Cette thèse est consacrée à l’étude théorique et numérique de plusieurs équations aux dérivées partielles non linéaires qui apparaissent dans la modélisation de la séparation de phase et des micro-systèmes électro-mécaniques (MSEM). Dans la première partie, nous étudions des modèles d’ordre élevé en séparation de phase pour lesquels nous obtenons le caractère bien posé et la dissipativité, ainsi que l’existence de l’attracteur global et, dans certains cas, des simulations numériques. De manière plus précise, nous considérons dans cette première partie des modèles de type Allen-Cahn et Cahn-Hilliard d’ordre élevé avec un potentiel régulier et des modèles de type Allen-Cahn d’ordre élevé avec un potentiel logarithmique. En outre, nous étudions des modèles anisotropes d’ordre élevé et des généralisations d’ordre élevé de l’équation de Cahn-Hilliard avec des applications en biologie, traitement d’images, etc. Nous étudions également la relaxation hyperbolique d’équations de Cahn-Hilliard anisotropes d’ordre élevé. Dans la seconde partie, nous proposons des schémas semi-discrets semi-implicites et implicites et totalement discrétisés afin de résoudre l’équation aux dérivées partielles non linéaire décrivant à la fois les effets élastiques et électrostatiques de condensateurs MSEM. Nous faisons une analyse théorique de ces schémas et de la convergence sous certaines conditions. De plus, plusieurs simulations numériques illustrent et appuient les résultats théoriques. / This thesis is devoted to the theoretical and numerical study of several nonlinear partial differential equations, which occur in the mathematical modeling of phase separation and micro-electromechanical system (MEMS). In the first part, we study higher-order phase separation models for which we obtain well-posedness and dissipativity results, together with the existence of global attractors and, in certain cases, numerical simulations. More precisely, we consider in this first part higher-order Allen-Cahn and Cahn-Hilliard equations with a regular potential and higher-order Allen-Cahn equation with a logarithmic potential. Moreover, we study higher-order anisotropic models and higher-order generalized Cahn-Hilliard equations, which have applications in biology, image processing, etc. We also consider the hyperbolic relaxation of higher-order anisotropic Cahn-Hilliard equations. In the second part, we develop semi-implicit and implicit semi-discrete, as well as fully discrete, schemes for solving the nonlinear partial differential equation, which describes both the elastic and electrostatic effects in an idealized MEMS capacitor. We analyze theoretically the stability of these schemes and the convergence under certain assumptions. Furthermore, several numerical simulations illustrate and support the theoretical results.

In this thesis, we have used a phase field model for studying the effect of atomic mobility inside the precipitate phase on coarsening behaviour in two dimensional (2D) systems. In all the available coarsening theories, the diffusivity inside the precipitate phase is not explicitly taken into account; this would imply that there is no chemical potential gradient inside the precipitate. This assumption is valid if (a) the atomic mobility inside the precipitate is much higher than that in the matrix, or (b) the precipitate volume fraction is small (i.e. the interparticle spacing is far higher than the average particle size). We undertook this study to evaluate the potential effect of diffusivity in the precipitate on coarsening in situations where conditions (a) and (b), above, do not hold, by studying systems with moderate volume fractions (20% and 30%) and with low atomic mobilities in the precipitate. In our study, we have fixed the atomic mobility in the matrix at a constant value. We have used the well known Cahn-Hilliard model in which the microstructure is described in terms of a composition field variable. The evolution of microstructure is studied by numerically solving a non-classical diffusion equation known as the Cahn-Hilliard equation. We have used a semi-implicit Fourier spectral technique for solving the CH equation using periodic boundary conditions. The coarsening behaviour is tracked and analyzed using number density of particles, their average size and their size distribution. The main conclusion from this study is that, contrary to expectations, the atomic mobility in the precipitate phase has only a small effect on coarsening behavior. Specifically, with decreasing atomic mobility in the precipitate phase, we report a small increase in the number density, a slightly wider size distribution and a slightly smaller coarsening rate. We also add that these effects are too small to allow experimental verification. These results indicate that the need for chemical potential equilibration within each precipitate is not an important factor during coarsening.

Universiẗat, Diss., 2002--Bonn.

Techn. Universiẗat, Diss., 2003--Dresden.

