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Dans cette thèse, nous nous intéressons à la simulation numérique d'écoulements compressibles comportant des interfaces. Ces interfaces peuvent séparer un fluide et un solide rigide, deux fluides de lois d'état différentes ou encore un fluide et un solide élastique. Dans un premier temps, nous avons élaboré une méthode de type frontières immergées afin d'imposer une condition de glissement au bord d'un obstacle rigide de manière précise. Nous avons ensuite étudié et validé un schéma de type interface non diffuse pour les écoulements multi-matériaux en vue d'appliquer la méthode de frontières immergées aux solides déformables.

The world of computing simulation has experienced great progresses in recent years and requires more exigent multidisciplinary challenges to satisfy the new upcoming demands. Increasing the importance of solving multi-disciplinary problems makes developers put more attention to these problems and deal with difficulties involved in developing software in this area. Conventional finite element codes have several difficulties in dealing with multi-disciplinary problems. Many of these codes are designed and implemented for solving a certain type of problems, generally involving a single field. Extending these codes to deal with another field of analysis usually consists of several problems and large amounts of modifications and implementations. Some typical difficulties are: predefined set of degrees of freedom per node, data structure with fixed set of defined variables, global list of variables for all entities, domain based interfaces, IO restriction in reading new data and writing new results and algorithm definition inside the code. A common approach is to connect different solvers via a master program which implements the interaction algorithms and also transfers data from one solver to another. This approach has been used successfully in practice but results duplicated implementation and redundant overhead of data storing and transferring which may be significant depending to the solvers data structure. The objective of this thesis is to design and implement a framework for building multi-disciplinary finite element programs. Generality, reusability, extendibility, good performance and memory efficiency are considered to be the main points in design and implementation of this framework. Preparing the structure for team development is another objective because usually a team of experts in different fields are involved in the development of multi-disciplinary code. Kratos, the framework created in this work, provides several tools for easy implementation of finite element applications and also provides a common platform for natural interaction of its applications in different ways. This is done not only by a number of innovations but also by collecting and reusing several existing works. In this work an innovative variable base interface is designed and implemented which is used at different levels of abstraction and showed to be very clear and extendible. Another innovation is a very efficient and flexible data structure which can be used to store any type of data in a type-safe manner. An extendible IO is also created to overcome another bottleneck in dealing with multi-disciplinary problems. Collecting different concepts of existing works and adapting them to coupled problems is considered to be another innovation in this work. Examples are using an interpreter, different data organizations and variable number of dofs per node. The kernel and application approach is used to reduce the possible conflicts arising between developers of different fields and layers are designed to reflect the working space of different developers also considering their programming knowledge. Finally several technical details are applied in order to increase the performance and efficiency of Kratos which makes it practically usable. This work is completed by demonstrating the framework's functionality in practice. First some classical single field applications like thermal, fluid and structural applications are implemented and used as benchmark to prove its performance. These applications are used to solve coupled problems in order to demonstrate the natural interaction facility provided by the framework. Finally some less classical coupled finite element algorithms are implemented to show its high flexibility and extendibility. / El mundo de la simulación computacional ha experimentado un gran avance en los últimos años y cada día requiere desafíos multidisciplinares más exigentes para satisfacer las nuevas demandas. El aumento de la importancia por resolver problemas multidisciplinares hizo poner más atención a la resolución de estos problemas y a los problemas que éstos implican en el área de desarrollo de software. Los códigos convencionales de elementos finitos tienen varias dificultades para enfrentar se con problemas multidisciplinares. Muchos de estos códigos se diseñan y desarrollan para solucionar ciertos tipos de problemas, implicando generalmente un solo campo. Ampliar estos códigos para resolver problemas en otros campos del análisis, normalmente es difícil y se necesitan grandes modificaciones. Los ejemplos más comunes son: grados de libertad predefinidos para los nodos, estructura de datos capaz de guardar sólo una serie de variables definidas, lista global de las variables para todas las entidades, interfaces basadas en los dominios, capacidad del Input/Ouput para leer nuevos datos o escribir nuevos resultados y definición del algoritmo dentro del código. Un método común para resolver estos problemas es conectar varios modulos de calculo a través de un programa principal que implemente los algoritmos de la interacción y también transfiera datos de un modulo de calculo a otro. Este método se ha utilizado en la práctica con éxito, pero resulta en muchas duplicaciones del código y exceso de almacenamiento y tiempo de ejecución, dependiendo de la estructura de datos de los modulos de calculo. El objetivo de esta tesis es diseñar e implementar un marco general para el desarrollo programas de elementos finitos multidisciplinares. La generalidad, la reutilización, la capacidad de ampliación, el buen rendimiento y la eficiencia en el uso de la memoria por parte del codigo son considerados los puntos principales para el diseño e implementación de este marco. La preparación de esta estructura para un fácil desarrollo en equipo es otro objetivo importante, porque el desarrollo de un código multidisciplinar generalmente requiere expertos en diferentes campos trabajando juntos. Kratos, el marco creado en este trabajo, proporciona distintas herramientas para una fácil implementación de aplicaciones basadas en el método de los elementos finitos. También proporciona una plataforma común para una interacción natural y de diferentes maneras entre sus aplicaciones. Esto no sólo está hecho innovando, sino que además se han recogido y usado varios trabajos existentes. En este trabajo se diseña y se implementa una interface innovadora basada en variables, que se puede utilizar a diferentes niveles de abstracción y que ha demostrado ser muy clara y extensible. Otra innovación es una estructura de datos muy eficiente y flexible, que se puede utilizar para almacenar cualquier tipo de datos de manera "type-safe". También se ha creado un Input/Ouput extensible para superar otras dificultades en la resolución de problemas multidisciplinares. Otra innovación de este trabajo ha sido recoger e integrar diversos conceptos de trabajos ya existentes, adaptándolos a problemas acoplado.Esto incluye el uso de un intérprete, diversas organizaciones de datos y distinto número de grados de libertad por nodo. El concepto de núcleo y aplicación se utiliza para separar secciones del codigo y reducir posibles conflictos entre desarrolladores de diversos campos. Varias capas en la estructura de Kratos han sido diseñadas considerando los distintos niveles de programación de diferentes tipos de desarrolladores. Por último, se aplican varios detalles técnicos para aumentar el rendimiento y la eficacia de Kratos, convirtiendo lo en una herramienta muy útil para la resolución de problemas prácticos. Este trabajo se concluye demostrando el funcionamiento de Kratos en varios ejemplos prácticos. Primero se utilizan algunas aplicaciones clásicas de un solo campo como prueba patrón de rendimiento. Después, estas aplicaciones se acoplan para resolver problemas multidisciplinares, demostrando la facilidad natural de la interacción proporcionada por Kratos. Finalmente se han implementado algunos algoritmos menos clásicos para demostrar su alta flexibilidad y capacidad.

