Nous présentons une classe de nouveaux modèles pour décrire les écoulements d’eau dans des aquifères peu profonds non confinés. Cette classe de modèles offre une alternative au modèle Richards 3d plus classique mais moins maniable. Leur dérivation est guidée par deux ambitions : le nouveau modèle doit d’une part être peu coûteux en temps de calcul et doit d’autre part donner des résultats pertinents à toute échelle de temps. Deux types d’écoulements dominants apparaissent dans ce contexte lorsque le rapport de l’épaisseur sur la longueur de l’aquifère est petit : le premier écoulement apparaît en temps court et est décrit par un problème vertical Richards 1d ; le second correspond aux grandes échelles de temps, la charge hydraulique est alors considérée comme indépendante de la variable verticale. Ces deux types d’écoulements sont donc modélisés de manière appropriée par le couplage d’une équation 1d pour la partie insaturée de l’aquifère et d’une équation 2d pour la partie saturée. Ces équations sont couplées au niveau d’une interface de profondeur h (t,x) en dessous de laquelle l’hypothèse de Dupuit est vérifiée. Le couplage est assuré de telle sorte que la masse globale du système soit conservée. Notons que la profondeur h (t,x) peut être une inconnue du problème ou être fixée artificiellement. Nous prouvons (dans le cas d’aquifères minces) en utilisant des développements asymptotiques que le problème Richards 3d se comporte de la même manière que les modèles de cette classe à toutes les échelles de temps considérées (courte, moyenne et grande). Nous décrivons un schéma numérique pour approcher le modèle couplé non linéaire. Une approximation par éléments finis est combinée à une méthode d’Euler implicite en temps. Ensuite, nous utilisons une reformulation de l’équation discrète en introduisant un opérateur de Dirichlet-to-Neumann pour gérer le couplage non linéaire en temps. Une méthode de point fixe est appliquée pour résoudre l’équation discrète reformulée. Le modèle couplé est testé numériquement dans différentes situations et pour différents types d’aquifère. Pour chacune des simulations, les résultats numériques obtenus sont en accord avec ceux obtenus à partir du problème de Richards original. Nous concluons notre travail par l’analyse mathématique d’un modèle couplant le modèle Richards 3d à celui de Dupuit. Il diffère du premier parce que nous ne supposons plus un écoulement purement vertical dans la frange capillaire supérieure. Ce modèle consiste donc en un système couplé non linéaire d’équation Richards 3d avec une équation parabolique non linéaire décrivant l’évolution de l’interface h (t,x) entre les zones saturées et non saturées de l’aquifère. Les principales difficultés à résoudre sont celles inhérentes à l’équation 3D-Richards, la prise en compte de la frontière libre h (t,x) et la présence de termes dégénérés apparaissant dans les termes diffusifs et dans les dérivées temporelles. / We present a class of new efficient models for water flow in shallow unconfined aquifers, giving an alternative to the classical but less tractable 3D-Richards model. Its derivation is guided by two ambitions : any new model should be low cost in computational time and should still give relevant results at every time scale.We thus keep track of two types of flow occurring in such a context and which are dominant when the ratio thickness over longitudinal length is small : the first one is dominant in a small time scale and is described by a vertical 1D-Richards problem ; the second one corresponds to a large time scale, when the evolution of the hydraulic head turns to become independent of the vertical variable. These two types of flow are appropriately modelled by, respectively, a one-dimensional and a two-dimensional system of PDEs boundary value problems. They are coupled along an artificial level below which the Dupuit hypothesis holds true (i.e. the vertical flow is instantaneous below the function h(t,x)) in away ensuring that the global model is mass conservative. Tuning the artificial level, which even can depend on an unknown of the problem, we browse the new class of models. We prove using asymptotic expansions that the 3DRichards problem and eachmodel of the class behaves the same at every considered time scale (short, intermediate and large) in thin aquifers. We describe a numerical scheme to approximate the non-linear coupled model. The standard Galerkin’s finite element approximation in space and Backward Euler method in time are used for discretization. Then we reformulate the discrete equation by introducing the Dirichlet to Neumann operator to handle the nonlinear coupling in time. The fixed point iterative method is applied to solve the reformulated discrete equation. We have examined the coupled model in different boundary conditions and different aquifers. In the every situations, the numerical results of the coupled models fit well with the original Richards problem. We conclude our work by the mathematical analysis of a model coupling 3D-Richards flow and Dupuit horizontal flow. It differs from the first one because we no longer assume a purely vertical flow in the upper capillary fringe. This model thus consists in a nonlinear coupled system of 3D-Richards equation with a nonlinear parabolic equation describing the evolution of the interface h(t,x) between the saturated and unsaturated zones of the aquifer. The main difficulties to be solved are those inherent to the 3D-Richards equation, the consideration of the free boundary h(t,x) and the presence of degenerate terms appearing in the diffusive terms and in the time derivatives.

