A preconditioner to improve the convergence properties of Krylov subspace solvers is derived and analyzed in this work. This method is adapted to linear systems arising from a finite-element discretization of a phase-field crystal equation.

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.

Polymer-modified bitumen (PMB) is a high-performance material for road construction and maintenance. But its storage stability and phase separation behaviour are still not sufficiently understood and need to be studied toward a more successful and sustainable application of PMB. In this thesis, the equilibrium thermodynamics and phase separation dynamics of PMB are investigated with the aim at a fundamental understanding on PMB storage stability and phase separation behaviour. The development of polymer modifiers for paving bitumen is reviewed. The phase separation process in unstable PMBs is captured by fluorescence microscopy at the storage temperature (180 °C). A coupled phase-field model of diffusion and flow is developed to simulate and predict the PMB storage stability and phase separation behaviour. The temperature dependency of PMB phase separation behaviour is modelled by introducing temperature-dependent model parameters between 140 °C and 180 °C. This model is implemented in a finite element software package and calibrated with the experimental observations of real PMBs. The results indicate that storage stability and phase separation behaviour of PMB are strongly dependent on the specific combination of the base bitumen and polymer. An unstable PMB starts to separate into two phases by diffusion, because of the poor polymer-bitumen compatibility. Once the density difference between the two phases becomes sufficiently significant, gravity starts to drive the flow of the two phases and accelerates the separation in the vertical direction. The proposed model, based on the Cahn-Hilliard equation, Flory-Huggins theory and Navier-Stokes equations, is capable of capturing the stability differences among the investigated PMBs and their distinct microstructures at different temperatures. The various material parameters of the PMBs determine the differences in the phase separation behaviour in terms of stability and temperature dependency. The developed model is able to simulate and explain the resulting differences due to the material parameters. The outcome of this study may thus assist in future efforts of ensuring storage stability and sustainable application of PMB. / Polymermodifierade bitumen (PMB) är ett högpresterande material för väganläggning och underhåll. Men PMB:s lagringsstabilitet och fassepareringsegenskaper är inte tillräckligt förstådda än och behöver studeras för en mer framgångsrik och hållbar användning av PMB. I denna avhandling studeras termodynamisk jämvikt och fasseparation av PMB med målsättning att uppnå en grundläggande förståelse av PMB:s lagringsstabilitet och fassepareringsegenskaper. Utvecklingen av polymermodifierade bitumen sammanfattas. Fasseparationsprocessen av instabil PMB:s studeras med hjälp av fluorescens mikroskopi vid lagringstemperatur (180 °C). En kopplad fas-fälts modell som beskriver diffusion och flöde har utvecklats för att simulera och förutsäga PMB:s lagringsstabilitet och fassepareringsegenskaper. Temperaturberoendet hos PMB:s fasseparation har beskrivits genom att införa temperaturberoende modellparametrar mellan 140 °C och 180 °C. Denna modell är införd i ett finit element program och kalibrerad med experimentella observationer på verkliga PMB. Resultaten indikerar att lagringsstabiliteten och fasseparationen hos PMB är starkt beroende av den specifika kombinationen av basbitumen och polymer. En instabil PMB börjar separera i två faser genom diffusion, beroende på dålig bitumen-polymer kompatibilitet. När skillnaden i densitet mellan de två faserna blir tillräckligt stor kommer gravitationen att driva flödet av de två faserna och accelerera separationen i vertikalled. Den föreslagna modellen, baserad på Cahn-Hilliards ekvation, Flory-Huggins teori och Navier-Stokes ekvation, kan beskriva stabilitetsskillnaderna mellan de undersökta PMB:erna och deras distinkta mikrostruktur vid olika temperaturer. De olika materialparametrarna hos PMB bestämmer skillnaden i fassepareringsegenskaper i termer av stabilitet och temperaturberoende. Den utvecklade modellen kan simulera och förklara de resulterande skillnaderna på grund av materialparametrarna. Resultatet av denna studie kan bidra till att säkerställa lagringsstabilitet och hållbara applikationer för PMB. / <p>QC 20161102</p>

