O tema deste trabalho é o estudo de métodos numéricos para a simulação de escoamentos incompressíveis com superfície livre a baixos Reynolds, por meio da metodologia GENSMAC e suas formulações explícita e implícita. Neste contexto, temos especial interesse na formulação implícita, objetivando o enfraquecimento da restrição de estabilidade parabólica no passo temporal em escoamentos com superfície livre. O trabalho pode ser dividido em duas partes: na primeira, algumas modificações são discutidas, propostas e testadas com o objetivo de tornar a formulação implícita mais eficiente e precisa; em seguida, aproveitamos os resultados obtidos e generalizamos o método numérico existente para simular escoamentos viscoelásticos modelados pela equação constitutiva SXPP. Em ambos os casos, a formulação explícita também é usada para comparação e teste. Resultados que demonstram a eficiência e robustez das técnicas desenvolvidas são apresentados por meio da simulação numérica de complexos problemas envolvendo superfície livre.O tema deste trabalho é o estudo de métodos numéricos para a simulação de escoamentos incom- pressíveis com superfície livre a baixos Reynolds, por meio da metodologia GENSMAC e suas formulações explícita e implícita. Neste contexto, temos especial interesse na formulação implícita, objetivando o en- fraquecimento da restrição de estabilidade parabólica no passo temporal em escoamentos com superfície livre. O trabalho pode ser dividido em duas partes: na primeira, algumas modificações são discutidas, propostas e testadas com o objetivo de tornar a formulação implícita mais eficiente e precisa; em seguida, aproveitamos os resultados obtidos e generalizamos o método numérico existente para simular escoamentos viscoelásticos modelados pela equação constitutiva SXPP. Em ambos os casos, a formulação explícita também é usada para comparação e teste. Resultados que demonstram a eficiência e robustez das técnicas desenvolvidas são apresentados por meio da simulação numérica de complexos problemas envolvendo superfície livre / The purpose of this work is the study of numerical methods for low Reynolds number incompressible free surface flows, including the GENSMAC methodology and its explicit and implicit formulations. In this context, we have special interest in the implicit formulation, in order to circumvent the parabolic stability restriction in the time step for free surface flows. The work can be divided into two parts: firstly, some modifications are proposed, discussed, and tested with the purpose of making the implicit formulation more e¢ cient and accurate; secondly, we take advantage of the results derived in the first part to generalize the numerical method for simulating viscoelastic free surface flows modeled by the constitutive equation SXPP. In both cases, the explicit formulation is also used for comparison and tests. Results that demonstrate the e¢ ciency and robustness of the developed techniques are presented and illustrated by the numerical simulation of complex problems involving free surface flows

