Neste trabalho é apresentado a modelagem mecânica e numérica de um escoamento elastoviscoplástico tixotrópico, em termos dos campos de velocidade, pressão, tensão e parâmetro de estrutura. A discretização numérica é feita pelo Método de Elementos Finitos Estabilizado, também conhecido como Galerkin Mínimos Quadrados (GMQ), através de elementos quadrangulares bilineares. O clássico problema da cavidade é utilizado nas simulações, a fim de comparar a formulação e o código utilizados com os resultados conhecidos na literatura. Além disso, apresenta-se o estudo de materiais elasto-viscoplástico tixotrópico em uma contração abrupta na escala 4:1, utilizando a formulação descrita e uma nova função viscosidade para fluidos viscoplásticos, denominada função Viscoplástica Harmônica (VPH). Resultados envolvendo a função VPH são introduzidos e discutidos pela primeira vez nesta tese e apresenta um ótimo ajuste de curva, quando comparada com outras funções disponíveis na literatura. Além da fácil implementação, essa função também apresenta um platô para as altas e baixas viscosidades que são fisicamente realistas, visto que não é possível uma viscosidade infinita ou nula. O menor tempo computacional é também uma característica perceptível nas simulações usando a nova função viscoplástica, isso é um atributo do seu equacionamento que não depende de um termo exponencial, como outros modelos. O estudo de qualidade de malha também é apresentado a fim de garantir a escolha do domínio discreto adequado. Apesar do uso de elementos de ordem inferior, o método GMQ mostrou-se estável na aproximação numérica de todos os problemas dispostos, garantindo até mesmo a análise sobre os efeitos da cinemática, da elasticidade e da tixotropia no escoamento dos fluidos dentro da contração abrupta. / In this work the mechanical and numerical modeling of a thixotropic elasto-viscoplastic flow in terms of the velocity, pressure, stress and structure parameter is presented. Numerical discretization is done by the Stabilized Finite Element Method, also known as Galerkin Least Squares (GLS), through bilinear quadrangular elements. The classical liddriven cavity problem is used in the simulations in order to compare the formulation and code used with the results in the literature. In addition, the study of elasto-viscoplastic thixotropic materials in a 4:1 abrupt contraction using the described formulation and a new viscosity function for viscoplastic fluids, called the Harmonic Viscoplastic Function (HVP), is presented. Results involving the HVP function are introduced and discussed for the first time in this thesis and present a better curve fit when compared to other functions available in the literature. Besides to easy implementation, this function also features a plateau for high and low viscosities that are physically realistic, since infinite or zero viscosity is not possible. The shortest computational time is also a perceptible feature in the simulations using the new viscoplastic function, this is an attribute of its equation that does not depend on an exponential term like other models. The mesh quality study is also presented in order to ensure the choice of the appropriate discrete domain. Despite the use of lower-order elements, the GLS method proved to be stable in the numerical approximation of all the problems, guaranteeing even the analysis of the effects of kinematics, elasticity and thixotropy on fluid flow within the abrupt contraction.