This thesis is devoted to the rigorous study of approximations for (multi-phase) mean curvature flow and related equations. We establish convergence towards weak solutions of the according geometric evolution equations in the BV-setting of finite perimeter sets. Our proofs are of variational nature in the sense that we use the gradient flow structure of (multi-phase) mean curvature flow. We study two classes of schemes, namely phase-field models and thresholding schemes. The starting point of our investigation is the fact that both, the Allen-Cahn Equation and the thresholding scheme, preserve this gradient flow structure. The Allen-Cahn Equation is a gradient flow itself, while the thresholding scheme is a minimizing movements scheme for an energy that Γ-converges to the total interfacial energy. In both cases we can incorporate external forces or a volume-constraint. In the spirit of the work of Luckhaus and Sturzenhecker (Calc. Var. Partial Differential Equations 3(2):253–271, 1995), our results are conditional in the sense that we assume the time-integrated energies to converge to those of the limit. Although this assumption is natural, it is not guaranteed by the a priori estimates at hand.

no / In recent years there have been major advances in our understanding of the mechanisms of phase separation in polymer and copolymer blends, to the extent that good control of phase-separated morphology is a real possibility. Many groups are studying the computational simulation of polymer phase separation. In the light of this, we are exploring methods which will give insight into the mechanical response of multiphase polymers. We present preliminary results from a process which allows the production of a two-dimensional finite-element mesh from the contouring of simulated composition data. We examine the stretching of two-phase structures obtained from a simulation of linear Cahn-Hilliard spinodal phase separation. In the simulations, we assume one phase to be hard, and the other soft, such that the shear modulus ratio ... is large (... ). We indicate the effect of varying composition on the material modulus and on the distribution of strains through the stretched material. We also examine in some detail the symmetric structures obtained at 50% composition, in which both phases are at a percolation threshold. Inspired by simulation results for the deformation of these structures, we construct a "scaling" theory, which reproduces the main features of the deformation. Of particular interest is the emergence of a lengthscale, below which the deformation is non-affine. This length is proportional to ... , and hence is still quite small for all reasonable values of this ratio. The same theory predicts that the effective composite modulus scales also as ..., which is supported by the simulations.

The phase-field method has become a popular numerical tool for moving boundary problems in recent years. In this method, the interface is intrinsically diffuse and stores a mixing energy that is equivalent to surface tension. The major advantage of this method is its energy formulation which makes it easy to incorporate different physics. Meanwhile, the energy decay property can be used to guide the design of energy stable numerical schemes. In this dissertation, we investigate the application of the Allen-Cahn model, a member of the phase-field family, in the simulation of phase change problems. Because phase change is usually accompanied with latent heat, heat transfer also needs to be considered. Firstly, we go through different theoretical aspects of the Allen-Cahn model for nonconserved interfacial dynamics. We derive the equilibrium interface profile and the connection between surface tension and mixing energy. We also discuss the well-known convex splitting algorithm, which is linear and unconditionally energy stable. Secondly, by modifying the free energy functional, we give the Allen-Cahn model for isothermal phase transformation. In particular, we explain how the Gibbs-Thomson effect and the kinetic effect are recovered. Thirdly, we couple the Allen-Chan and heat transfer equations in a way that the whole system has the energy decay property. We also propose a convex-splitting-based numerical scheme that satisfies a similar discrete energy law. The equations are solved by a finite-element method using the deal.ii library. Finally, we present numerical results on the evolution of a liquid drop in isothermal and non-isothermal settings. The numerical results agree well with theoretical analysis. / Master of Science / Phase change phenomena, such as freezing and melting, are ubiquitous in our everyday life. Mathematically, this is a moving boundary problem where the phase front evolves based on the local temperature. The phase change is usually accompanied with the release or absorption of latent heat, which in turn affects the temperature. In this work, we develop a phase-field model, where the phase front is treated as a diffuse interface, to simulate the liquid-solid transition. This model is consistent with the second law of thermodynamics. Our finite-element simulations successfully capture the solidification and melting processes including the interesting phenomenon of recalescence.