La thèse porte sur la modélisation de la transformation de l'austénite en ferrite dans les aciers en mettant l'accent sur les conditions thermodynamiques et cinétiques aux interfaces alpha/gamma en cours de croissance de la ferrite. Dans une première partie, la thèse se concentre sur la description des équilibres thermodynamiques entre alpha et gamma à l'aide de la méthode CalPhad. Nous avons développé un nouvel algorithme hybride combinant la construction d'une enveloppe convexe avec la méthode classique de Newton-Raphson. Nous montrons ses possibilités pour des aciers ternaire Fe-C-Cr et quaternaire Fe-C-Cr-Mo dans des cas particulièrement difficiles. Dans un second chapitre, un modèle à interface épaisse a été développé. Il permet de prédire l'ensemble du spectre des conditions à l'interface alpha/gamma au cours de la croissance de la ferrite, de l'équilibre complet au paraéquilibre avec des cas intermédiaires des plus intéressants. Nous montrons que de nombreux régimes cinétiques particuliers dans les systèmes Fe-C-X peuvent être prévus avec un minimum de paramètres d'ajustement, principalement le rapport entre les diffusivités de l'élément substitutionnel dans l'interface épaisse et dans le volume d'austénite. Le troisième chapitre porte sur l'étude d'un modèle de champ de phase. Une analyse approfondie des conditions à l'interface données par le modèle est réalisée en utilisant la technique des développements asymptotiques. En utilisant les connaissances fournies par cette analyse, le rôle de la mobilité intrinsèque d'interface sur la cinétique et les régimes de croissance est étudié, à la fois dans le cas simple d'alliages binaires Fe-C et dans le cas plus complexe d'alliages Fe-C-Mn / Transformation in steels focusing on the thermodynamic and kinetics conditions at the alpha/gamma interfaces during the ferrite growth. The first chapter deals with the determination of thermodynamic equilibria between alpha and gamma with CalPhad thermodynamic description. We have developed a new hybrid algorithm combining the construction of a convex hull to the more classical Newton-Raphson method to compute two phase equilibria in multicomponent alloys with two sublattices. Its capabilities are demonstrated on ternary Fe-C-Cr and quaternary Fe-C-Cr-Mo steels. In the second chapter, we present a thick interface model aiming to predict the whole spectrum of conditions at an alpha/gamma interface during ferrite growth, from full equilibrium to paraequilibrium with intermediate cases as the most interesting feature. The model, despite its numerous simplifying assumptions to facilitate its numerical implementation, allows to predict some peculiar kinetics in Fe-C-X systems with a minimum of fitting parameters, mainly the ratio between the diffusivities of the substitutional element inside the thick interface and in bulk austenite. The third chapter deals with the phase field model of austenite to ferrite transformation in steels. A thorough analysis on the conditions at the interface has been performed using the technique of matched asymptotic expansions. Special attention is given to clarify the role of the interface mobility on the growth regimes both in simple Fe-C alloys and in more complex Fe-C-Mn alloys

L'ensemble des travaux présentés dans cette thèse porte sur la simulation numérique des procédés de coextrusion par un modèle d'écoulement stratifié basé sur la méthode du champ de phase. L'avantage technologique offert par la coextrusion réside dans la possibilité de combiner des matériaux ayant des propriétés physiques très spécifiques dans un produit unique. Toutefois, les différences rhéologiques entre les divers matériaux sont elles-mêmes responsables d'un phénomène de distorsion de l'interface séparant deux couches adjacents. Les données expérimentales en coextrusion bicouches montrent que, en raison des différences de viscosité et d'élasticité entre le deux composants, le fluide le moins visqueux encapsule le fluide plus visqueux et le passage d'une configuration stratifiée à une encapsulée comporte une perte de qualité du produit final. Ce phénomène, dit d'enrobage représente donc un sujet de très grande actualité dans la recherche industrielle et la compréhension des mécanismes le générant sera utile pour l'amélioration des procédés de mise en forme des polymères. La nature intrinsèquement tridimensionnelle de l'enrobage a requis le développement d'un code pour la simulation tridimensionnelle basée sur la méthode des volumes finis pour la discrétisation des équations de Navier-Stokes pour les écoulement incompressibles et isothermes couplées avec une loi constitutive différentielle non linéaire (modèles de Giesekus ou PTT). La présence de deux fluides est prise en compte par une équation scalaire supplémentaire décrivant l'évolution de l'interface sur un maillage fixe. Cette équation offre une interprétation physique précise car elle est dérivée de la thermodynamique de séparation de phase d'un fluide binaire. Le modèle proposé est validé par confrontation avec les résultats expérimentaux et numériques disponibles dans la littérature. Une étude numérique de la coextrusion en filière rectangulaire est effectuée afin de mettre en évidence les facteurs influençant l'enrobage et la nature de son origine / The objective of the present work is the analysis of coextrusion processes by numerical simulation based on phase-field modeling of stratified confined flows. The study of such flows is motivated by the presence of complex phenomena appearing in a vast range of industrial operational coextrusion conditions due to the differences in the components properties and their viscoelastic behavior. The basic idea in coextrusion is to combine several layers of different polymers in a common die, to form a unique product with enhanced properties. However, the existence of fluid stratification in the die is responsible of a severe distortion of the interface between the fluid components, causing a loss of efficiency for the whole process. Experimental data show that, even if a stratified initial configuration is imposed at the die entry, one fluid eventually encapsulates the other in most of the flow condition analyzed. The intrinsically three-dimensional nature of this phenomenon has required the development of a three-dimensional flow solver based on the finite volume discretization of the Navier-Stokes equations for incompressible and isothermal flow, together with differential nonlinear constitutive equations (Giesekus, PTT models). The presence of two fluid phases is taken into account by a phase field model that implies the solution of an additional scalar equation to describe the evolution of the interface on a fixed Eulerian grid. This model, unlike others of the same family, has a thermodynamic derivation and can be physically interpreted. The proposed method is tested against experimental data and solutions already available in literature and a study of coextrusion in rectangular dies is performed to identify the dependence of encapsulation on the flow parameters