The work developed in this thesis focusses on the electrical activity of the heart, from the modeling of the action potential originating from cardiac cells and propagating through the heart, as well as its electrical manifestation at the body surface. The study is divided in two main parts: modeling the action potential, and numerical simulations. For modeling the action potential a dimensional and asymptotic analysis is done. The key advance in this part of the work is that this analysis gives the steps to reliably control the action potential. It allows predicting the time/space scales and speed of any action potential that is to say the shape of the action potential and its propagation. This can be done as the explicit relations on all the physiological constants are defined precisely. This method facilitates the integrative modeling of a complete human heart with tissue-specific ionic models. It even proves that using a single model for the cardiac action potential is enough in many situations. For efficient numerical simulations, a numerical method for solving the heart-torso coupling problem is explored according to a level set description of the domains. This is done in the perspective of using directly medical images for building computational domains. A finite element method is then developed to manage meshes not adapted to internal interfaces. Finally, an anisotropic adaptive remeshing methods for unstructured finite element meshes is used to efficiently capture propagating action potentials within complex, realistic two dimensional geometries.

The work developed in this thesis focusses on the electrical activity of the heart, from the modeling of the action potential originating from cardiac cells and propagating through the heart, as well as its electrical manifestation at the body surface. The study is divided in two main parts: modeling the action potential, and numerical simulations. For modeling the action potential a dimensional and asymptotic analysis is done. The key advance in this part of the work is that this analysis gives the steps to reliably control the action potential. It allows predicting the time/space scales and speed of any action potential that is to say the shape of the action potential and its propagation. This can be done as the explicit relations on all the physiological constants are defined precisely. This method facilitates the integrative modeling of a complete human heart with tissue-specific ionic models. It even proves that using a single model for the cardiac action potential is enough in many situations. For efficient numerical simulations, a numerical method for solving the heart-torso coupling problem is explored according to a level set description of the domains. This is done in the perspective of using directly medical images for building computational domains. A finite element method is then developed to manage meshes not adapted to internal interfaces. Finally, an anisotropic adaptive remeshing methods for unstructured finite element meshes is used to efficiently capture propagating action potentials within complex, realistic two dimensional geometries.

<p>Many types of services are provided using some equipment or machines, e.g. transportation systems using vehicles. Designs of these systems require capacity decisions, e.g., the number of vehicles. In my dissertation, I use many-server and conventional heavy-traffic limit theory to derive asymptotically optimal capacity decisions, giving the desired level of delay and availability performance with minimum investment. The results provide near-optimal solutions and insights to otherwise analytically intractable problems.</p> <p>The dissertation will comprise two essays. In the first essay, &ldquoAsymptotic Analysis of Delay-based Performance Metrics and Optimal Capacity Decisions for the Machine Repair Problem with Spares,&rdquo I study the M/M/R machine repair problem with spares. This system can be represented by a closed queuing network. Applications include fleet vehicles' repair and backup capacity, warranty services' staffing and spare items investment decisions. For these types of systems, customer satisfaction is essential; thus, the delays until replacements of broken units are even more important than delays until the repair initiations of the units. Moreover, the service contract may include conditions on not only the fill rate but also the probability of acceptable delay (delay being less than a specified threshold value).