Neste trabalho é apresentado a modelagem mecânica e numérica de um escoamento elastoviscoplástico tixotrópico, em termos dos campos de velocidade, pressão, tensão e parâmetro de estrutura. A discretização numérica é feita pelo Método de Elementos Finitos Estabilizado, também conhecido como Galerkin Mínimos Quadrados (GMQ), através de elementos quadrangulares bilineares. O clássico problema da cavidade é utilizado nas simulações, a fim de comparar a formulação e o código utilizados com os resultados conhecidos na literatura. Além disso, apresenta-se o estudo de materiais elasto-viscoplástico tixotrópico em uma contração abrupta na escala 4:1, utilizando a formulação descrita e uma nova função viscosidade para fluidos viscoplásticos, denominada função Viscoplástica Harmônica (VPH). Resultados envolvendo a função VPH são introduzidos e discutidos pela primeira vez nesta tese e apresenta um ótimo ajuste de curva, quando comparada com outras funções disponíveis na literatura. Além da fácil implementação, essa função também apresenta um platô para as altas e baixas viscosidades que são fisicamente realistas, visto que não é possível uma viscosidade infinita ou nula. O menor tempo computacional é também uma característica perceptível nas simulações usando a nova função viscoplástica, isso é um atributo do seu equacionamento que não depende de um termo exponencial, como outros modelos. O estudo de qualidade de malha também é apresentado a fim de garantir a escolha do domínio discreto adequado. Apesar do uso de elementos de ordem inferior, o método GMQ mostrou-se estável na aproximação numérica de todos os problemas dispostos, garantindo até mesmo a análise sobre os efeitos da cinemática, da elasticidade e da tixotropia no escoamento dos fluidos dentro da contração abrupta. / In this work the mechanical and numerical modeling of a thixotropic elasto-viscoplastic flow in terms of the velocity, pressure, stress and structure parameter is presented. Numerical discretization is done by the Stabilized Finite Element Method, also known as Galerkin Least Squares (GLS), through bilinear quadrangular elements. The classical liddriven cavity problem is used in the simulations in order to compare the formulation and code used with the results in the literature. In addition, the study of elasto-viscoplastic thixotropic materials in a 4:1 abrupt contraction using the described formulation and a new viscosity function for viscoplastic fluids, called the Harmonic Viscoplastic Function (HVP), is presented. Results involving the HVP function are introduced and discussed for the first time in this thesis and present a better curve fit when compared to other functions available in the literature. Besides to easy implementation, this function also features a plateau for high and low viscosities that are physically realistic, since infinite or zero viscosity is not possible. The shortest computational time is also a perceptible feature in the simulations using the new viscoplastic function, this is an attribute of its equation that does not depend on an exponential term like other models. The mesh quality study is also presented in order to ensure the choice of the appropriate discrete domain. Despite the use of lower-order elements, the GLS method proved to be stable in the numerical approximation of all the problems, guaranteeing even the analysis of the effects of kinematics, elasticity and thixotropy on fluid flow within the abrupt contraction.

There are two big challenges which restrict the extensive application of fully coupled geomechanics-reservoir modeling. The first challenge is computational effort. Consider a 3-D simulation combining pressure and heat diffusion, elastoplastic mechanical response, and saturation changes; each node has at least 5 degrees of freedom, each leading to a separate equation. Furthermore, regions of large p, T and σ′ gradients require small-scale discretization for accurate solutions, greatly increasing the number of equations. When the rock mass surrounding the reservoir region is included, it is represented by many elements or nodes. These factors mean that accurate analysis of realistic 3-D problems is challenging, and will so remain as we seek to solve larger and larger coupled problems involving nonlinear responses. To overcome the first challenge, the displacement discontinuity method is introduced wherein a large-scale 3-D case is divided into a reservoir region where Δp, ΔT and non-linear effects are critical and analyzed using FEM, and an outside region in which the reservoir is encased where Δp and ΔT effects are inconsequential and the rock may be treated as elastic, analyzed with a 3D displacement discontinuity formulation. This scheme leads to a tremendous reduction in the degrees of freedom, yet allows for reasonably rigorous incorporation of the reactions of the surrounding rock. The second challenge arises from some forms of numerical instability. There are actually two types of sharp gradients implied in the transient advection-diffusion problem: one is caused by the high Peclet numbers, the other by the sharp gradient which appears during the small time steps due to the transient solution. The way to eliminate the spurious oscillations is different when the sharp gradients are induced by the transient evolution than when they are produced by the advective terms, and existing literature focuses mainly on eliminating the spurious spatial temperature oscillations caused by advection-dominated flow. To overcome the second challenge, numerical instability sources are addressed by introducing a new stabilized finite element method, the subgrid scale/gradient subgrid scale (SGS/GSGS) method.