Kolloidpartikel die von zwei nicht mischbaren Fluiden benetzt werden, tendieren dazu sich an der fluiden Grenzfläche aufzuhalten um die Oberflächenspannung zu minimieren. Bei genügender Anzahl solcher Kolloide werden diese zusammengedrückt und lassen die fluide Grenzfläche erstarren. Das gesamte System aus Fluiden und Kolloiden bildet dann eine spezielle Emulsion mit interessanten Eigenschaften. In dieser Arbeit wird ein kontinuum Model für solche Systeme entwickelt, basierend auf den Prinzipien der Massenerhaltung und der themodynamischen Konsistenz. Dabei wird die makroskopische Zwei-Phasen-Strömung durch eine Navier-Stokes Cahn-Hilliard Gleichung modelliert und die mikroskopischen Partikel an der fluiden Grenzfläche durch einen Phase-Field-Crystal Ansatz beschrieben. Zur Evaluation des verwendeten Strömungsmodells wird ein Test verschiedener Navier-Stokes Cahn-Hilliard Modelle anhand eines bekannten Benchmark Szenarios durchgeführt. Die Ergebnisse werden mit denen von anderen Methoden zur Simulation von Zwei-Phasen-Strömungen verglichen. Desweiteren wird eine neue Methode zur Simulation von Zwei-Phasen-Strömungen in komplexen Gebieten vorgestellt. Dabei wird die komplexe Geometrie implizit durch eine Phasenfeldvariable beschrieben, welche die charakteristische Funktion des Gebietes approximiert. Die Strömungsgleichungen werden dementsprechend so umformuliert, dass sie in einem größeren und einfacheren Gebiet gelten, wobei die Randbedingungen implizit durch zusätzliche Quellterme eingebracht werden. Zur Einarbeitung der Oberflächenkolloide in das Strömungsmodell wird schließlich die Variation der freien Energie des Gesamtsystems betrachtet. Dabei wird die Energie der Partikel durch die Phase-Field-Crystal Energie approximiert und die Energie der Oberfläche durch die Ginzburg-Landau Energie. Eine Variation der Gesamtenergie liefert dann die Phase-Field-Crystal Gleichung und die Navier-Stokes Cahn-Hilliard Gleichungen mit zusätzlichen elastischen Spannunngen. Zur Validierung des Ansatzes wird auch eine sharp interface Version der Gleichungen hergeleitet und mit der zuvor hergeleiteten diffuse interface Version abgeglichen. Die Diskretisierung der erhaltenen Gleichungen erfolgt durch Finiten Elemente in Kombination mit einem semi-impliziten Euler Verfahren. Durch numerische Simulationen wird die Anwendbarkeit des Modells gezeigt und bestätigt, dass die oberflächenaktiven Kolloide die fluide Grenzfläche hinreichend steif machen können um externen Kräften entgegenzuwirken und das gesamte System zu stabilisieren. / Colloid particles that are partially wetted by two immiscible fluids can become confined to fluidfluid interfaces. At sufficiently high volume fractions, the colloids may jam and the interface may crystallize. The fluids together with the interfacial colloids compose an emulsion with interesting new properties and offer an important route to new soft materials. Based on the principles of mass conservation and thermodynamic consistency, we develop a continuum model for such systems which combines a Cahn-Hilliard-Navier-Stokes model for the macroscopic two-phase fluid system with a surface Phase-Field-Crystal model for the microscopic colloidal particles along the interface. We begin with validating the used flow model by testing different diffuse interface models on a benchmark configuration for a two-dimensional rising bubble and compare the results with reference solutions obtained by other two-phase flow models. Furthermore, we present a new method for simulating two-phase flows in complex geometries, taking into account contact lines separating immiscible incompressible components. In this approach, the complex geometry is described implicitly by introducing a new phase-field variable, which is a smooth approximation of the characteristic function of the complex domain. The fluid and component concentration equations are reformulated and solved in larger regular domain with the boundary conditions being implicitly modeled using source terms. Finally, we derive the thermodynamically consistent diffuse interface model for two-phase flow with interfacial particles by taking into account the surface energy and the energy associated with surface colloids from the surface PFC model. The resulting governing equations are the phase field crystal equations and Navier-Stokes Cahn-Hilliard equations with an additional elastic stress. To validate our approach, we derive a sharp interface model and show agreement with the diffuse interface model. We demonstrate the feasibility of the model and present numerical simulations that confirm the ability of the colloids to make the interface sufficiently rigid to resist external forces and to stabilize interfaces for long times.