In this study, dynamic crack propagation in brittle materials has been studied using a regularized phase field approach.The phase field model used has been validated using specific experimental results of a dynamic in-plane fracture. The crack branching phenomena and existence of a limiting crack tip velocity has been validated using a mode I simulation set-up. A parametric study has also been performed so as to normalize the various numerical parameters that affect the velocity at the crack tip. Following the validation of the phase field model a stochastic analysis of a PBX microstructure has been performed. The microstructure has a high HMX volume fraction of 79\%. The energetic material is HMX and the binder used is Sylgard. Artificial defects are introduced in the system using phase field cracks. The analysis uses a finite element framework that accounts for various thermal-mechanical processes like deformation, heat generation, conduction, fracture and frictional heating at the crack surfaces. The effect on the temperature and damage field due to varying parameters like loading velocities and critical energy release rates is studied. Critical hotspot formation due to localized frictional heating is also studied. A concept of dirty binder is introduced to increase the grain volume fraction of the energetic in the composite. This amounts to a homogenized binder that accounts for the influence of the subsume particles that do not contribute to fracture but affect material properties of the binder.

<div>A physics-based, multiscale framework is presented to describe the degradation in rechargeable lithium-ion batteries. The framework goes beyond traditional (empirical) coulomb-counting approaches and enables the identification of different degradation regimes of behavior. Macroscopically, five degradation mechanisms: (i) solid electrolyte interphase (SEI) growth on the anode, (ii) electrolyte oxidation on the cathode, (iii) anode active material loss and (iv) cathode active material loss due to chemomechanical fracture, and (v) dendrite growth were identified and modeled. Great emphasis was placed on describing the physics of lithium dendrite growth in order to demonstrate five distinguishable regimes: thermodynamic suppression regime, incubation regime, tip-controlled growth regime, base-controlled growth regime, and mixed growth regime. Mesoscopically, three local dendrite growth mechanisms are identified: 1) electrochemical shielding, where there is practically no electrodeposition/electrodissolution, 2) stress-induced electrodissolution and electrodeposition on those interfaces directly facing each other, generating a self-sustained overpotential that pushes the dendrites towards the counter electrode, and 3) lateral plastic extrusion in those side branches experiencing non-hydrostatic stresses. Overall, the experimentally validated theoretical framework allows to fundamentally understand battery degradation and sets the stage to design high energy density and fast charging rechargeable batteries. </div><div><br></div>

