Ce travail est dédié à la mise en place de modèles d'écoulements de fluides complexes, à leur analyse théorique ainsi qu'au développement et à l'analyse de convergence de schémas numériques appropriés. <br /><br />Une première partie du travail concerne l'étude de modèles dits à interface diffuse pour les écoulements incompressibles multiphasiques. Après une étude assez précise du cadre diphasique, on propose la généralisation au cadre triphasique, ce qui nécessite d'introduire la notion importante de consistance des modèles. Des résultats numériques confirment la pertinence des modèles proposés. Ensuite, on s'intéresse au modèle plus classique de Navier-Stokes non-homogène incompressible pour lequel on établit le caractère bien posé du problème pour des conditions aux limites ouvertes non-linéaires en sortie d'un écoulement. Une brique essentielle de ce travail est l'étude détaillée du problème de traces pour l'équation de transport associée à un champ de vitesse peu régulier. Ce travail, dont l'intérêt dépasse le cadre applicatif décrit ci-dessus, fait l'objet d'un chapitre à part entière.<br /><br />Dans une seconde partie, on s'intéresse à l'approximation numérique par des méthodes de volumes finis des solutions de problèmes elliptiques non-linéaires monotones (du type p-laplacien). Un premier chapitre décrit un certain nombre de résultats obtenus dans le contexte de maillages cartésiens. Un second chapitre est consacré à l'étude d'un cadre géométrique plus général par le biais de méthodes dites en dualité discrète. Une attention particulière est portée au cas où les coefficients du problème présentent des discontinuités spatiales, ce qui mène à des problèmes de transmission non-linéaire entre deux milieux.<br /><br />Le mémoire s'achève par la description de quelques travaux connexes, d'une part sur une classe de schémas VF pour les équations elliptiques linéaires adaptés à des maillages non orthogonaux, et d'autre sur l'étude numérique de problèmes elliptiques couplés 2D/1D issus de la description asymptotique d'écoulements dans des milieux poreux fracturés.

L'objet de ce mémoire est de rendre compte de travaux portant sur les phases spatialement modulées ou phases lamellaires, leur thermodynamique (diagramme de phases, profi l des interfaces) et la dynamique des transitions de phase qui les font apparaître. En particulier, je me suis intéressé aux non-linéarités de cette dynamique de transition de phase particulière en me basant sur les modèles de Swift-Hohenberg, Cahn-Hilliard et Oono, d'après les noms de leurs auteurs.

The development, analysis, and implementation of efficient methods to solve algebraic systems of equations are main research directions in the field of numerical simulation and are the focus of this thesis. Due to their lesser demands for computer resources, iterative solution methods are the choice to make, when very large scale simulations have to be performed. To improve their efficiency, iterative methods are combined with proper techniques to accelerate convergence. A general technique to do this is to use a so-called preconditioner. Constructing and analysing various preconditioning methods has been an active field of research already for decades. Special attention is devoted to the class of the so-called optimal order preconditioners, that possess both optimal convergence rate and optimal computational complexity. The preconditioning techniques, proposed and studied in this thesis, utilise the block structure of the underlying matrices, and lead to methods that are of optimal order. In the first part of the thesis, we construct an Algebraic MultiLevel Iteration (AMLI) method for systems arising from discretizations of parabolic problems, using Crouzeix-Raviart finite elements. The developed AMLI method is based on an approximated block factorization of the original system matrix, where the partitioning is associated with a sequence of nested discretization meshes. In the second part of the thesis we develop solution methods for the numerical simulation of multiphase flow problems, modelled by the Cahn-Hilliard (C-H) equation. We consider the discrete C-H problem, obtained via finite element discretization in space and implicit schemes in time. We propose techniques to precondition the Jacobian of the discrete nonlinear system, based on its natural two-by-two block structure. The preconditioners are used in the framework of inexact Newton methods. We develop two nonlinear solution algorithms for the Cahn-Hilliard problem. Both lead to efficient optimal order methods. One of the main advantages of the proposed methods is that they are implemented using available software toolboxes for both sequential and distributed execution. The theoretical analysis of the solution methods presented in this thesis is combined with numerical studies that confirm their efficiency.

