire la propagation des flammes turbulentes indépendamment des conditions thermodynamiques initiales, de l’intensité de la turbulence et de la nature du mélange réactif est notamment proposée. / Engine downsizing is a promising way explored for reducing carbon dioxide and pollutant emissions of spark ignition engines. Its principle is to reduce the engine size and to increase its specific power with the use of a turbocharger, especially at middle and high loads. Its development leads to the appearance of abnormal combustions, whose control by a high dilution rate can also lead to important cyclic variabilities. The understanding of the multiple parameters controlling these phenomena and their interactions, is still very limited. In this context, the present thesis addresses the issue related to the understanding of the mechanisms implied in turbulent flame propagation processes. This study is carried out in a fan-stirred high pressure high temperature spherical combustion vessel where turbulence is homogeneous and isotropic. The first part of this study concerns the study of laminar premixed flames of isooctane/air mixtures. Second, the turbulent flow in the vessel is accurately characterized using Laser Doppler Velocimetry and Particle Image Velocimetry measurements. Then, turbulent flame propagation is investigated in terms of flame velocity using shadowgraph visualizations. A unified scaling law for the turbulent propagation speed is especially proposed, regardless of the thermodynamic initial conditions, the intensity of the turbulence and the nature of the air/fuel mixture.

A duração da combustão tem implicação direta na determinação do trabalho negativo no ciclo motor. A duração da combustão tem relação inversa com a velocidade de propagação da chama. Assim, quanto maior a velocidade de propagação da chama, menor a duração da combustão. Além disto, a duração da combustão está relacionada com a relação volumétrica de compressão, com a razão de mistura e com o tipo de combustível, entre outros fatores. Neste sentido, quanto maior a velocidade de queima da mistura ar-combustível, menor o trabalho negativo no ciclo. Neste contexto, foram ensaiadas 3 relações de compressão e 3 razões de mistura, para 6 combustíveis de composição conhecida: gasolina comum tipo C, álcool etílico hidratado combustível (AEHC) e AEHC adicionado à gasolina comum tipo C nas proporções de 20%, 40%, 60% e 80%. A variação da pressão no interior da câmara de combustão, para todos os combustíveis, e a posição da centelha de ignição foram determinadas com a utilização da base angular e com variação da relação volumétrica de compressão e da razão de mistura. O diagrama , para fins de cálculo de área, foi traçado por pós-processamento. Os resultados obtidos indicam que a duração da combustão aumenta com o aumento da razão de mistura e diminui com o aumento da relação volumétrica de compressão, para todos os combustíveis ensaiados. Verificou-se que a amplitude da pressão, no interior da câmara de combustão, se comporta inversamente à duração da combustão. Mostrou-se que o aumento da razão de mistura implica na redução do trabalho líquido entregue ao êmbolo ao longo de todo o ciclo do motor. Opostamente ocorre com o aumento da relação volumétrica de compressão, que implica no aumento do trabalho líquido por ciclo. Concluiu-se que o máximo valor para o trabalho líquido ao longo de todo o ciclo foi obtido para a maior relação volumétrica de compressão (8:1) e a menor razão de mistura (λ = 0,9). O combustível que representou este resultado foi o álcool etílico hidratado combustível adicionado à gasolina comum tipo C em 80% (AEHC80). O mínimo valor para o trabalho líquido, ao longo de todo o ciclo foi obtido, para a menor relação volumétrica de compressão (6:1) e a maior razão de mistura dos ensaios (λ = 1,1). O combustível que representou este resultado foi o álcool etílico hidratado combustível (AEHC). / The duration of the combustion is directly implicative on determining the negative work of the motor cycle. The duration of the combustion has an inverse relation with the flame propagation speed. Thus, the higher propagation of the flame, the shorter duration of the combustion. Besides that, the duration of the combustion is related to the relation of compression, to the air/fuel relation and to the type of fuel, among other factors. In this way, the higher burning speed of the air/fuel mixture, the smallest negative work in the cycle. In such a context, there were tried three compression relations and three mixture ratios for six fuels of know compositions: ordinary gasoline C type, combustible hydrated ethyl alcohol and combustible hydrated ethyl alcohol added to ordinary gasoline C type in the proportions of 20%, 40%, 60% and 80%. The pressure variation inside the combustion chamber, for all fuels, and the position of the ignition spark were determined using the angular base and with variation of the compression relation and the mixture ratio. The diagram, for area calculation, was traced by post processing. The obtained results indicate that the combustion duration increases with the rising of the mixture ratio and decreases with the rising of the compression relation, for all the tried combustibles. It was found that the pressure amplitude inside the combustion chamber behaves inversely to the combustion duration behavior. It was shown that the rising of the mixture ratio implies on the reduction of the net work delivered by the piston all through the cycle of the motor. The opposite occurs with the rising of the compression relation, that implies on the increase of the net work by cycle. It was concluded that the maximum value for the net work all through the cycle was obtained for the higher compression relation (8:1) and the least mixture ratio of the tests (λ = 0,9). The combustible that achieved such a result was the combustible hydrated ethyl alcohol added to ordinary gasoline C type in 80%. The least value for the net work all through the cycle was obtained by the least compression relation (6:1) and the highest mixture ratio of the tests (λ = 1,1). The fuel that achieved that result was the combustible hydrated ethyl alcohol.

