This report describes the development of a software for computing viscous incompressiblemultiphase ows. The software does this with solving the coupled non-linear Navier-Stokes(Fluid) and the Cahn-Hilliard (Phase-Field) equations using a Finite Element Method. Thereason for the development is to produce a simulation tool, which eventually is capable ofsimulating the ow of uids inside the OptusAir aeronator manufactured by the Sorubincompany. The solving software developed is built on the ParMetis, PETSc and OpenMPIframeworks. Our primary benchmark has been a geometry resembling the OptusAir product,a cylinder with a rotating bottom. We have made comparisons between simulation resultsand the theory of a free surface in a uniform rotating ow.This thesis shows that the shape of the interface between two uids coincides with theoryto some extent, while the approximate boundary conditions prevent it from coinciding fully.

Doctor en Ciencias de la Ingeniería, Mención Modelación Matemática / En esta tesis doctoral se construyen soluciones rotacionalmente simétricas de la ecuación de Cahn-Hilliard en $ \R{^d} $ y se estudian sus propiedades de estabilidad. En el Capítulo \ref{ch1} se presenta la ecuación de Cahn-Hilliard y se explica su origen e interpretación física. Además se repasan varios resultados conocidos, se presenta la notación y se exponen los dos resultados más importantes de esta tesis: el primero establece la existencia de soluciones rotacionalmente simétricas cuyos conjuntos de nivel se aproximan a los unduloides de Delaunay. El segundo resultado afirma que las propiedades de estabilidad de los unduloides de Delanay heredan propiedades de estabilidad de las soluciones encontradas, en el sentido que son no degeneradas y tienen 6 campos de Jacobi con crecimiento moderado. En el Capítulo \ref{prel} se presentan en detalle los principales ingredientes que se necesitan para probar los Teoremas \ref{teo 1} y \ref{teo 2}, a saber las coordenadas de Fermi cerca de una superficie de curvatura media constante, los unduloides de Delaunay y su operador de Jacobi. También se muestra la primera aproximación de la solución anunciada en el Teorema \ref{teo 1}. En el Capítulo \ref{chap proof teo 1} se demuestra el Teorema \ref{teo 1}. Usamos una versión refinada del método de reducción del Lyapunov-Schmidt que simplifica varios aspectos técnicos de construcciones de problemas similares. Los resultados de este capítulo fueron obtenidos en colaboración con mi Profesor Guía, Dr. Micha\l\ Kowalczyk y fueron publicados en la revista \emph{Discrete and Continous Dynamical Systems} bajo el título \emph{Rotationally Symmetric Solutions to the Cahn-Hillard Equation}. Una demostración del Teorema \ref{teo 2} se da el Capítulo \ref{chap proof teo 2}. La clave es relacionar el núcleo del operador linearizado alrededor de nuestra solución con los campos de Jacobi que provienen de invariancias geométricas. Esta relación se puede realizar debido a que es posible separar las variables una vez que se ha aplicado la transformada de Laplace-Fourier. Los resultados de este capítulo también fueron obtenidos con mi profesor Guía y han sido aceptados para su publicación en la revista \emph{Indiana University Mathematics Journal} bajo el título \emph{Nondegeneracy and the Jacobi Fields of Rotationally Symmetric Solutions to the Cahn-Hillard Equation}. In this PhD thesis rotationally symmetric solutions to the Cahn-Hilliard equation are constructed. Also we study its stability properties. In Chapter \ref{ch1} we present the Cahn-Hilliard equation in $ \R^d $ and explain its origin and physical interpretation. We also review several known results, introduce some basic notation and present the two main results of this thesis. The first one states the existence of radially symmetric solutions to the Cahn-Hilliard equation which nodal sets approaches to Delaunay unduloids, and the second one claims that stability properties of the Delaunay unduloids inherit stability properties of the solutions we found in the sense that our solutions are non degenerated and have 6 Jacobi fields with temperate growth. Chapter \ref{prel} is devoted to present in detail the main ingredients we need to prove Theorem \ref{teo 1} and Theorem \ref{teo 2}, namely Fermi coordinates near a constant mean curvature (CMC), the Delaunay unduloids and its Jacobi operator. We also present the construction of the first approximation of the solutions announced in Theorem \ref{teo 1}. In Chapter \ref{chap proof teo 1} we prove Theorem \ref{teo 1}. We use a refined version of the Lyapunov-Schmidt reduction method which simplifies very technical aspects of previous constructions for similar problems. The results of this chapter were obtained in collaboration with my thesis advisor Dr. Micha\l\ Kowalczyk and published in \emph{Discrete and Continuous Dynamical Systems}. A proof of Theorem \ref{teo 2} is given in Chapter \ref{chap proof teo 2}. The key is to relate the kernel of the linearized operator about our solution with the Jacobi fields that comes from the geometric invariances. This relation can be performed since we are able to separate the variables once the Laplace-Fourier transform is applied. The results of this chapter were obtained in collaboration with my thesis advisor Dr. Micha\l\ Kowalczyk and admitted for publication in \emph{Indiana University Mathematics Journal}.