Cette thèse de mathématiques appliquées traite de problèmes théoriques, numériques et asymptotiques en transport motivés par la physiologie rénale. Plus précisément, elle vise à comprendre et quantifier les échanges de solutés qui peuvent mener dans des cas pathologiques à des néphrocalcinoses, qui se caractérisent par des dépôts calciques dans le parenchyme rénal. Le manuscrit est constitué de deux parties. La première partie concerne le développement et l'analyse mathématique d'un modèle simplifié du rein. Il s'agit d'un système de 3 EDP hyperboliques à vitesses constantes, couplées par leur terme source non linéaire et assorti de conditions aux bords spécifiques. Le modèle rentre dans le cadre des modèles cinétiques avec un nombre fini de vitesses et des conditions aux bords de type réflexion. Nous montrons que ce système est bien posé, qu'il tend en temps grand vers un état stationnaire. On montre que le taux de convergence est exponentiel avec des éléments spectraux. Nous proposons l'étude du rôle de deux paramètres à travers une analyse asymptotique. L'une d'entre elles nous place dans le cadre de la relaxation hyperbolique vers une loi de conservation scalaire avec un flux hétérogène en espace sur un domaine borné. La deuxième partie concerne le développement et l'analyse numérique d'un modèle réaliste du rein. Il s'agit d'un système de 27 équations aux dérivées partielles de type hyperboliques dont les vitesses sont les solutions de 8 équations différentielles non linéaires, et toutes ces équations sont couplées par leur terme source. Les conditions aux bords sont là aussi spécifiques au modèle. Nous interprétons ensuite les résultats obtenus d'un point de vue physiologique, en proposant des prédictions de profils de concentration calciques dans le rein, dans le cas normal et dans certains cas pathologiques.