A duração da combustão tem implicação direta na determinação do trabalho negativo no ciclo motor. A duração da combustão tem relação inversa com a velocidade de propagação da chama. Assim, quanto maior a velocidade de propagação da chama, menor a duração da combustão. Além disto, a duração da combustão está relacionada com a relação volumétrica de compressão, com a razão de mistura e com o tipo de combustível, entre outros fatores. Neste sentido, quanto maior a velocidade de queima da mistura ar-combustível, menor o trabalho negativo no ciclo. Neste contexto, foram ensaiadas 3 relações de compressão e 3 razões de mistura, para 6 combustíveis de composição conhecida: gasolina comum tipo C, álcool etílico hidratado combustível (AEHC) e AEHC adicionado à gasolina comum tipo C nas proporções de 20%, 40%, 60% e 80%. A variação da pressão no interior da câmara de combustão, para todos os combustíveis, e a posição da centelha de ignição foram determinadas com a utilização da base angular e com variação da relação volumétrica de compressão e da razão de mistura. O diagrama , para fins de cálculo de área, foi traçado por pós-processamento. Os resultados obtidos indicam que a duração da combustão aumenta com o aumento da razão de mistura e diminui com o aumento da relação volumétrica de compressão, para todos os combustíveis ensaiados. Verificou-se que a amplitude da pressão, no interior da câmara de combustão, se comporta inversamente à duração da combustão. Mostrou-se que o aumento da razão de mistura implica na redução do trabalho líquido entregue ao êmbolo ao longo de todo o ciclo do motor. Opostamente ocorre com o aumento da relação volumétrica de compressão, que implica no aumento do trabalho líquido por ciclo. Concluiu-se que o máximo valor para o trabalho líquido ao longo de todo o ciclo foi obtido para a maior relação volumétrica de compressão (8:1) e a menor razão de mistura (λ = 0,9). O combustível que representou este resultado foi o álcool etílico hidratado combustível adicionado à gasolina comum tipo C em 80% (AEHC80). O mínimo valor para o trabalho líquido, ao longo de todo o ciclo foi obtido, para a menor relação volumétrica de compressão (6:1) e a maior razão de mistura dos ensaios (λ = 1,1). O combustível que representou este resultado foi o álcool etílico hidratado combustível (AEHC). / The duration of the combustion is directly implicative on determining the negative work of the motor cycle. The duration of the combustion has an inverse relation with the flame propagation speed. Thus, the higher propagation of the flame, the shorter duration of the combustion. Besides that, the duration of the combustion is related to the relation of compression, to the air/fuel relation and to the type of fuel, among other factors. In this way, the higher burning speed of the air/fuel mixture, the smallest negative work in the cycle. In such a context, there were tried three compression relations and three mixture ratios for six fuels of know compositions: ordinary gasoline C type, combustible hydrated ethyl alcohol and combustible hydrated ethyl alcohol added to ordinary gasoline C type in the proportions of 20%, 40%, 60% and 80%. The pressure variation inside the combustion chamber, for all fuels, and the position of the ignition spark were determined using the angular base and with variation of the compression relation and the mixture ratio. The diagram, for area calculation, was traced by post processing. The obtained results indicate that the combustion duration increases with the rising of the mixture ratio and decreases with the rising of the compression relation, for all the tried combustibles. It was found that the pressure amplitude inside the combustion chamber behaves inversely to the combustion duration behavior. It was shown that the rising of the mixture ratio implies on the reduction of the net work delivered by the piston all through the cycle of the motor. The opposite occurs with the rising of the compression relation, that implies on the increase of the net work by cycle. It was concluded that the maximum value for the net work all through the cycle was obtained for the higher compression relation (8:1) and the least mixture ratio of the tests (λ = 0,9). The combustible that achieved such a result was the combustible hydrated ethyl alcohol added to ordinary gasoline C type in 80%. The least value for the net work all through the cycle was obtained by the least compression relation (6:1) and the highest mixture ratio of the tests (λ = 1,1). The fuel that achieved that result was the combustible hydrated ethyl alcohol.