This work presents a novel method to model immiscible incompressible fluids in a stable manner. Here, the immiscible behavior of the flow is described by the incompressible Navier-Stokes-Cahn-Hilliard model, which is based on a diffuse interface method. We introduce buoyancy effects in the model through the Boussinesq approximation in a consistent manner. A structure-preserving discretization is used to guarantee the linear stability of the discrete problem and to satisfy the incompressibility of the discrete solution at every point in space by construction. For the solution of the model, we developed the Portable Extensible Toolkit for Isogeometric Analysis with Multi-Field discretizations (PetIGA-MF), a high-performance framework that supports structure-preserving spaces. PetIGA-MF is built on top of PetIGA and the Portable Extensible Toolkit for Scientific Computation (PETSc), sharing all their user-friendly, performance, and flexibility features. Herein, we describe the implementation of our model in PetIGA-MF and the details of the numerical solution. With several numerical tests, we verify the convergence, scalability, and validity of our approach. We use highly-resolved numerical simulations to analyze the merging and rising of droplets. From these simulations, we detailed the energy exchanges in the system to evaluate quantitatively the quality of our simulations. The good agreement of our results when compared against theoretical descriptions of the merging, and the small errors found in the energy analysis, allow us to validate our approach. Additionally, we present the development of an unconditionally energy-stable generalized-alpha method for the Swift-Hohenberg model that offers control over the numerical dissipation. A pattern formation example demonstrates the energy-stability and convergence of our method.

L'endommagement par irradiation mène à la ruine d'un matériaux, il est donc impératif de savoir prévoir son évolution afin de garantir la sécurité des réacteurs nucléaires.Bien que le comportement mécanique sous irradiation ait fait l'objet de nombreuses recherches, les capacités de prédiction actuelles restent limitées.L'agrégation des défauts ponctuels, tels que les lacunes et les auto-interstitiels, provoque du fluage, du gonflement et fragilise le matériau.Les nano-composites multicouches métalliques cristallins sont capables d'évacuer ces défauts ponctuels grâce à leur densité d'interface élevée, et permettent de retarder les phénomènes délétères précédemment cités.Ils ont, de plus, une résistance mécanique élevée.L'objectif de cette thèse est de développer un cadre thermodynamique à l'échelle continue meso et nano-scopique, rendant compte des principaux phénomènes physiques à l'oeuvre dans ces laminés irradiés.Principalement trois points sont abordés: le couplage diffusion-mécanique et le fluage, la nucléation et croissance de cavités sous irradiation, et le comportement mécanique des multicouches.La micro-structure du matériau est ici entièrement modélisée, afin de précisément rendre compte de son influence sur le comportement du matériau.Le fluage par diffusion est traité via une approche originale où le tenseur des vitesses de déformation est directement relié au gradient du flux de lacunes.Un modèle de type Cahn-Hilliard est utilisé afin de prédire la nucléation et la croissance des cavités.Les équations de diffusion y sont complétées pour prendre en compte la production des défauts ponctuels dus à l'irradiation, ainsi que leur recombinaison.Dans les systèmes multicouches, une zone affecté