There are two big challenges which restrict the extensive application of fully coupled geomechanics-reservoir modeling. The first challenge is computational effort. Consider a 3-D simulation combining pressure and heat diffusion, elastoplastic mechanical response, and saturation changes; each node has at least 5 degrees of freedom, each leading to a separate equation. Furthermore, regions of large p, T and σ′ gradients require small-scale discretization for accurate solutions, greatly increasing the number of equations. When the rock mass surrounding the reservoir region is included, it is represented by many elements or nodes. These factors mean that accurate analysis of realistic 3-D problems is challenging, and will so remain as we seek to solve larger and larger coupled problems involving nonlinear responses. To overcome the first challenge, the displacement discontinuity method is introduced wherein a large-scale 3-D case is divided into a reservoir region where Δp, ΔT and non-linear effects are critical and analyzed using FEM, and an outside region in which the reservoir is encased where Δp and ΔT effects are inconsequential and the rock may be treated as elastic, analyzed with a 3D displacement discontinuity formulation. This scheme leads to a tremendous reduction in the degrees of freedom, yet allows for reasonably rigorous incorporation of the reactions of the surrounding rock. The second challenge arises from some forms of numerical instability. There are actually two types of sharp gradients implied in the transient advection-diffusion problem: one is caused by the high Peclet numbers, the other by the sharp gradient which appears during the small time steps due to the transient solution. The way to eliminate the spurious oscillations is different when the sharp gradients are induced by the transient evolution than when they are produced by the advective terms, and existing literature focuses mainly on eliminating the spurious spatial temperature oscillations caused by advection-dominated flow. To overcome the second challenge, numerical instability sources are addressed by introducing a new stabilized finite element method, the subgrid scale/gradient subgrid scale (SGS/GSGS) method.

Les procédés de trempe sont largement répandus dans l'industrie en particulier dans le domaine de l'automobile, du nucléaire et de l'aérospatiale car ils ont un impact direct sur la microstructure, les propriétés mécaniques et les contraintes résiduelles de pièces critiques. La trempe est un processus fortement non-linéaire à cause des couplages forts entre la mécanique des fluides, les transferts thermiques aux différentes interfaces, les transformations de phase du solide et l'ébullition du milieu de trempe. Malgré les progrès effectués par la simulation numérique, ce procédé reste extrêmement difficile à modéliser.Dans ce travail, nous proposons le développement d'outils numériques permettant la simulation réaliste de ce procédé à l'échelle industrielle. La mécanique des fluides est simulée en utilisant une méthode d'Elements Finis stabilisés permettant de considérer des écoulements à haut nombre de Reynolds. Les transferts thermiques sont calculés directement sans l'utilisation de coefficients de transferts empiriques, en utilisant le couplage fort entre le fluide et le solide. Nous avons développé un modèle de changement de phase pour l'eau permettant de considérer les différents régimes d'ébullition. Une formulation unifiée des équations de Navier-Stokes, considérant une phase compressible et une phase incompressible a été développée afin de prendre en compte plus précisément la dynamique de la vapeur et de l'eau. Une procédure dynamique d'adaptation anisotrope de maillage, permettant une description plus fine des interfaces et une prise en compte plus précise des caractéristiques des écoulements est utilisée.Des exemples numériques exigeants ainsi qu'une validation expérimentale permettent d'évaluer la précision et la robustesse des outils proposés.Les outils développés permettent ainsi l'optimisation du mode opératoire du procédé, des ressources consommées et servent ainsi d'outils prospectifs pour la conception de produits. / Quenching processes of metals are widely adopted procedures in the industry, in particular automotive, nuclear and aerospace industries, since they have direct impacts on changing mechanical properties, controlling microstructure and releasing residual stresses of critical parts. Quenching is a highly nonlinear process because of the strong coupling between the fluid mechanics, heat transfer at the interface solid-fluid, phase transformation in the metal and boiling. In spite of the maturity and the popularity of numerical formulations, several involved mechanisms are still not well resolved.Therefore we propose a Direct Numerical Simulation of quenching processes at the industrial scale dealing with these phenomena. The fluid mechanics is simulated using a Finite Element Method adapted for high convective flows allowing the use of high stirring velocity in the quenching bath. Heat transfers are computed directly without the use of heat transfer coefficients but using the strong coupling between the fluid and the solid. We use a phase change model for the water that models all boiling regimes. A unified formulation of the Navier-Stokes equations, taking into account a compressible gas and an incompressible liquid is developed to model more accurately the vapor-water dynamics. A dynamic mesh adaptation procedure is used, increasing the resolution in the description of the interfaces and capturing more accurately the features of the flows.We assess the behavior and the accuracy of the proposed formulation in the simulation of time-dependent challenging numerical examples and experimental results.These recent developments enable the optimization of the process in terms of operating conditions, resources consumed and products conception.