In this dissertation we study non-negative self-adjoint Laplace type operators acting on smooth sections of a vector bundle. First, we assume base manifolds are compact, boundaryless, and Riemannian. We start from the Fourier integral operator representation of half-wave operators, continue with spectral zeta functions, heat and resolvent trace asymptotic expansions, and end with the quantitative Wodzicki residue method. In particular, all of the asymptotic coefficients of the microlocalized spectral counting function can be explicitly given and clearly interpreted. With the auxiliary pseudo-differential operators ranging all smooth endomorphisms of the given bundle, we obtain certain asymptotic estimates about the integral kernel of heat operators. As applications, we study spectral asymptotics of Dirac type operators such as characterizing those for which the second coefficient vanishes. Next, we assume vector bundles are trivial and base manifolds are Euclidean domains, and study non-negative self-adjoint extensions of the Laplace operator which acts component-wise on compactly supported smooth functions. Using finite propagation speed estimates for wave equations and explicit Fourier Tauberian theorems obtained by Yuri Safarov, we establish the principle of not feeling the boundary estimates for the heat kernel of these operators. In particular, the implied constants are independent of self-adjoint extensions. As a by-product, we affirmatively answer a question about upper estimate for the Neumann heat kernel. Finally, we study some specific values of the spectral zeta function of two-dimensional Dirichlet Laplacians such as spectral determinant and Casimir energy. For numerical purposes we substantially improve the short-time Dirichlet heat trace asymptotics for polygons. This could be used to measure the spectral determinant and Casimir energy of polygons whenever the first several hundred or one thousand Dirichlet eigenvalues are known with high precision by other means.

A duração da combustão tem implicação direta na determinação do trabalho negativo no ciclo motor. A duração da combustão tem relação inversa com a velocidade de propagação da chama. Assim, quanto maior a velocidade de propagação da chama, menor a duração da combustão. Além disto, a duração da combustão está relacionada com a relação volumétrica de compressão, com a razão de mistura e com o tipo de combustível, entre outros fatores. Neste sentido, quanto maior a velocidade de queima da mistura ar-combustível, menor o trabalho negativo no ciclo. Neste contexto, foram ensaiadas 3 relações de compressão e 3 razões de mistura, para 6 combustíveis de composição conhecida: gasolina comum tipo C, álcool etílico hidratado combustível (AEHC) e AEHC adicionado à gasolina comum tipo C nas proporções de 20%, 40%, 60% e 80%. A variação da pressão no interior da câmara de combustão, para todos os combustíveis, e a posição da centelha de ignição foram determinadas com a utilização da base angular e com variação da relação volumétrica de compressão e da razão de mistura. O diagrama , para fins de cálculo de área, foi traçado por pós-processamento. Os resultados obtidos indicam que a duração da combustão aumenta com o aumento da razão de mistura e diminui com o aumento da relação volumétrica de compressão, para todos os combustíveis ensaiados. Verificou-se que a amplitude da pressão, no interior da câmara de combustão, se comporta inversamente à duração da combustão. Mostrou-se que o aumento da razão de mistura implica na redução do trabalho líquido entregue ao êmbolo ao longo de todo o ciclo do motor. Opostamente ocorre com o aumento da relação volumétrica de compressão, que implica no aumento do trabalho líquido por ciclo. Concluiu-se que o máximo valor para o trabalho líquido ao longo de todo o ciclo foi obtido para a maior relação volumétrica de compressão (8:1) e a menor razão de mistura (λ = 0,9). O combustível que representou este resultado foi o álcool etílico hidratado combustível adicionado à gasolina comum tipo C em 80% (AEHC80). O mínimo valor para o trabalho líquido, ao longo de todo o ciclo foi obtido, para a menor relação volumétrica de compressão (6:1) e a maior razão de mistura dos ensaios (λ = 1,1). O combustível que representou este resultado foi o álcool etílico hidratado combustível (AEHC). / The duration of the combustion is directly implicative on determining the negative work of the motor cycle. The duration of the combustion has an inverse relation with the flame propagation speed. Thus, the higher propagation of the flame, the shorter duration of the combustion. Besides that, the duration of the combustion is related to the relation of compression, to the air/fuel relation and to the type of fuel, among other factors. In this way, the higher burning speed of the air/fuel mixture, the smallest negative work in the cycle. In such a context, there were tried three compression relations and three mixture ratios for six fuels of know compositions: ordinary gasoline C type, combustible hydrated ethyl alcohol and combustible hydrated ethyl alcohol added to ordinary gasoline C type in the proportions of 20%, 40%, 60% and 80%. The pressure variation inside the combustion chamber, for all fuels, and the position of the ignition spark were determined using the angular base and with variation of the compression relation and the mixture ratio. The diagram, for area calculation, was traced by post processing. The obtained results indicate that the combustion duration increases with the rising of the mixture ratio and decreases with the rising of the compression relation, for all the tried combustibles. It was found that the pressure amplitude inside the combustion chamber behaves inversely to the combustion duration behavior. It was shown that the rising of the mixture ratio implies on the reduction of the net work delivered by the piston all through the cycle of the motor. The opposite occurs with the rising of the compression relation, that implies on the increase of the net work by cycle. It was concluded that the maximum value for the net work all through the cycle was obtained for the higher compression relation (8:1) and the least mixture ratio of the tests (λ = 0,9). The combustible that achieved such a result was the combustible hydrated ethyl alcohol added to ordinary gasoline C type in 80%. The least value for the net work all through the cycle was obtained by the least compression relation (6:1) and the highest mixture ratio of the tests (λ = 1,1). The fuel that achieved that result was the combustible hydrated ethyl alcohol.