Fluid flows mainly driven by capillary forces are presented in this thesis. By means of modeling and simulations, interesting dynamics in capillary-driven flows are revealed such as coalescences, breakups, precursor films, flow instabilities, rapid spreading, rigid body motions, and reactive wetting. Diffuse-interface methods model a fluid interface as having a finite thickness endowed with physical properties such as surface tension. Two diffuse-interface models that are based on the free energy of the system are presented. The binary model, more specifically the coupled Navier-Stokes/Cahn-Hilliard equations, was used to study different two-phase flows including problems related to microfluidics. Numerical issues using this model have been addressed such as the need for mesh adaptivity and time-step restrictions. Moreover, the flexibility of this model to simulate 2D, axisymmetric, and 3D flows has been demonstrated. The factors affecting reproducibility of microdroplet depositions performed under a liquid medium are investigated. In the deposition procedure, sample solution is dispensed from the end of a capillary by the aid of a pressure pulse onto a substrate with pillar-shaped sample anchors. In both the experimental and numerical study it was shown that the deposited volume mainly depends on the capillary-substrate distance and anchor surface wettability. Furthermore, a critical equilibrium contact angle has been identified below which reproducible depositions are facilitated. The ternary model is developed for more complicated flows such as liquid phase sintering. With the introduction of a Gibbs energy functional, the governing equations are derived, consisting of convective concentration and phase-field equations which are coupled to the Navier-Stokes equations with surface tension forces. Arbitrary phase diagrams, surface energies, and typical dimensionless numbers are some input parameters into the model. Detailed analysis of the important capillary phenomena in liquid phase sintering such as reactive and nonreactive wetting and motion of two particles connected by a liquid bridge are presented. The dynamics of the wetting is found to match with a known hydrodynamic theory for spreading liquids. Factors affecting the equilibrium configuration of the particles such as equilibrium contact angles and volume ratios are also investigated. / QC 20100823

Nanoskalige Multischichten besitzen attraktive physikalische Eigenschaften wie den Riesenmagnetwiderstand, die sehr sensibel von der Struktur der Grenzfläche und der Einzelschichtdicke abhängen. Mit Hilfe einer Wärmebehandlung wird versucht, den Riesenmagnetwiderstand der durch Sputtern abgeschiedenen Schichten zu erhöhen. In entmischenden System Co/Cu wird bei Einzelschichtdicken von 2nm eine Erhöhung des Riesenmagnetwiderstandes gemessen. Allerdings verringert sich der Widerstand bei höheren Temperaturen drastisch. Die Verringerung wird begleitet von einem Zerfall der Schichtstruktur. Diese Arbeit untersucht die thermische Stabilität und morphologische Entwicklung nanoskaliger Multischichten mit binären nichtmischbaren Komponenten während der Wärmebehandlung mit Hilfe der Monte-Carlo-Methode und im Rahmen der Cahn-Hilliard-Theorie. Es wird gezeigt, dass bei einer Wärmebehandlung abgeschiedener Schichten die chemische Unschärfe der Phasengrenzfläche verringert werden kann. Bei der Wärmebehandlung bildet sich abhängig von der Temperatur eine morphologische Rauigkeit an der Phasengrenzfläche. Oberhalb einer kritische Temperatur findet ein Rauigkeitsübergang statt, bei der langwellige Rauigkeiten mit ständig wachsender Amplitude entstehen. Die Überlappung der Undulationen der morphologischen Rauigkeit von Unter- und Oberseite einer dünnen Schicht wurde als ein Mechanismus für die Bildung von Schichtdurchbrüchen identifiziert. In polykristallinen Schichten verursachen Korngrenzen thermische Instabilitäten der Multischicht. &amp;quot;Grain boundary grooving&amp;quot; wird als ein Mechanismus für die Bildung eines Schichtdurchbruchs an Korngrenzen vorgeschlagen. Durchbrochen Schichten ziehen sich getrieben durch Kapillarkräfte zurück. An den Endstellen bilden sich Wulste aus. Je nach Schichtdickenverhältnissesn kommt es zu einer Verschmelzung mit benachbarten Schichten gleicher Phase. Die ursprüngliche Schichtstruktur wird zerstört.