Cette thèse se situe dans le cadre de l'analyse théorique et numérique de quelques généralisations de l'équation de Cahn-Hilliard. On étudie l'existence, l'unicité et la régularité de la solution de ces modèles ainsi que son comportement asymptotique en terme d'existence d'un attracteur global de dimension fractale finie. La première partie de la thèse concerne des modèles appliqués à la retouche d'images. D'abord, on étudie la dynamique de l'équation de Bertozzi-Esedoglu-Gillette-Cahn-Hilliard avec des conditions de type Neumann sur le bord et une nonlinéarité régulière de type polynomial et on propose un schéma numérique avec une méthode de seuil efficace pour le problème de la retouche et très rapide en terme de temps de convergence. Ensuite, on étudie ce modèle avec des conditions de type Neumann sur le bord et une nonlinéarité singulière de type logarithmique et on donne des simulations numériques avec seuil qui confirment que les résultats obtenus avec une nonlinéarité de type logarithmique sont meilleurs que ceux obtenus avec une nonlinéarité de type polynomial. Finalement, on propose un modèle basé sur le système de Cahn-Hilliard pour la retouche d'images colorées. La deuxième partie de la thèse est consacrée à des applications en biologie et en chimie. On étudie la convergence de la solution d'une généralisation de l'équation de Cahn-Hilliard avec un terme de prolifération, associée à des conditions aux limites de type Neumann et une nonlinéarité régulière. Dans ce cas, on démontre que soit la solution explose en temps fini soit elle existe globalement en temps. Par ailleurs, on donne des simulations numériques qui confirment les résultats théoriques obtenus. On termine par l'étude de l'équation de Cahn-Hilliard avec un terme source et une nonlinéarité régulière. Dans cette étude, on considère le modèle à la fois avec des conditions aux limites de type Neumann et de type Dirichlet. / This thesis is situated in the context of the theoretical and numerical analysis of some generalizations of the Cahn-Hilliard equation. We study the well-possedness of these models, as well as the asymptotic behavior in terms of the existence of finite-dimenstional (in the sense of the fractal dimension) attractors. The first part of this thesis is devoted to some models which, in particular, have applications in image inpainting. We start by the study of the dynamics of the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation with Neumann boundary conditions and a regular nonlinearity. We give numerical simulations with a fast numerical scheme with threshold which is sufficient to obtain good inpainting results. Furthermore, we study this model with Neumann boundary conditions and a logarithmic nonlinearity and we also give numerical simulations which confirm that the results obtained with a logarithmic nonlinearity are better than the ones obtained with a polynomial nonlinearity. Finally, we propose a model based on the Cahn-Hilliard system which has applications in color image inpainting. The second part of this thesis is devoted to some models which, in particular, have applications in biology and chemistry. We study the convergence of the solution of a Cahn-Hilliard equation with a proliferation term and associated with Neumann boundary conditions and a regular nonlinearity. In that case, we prove that the solutions blow up in finite time or exist globally in time. Furthermore, we give numericial simulations which confirm the theoritical results. We end with the study of the Cahn-Hilliard equation with a mass source and a regular nonlinearity. In this study, we consider both Neumann and Dirichlet boundary conditions.

Ce manuscrit de thèse porte sur l'analyse numérique de schémas volumes finis pour la discrétisation de deux systèmes particuliers d'équations. Dans un premier temps nous étudions l'équation de Cahn-Hilliard associée à des conditions aux limites dynamiques dont l'une des principales difficultés est que cette condition aux limites est une équation parabolique, non linéaire, posée sur le bord et couplée avec l'intérieur du domaine. Nous proposons une discrétisation de type volumes finis en espace qui permet de coupler naturellement l'équation dans le domaine et celle sur sa frontière par un terme de flux et qui s'adapte facilement à la géométrie courbe du domaine. Nous montrons l'existence et la convergence des solutions discrètes vers une solution faible du système. Dans un second temps nous étudions la stabilité Inf-Sup du problème de Stokes pour un schéma volumes finis de type dualité discrète (DDFV). Nous donnons une analyse complète de la stabilité Inf-Sup inconditionnelle dans certains cas et de la stabilité de codimension 1 dans le cas de maillages cartésiens. Nous mettons également en place une méthode numérique permettant de calculer la constante Inf-Sup associée à ce schéma pour un maillage donné. On peut ainsi observer le comportement stable ou instable selon les cas en fonction de la géométrie des maillages. Dans une dernière partie nous proposons un schéma DDFV pour un modèle couplé Cahn-Hilliard/Stokes ce qui nécessite l'introduction de nouveaux opérateurs discrets. Nous démontrons la décroissance de l'énergie au niveau discret ainsi que l'existence d'une solution au problème discret. L'ensemble de ces travaux est validé par de nombreux résultats numériques. / This manuscript is devoted to the numerical analysis of finite-volume schemes for the discretization of two particular equations. First, we study the Cahn-Hilliard equation with dynamic boundary conditions whose one of the main difficulties is that this boundary condition is a non-linear parabolic equation on the boundary coupled with the interior of the domain. We propose a spatial finite-volume discretization which is well adapted to the coupling of the dynamics in the domain and those on the boundary by the flux term. Moreover this kind of scheme accounts naturally for the non-flat geometry of the boundary. We prove the existence and the convergence of the discrete solutions towards a weak solution of the system. Second, we study the Inf-Sup stability of the discrete duality finite volume (DDFV) scheme for the Stokes problem. We give a complete analysis of the unconditional Inf-Sup stability in some cases and of codimension 1 Inf-Sup stability for Cartesian meshes. We also implement a numerical method which allows us to compute the Inf-Sup constant associated with this scheme for a given mesh. Thus, we can observe the stable or unstable behaviour that can occur depending on the geometry of the meshes. In a last part we propose a DDFV scheme for a Cahn-Hilliard/Stokes phase field model that required the introduction of new discrete operators. We prove the dissipation of the energy in the discrete case and the existence of a solution to the discrete problem. All these research results are validated by extensive numerical results.