L’accent mis actuellement par l’industrie pétrolière sur l’augmentation de l’efficacité des réservoirs, ainsi que sur l’intérêt grandissant pour l’exploitation d’autres sources d’énergie enfouies profondément sous terre a suscité un regain d’intérêt pour la mécanique de la fracturation des roches en général et la fracturation hydraulique en particulier. La fracturation hydraulique, appelée de manière informelle “fracturation”, est un processus qui consiste généralement à injecter de l’eau, sous haute pression dans une formation rocheuse via le puits. Ce processus vise à créer de nouvelles fractures dans la roche et à augmenter la taille, l’étendue et la connectivité des fractures existantes. Des avancées récentes dans la modélisation et la simulation de fractures hydrauliques ont eu lieu, au confluent de facteurs qui incluent une activité accrue, une tendance vers une complexité accrue et une compréhension approfondie du modèle mathématique sous-jacent et de ses défis intrinsèques. Cependant, certaines des caractéristiques très importantes de ce processus ont été négligées. Parmi les caractéristiques négligées, on peut citer l’incapacité de la grande majorité des modèles existants de s’attaquer à la fois à la propagation de fractures hydrauliques dans la roche intacte, à l’inititation de nouvelles fractures ainsi qu’à la réactivation des fractures existantes. Une autre caractéristique qui a été ignorée est sa dimension intrinsèque en trois dimensions, négligée par la plupart des modèles actuallement proposés. Parmi tous les différents types de méthodes numériques développées pour évaluer le mécanisme du phénomène de fracturation, très peu sont capables de représenter la totalité des mecanismes mis en jeu. Dans la présente thèse, l’initiation et la propagation de fissures induites par les fluides dans des roches isotropes transversales sont simulées à l’aide d’un modèle hydromécanique (HM) couplé basé la méthode XFEM (eXtended Finite Element Method) et un modèle de zones cohésives. Le HM-XFEM développé dans cette thèse est une extension des modèles précédemment développés dans l’équipe hydro-géomécanique multi échelle de GeoRessources. L’accent a été porté plus particulièrement sur la prise en compte de l’anisotropie du milieu et sur son influence sur le chemin de propagation. Ce dernier est défini à partir du le concept d’angle de bifurcation introduit auparavant dans la littérature. En complément des développements réalisés dans le modèle HM-XFEM, effort a été fait pour mieux comprendre l’initiation de la fissure en utilisant la méthode des éléments discrets (DEM) à l’aide du logiciel open source YADE Open DEM. La nature différente des deux méthodes, DEM étant une méthode discontinue et XFEM, une méthode continue, révèle les potentiels des deux méthodes et permet de comparer correctement la méthode qui convient le mieux au problème à résoudre, compte tenu des objectifs de la conception / Current emphasis in petroleom industry toward increasing the reservoirs efficiency, along with the interest in exploitation of other sources of energy buried deep underground created a renewed interest in rock fracture mechanics in general and hydraulic fracturing specifically. Hydraulic fracturing, informally referred to as “fracking,” is an oil and gas well development process that typically involves injecting water, under high pressure into a bedrock formation via the well. This process is intended to create new fractures in the rock as well as increase the size, extent, and connectivity of existing fractures. However some of the very important features of this process have been overlooked. Among these neglected features one can name of inability of the vast majority of existing models to tackle at once the propagation of hydraulic fractures in fractured rocks-masses where a competing dipole mechanism exists between fracturing of the intact rock and re-activation of exiting fracture networks. Another feature that has been ignored is its intrinsically three dimensionality which is neglected by most models. Among all different types of numerical methods that have been developed in order to assess the mechanism of fracturing phenomenon very few, if any, can handle the entire complexity of such process. In the present thesis, fluid-driven crack initiation and propagation in transverse isotropic rocks is simulated using a coupled model comprising of eXtended Finite Element Method (XFEM) and cohesive zone models. The HM XFEM developed in this thesis is an extension to previous models developed introduced in multiscale hydrogeomechanics team of GeoRessources. An emphasis is put on considering the anisotropic nature of the medium and on studying its influence on the propagation path. This latter is investigated by the concept of bifurcation angle previously introduced in literature. In complementary efforts was made to have a better understanding of crack initiation in transversely isotropic media, we also used the discrete element method (DEM) in order to gain insights into the mechanisms at stake. Both methods exhibit their advantages and disadvantages in modeling fracturing phenomenon. The different nature of two methods, DEM being a discontinuous and XFEM being a continuous method, reveals potentials of both methods and renders a good comparison of which method suits the problem in hand the best, considering the the objectives of the design

Bubbly flows around ships have been studied for years, mostly in relation with ship acoustic signatures. Bubbles are generated at the bow and shoulder breaking waves, at the hull/free surface contact line, the propeller and the highly turbulent stern flow. These bubbles are further transported downstream by the flow forming a two-phase mixture in the wake that can be kilometers long. The presence of bubbles in the wake of a ship significantly affects the acoustic response of the medium and can be detected by measuring acoustic attenuation and backscattering making a ship vulnerable to detection. Additionally, the bubbly wake shows at the surface as a characteristic signature of white water, and given the length of the bubbly wake, it makes a ship visible from satellites. Therefore, the bubbly wake can be used to detect and identify surface ships. Bubbly flows do not scale to model scale experiments, and experiments on full scale ships are scarce mostly due to difficult access areas and the high speeds involved. It is therefore of interest to simulate the bubbly flow around ships to provide information difficult, if not impossible, to obtain with experiments. This work presents the development of a code for the simulation of polydispersed bubbly flows with a focus on ship hydrodynamics. The mathematical model implemented is based on a two-fluid formulation coupled with a Boltzmann-like transport equation describing the bubbly phase. The tool developed attempts to include most of the relevant physics of the problem to represent better the conditions of real scenarios. The resulting code allows the simulation of polydispersed bubbly flows in situations including free surface and air entrainment, high void fraction levels and moving control surfaces and propulsors. The code is two-way coupled, with a strong coupling between the two phases and between the bubble sizes. The complexity of the problems tackled in this research required the development of novel numerical methods solving issues never identified before or simply neglected. These methods play an essential role in the accuracy, robustness and efficiency of the code and include: a two-phase projection method that not only couples pressure and velocity but also implicitly couples void fraction, a time splitting marching scheme to solve separately coupling in space and in bubble sizes, and a stable numerical method to integrate the strong coupling introduced by collision forces. The implemented code is applied to the simulation of the bubbly flow around a full scale ship using the latest available models and computational techniques. A study is performed on the influence of several mechanisms on the predicted bubbly wake and comparisons with available experimental data are presented. The influence of breakup in the boundary layer is analyzed in detail as well. In addition, this work identifies several modeling and implementations issues and attempts to provide a path for future studies. To illustrate the flexibility and robustness of the code, a final demonstration case is presented that includes rotating propellers. The computation is performed at full scale, with the fully appended geometry of the vessel and includes incoming waves, oceanic background and rectified diffusion models. Many of these features are unique to this computation and make it the first of its kind.