We have studied precipitate growth and coarsening in ternary alloys using two different phase held models. The first one is a ternary extension of the classical Cahn-Hilliard (C-H) model in which both the phases are characterized using conserved held variables i.e. composition (cB; cC ); mobility matrix and gradient energy efficient are the other input parameters in this model. In the second model, each phase is treated as separate, and phase identify cation is through a (non-conserved) phase held variable ; we have used a grand potential-based (GP) formulation, due to Plapp [1], Choudhury and Nestler [2], where interfacial energy and interface width, as well as free energy and diffusivity matrix for the relevant phases are the input parameters. The first model i.e. the Cahn-Hilliard (C-H) type model is conceptually simple. The model for ternary is a straight forward extension of the binary. The grand potential (GP) formulation has the advantage of being able to incorporate thermodynamic database like Thermocalc in it. We present below a summary of the findings of our research on (a) precipitate growth, precipitate coarsening, and (c) a critical comparison between results from phase held simulations and those from experiments on an Ni-Al-Mo alloy Precipitate growth In our study of precipitate growth in ternary alloys, we end that when both the solute elements have the same diffusivity, precipitate growth behaviour in ternary alloys is identical to that binary alloys; specifically, we recover the temporal power law r2 = kgt relating the particle radius to time, and the growth kg depends only on supersaturation (i.e., equilibrium volume fraction of the precipitate phase), and is independent of the slope of the tie line. However, when one solute element, (say, C) di uses slower than the other (i.e. (DCC =DBB) < 1,(where DBB, DCC are intertie suavities’ in the lab frame of reference), the ux of C at the interface is smaller than that of species B, causing the precipitate to become depleted in C and enriched in B; this process continues until the growth phase enters a scaling regime where we recover the temporal law for growth: r2 = kgt. In this regime, the tie line selected by the precipitate and matrix interfacial compositions is different from the thermodynamic tie line containing the alloy, a result first reported by Coates [3]. After validating our phase held model quantitatively through a critical comparison with Coates' theory of tie line selection, we have characterized the growth behaviour: specifically, we end that growth kg drops with decreasing value of DCC ; the magnitude of this drop is stronger for alloys which (a) are on higher-C tie lines (i.e., the slope of the tie line is higher), and (b) have smaller precipitate volume fractions. Precipitate coarsening In our simulations, we end that precipitate coarsening does indeed enter a scaling regime where the temporal power law r3 = kt (which relates the average precipitate radius r to (b) time t) is valid; the coarsening rate k depends, as expected, not only on precipitate volume fraction, but also on the slope of the tie line and diffusivity ratio (DCC =DBB). (c) (d) When the solutes have equal diffusivity (i.e., (DCC =DBB) = 1), the coarsening behaviour is essentially the same as that in a binary alloy. However, when solute C (say) is the slower di using species, the coarsening rate k drops, with a deeper drop in alloys on higher-C tie lines. Both these conclusions are similar to those from our study of precipitate growth. (e) (f) However, there is a crucial difference between precipitate growth and coarsening in ternary alloys: The suppression in coarsening rate (for DCC < DBB) in ternary alloys is accompanied by another e ect: larger (and growing) precipitates are richer in the faster di using species B, while the smaller and shrinking precipitates are richer in the slower di using C. In other words, during coarsening in ternary alloys, the tie line selected by precipitate and matrix interfacial components depends on precipitate size; during growth, however, the scaling regime is characterized by the same tie line, independent of precipitate size. (g) (h) (i) Critical comparison between theory and experiment (j) (k) (l) We have used the grand potential based phase held model [1] [2] to study coarsening in Ni-Al-Mo alloys. This model has the advantage of ease with which we can incorporate the thermodynamic and kinetic data on real alloys. (m) (n) A comparison of coarsening rate from our 3D simulations with the experimentally observed rate reveals that diffusivity of the faster di using species (which, in Ni-Al-Mo alloys, is aluminium) from our simulations is within an order of magnitude from the experimental value. However the dominant term in the (@ =@c) matrix is underestimated by 2 to 3 orders of magnitude (compared to its value computed from CALPHAD-based thermodynamic data).