par l'interface est définie, dans laquelle les dislocations peuvent être annihilées.De plus, l'interface elle même est traitée comme un plan de glissement cristallographique.Le modèle est implémenté numériquement via la méthode des éléments finis.Des simulations de fluage couplé à la diffusion de lacunes sont pour la première fois réalisé sur des agrégats polycristallins, prédisant des champs de déformation intra-granulaire fortement hétérogènes.De plus, la vitesse de fluage macroscopique obtenue met en évidence les dépendances classiques à la taille de grain ainsi qu'à la contrainte appliquée.Lors des simulations d'irradiation de multicouches, des zones libres de cavités sont prédites de part et d'autre des interfaces, en accord avec les observations expérimentales.Enfin, des essais de traction sont simulés sur des systèmes Cu-Nb en 3D, mettant en évidence un mode de déformation complexe, et un effet moindre de l'anisotropie élastique. / Radiation damage is known to lead to material failure and thus is of critical importance to lifetime and safety within nuclear reactors.While mechanical behaviour of materials under irradiation has been the subject of numerous studies, the current predictive capabilities of such phenomena appear limited.The clustering of point defects such as vacancies and self interstitial atoms gives rise to creep, void swelling and material embrittlement.Nanoscale metallic multilayer systems have be shown to have the ability to evacuate such point defects, hence delaying the occurrence of critical damage.In addition, they exhibit outstanding mechanical properties.The objective of this work is to develop a thermodynamically consistent continuum framework at the meso and nano-scales, which accounts for the major physical processes encountered in such metallic multilayer systems and is able to predict their microstuctural evolution and behavior under irradiation.Mainly three physical phenomena are addressed in the present work: stress-diffusion coupling and diffusion induced creep, the void nucleation and growth in multilayer systems under irradiation, and the interaction of dislocations with the multilayer interfaces.In this framework, the microstructure is explicitly modeled, in order to account accurately for their effects on the system behavior.The diffusion creep strain rate is related to the gradient of the vacancy flux.A Cahn-Hilliard approach is used to model void nucleation and growth, and the diffusion equations for vacancies and self interstitial atoms are complemented to take into account the production of point defects due to irradiation cascades, the mutual recombination of defects and their evacuation through grain boundaries.In metallic multilayers, an interface affected zone is defined, with an additional slip plane to model the interface shearable character, and where dislocations cores are able to spread.The model is then implemented numerically using the finite elements method.Simulations of biaxial creep of polycrystalline aggregates coupled with vacancy diffusion are performed for the first time, and predict strongly heterogeneous viscoplastic strain fields.The classical macroscopic strain rate dependence on the stress and grain size is also retrieved.Void denuded zones close to the multilayer interfaces are obtained in irradiation simulations of a multilayer, in agreement with experimental observations.Finally, tensile tests of Cu-Nb multilayers are simulated in 3D, where it is shown that the effect of elastic anisotropy is negligible, and evidencing a complex deformation mode.