Les phénomènes liés au comportement à la rupture de la glace sous impact sont fréquents dans le génie civil, pour les structures offshore, et les processus de dégivrage. Pour réduire les dommages causés par l'impact de la glace et optimiser la conception des structures ou des machines, l'étude sur le comportement à la rupture dynamique de la glace sous impact est nécessaire. Ces travaux de thèse portent donc sur la propagation dynamique des fissures dans la glace sous impact. Une série d'expériences d'impact est réalisée avec un dispositif de barres de Hopkinson. La température est contrôlée par une chambre de refroidissement. Le processus dynamique de la rupture de la glace est enregistré avec une caméra à grande vitesse et ensuite analysé par des méthodes d'analyse d'images. La méthode des éléments finis étendus complète cette analyse pour évaluer la ténacité dynamique. Au premier abord, le comportement dynamique de la glace sous impact est étudié avec des échantillons cylindriques afin d'établir la relation contrainte-déformation dynamique qui sera utilisée dans les simulations numériques plus tard. Nous avons observé de multi-fissuration dans les expériences sur les échantillons cylindriques mais son étude est trop difficile à mener. Pour mieux comprendre la propagation des fissures dans la glace, des échantillons rectangulaires avec une pré-fissure sont employés. En ajustant la vitesse d'impact on aboutit à la rupture des spécimens avec une fissure principale à partir de la pré-fissure. L'histoire de la propagation de fissure et de sa vitesse sont évaluées par analyse d'images basée sur les niveaux de gris et par corrélation d'images. La vitesse de propagation de la fissure principale est identifiée dans la plage de 450 à 610 m/s ce qui confirme les résultats précédents. Elle varie légèrement au cours de la propagation, dans un premier temps elle augmente et se maintient constante ensuite et diminue à la fin. Les paramètres obtenus expérimentalement, tels que la vitesse d'impact et la vitesse de propagation de fissure, sont utilisés pour la simulation avec la méthode des éléments finis étendus. La ténacité d'initiation dynamique et la ténacité dynamique en propagation de fissure sont déterminées lorsque la simulation correspond aux expériences. Les résultats indiquent que la ténacité dynamique en propagation de fissure est linéaire vis à vis de la vitesse de propagation et semble indépendante de la température dans l'intervalle -15 à -1 degrés. / The phenomena relating to the fracture behaviour of ice under impact loading are common in civil engineering, for offshore structures, and de-ice processes. To reduce the damage caused by ice impact and to optimize the design of structures or machines, the investigation on the dynamic fracture behaviour of ice under impact loading is needed. This work focuses on the dynamic crack propagation in ice under impact loading. A series of impact experiments is conducted with the Split Hopkinson Pressure Bar. The temperature is controlled by a cooling chamber. The dynamic process of the ice fracture is recorded with a high speed camera and then analysed by image methods. The extended finite element method is complementary to evaluate dynamic fracture toughness at the onset and during the propagation. The dynamic behaviour of ice under impact loading is firstly investigated with cylindrical specimen in order to obtain the dynamic stress-strain relation which will be used in later simulation. We observed multiple cracks in the experiments on the cylindrical specimens but their study is too complicated. To better understand the crack propagation in ice, a rectangular specimen with a pre-crack is employed. By controlling the impact velocity, the specimen fractures with a main crack starting from the pre-crack. The crack propagation history and velocity are evaluated by image analysis based on grey-scale and digital image correlation. The main crack propagation velocity is identified in the range of 450 to 610 m/s which confirms the previous results. It slightly varies during the propagation, first increases and keeps constant and then decreases. The experimentally obtained parameters, such as impact velocity and crack propagation velocity, are used for simulations with the extended finite element method. The dynamic crack initiation toughness and dynamic crack growth toughness are determined when the simulation fits the experiments. The results indicate that the dynamic crack growth toughness is linearly associated with crack propagation velocity and seems temperature independent in the range -15 to -1 degrees.