The study of finite-sized, deformable droplets adverted by turbulent flows is an active area of research. It spans many streams of sciences and engineering, which include chemical engineering, fluid mechanics, statistical physics, nonlinear dynamics, and also biology. Advances in experimental techniques and high-performance computing have made it possible to investigate the properties of turbulent fluids laden with droplets. The main focus of this thesis is to study the statistical properties of the dynamics of such finite-size droplets in turbulent flows by using direct numerical simulations (DNSs). The most important feature of the model we use is that the droplets have a back-reaction on the advecting fluid: the turbulent fluid affects the droplets and they, in turn, affect the turbulence of the fluid. Our study uncovers (a) statistical properties that characterize the spatiotemporal evolution of droplets in turbulent flows, which are statistically homogeneous and isotropic, and (b) the modification of the statistical properties of this turbulence by the droplets. This thesis is divided into seven Chapters. Chapter 1 contains an introduction to the background material that is required for this thesis, especially the details about the equations we use; it also contains an outline of the problems we study in subsequent Chapters. Chapter 2 contains our study of “Droplets in Statistically Homogeneous Turbulence: From Many Droplets to a few Droplets”. Chapter 3 is devoted to our study of “Coalescence of Two Droplets”. Chapter 4 deals with “Binary-Fluid Turbulence: Signatures of Multifractal Droplet Dynamics and Dissipation Reduction”. Chapter 5 deals with “A BKM-type theorem and associated computations of solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations”. Chapter 6 is devoted to our study of “Turbulence-induced Suppression of Phase Separation in Binary-Fluid Mixtures”. Chapter 7 is devoted to our study of “Antibubbles: Insights from the Cahn-Hilliard-Navier-Stokes Equations”.

Nanoskalige Multischichten besitzen attraktive physikalische Eigenschaften wie den Riesenmagnetwiderstand, die sehr sensibel von der Struktur der Grenzfläche und der Einzelschichtdicke abhängen. Mit Hilfe einer Wärmebehandlung wird versucht, den Riesenmagnetwiderstand der durch Sputtern abgeschiedenen Schichten zu erhöhen. In entmischenden System Co/Cu wird bei Einzelschichtdicken von 2nm eine Erhöhung des Riesenmagnetwiderstandes gemessen. Allerdings verringert sich der Widerstand bei höheren Temperaturen drastisch. Die Verringerung wird begleitet von einem Zerfall der Schichtstruktur. Diese Arbeit untersucht die thermische Stabilität und morphologische Entwicklung nanoskaliger Multischichten mit binären nichtmischbaren Komponenten während der Wärmebehandlung mit Hilfe der Monte-Carlo-Methode und im Rahmen der Cahn-Hilliard-Theorie. Es wird gezeigt, dass bei einer Wärmebehandlung abgeschiedener Schichten die chemische Unschärfe der Phasengrenzfläche verringert werden kann. Bei der Wärmebehandlung bildet sich abhängig von der Temperatur eine morphologische Rauigkeit an der Phasengrenzfläche. Oberhalb einer kritische Temperatur findet ein Rauigkeitsübergang statt, bei der langwellige Rauigkeiten mit ständig wachsender Amplitude entstehen. Die Überlappung der Undulationen der morphologischen Rauigkeit von Unter- und Oberseite einer dünnen Schicht wurde als ein Mechanismus für die Bildung von Schichtdurchbrüchen identifiziert. In polykristallinen Schichten verursachen Korngrenzen thermische Instabilitäten der Multischicht. &amp;quot;Grain boundary grooving&amp;quot; wird als ein Mechanismus für die Bildung eines Schichtdurchbruchs an Korngrenzen vorgeschlagen. Durchbrochen Schichten ziehen sich getrieben durch Kapillarkräfte zurück. An den Endstellen bilden sich Wulste aus. Je nach Schichtdickenverhältnissesn kommt es zu einer Verschmelzung mit benachbarten Schichten gleicher Phase. Die ursprüngliche Schichtstruktur wird zerstört.