</p> <p>I address these concerns by expressing delays in terms of the broken-machines process; scaling this process by the number of required operating machines (or the number of customers in the system); and using many-server limit theorems (limits taken as the number of customers goes to infinity) to obtain the limiting expected delay and probability of acceptable delay for both delay until replacement and repair initiation. These results lead to an approximate optimization problem to decide on the repair and backup-capacity investment giving the minimum expected cost rate, subject to a constraint on the acceptable delay probability. Using the characteristics of the scaled broken-machines process, we obtain insights about the relationship between quality of service and capacity decisions. Inspired by the call-center literature, we categorize capacity level choice as efficiency-driven, quality-driven, or quality- and efficiency-driven. Hence, our study extends the conventional call center staffing problem to joint staffing and backup provisioning. Moreover, to our knowledge, the machine-repair problem literature has focused mainly on mean and fill rate measures of performance for steady-state cost analysis. This approach provides complex, nonlinear expressions not possible to solve analytically. The contribution of this essay to the machine-repair literature is the construction of delay-distribution approximations and a near-optimal analytical solution. Among the interesting results, we find that for capacity levels leading to very high utilization of both spares and repair capacity, the limiting distribution of delay until replacement depends on one type of resource only, the repair capacity investment.</p> <p>In the second essay, &ldquoDiffusion Approximations and Near-Optimal Design of a Make-to-Stock Queue with Perishable Goods and Impatient Customers,&rdquo I study a make-to-stock system with perishable inventory and impatient customers as a two-sided queue with abandonment from both sides. This model describes many consumer goods, where not only spoilage but also theft and damage can occur. We will refer to positive jobs as individual products on the shelf and negative jobs as backlogged customers. In this sense, an arriving negative job provides the service to a waiting positive job, and vice versa. Jobs that must wait in queue before potential matching are subject to abandonment. Under certain assumptions on the magnitude of the abandonment rates and the scaled difference between the two arrival rates (products and customers), we suggest approximations to the system dynamics such as average inventory, backorders, and fill rate via conventional heavy traffic limit theory.</p> <p>We find that the approximate limiting queue length distribution is a normalized weighted average of two truncated normal distributions and then extend our results to analyze make-to-stock queues with/without perishability and limiting inventory space by inducing thresholds on the production (positive) side of the queue. Finally, we develop conjectures for the queue-length distribution for a non-Markovian system with general arrival streams. We take production rate as the decision variable and suggest near-optimal solutions.</p> / Dissertation

Functionally graded materials (FGMs) are composites with continuous material property variations. The distinct interfaces between the reinforcement and the matrix in classical composites are potential damage initiation sites. The concept of FGM aims at avoiding the material mismatch at the interfaces. Functionally graded materials originated from the need for a material that has high-toughness at very high operating temperatures that occur in rocket nozzles and aeroplane engines. One of the early applications of graded materials can be thus found in thermal barrier coatings of gas turbine blades. Recent applications of FGMs include optoelectronics, ballistic impact resistance structures, wear resistant coatings and others. Although the manufacturing and applications of FGMs are well developed the basic mechanics of failure is not well understood, which is important in developing engineering design methodologies. Modern day design practice uses the concepts of fracture mechanics and the fracture properties of graded materials is not well understood. Most studies in the literature have assumed that the material response of the bulk functionally graded material to be elastic even though the constituents are nominally ductile. Some asymptotic analysis available in the literature have described the effect of ductility on the fracture parameters. However, these analysis are not complete in the sense that they have some undetermined constants. The present thesis aims at performing whole-field finite element (FE) simulations of a crack embedded in a ductile functionally graded material subjected to an anti-plane shear (mode-3) loading. A J2-deformation theory based power-law hardening nonlinear material response is assumed. The material property variation is assumed to be in the radial-direction (r-FGM), tangential to the crack (x-FGM), normal to the crack plane (y-FGM) and also at an arbitrary angle to the crack-plane (xy-FGM). Yet another power law described the material property variation. The competition between the indices of the hardening and material property variation is understood by performing a parametric analysis by varying both systematically. Our results indicate that the first most singular term of the asymptotic series remains unaffected. For some values of the material property variation index, the second asymptotic term is affected. The semi-closed form solutions available in the literature were unable to decipher the relative range of dominance of the first and second terms. From the present whole-field FEM analysis were able to extract this relative range of dominance. Our results indicate the range of dominance of the first term is least for FGMs when the material property variation is in the direction to the crack (x-FGM), and it is more for y-FGM.