Numerical schemes for the partial differential equations used to characterize stiffly forced conservation laws are constructed and analyzed. Partial differential equations of this form are found in many physical applications including modeling gas dynamics, fluid flow, and combustion. Many difficulties arise when trying to approximate solutions to stiffly forced conservation laws numerically. Some of these numerical difficulties are investigated. A new class of numerical schemes is developed to overcome some of these problems. The numerical schemes are constructed using an infinite sequence of conservation laws. Restrictions are given on the schemes that guarantee they maintain a uniform bound and satisfy an entropy condition. For schemes meeting these criteria, a proof is given of convergence to the correct physical solution of the conservation law. Numerical examples are presented to illustrate the theoretical results.

Les géomatériaux tels les roches et le béton ont la particularité de s’amollir sous chargement dynamique, c.-à-d. que la vitesse du son diminue avec l’amplitude de forçage. Afin de reproduire ce comportement, un modèle de milieu continu à variables internes est proposé. Il est composé d’une loi de comportement donnant l’expression de la contrainte, et d’une équation d’évolution pour la variable interne. La viscoélasticité non linéaire de type Zener est prise en compte par l’ajout de variables internes supplémentaires. Les équations du mouvement forment un système de lois de conservation non linéaire et non homogène. Le système d’équations aux dérivées partielles est résolu numériquement à l’aide de la méthode des volumes finis. Une solution analytique du problème de Riemann de l’élastodynamique non linéaire est explicitée. Elle est utilisée pour évaluer les performances des méthodes numériques. Les résultats numériques sont en accord qualitatif avec les résultats expérimentaux d’expériences de résonance (NRUS) et d’acousto-élasticité dynamique (DAET). Des méthodes similaires sont développées en 2D pour réaliser des simulations de propagation d’ondes. Dans le cadre des méthodes de continuation reposant sur la décomposition en harmoniques, une méthode numérique est développée pour le calcul de solutions périodiques. Sur la base d’une discrétisation éléments finis des équations du mouvement, cette méthode fréquentielle donne des simulations de résonance rapides, ce qui est utile pour mener des validations expérimentales. / Geomaterials such as rocks and concrete are known to soften under a dynamic loading, i.e., the speed of sound diminishes with forcing amplitudes. To reproduce this behavior, an internal-variable model of continuum is proposed. It is composed of a constitutive law for the stress and an evolution equation for the internal variable. Nonlinear viscoelasticity of Zener type is accounted for by using additional internal variables. The equations of motion write as a nonlinear and nonhomogeneous system of conservation laws. This system of partial differential equations is solved numerically using finite-volume methods. An analytical solution to the Riemann problem of nonlinear elastodynamics is provided, which is used to benchmark the performances of the numerical methods. Numerical results are in qualitative agreement with experimental results from resonance experiments (NRUS) and dynamic acousto-elastic testing (DAET). Similar methods are developed in 2D to perform wave propagation simulations. In the framework of harmonic-based continuation methods, a numerical method is developed for the computation of periodic solutions. Based on a finite element discretization of the equations of motion, this frequency-domain method provides fast resonance simulations, which is useful to carry out experimental validations.