<p dir="ltr">This thesis presents advances in numerical methods for partial differential equations (PDEs) and optimization problems, with a focus on improving efficiency, stability, and accuracy across various applications. We begin by addressing 3D Poisson-type equations, developing a GPU-accelerated spectral-element method that utilizes the tensor product structure to achieve extremely fast performance. This approach enables solving problems with over one billion degrees of freedom in less than one second on modern GPUs, with applications to Schrödinger and Cahn<i>–</i>Hilliard equations demonstrated. Next, we focus on parabolic PDEs, specifically the Cahn<i>–</i>Hilliard equation with dynamical boundary conditions. We propose an efficient energy-stable numerical scheme using a unified framework to handle both Allen<i>–</i>Cahn and Cahn<i>–</i>Hilliard type boundary conditions. The scheme employs a scalar auxiliary variable (SAV) approach to achieve linear, second-order, and unconditionally energy stable properties. Shifting to a machine learning perspective for PDEs, we introduce an unsupervised learning-based numerical method for solving elliptic PDEs. This approach uses deep neural networks to approximate PDE solutions and employs least-squares functionals as loss functions, with a focus on first-order system least-squares formulations. In the realm of optimization, we present an efficient and robust SAV based algorithm for discrete gradient systems. This method modifies the standard SAV approach and incorporates relaxation and adaptive strategies to achieve fast convergence for minimization problems while maintaining unconditional energy stability. Finally, we address optimization in the context of machine learning by developing a structure-guided Gauss<i>–</i>Newton method for shallow ReLU neural network optimization. This approach exploits both the least-squares and neural network structures to create an efficient iterative solver, demonstrating superior performance on challenging function approximation problems. Throughout the thesis, we provide theoretical analysis, efficient numerical implementations, and extensive computational experiments to validate the proposed methods. </p>

Lors d'un hypothétique accident grave de réacteur à eau sous pression, un mélange de matériaux fondus, appelé corium, issu de la fusion du cœur peut se relocaliser dans le puits de cuve constitué par un radier en béton. Les codes d'évaluation réacteur pour simuler la phénoménologie de l'interaction corium-béton sont basés sur une description à grande échelle des échanges qui soulève de nombreuses questions, tant sur la prise en compte des phénomènes multi-échelles mis en jeu que sur la structure adoptée de la couche limite au voisinage du front d'ablation. Dans ce contexte, l'objectif principal de ce travail consiste à aborder le problème de la structure de la couche limite par simulation numérique directe. Ce travail s'inscrit dans le cadre plus général d'une description et d'une modélisation multi-échelle des échanges, c'est-à-dire de l'échelle locale associée au voisinage du front d'ablation jusqu'à l'échelle du code d'évaluation réacteur. Une telle description multi-échelle des échanges soulève le problème de la description locale de l'écoulement multiphasique multiconstituant mais aussi le problème du changement d'échelle et en particulier le passage de l'échelle locale à l'échelle de description supérieure dite macroscopique associée aux mouvements convectifs dans le bain de corium. Parmi les difficultés associées au changement d'échelle, nous nous intéressons à la problématique de la construction de conditions aux limites effectives ou lois de parois pour les modèles macroscopiques. Devant la complexité du problème multiphasique multiconstituant posé au voisinage du front, cette contribution a été abordée sur un problème modèle. Des conditions aux limites dites effectives ont été construites dans le cadre d'une méthode de décomposition de domaine puis testées pour un problème d'écoulement laminaire de convection naturelle sur parois rugueuses. Mˆeme si le problème traité reste encore éloigné des applications visées, cette contribution offre de nombreuses perspectives et constitue une première étape d'une modélisation multiéchelle