Anticiper l’évolution de la microstructure d’un matériau en condition d’usage est d’une importance cruciale pour l’industrie. Cette maîtrise du vieillissement nécessite une compréhension claire des mécanismes sous-jacents, qui agissent sur une large gamme d’échelles spatiales et temporelles. Dans cette optique, ce travail de thèse a choisi d’appliquer la méthode de champ de phase qui, en raison du saut d’échelle qu’elle réalise naturellement, est un outil intensivement employé dans le domaine des matériaux, pour prédire l’évolution en temps long de la microstructure. L’enjeu de l’étude a été d’étendre cette méthode à un système porté loin de l’équilibre thermodynamique, en particulier en présence d’irradiation. Nous avons ainsi adopté le formalisme du mélange ionique, introduit par Gras-Marti pour décrire le mélange balistique au sein d’une cascade de déplacements. Par l’utilisation conjointe d’un schéma numérique et d’une approche analytique, il nous a été possible d’établir le diagramme de phase générique d’un matériau irradié. Nous avons ensuite étudié le vieillissement de l’alliage binaire test AgCu sous irradiation, par l’utilisation conjointe de la méthode du champ de phase et d’approches atomistiques, dans une démarche multi-échelles. En fixant les paramètres de contrôle que sont le flux d’irradiation et la température, il nous a ainsi été possible de prédire la taille,la concentration ainsi que la distribution spatiale des nodules de cuivre produits sous irradiation dans cet alliage. La connaissance de ces informations a permis de simuler un diagramme de diffraction en incidence rasante, directement comparable aux diagrammes expérimentaux. / It is of dramatic matter for industry to be able to predict the evolution of material microstructure under working conditions. This requires a clear understanding of the underlying mechanisms, which act on numerous space and time scales. Because it intrinsically performs a scale jump, we chose to use a phase field approach, which is widely used amidst the condensed matter community to study the aging of materials. The first challenge of this work was to extend this formalism beyond its thermodynamic scope and embrace the case of far from equilibrium systems when subjected to irradiation. For that purpose, we adopted the model of ion mixing, developed by Gras Marti to account for ballistic exchanges within a displacements cascade. Based on a numerical scheme and ananalytical method, we were able to describe the generic microstructure signature for materials under irradiation.We then applied this formalism to the particular case of the immiscible binary alloy AgCu.With the joined use of the phase field approach and atomistic methods, we managed to predict how the temperature and the irradiation flux tailor the main microstructure features such as the size, the concentration and the distribution of copper precipitates. This eventually allowed us to simulate a diffraction pattern in grazing incidence, which is directly comparable to experimental ones.

[pt] Neste trabalho estudamos o comportamento dinâmico de uma mistura binária granular submetida a uma equação do tipo Cahn-Hilliard com o objetivo de observar o fenômeno da segregação. Construímos um modelo de sistema de células dinâmicas para simularmos computacionalmente esses sistemas e observarmos o seu comportamento, a fim de comparar nosso modelo com o que é observado experimentalmente. / [en] In this work we study the dynamic behavior of a granular binary mixture with the aim of investigation phenomenom. We construct also a cell dynamic systems model to simulate computationally these systems and observe its behaviour, in order to compare our model with the experiments.

In this thesis, we consider two models from physics, which are characterized by the interplay of thermodynamical and fluid mechanical phenomena: demixing (spinodal decomposition) and Rayleigh--Bénard convection. In both models, we investigate the dependencies of certain intrinsic quantities on the system parameters. The first model describes a thermodynamically driven demixing process of a binary viscous fluid. During the evolution, the two components of the mixture separate into two domains of the different equilibrium volume fractions. One observes a clear tendency: Larger domains grow at the expense of smaller ones, and thus, the average domain sizes increases --- a phenomenon called coarsening. It turns out that two mechanisms are relevant for the coarsening process. At an early stage of the evolution, material transport is essentially mediated by diffusion; at a later stage, when the typical domain size exceeds a certain value, due to the viscosity of the mixture, a fluid flow sets in and becomes the relevant transport mechanism. In both regimes, the growth rates of the typical domain size obey certain power laws. In this thesis, we rigorously establish one-sided bounds on these growth rates via a priori estimates. The second model, Rayleigh--Bénard convection, describes the behavior of a fluid between two rigid horizontal plates that is heated from below and cooled from above. There are two competing heat transfer mechanisms in the system: On the one hand, thermodynamics favors a state in which temperature variations are locally minimized. Thus, in our model, the thermodynamical equilibrium state is realized by a temperature with a linearly decreasing profile, corresponding to pure conduction. On the other hand, due to differences in the densities of hot and cold fluid parcels, buoyancy forces act on the fluid. This results in an upward motion of hot parcels and a downward motion of cold parcels. We study the dependence of the average upward heat flux, measured in the so-called Nusselt number, on the temperature forcing encoded by the container height. It turns out that the efficiency of the heat transport is independent of the height of the container, and thus, the Nusselt number is a constant function of height. Using a priori estimates, we prove an upper bound on the Nusselt number that displays this dependency --- up to logarithmic errors. Further investigations on the flow pattern in Rayleigh--Bénard convection show a clear separation of length scales: Along the horizontal top and bottom plates one observes thin boundary layers in which heat is essentially conducted, whereas the large bulk is characterized by a convective heat flow. We give first rigorous results in favor of linear temperature profiles in the boundary layers, which indicate that heat is indeed essentially conducted close to the boundaries.