Les travaux de thèse gravitent autour de la thématique des matériaux poreux en aéronautique, et de la prise en compte de l'incertitude sur les caractérisations réalisées. Est envisagé l'ajout de matériaux poreux au sein des cavités de liners acoustiques, matériaux constitués d'une plaque perforée et d'une cavité fonctionnant sur le principe du résonateur de Helmholtz et majoritairement utilisés dans l'industrie aéronautique. Cet ajout est réalisé avec pour objectif d'augmenter l'étendue spectrale de l'absorption acoustique de tels matériaux et d'en améliorer le fonctionnement en présence d'un fort niveau sonore et d'un écoulement rasant.Pour répondre à cette problématique générale, deux grandes pistes sont suivies. Plusieurs études sont d'abord menées sur des matériaux poreux seuls, afin de déterminer les propriétés intrinsèques de leur micro-géométrie, nécessaires à l'utilisation des modèles semi-phénoménologiques de fluide équivalent adoptés par la suite. Pour cela, un outil statistique d'inférence Bayésienne est utilisé afin d'extraire l'information sur ces propriétés, contenue dans les signaux réfléchis ou transmis par un matériau poreux, et ce dans trois régimes fréquentiels distincts. De plus, une extension de la modélisation des matériaux poreux rigides est proposée, par l'ajout de deux paramètres intrinsèques reliés au comportement visco-inertiel du fluide intra-pores dans le régime des basses fréquences.Dans un second temps, l'impédance d'un liner, une propriété globale représentant le comportement acoustique de matériaux, est identifiée par inférence Bayésienne. Des données issues d'un benchmark de la NASA sont utilisées pour valider l'outil d'inférence développé, lorsque le matériau est en présence d'un écoulement rasant. Une extension des résultats au cas du banc B2A de l'ONERA est également réalisée, avec des mesures des champs de vitesses au dessus du liner, obtenues par LDV. Cette technique d'identification est par la suite utilisée sur un cas issu du B2A où un matériau poreux est présent au sein des cavités du liner, afin de mettre à jour l'influence du matériau poreux sur la réponse acoustique du liner en présence d'un écoulement rasant. Des mesures complémentaires en tube à impédance, sans écoulement et en incidence normale, sont également réalisées à différents niveaux sonores et pour diverses combinaisons de plaques perforées et de matériaux poreux, de façon à mettre en évidence l'influence de la présence d'un matériau poreux sur le comportement acoustique d'un liner soumis à de forts niveaux sonores. / The present work focuses on porous materials in aeronautics and the uncertainty considerations on the performed identifications. Porous materials are added inside the cavities of acoustic liners, materials formed with perforated plates and cavities, behaving as Helmholtz resonators, which are widely used in the industry. The aim is to increase the frequency range of the absorption spectrum, while improving the behaviour of liners to grazing flow and high sound intensity.This general topic is addressed by following two different leads.Porous materials were first considered in order to identify the intrinsic properties of their micro-geometry, necessary to the equivalent fluid semi-phenomenological models used later on. To achieve this, a statistical Bayesian inference tool is used to extract information on these properties, contained in reflected or transmitted signals, in three distinct frequency regimes. Furthermore, a modelling extension of rigid porous media is introduced, by adding two new intrinsic parameters related to the pore micro-structure and linked to the visco-inertial behaviour of the intra-pore fluid, at low frequencies.Then, the liner impedance, a global property representing the acoustic behaviour of materials, is identified through a Bayesian inference process. Data from a NASA benchmark are used to validate the developed tool, when the liner is subject to a shear grazing flow. An extension of these results to ONERA's B2A aeroacoustic bench is also performed, with measurements of the velocity profiles above the liner, obtained with a Laser Doppler Velocimetry technique. This identification technique is then further used for liner materials filled with porous media, to highlight the eventual influence of such a porous media on the acoustic response of the liner, when subject to a shear grazing flow. Additional measurements are permed without flow, at normal incidence, in a classical impedance tube. Different combinations of perforated plates and porous materials are tested at different sound pressure level, to evaluate the influence of the presence of porous media on the non-linear behaviour of liners when high sound pressure levels are present.