des échanges pour la problématique de l'interaction corium-béton. Dans le cas plus complexe des écoulements multiphasiques multiconstituants et devant les difficultés expérimentales associées, le développement de lois de parois pour les outils existants aux échelles de description supérieures nécessite, au préalable, de disposer d'un outil de simulation numérique directe de l'écoulement au voisinage du front d'ablation. L'outil développé dans ce travail correspond à un modèle de Cahn-Hilliard/Navier-Stokes pour un mélange diphasique (liquide-gaz) compositionnel (corium-béton fondu) s'appuyant sur une description du système selon trois paramètres d'ordre associés respectivement aux fractions volumiques du gaz et aux deux espèces miscibles de la phase liquide ainsi que sur une décomposition de l'énergie libre selon une contribution diphasique et compositionnelle. Les équations de transport sont dérivées dans le cadre de la thermodynamique des processus irréversibles et résolues sur la base d'une application éléments finis de la plate-forme PELICANS. Plusieurs expériences numériques illustrent la validité et les potentialités d'application de cet outil sur des problèmes diphasiques et/ou compositionnels. Enfin, à partir de l'outil développé, nous abordons par simulation numérique directe une étude de la structure de la couche limite au voisinage du front d'ablation pour des bétons siliceux et silico-calcaire. / In the late phases of some scenario of hypothetical severe accident in Pressurized Water Reactors, a molten mixture of core and vessel structures, called corium, comes to interact with the concrete basemat. The safety numerical tools are lumped parameter codes. They are based on a large averaged description of heat and mass transfers which raises some uncertainties about the multi-scale description of the exchanges but also about the adopted boundary layer structure in the vicinity of the ablation front. In this context, the aim of this work is to tackle the problem of the boundary layer structure by means of direct numerical simulation. This work joins within the more general framework of a multi-scale description and a multi-scale modeling, namely from the local scale associated with the vicinity of the ablation front to the scale associated with the lumped parameter codes. Such a multi-scale description raises not only the problem of the local description of the multiphase multicomponent flow but also the problem of the upscaling between the local- and the macro-scale which is associated with the convective structures within the pool of corium. Here, we are particularly interested in the building of effective boundary conditions or wall laws for macro-scale models. The difficulty of the multiphase multicomponent problem at the local scale leads us to consider a relatively simplified problem. Effective boundary conditions are built in the frame of a domain decomposition method and numerical experiments are performed for a natural convection problem in a stamp shaped cavity to assess the validity of the proposed wall laws. Even if the treated problem is still far from the target applications, this contribution can be viewed as a first step of a multi-scale modeling of the exchanges for the molten core concrete issue. In the more complicated case of multiphase multicomponent flows, it is necessary to have a direct numerical simulation tool of the flow at the local scale to build wall laws for macro-scale models. Here, the developed tool corresponds to a Cahn-Hilliard/Navier-Stokes model for a two-phase compositional system. It relies on a description of the system by three volume fractions and on a free energy composed by a two-phase part and a compositional part. The governing equations are derived in the frame of the thermodynamic of irreversible processes. They are solved on the basis of a finite element application of the object-oriented software component library PELICANS. Several numerical experiments illustrate the validity and the potentialities of application of this tool on two-phase compositional problems. Finally, using the developed tool, we tackle by means of direct numerical simulation the problem boundary layer structure in the vicinity of the ablation front for limestone-sand and siliceous concretes.