The work developed in this thesis focusses on the electrical activity of the heart, from the modeling of the action potential originating from cardiac cells and propagating through the heart, as well as its electrical manifestation at the body surface. The study is divided in two main parts: modeling the action potential, and numerical simulations. For modeling the action potential a dimensional and asymptotic analysis is done. The key advance in this part of the work is that this analysis gives the steps to reliably control the action potential. It allows predicting the time/space scales and speed of any action potential that is to say the shape of the action potential and its propagation. This can be done as the explicit relations on all the physiological constants are defined precisely. This method facilitates the integrative modeling of a complete human heart with tissue-specific ionic models. It even proves that using a single model for the cardiac action potential is enough in many situations. For efficient numerical simulations, a numerical method for solving the heart-torso coupling problem is explored according to a level set description of the domains. This is done in the perspective of using directly medical images for building computational domains. A finite element method is then developed to manage meshes not adapted to internal interfaces. Finally, an anisotropic adaptive remeshing methods for unstructured finite element meshes is used to efficiently capture propagating action potentials within complex, realistic two dimensional geometries.

We study asymptotic properties of evolution partial differential equations posed in unbounded spatial domain in the context of locally uniform spaces. This context allows the use of non-integrable data and carries an inherent non-compactness and non-separability. We establish the existence of a lo- cally compact attractor for non-local parabolic equation and weakly damped semilinear wave equation and provide an upper bound on the Kolmogorov's ε-entropy of these attractors and the attractor of strongly damped wave equation in the subcritical case using the method of trajectories. Finally we also investigate infinite dimensional exponential attractors of nonlinear reaction-diffusion equation in its natural energy setting. 1

Cette thèse porte sur l’analyse mathématique de l’interaction d’un fluide non-visqueux avec des structures immergées. Plus précisément, elle est structurée autour de deux axes principaux. L’un d’eux est l’analyse asymptotique du mouvement d’une particule infinitésimale en milieu liquide. L’autre concerne l’interaction entre des vagues et une structure immergée. La première partie de la thèse repose sur l’analyse mathématique d’un système d’équations différentielles ordinaires non-linéaires d’ordre 2 modélisant le mouvement d’un solide infiniment petit dans un fluide incompressible en 2D. Les inconnues du modèle décrivent la position du solide, c’est-à-dire la position du centre de masse et son angle de rotation. Les équations proviennent de la deuxième loi de Newton avec un prototype de force de type Kutta-Joukowski. Plus précisément, nous étudions la dynamique de ce système lorsque l’inertie du solide tend vers 0. Les principaux outils utilisés sont des développements asymptotiques multiéchelles en temps. Pour la dynamique de la position du centre de masse, l’étude met en évidence des analogies avec le mouvement d’une particule chargée dans un champ électromagnétique et la théorie du centre-guide. En l’occurrence, le mouvement du centreguide est donné par une équation de point-vortex. La dynamique de l’angle est quant à elle donnée par une équation de pendule non-linéaire lentement modulée. Des régimes très différents se distinguent selon les données initiales. Pour de petites vitesses angulaires initiales la méthode de Poincaré-Lindstedt fait apparaitre une modulation des oscillations rapides, alors que pour de grandes vitesses angulaires initiales, un movement giratoire bien plus irrégulier est observé. C’est une conséquence particulière et assez spectaculaire de l’enchevêtrement des trajectoires homocliniques. La deuxième partie de la thèse porte sur le problème des vagues dans le cas où le domaine occupé par le fluide est à surface libre et avec un fond plat sur lequel un objet solide se translate horizontalement sous l’effet des forces de pression du fluide. Nous avons étudié deux systèmes asymptotiques qui décrivent le cas d’un fluide parfait incompressible en faible profondeur. Ceux-ci correspondent respectivement aux équations de Saint-Venant et de Boussinesq. Grâce à leur caractère bien-posé en temps long, les modèles traités permettent de prendre en compte certains effets de la mécanique du solide, comme les forces de friction, ainsi que les effets non-hydrostatiques. Notre analyse théorique a été complétée par des études numériques. Nous avons développé un schéma de différences finies d’ordre élevé et nous l’avons adapté à ce problème couplé afin de mettre en évidence les effets d’un solide (dont le mouvement est limité à des translations sur le fond) sur les vagues qui passent au dessus de lui. A la suite de ces travaux, nous avons souligné