For over 30 years, the Implicit Monte Carlo (IMC) method has been used to solve challenging problems in thermal radiative transfer. These problems are typically optically thick and di ffusive, as a consequence of the high degree of "pseudo-scattering" introduced to model the absorption and reemission of photons from a tightly-coupled, radiating material. IMC has several well-known features which could be improved: a) it can be prohibitively computationally expensive, b) it introduces statistical noise into the material and radiation temperatures, which may be problematic in multiphysics simulations, and c) under certain conditions, solutions can be unphysical and numerically unstable, in that they violate a maximum principle - IMC calculated temperatures can be greater than the maximum temperature used to drive the problem. We have developed a variant of IMC called "iterative thermal emission" IMC, which is designed to be more stable than IMC and have a reduced parameter space in which the maximum principle is violated. ITE IMC is a more implicit method version of the IMC in that it uses the information obtained from a series of IMC photon histories to improve the estimate for the end of time-step material temperature during a time step. A better estimate of the end of time-step material temperature allows for a more implicit estimate of other temperature dependent quantities: opacity, heat capacity, Fleck Factor (probability that a photon absorbed during a time step is not reemitted) and the Planckian emission source. The ITE IMC method is developed by using Taylor series expansions in material temperature in a similar manner as the IMC method. It can be implemented in a Monte Carlo computer code by running photon histories for several sub-steps in a given time-step and combining the resulting data in a thoughtful way. The ITE IMC method is then validated against 0-D and 1-D analytic solutions and compared with traditional IMC. We perform an in finite medium stability analysis of ITE IMC and show that it is slightly more numerically stable than traditional IMC. We find that significantly larger time-steps can be used with ITE IMC without violating the maximum principle, especially in problems with non-linear material properties. We also compare ITE IMC to IMC on a two-dimensional, orthogonal mesh, x-y geometry problem called the "crooked pipe" and show that our new method reproduces the IMC solution. The ITE IMC method yields results with larger variances; however, the accuracy of the solution is improved in comparison with IMC, for a given choice of spatial and temporal grid. / Graduation date: 2013

The discontinuities in the solutions of systems of conservation laws are widely considered as one of the difficulties in numerical simulation. A numerical method is proposed for solving these partial differential equations with discontinuities in the solution. The method is able to track these sharp discontinuities or interfaces while still fully maintain the conservation property. The motion of the front is obtained by solving a Riemann problem based on the state values at its both sides which are reconstructed by using weighted essentially non oscillatory (WENO) scheme. The propagation of the front is coupled with the evaluation of "dynamic" numerical fluxes. Some numerical tests in 1D and preliminary results in 2D are presented. / Singapore-MIT Alliance (SMA)

A mathematical model for a consolidation process of a highlyconcentrated, flocculated suspension is developed.Thesuspension is treated as a mixture of a fluid and solidparticles by an Eulerian two-phase fluid model.W e characterizethe suspension by constitutive relations correlating thestresses, interaction forces, and inter-particle forces toconcentration and velocity gradients.This results in threeempirically determined material functions: a hystereticpermeability, a non-Newtonian viscosity and a non-reversibleparticle interaction pressure.P arameters in the models arefitted to experimental data. A simulation program using finite difference methods both intime and space is applied to one and two dimensional testcases.Numer ical experiments are performed to study the effectof different viscosity and permeability models. The effect ofshear on consolidation rate is studied and it is significantwhen the permeability hysteresis model is employed.

In this thesis we focus mainly on special finite differences and finite volume methods and apply them to the pricing of barrier options.The structure of this work is the following: in Chapter 1 we introduce the definitions of options and illustrate some properties of vanilla European options and exotic options.Chapter 2 describes a classical model used in the financial world, the  Black-Scholes model. We derive theBlack-Scholes formula and show how stochastic differential equations model financial instruments prices.The aim of this chapter is also to present the initial boundary value problem and the maximum principle.We discuss boundary conditions such as: the first boundary value problem, also called  Dirichlet problem that occur in pricing ofbarrier options and European options. Some kinds of put options lead to the study of a second boundary value problem (Neumann, Robin problem),while the Cauchy problem is associated with one-factor European and American options.Chapter 3 is about finite differences methods such as theta, explicit, implicit and Crank-Nicolson method, which are used forsolving partial differential equations.The exponentially fitted scheme is presented in Chapter 4. It is one of the new classesof a robust difference scheme that is stable, has good convergence and does not produce spurious oscillations.The stability is also advantage of the box method that is presented in Chapter 5.In the beginning of the Chapter 6 we illustrate barrier options and then we consider a novel finite volume discretization for apricing the above options.Chapter 7 describes discretization of the Black-Scholes equation by the fitted finite volume scheme. In  Chapter 8 we present and describe numerical results obtained by using  the finite difference methods illustrated in the previous chapters.