In this dissertation, a numerical method to design a curved surface for accurately flank milling with a general tool of revolution is presented. Instead of using the ruled surface as the design surface, the flank millable surface can better match the machined surface generated by flank milling techniques, and provide an effective tool to the designer to control the properties and the specifications of the design surface. A method using the least squares surface fitting to design the flank millable surface is first discussed. Grazing points on the envelope of the moving tool modeled by the grazing surface are used as the sample points and a NURBS surface is used to approximate the given grazing surface. The deviation between the grazing surface and the NURBS surface can be controlled by increasing the number of the control points. The computation process for this method is costly in time and effort. In engineering design, there is a need for fast and effortless methods to simplify the flank millable surface design procedure. A technique to approximate the grazing curve with NURBS at each tool position is developed. Based on the characteristics of the grazing surface and the geometries of the cutting tool, these NURBS representations at a few different tool positions, namely at the start, interior and end, are lofted to generate a NURBS surface. This NURBS surface represents the grazing surface and is treated as the design surface. Simulation results show that this design surface can accurately match the machined surface. The accuracy of the surface can be controlled by adding control points to the control net of the NURBS surface. A machining test on a 5-axis machine was done to verify the proposed flank millable surface design method. The machined surface was checked on a CMM and the obtained results were compared with the designed flank millable surface. The comparison results show that the machined surface closely matches the design surface. The proposed flank millable surface design method can be accurately used in the surface design.

In this thesis a finite element model of a household washing machine cylinder is built and analysed in ABAQUS 6.9-2. The aim is to help Asko appliances in conducting similar analysis for future manufacturing of high capacity cylinders by reducing experimentation. The analysis is mainly concerned with an evenly distributed load at a constant angular velocity. The load is applied with the help of lead plates instead of clothes. The cylinder is loaded with three thin (2 mm) lead plates weighing 2 kg each. The plates with dimensional 370x240x2 mm are mounted with one strip of double sided foam tape inside the cylinder. To estimate the behavior of the cylinder the strains are measured when the cylinder is rotating at 1620 and 2200 revolution per minute (rpm). To validate the model the numerical analyses are compared with experimental results. The results clearly show that the numerical strain values fit with experimental strain values.

In this thesis we investigate Asian oating strike options. We particu-larly focus on options with early exercise - American options. This typeof options are very lucrative to the end-users of commodities or ener-gies who are tend to be exposed to the average prices over time. Asianoptions are also very popular with corporations, who have ongoing cur-rency exposures. The main idea of the pricing is to examine the freeboundary position on which the value of the option is depending. Wefocus on developing a ecient numerical algorithm for this boundary.In the rst Chapter we give an informative description of the nancialderivatives including Asian options. The second Chapter is devoted tothe analytical derivation of the corresponding partial dierential equa-tion coming from the original Black - Scholes equation. The problemis simplied using transformation methods and dimension reduction. Inthe third and fourth Chapter we describe important numerical methodsand discretize the problem. We use the rst order Lie splitting and thesecond order Strang splitting. Finally, in the fth Chapter we makenumerical experiments with the free boundary and compare the resultwith other known methods.

Asian options are exotic financial derivative products which price must be calculated by numerical evaluation. In this thesis, we study certain ways of solving partial differential equations, which are associated with these derivatives. Since standard numerical techniques for Asian options are often incorrect and impractical, we discuss their variations, which are efficiently applicable for handling frequent numerical instabilities reflected in form of oscillatory solutions. We will show that this crucial problem can be treated and eliminated by adopting flux limiting techniques, which are total variation dimishing.