Este trabalho apresenta um estudo da solidificação de metais puros utilizando o modelo de campo de fases. O modelo é utilizado para simular a solidificação com o intuito de obter a morfologia da interface sólido-líquido sob diversas condições de transferência de calor. Foram realizados testes de validação comparando as morfologias da interface sólido-líquido obtida com as morfologias apresentadas em trabalhos anteriores para os casos bi e tridimensionais. O modelo do campo de fases adotado consiste principalmente de duas equações diferenciais: uma para calcular a variável de campo de fases e outra para calcular o campo de temperaturas. As equações foram solucionadas numericamente para um oitavo do domínio devido a simetria do problema. Os cálculos do modelo indicam que um sólido esférico com um raio inicial menor que o raio crítico de nucleação refunde. Entretanto uma esfera de raio maior cresce. Quando o sólido inicial cresce em uma malha numérica relativamente grosseira, a forma do sólido desvia da forma esférica devido perturbações na interface sólido-líquido. Quando a malha é refinada, as perturbações não são detectadas; contudo, quando introduzidas artificialmente as perturbações crescem e distorcem o formato esférico. / This work presents a study of the solidification of pure metals using the phase field model. The model is used to simulate solidification in order to obtain the morphology of the solid-liquid interface under different heat transfer conditions. Validation tests were performed comparing the morphology of the solid-liquid interface with the morphologies obtained from previous works for two and three dimensional cases. The adopted phase-field model consisted mainly of two differential equations: one to calculate the field of phase variable and another for the temperature field. The equations were solved numerically in only one eighth of the domain owing to the symmetry of the problem. Model calculations show that a solid sphere with an initial radius smaller than the critical radius for nucleation shrinks, whereas a sphere with a larger radius grows. When it grows in a relatively coarse numerical mesh, the initial solid shape deviates from a sphere owing to perturbations at the solid-liquid interface. When the numerical mesh is refined, the growth of perturbations is not detected, but artificially introduced perturbations grow and distort the spherical shape.

This thesis presents numerical techniques for solving problems of incompressible flow coupled to scalar transport equations using finite element discretizations in space. The two applications considered in this thesis are multi-phase flow, modeled by level set or phase field methods, and planetary mantle convection based on the Boussinesq approximation. A systematic numerical study of approximation errors in evaluating the surface tension in finite element models for two-phase flow is presented. Forces constructed from a gradient in the same discrete function space as used for the pressure are shown to give the best performance. Moreover, two approaches for introducing contact line dynamics into level set methods are proposed. Firstly, a multiscale approach extracts a slip velocity from a micro simulation based on the phase field method and imposes it as a boundary condition in the macro model. This multiscale method is shown to provide an efficient model for the simulation of contact-line driven flow. The second approach combines a level set method based on a smoothed color function with a the phase field method in different parts of the domain. Away from contact lines, the additional information in phase field models is not necessary and it is disabled from the equations by a switch function. An in-depth convergence study is performed in order to quantify the benefits from this combination. Also, the resulting hybrid method is shown to satisfy an a priori energy estimate. For the simulation of mantle convection, an implementation framework based on modern finite element and solver packages is presented. The framework is capable of running on today's large computing clusters with thousands of processors. All parts in the solution chain, from mesh adaptation over assembly to the solution of linear systems, are done in a fully distributed way. These tools are used for a parallel solver that combines higher order time and space discretizations. For treating the convection-dominated temperature equation, an advanced stabilization technique based on an artificial viscosity is used. For more efficient evaluation of finite element operators in iterative methods, a matrix-free implementation built on cell-based quadrature is proposed. We obtain remarkable speedups over sparse matrix-vector products for many finite elements which are of practical interest. Our approach is particularly efficient for systems of differential equations.

This work is a thermodynamic and kinetic study of the Fe-Cr-Ni system as the core of stainless steels. The Fe-Cr, Fe-Ni and Cr-Ni systems were studied intensively using both computational and experimental techniques, including CALPHAD (CALculation of PHAse Diagrams), phase field simulation, ab initio modeling, calorimetry, and atom probe tomography. The purpose of this thesis is to reveal the complexity of the phase transformations in the Fe-Cr-Ni system via the integrated techniques. Due to the importance of the binary Fe-Cr system, it was fully reassessed using the CALPHAD technique by incorporating an updated description of the lattice stability for Fe down to zero kelvin. The improved thermodynamic description was later adopted in a phase field simulation for studying the spinodal decomposition in a series of Fe-Cr binary alloys. Using atom probe tomography and phase field simulation, a new approach to analyze the composition amplitude of the spinodal decomposition was proposed by constructing an amplitude density spectrum. The magnetic phase diagram of the Fe-Ni system was reconstructed according to the results from both ab initio calculations and reported experiments. Based on the Inden-Hillert-Jarl magnetic model, the thermodynamic reassessment of the Fe-Ni system demonstrated the importance of magnetism in thermodynamic and kinetic investigations. Following this, the current magnetic model adopted in the CALPHAD community was further improved. Case studies were performed showing the advantages of the improved magnetic model. Additionally, the phase equilibria of the Fe-Cr-Ni ternary were discussed briefly showing the need of thermodynamic and kinetic studies at low temperatures. The “low temperature CALPHAD” concept was proposed and elucidated in this work showing the importance of low temperature thermodynamics and kinetics for designing the new generation of stainless steels. / <p>QC 20120612</p> / Hero-m