l’influence des forces de friction sur ce genre de systèmes couplés ainsi que sur le déferlement des vagues. Quant à l’amortissement dû aux effets hydrodynamiques, une vague ressemblance avec le phénomène de l’eau morte est mise en évidence. / This PhD thesis concerns the mathematical analysis of the interaction of an inviscid fluid with immersed structures. More precisely it revolves around two main problems: one of them is the asymptotic analysis of an infinitesimal immersed particle, the other one being the interaction of water waves with a submerged solid object. Concerning the first problem, we studied a system of second order non-linear ODEs, serving as a toy model for the motion of a rigid body immersed in a two-dimensional perfect fluid. The unknowns of the model describe the position of the object, that is the position of its center of mass and the angle of rotation; the equations arise from Newton’s second law with the consideration of a Kutta-Joukowski type lift force. It concerns the detailed analysis of the dynamic of this system when the solid inertia tends to 0. For the evolution of the position of the solid’s center of mass, the study highlights similarities with the motion of a charged particle in an electromagnetic field and the wellknown “guiding center approximation”; it turns out that the motion of the corresponding guiding center is given by a point-vortex equation. As for the angular equation, its evolution is given by a slowly-in-time modulated non-linear pendulum equation. Based on the initial values of the system one can distinguish qualitatively different regimes: for small angular velocities, by the Poincaré-Lindstedt method one observes a modulation in the fast time-scale oscillatory terms, for larger angular velocities however erratic rotational motion is observed, a consequence of Melnikov’s observations on the presence of a homoclinic tangle. About the other problem, the Cauchy problem for the water waves equations is considered in a fluid domain which has a free surface on the upper vertical limit and a flat bottom on which a solid object moves horizontally, its motion determined by the pressure forces exerted by the fluid. Two shallow water asymptotic regimes are detailed, well-posedness results are obtained for both the Saint-Venant and the Boussinesq system coupled with Newton’s equation characterizing the solid motion. Using the particular structure of the coupling terms one is able to go beyond the standard scale for the existence time of solutions to the Boussinesq system with a moving bottom. An extended numerical study has also been carried out for the latter system. A high order finite difference scheme is developed, extending the convergence ratio of previous, staggered grid based models. The discretized solid mechanics are adapted to represent important features of the original model, such as the dissipation due to the friction term. We observed qualitative differences for the transformation of a passing wave over a moving solid object as compared to an immobile one. The movement of the solid not only influences wave attenuation but it affects the shoaling process as well as the wave breaking. The importance of the coefficient of friction is also highlighted, influencing qualitative and quantitative properties of the coupled system. Furthermore, we showed the hydrodynamic damping effects of the waves on the solid motion, reminiscent of the so-called dead water phenomenon.

The Topological Sensitivity Analysis results in a scalar function, denoted as Topological Derivative, that supplies for each point of the domain of definition of the problem the sensitivity of a given cost function when a small hole is created. However, when a hole is introduced, it is no longer possible to stablish a homeomorphism between the domains. Due to this mathematical difficulty the Topological Derivative may become restrictive, nevertheless be extremely general. Thus, in the present work it is proposed a new method to calculte the Topological Derivative via Shape Sensitivity Analysis. This result, formally proved through a theorem, leads to a simpler and more general methodology than the others found in the literature. The Topological Sensitivity Analysis is performed for several Engineering problems, and the obtained results are used to improve the design of mechanical devices by introducing holes. The same theory developed to calculate the Topological Derivative is used to determine the sensitivity of the cost function when a small incrustation is introduced in each position of the domain, resulting in a novel concept denoted as Configurational Sensitivity Analysis, being discussed some possible applications in the context of Inverse Problems and modelling of phenomena that experiment changes in the physical properties of the medium. Thus, the methodology developed in the present work results in a framework with potential applications in Topology Optimization, Inverse Problems and Mechanical Modelling, which may be seen, from now on, not only as a method to calculate the Topological Derivative, but as a promising research area in Computational Modelling. / A análise de Sensibilidade Topológica resulta