Anticiper l’évolution de la microstructure d’un matériau en condition d’usage est d’une importance cruciale pour l’industrie. Cette maîtrise du vieillissement nécessite une compréhension claire des mécanismes sous-jacents, qui agissent sur une large gamme d’échelles spatiales et temporelles. Dans cette optique, ce travail de thèse a choisi d’appliquer la méthode de champ de phase qui, en raison du saut d’échelle qu’elle réalise naturellement, est un outil intensivement employé dans le domaine des matériaux, pour prédire l’évolution en temps long de la microstructure. L’enjeu de l’étude a été d’étendre cette méthode à un système porté loin de l’équilibre thermodynamique, en particulier en présence d’irradiation. Nous avons ainsi adopté le formalisme du mélange ionique, introduit par Gras-Marti pour décrire le mélange balistique au sein d’une cascade de déplacements. Par l’utilisation conjointe d’un schéma numérique et d’une approche analytique, il nous a été possible d’établir le diagramme de phase générique d’un matériau irradié. Nous avons ensuite étudié le vieillissement de l’alliage binaire test AgCu sous irradiation, par l’utilisation conjointe de la méthode du champ de phase et d’approches atomistiques, dans une démarche multi-échelles. En fixant les paramètres de contrôle que sont le flux d’irradiation et la température, il nous a ainsi été possible de prédire la taille,la concentration ainsi que la distribution spatiale des nodules de cuivre produits sous irradiation dans cet alliage. La connaissance de ces informations a permis de simuler un diagramme de diffraction en incidence rasante, directement comparable aux diagrammes expérimentaux. / It is of dramatic matter for industry to be able to predict the evolution of material microstructure under working conditions. This requires a clear understanding of the underlying mechanisms, which act on numerous space and time scales. Because it intrinsically performs a scale jump, we chose to use a phase field approach, which is widely used amidst the condensed matter community to study the aging of materials. The first challenge of this work was to extend this formalism beyond its thermodynamic scope and embrace the case of far from equilibrium systems when subjected to irradiation. For that purpose, we adopted the model of ion mixing, developed by Gras Marti to account for ballistic exchanges within a displacements cascade. Based on a numerical scheme and ananalytical method, we were able to describe the generic microstructure signature for materials under irradiation.We then applied this formalism to the particular case of the immiscible binary alloy AgCu.With the joined use of the phase field approach and atomistic methods, we managed to predict how the temperature and the irradiation flux tailor the main microstructure features such as the size, the concentration and the distribution of copper precipitates. This eventually allowed us to simulate a diffraction pattern in grazing incidence, which is directly comparable to experimental ones.

Dans cette thèse, on étudie l'existence, l'unicité et la régularité des solutionsd'équation de type Cahn-Hilliard ainsi que son comportement asymptotiqueen termes d'existence de l'attracteur global et d'un attracteur exponentiel. Cetteéquation est considérée dans un domaine borné et régulier pour différents types denonlinéarités et de conditions au bord.D'abord, on étudie l'équation avec des conditions de type Dirichlet sur le bord etune nonlinéarité régulière. Après, on considère une perturbation du problème et ondémontre l'existence d'une famille robuste d'attracteurs exponentiels lorsque ε tendvers 0.Ensuite, on étudie l'équation avec des conditions dynamiques sur le bord. On considèretout d'abord une nonlinéarité régulière et on donne une étude théorique etnumérique. Après, on illustre ces résultats par des simulations numériques en dimensiondeux d'espace qui permettent d'étudier l'influence des différents paramètres.On termine par une étude du modèle considéré avec une nonlinéarité singulière quel'on approche par des fonctions régulières et on introduit une notion de solutionappropriée. / This thesis is devoted to the study of the existence, uniqueness andregularity of solutions for a Cahn-Hilliard type equation, as well as the asymptoticbehavior in terms of existence of the global attractor and of an exponential attractor.This equation is considered in a bounded and smooth domain under variousassumptions on the nonlinear terms and with different boundary conditions.We start by studying the equation with Dirichlet boundary conditions and a regularnonlinearity. Then, we consider a perturbation of the problem and we prove theexistence of a robust family of exponential attractors as ε tends to 0.For the equation endowed with dynamic boundary conditions, we first consider aregular nonlinearity and we treat the theoretical and numerical analysis. Then, weillustrate the results by numerical simulations in two space dimension which allow usto study the influence of different parameters. Finally, we treat the problem consideredwith a singular nonlinearity which is approximated by regular functions andwe give a suitable notion of solutions.