L'Interaction Fluide-Structure (IFS) décrit une classe très générale de problème physique, ce qui explique la nécessité de développer une méthode numérique capable de simuler le problème FSI. Pour cette raison, un solveur IFS est développé qui peut traiter un écoulement de fluide incompressible en interaction avec des structures différente: élastique ou rigide. Dans cet aspect, le solveur peut couvrir une large gamme d'applications.La méthode proposée est développée dans le cadre d'une formulation monolithique dans un contexte Eulérien. Cette méthode consiste à considérer un seul maillage et résoudre un seul système d'équations avec des propriétés matérielles différentes. La fonction distance permet de définir la position et l'interface de tous les objets à l'intérieur du domaine et de fournir les propriétés physiques pour chaque sous-domaine. L'adaptation de maillage anisotrope basé sur la variation de la fonction distance est ensuite appliquée pour assurer une capture précise des discontinuités à l'interface fluide-solide.La formulation monolithique est assurée par l'ajout d'un tenseur supplémentaire dans les équations de Navier-Stokes. Ce tenseur provient de la présence de la structure dans le fluide. Le système est résolu en utilisant une méthode élément fini et stabilisé suivant la formulation variationnelle multiéchelle. Cette formulation consiste à décomposer les champs de vitesse et pression en grande et petite échelles. La particularité de l'approche proposée réside dans l'enrichissement du tenseur de l'extra contraint.La première application est la simulation IFS avec un corps rigide. Le corps rigide est décrit en imposant une valeur nul du tenseur des déformations, et le mouvement est obtenu par la résolution du mouvement de corps rigide. Nous évaluons le comportement et la précision de la formulation proposée dans la simulation des exemples 2D et 3D. Les résultats sont comparés avec la littérature et montrent que la méthode développée est stable et précise.La seconde application est la simulation IFS avec un corps élastique. Dans ce cas, une équation supplémentaire est ajoutée au système précédent qui permet de résoudre le champ de déplacement. Et la contrainte de rigidité est remplacée par la loi de comportement du corps élastique. La déformation et le mouvement du corps élastique sont réalisés en résolvant l'équation de convection de la Level-Set. Nous illustrons la flexibilité de la formulation proposée par des exemples 2D. / Numerical simulations of fluid-structure interaction (FSI) are of first interest in numerous industrial problems: aeronautics, heat treatments, aerodynamic, bioengineering... Because of the high complexity of such problems, analytical study is in general not sufficient to understand and solve them. FSI simulations are then nowadays the focus of numerous investigations, and various approaches are proposed to treat them. We propose in this thesis a novel monolithic approach to deal with the interaction between an incompressible fluid flow and rigid/ elastic material. This method consists in considering a single grid and solving one set of equations with different material properties. A distance function enables to define the position and the interface of any objects with complex shapes inside the volume and to provide heterogeneous physical properties for each subdomain. Different anisotropic mesh adaptation algorithms based on the variations of the distance function or on using error estimators are used to ensure an accurate capture of the discontinuities at the fluid-solid interface. The monolithic formulation is insured by adding an extra-stress tensor in the Navier-Stokes equations coming from the presence of the structure in the fluid. The system is then solved using a finite element Variational MultiScale (VMS) method, which consists of decomposition, for both the velocity and the pressure fields, into coarse/resolved scales and fine/unresolved scales. The distinctive feature of the proposed approach resides in the efficient enrichment of the extra constraint. In the first part of the thesis, we use the proposed approach to assess its accuracy and ability to deal with fluid-rigid interaction. The rigid body is prescribed under the constraint of imposing the nullity of the strain tensor, and its movement is achieved by solving the rigid body motion. Several test case, in 2D and 3D with simple and complex geometries are presented. Results are compared with existing ones in the literature showing good stability and accuracy on unstructured and adapted meshes. In the second, we present different routes and an extension of the approach to deal with elastic body. In this case, an additional equation is added to the previous system to solve the displacement field. And the rigidity constraint is replaced with a corresponding behaviour law of the material. The elastic deformation and motion are captured using a convected level-set method. We present several 2D numerical tests, which is considered as classical benchmarks in the literature, and discuss their results.