em uma função escalar, denominada Derivada Topológica, que fornece para cada ponto do domínio de definição do problema a sensibilidade de uma dada função custo quando um pequeno furo é criado. No entanto, ao introduzir um furo, não é mais possível estabelecer um homeomorfismo entre os domínios envolvidos. Devido a essa dificuldade matemática a Derivada Topológica pode se tornar restritiva, não obstante seja extremamente geral. No presente trabalho, portanto, é proposto um novo método de cálculo da Derivada Topológica via Análise de Sensibilidade à Mudança de Forma. Este resultado, formalmente demonstrado através de um teorema, conduz a uma metodologia mais simples e geral do que as demais encontradas na literatura. A Análise de Sensibilidade Topológica é então realizada em diversos problemas da Engenharia e os resultados obtidos são empregados para melhorar o projeto de componentes mecânicos mediante a introdução de furos. A mesma teoria desenvolvida para calcular a Derivada Topológica é utilizada para determinar a sensibilidade da função custo ao introduzir uma pequena incrustação numa dada posição do domínio, resultando em um novo conceito denominado Análise de Sensibilidade Configuracional, sendo discutidas suas possíveis aplicações no contexto de Problemas Inversos e de modelagem de fenômenos que experimentam mudanças nas propriedades físicas do meio. Assim, a metodologia aqui desenvolvida é uma ferramenta em potencial tanto de Otimização Topológica quanto de Problemas Inversos e de Modelagem Mecânica, podendo ser vista, a partir de agora, não somente como um método de cálculo da Derivada Topológica, mas como uma promissora área de pesquisa em Modelagem Computacional.

The topological derivative provides the sensitivity of a given shape functional with respect to an infinitesimal non-smooth domain pertubation (insertion of hole or inclusion, for instance). Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with inifinitesimal pertubations. However, for pratical applications, we need to insert pertubations of finite sizes.Therefore, we consider other terms in the expansion, leading to the concept of higher-order topological derivatives. In a particular, we observe that the topological-shape sensitivity method can be naturally extended to calculate these new terms, resulting in a systematic methodology to obtain higher-order topological derivatives. In order to present these ideas, initially we apply this technique in some problems with exact solution, where the topological asymptotic expansion is obtained until third order. Later, we calculate first as well as second order topological derivative for the total potential energy associated to the Laplace equation in two-dimensional domain pertubed with the insertion of a hole, considering homogeneous Neumann or Dirichlet boundary conditions, or an inclusion with thermal conductivity coefficient value different from the bulk material. With these results, we present some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion, which has two main features:it allows us to deal with pertubations of finite sizes and provides a better descent direction in optimization and reconstruction algorithms. / A derivada topológica fornece a sensibilidade de uma dada função custo quando uma pertubação não suave e infinitesimal (furo ou inclusão, por exemplo) é introduzida. Classicamente, esta derivada vem do segundo termo da expansão assintótica topológica considerando-se apenas pertubações infinitesimais. No entanto, em aplicações práticas, é necessário considerar pertubação de tamanho finito. Motivado por este fato, o presente trabalho tem como objetivo fundamental introduzir o conceito de derivadas topológicas de ordem superiores, o que permite considerar mais termos na expansão assintótica topológica. Em particular, observa-se que o topological-shape sensitivity method pode ser naturalmente estendido para o cálculo destes novos termos, resultando em uma metodologia sistemática de análise de sensibilidade topológica de ordem superior. Para se apresentar essas idéias, inicialmente essa técnica é verificada através de alguns problemas que admitem solução exata, onde se calcula explicitamente a expansão assintótica topológica até terceira ordem. Posteriormente, considera-se a equação de Laplace bidimensional, cujo domínio é topologicamente pertubado pela introdução de um furo com condição de contorno de Neumann ou de Dirichlet homogêneas, ou ainda de uma inclusão com propriedade física distinta do meio. Nesse caso, são calculadas explicitamente as derivadas topológicas de primeira e segunda ordens. Com os resultados obtidos em todos os casos, estuda-se a influência dos termos de ordem superiores na expansão assintótica topológica, através de experimentos numéricos. Em particular, observa-se que esses novos termos, além de permitir considerar pertubações de tamanho finito, desempenham ainda um importante papel tanto como fator de correção da expansão assintótica topológica, quanto como direção de descida em processos de otimização. Finalmente, cabe mencionar que a metodologia desenvolvida neste trabalho apresenta um grande potencial para aplicação na otimização e em algoritimos de reconstrução.