Die vorliegende Doktorarbeit befasst sich mit der optimalen Steuerung von einem Cahn–Hilliard–Navier–Stokes-System mit variablen Flüssigkeitsdichten. Dabei konzentriert sie sich auf das Doppelhindernispotential, was zu einem optimalen Steuerungsproblem einer Gruppe von gekoppelten Systemen, welche eine Variationsungleichung vierter Ordnung sowie eine Navier–Stokes-Gleichung beinhalten, führt. Eine geeignete Zeitdiskretisierung wird präsentiert und zugehörige Energieabschätzungen werden bewiesen. Die Existenz von Lösungen zum primalen System und von optimalen Steuerungen für das ursprüngliche Problem sowie für eine Gruppe von regularisierten Problemen wird etabliert. Die Optimalitätsbedingungen erster Ordnung für die regularisierten Probleme werden hergeleitet. Mittels eines Grenzübergangs in Bezug auf den Regularisierungsparameter werden Stationaritätsbedingungen für das ursprüngliche Problem etabliert, welche einer Form von C-Stationarität im Funktionenraum entsprechen. Weiterhin wird ein numerischer Lösungsalgorithmus für das Steuerungsproblem basierend auf einer Strafmethode entwickelt, welche die Moreau–Yosida-artigen Approximationen des Doppelhindernispotentials einschließt. In diesem Zusammenhang wird ein dual-gewichteter Residuenansatz für zielorientierte adaptive finite Elemente präsentiert, welcher auf dem Konzept der C-Stationarität beruht. Die numerische Realisierung des adaptiven Konzepts und entsprechende numerische Testergebnisse werden beschrieben. Die Lipschitzstetigkeit des Steuerungs-Zustandsoperators des zugehörigen instantanen Steuerungsproblems wird bewiesen und dessen Richtungsableitung wird charakterisiert. Starke Stationaritätsbedingungen für dieses Problem werden durch die Anwendung einer Technick von Mignot und Puel hergeleitet. Basierend auf der primalen Form der Bouligard-Ableitung wird ein impliziter numerischer Löser entwickelt, dessen Implentierung erläutert und anhand von numerischen Resultaten illustriert wird. / This thesis is concerned with the optimal control of a Cahn–Hilliard–Navier–Stokes system with variable fluid densities. It focuses on the double-obstacle potential, which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier–Stokes equation. A suitable time-discretization is presented and associated energy estimates are proven. The existence of solutions to the primal system and of optimal controls is established for the original problem as well as for a family of regularized problems. The consistency of these approximations is shown and first order optimality conditions for the regularized problems are derived. Through a limit process with respect to the regularization parameter, a stationarity system for the original problem is established, which corresponds to a function space version of ε-almost C-stationarity. Moreover, a numerical solution algorithm for the optimal control problem is developed based on a penalization method involving the Moreau–Yosida type approximations of the double-obstacle potential. A dual-weighted residual approach for goal-oriented adaptive finite elements is presented, which is based on the concept of C-stationarity. The overall error representation depends on dual weighted primal residuals and vice versa, supplemented by additional terms corresponding to the complementarity mismatch. The numerical realization of the adaptive concept is described and a report on numerical tests is provided. The Lipschitz continuity of the control-to-state operator of the corresponding instantaneous control problem is verified and its directional derivative is characterized. Strong stationarity conditions for the instantaneous control problem are derived. Utilizing the primal notion of B-differentiability, a bundle-free implicit programming method is developed. Details on the numerical implementation are given and numerical results are included.