Aujourd'hui, avec l'amélioration des puissances de calcul informatique, la simulation numérique est devenue un outil essentiel pour la prédiction des phénomènes physiques et l'optimisation des procédés industriels. La modélisation de ces phénomènes pose des difficultés scientifiques car leur résolution implique des temps de calcul très longs malgré l'utilisation d'importantes ressources informatiques.Dans cette thèse, on s'intéresse à la résolution de problèmes complexes couplant écoulements et transferts thermiques. Les problèmes physiques étant fortement anisotropes, il est nécessaire d'avoir un maillage avec une résolution très élevée pour obtenir un bon niveau de précision. Cela implique de longs temps de calcul. Ainsi il faut trouver un compromis entre précision et efficacité. Le développement de méthodes d'adaptation en temps et en espace est motivé par la volonté de faire des applications réelles et de limiter les inconvénients inhérents aux méthodes de résolution non adaptatives en terme de précision et d'efficacité. La résolution de problèmes multi-échelles instationnaires sur un maillage uniforme avec un nombre de degrés de liberté limité est souvent incapable de capturer les petites échelles, nécessite des temps de calcul longs et peut aboutir à des résultats incorrects. Ces difficultés ont motivé le développement de méthodes de raffinement local avec une meilleure précision aux endroits adéquats. L'adaptation en temps et en espace peut donc être considérée comme une composante essentielle de ces méthodes.L'approche choisie dans cette thèse consiste en l'utilisation de méthodes éléments finis stabilisées et le développement d'outils d'adaptation espace-temps pour améliorer la précision et l'efficacité des simulations numériques.Le développement de la méthode adaptative est basé sur un estimateur d'erreur sur les arrêtes du maillage afin de localiser les régions du domaine de calcul présentant de forts gradients ainsi que les couches limites. Ensuite une métrique décrivant la taille de maille en chaque noeud dans les différentes directions est calculée. Afin d'améliorer l'efficacité des calculs la construction de cette métrique prend en compte un nombre fixe de noeuds et aboutit à une répartition et une orientation optimale des éléments du maillage. Cette approche est étendue à une formulation espace-temps où les maillages et les pas de temps optimaux sont prédits sur des intervalles de temps en vue de contrôler l'erreur d'interpolation sur la domaine de calcul. / Nowadays, with the increase in computational power, numerical modeling has become an intrinsic tool for predicting physical phenomena and developing engineering designs. The modeling of these phenomena poses scientific complexities the resolution of which requires considerable computational resources and long lasting calculations.In this thesis, we are interested in the resolution of complex long time and large scale heat transfer and fluid flow problems. When the physical phenomena exhibit sharp anisotropic features, a good level of accuracy requires a high mesh resolution, hence hindering the efficiency of the simulation. Therefore a compromise between accuracy and efficiency shall be adopted. The development of space and time adaptive adaptation techniques was motivated by the desire to devise realistic configurations and to limit the shortcomings of the traditional non-adaptive resolutions in terms of lack of solution's accuracy and computational efficiency. Indeed, the resolution of unsteady problems with multi-scale features on a prescribed uniform mesh with a limited number of degrees of freedom often fails to capture the fine scale physical features, have excessive computational cost and might produce incorrect results. These difficulties brought forth investigations towards generating meshes with local refinements where higher resolution was needed. Space and time adaptations can thus be regarded as essential ingredients in this recipe.The approach followed in this work consists in applying stabilized finite element methods and the development of space and time adaptive tools to enhance the accuracy and efficiency of the numerical simulations.The derivation process starts with an edge-based error estimation for locating the regions, in the computational domain, presenting sharp gradients, inner and boundary layers. This is followed by the construction of nodal metric tensors that prescribe, at each node in the spatial mesh, mesh sizes and the directions along which these sizes are to be imposed. In order to improve the efficiency of computations, this construction takes into account a fixed number of nodes and generates an optimal distribution and orientation of the mesh elements. The approach is extended to a space-time adaptation framework, whereby optimal meshes and time-step sizes for slabs of time are constructed in the view of controlling the global interpolation error over the computation domain.