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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.

We examine the Rayleigh–Bénard convection as modelled by the Boussinesq equation. Our aim is at deriving bounds for the heat enhancement factor in the vertical direction, the Nusselt number, which reproduce physical scalings. In the first part of the dissertation, we examine the the simpler model when the acceleration of the fluid is neglected (Pr=∞) and prove the non-optimality of the temperature background field method by showing a lower bound for the Nusselt number associated to it. In the second part we consider the full model (Pr<∞) and we prove a new upper bound which improve the existing ones (for large Pr numbers) and catches a transition at Pr~Ra^(1/3).

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 study focuses ontransient heat flow behavior in which centralizing energy on themetal makes metal surface come to aheat molten state with centralized heat source . This flow field is two-dimensional transient model, using Phase-field method and Two-phase flow to simulatemetal surface. In this study is under considerations of the mass conservation equation, momentum equation, energy equation and the level-set equation, regardless of the impact due to the concentration diffusion. At last it will show the flow of the molten zone caused by temperature, and the flows in molten zone forced by thermocapillary which is caused byvariation of temperature.

Steadily increasing heat dissipation in electronic devices has generated renewed interest in direct immersion cooling. The ideal heat transfer fluid for direct immersion cooling applications should be chemically and thermally stable, and compatible with the electronic components. These constraints have led to the use of Novec fluids and fluroinerts as coolants. Although these fluids are chemically stable and have low dielectric constants, they are plagued by poor thermal properties. These factors necessitate the development of new heat transfer fluids with improved heat transfer properties and applicability. Computer Aided Molecular Design (CAMD) approach was used in this work to systematically design novel heat transfer fluids that exhibit significantly better properties than those of current high performance electronic coolants. The candidate fluids generated by CAMD were constrained by limiting their boiling points, latent heat of vaporization and thermal conductivity. The selected candidates were further screened using a figure of merit (FOM) analysis. Some of the fluids/additives that have been identified after the FOM analysis include C₄H₅F₃O, C₄H₄F₆O, C₆H₁₁F₃, C₄ H₁₂O₂Si, methanol, and ethoxybutane. The heat transfer performance of these new fluids/fluid mixtures was analyzed through pool boiling and flow boiling experiments. All the fluid mixtures tested showed an improvement in the critical heat flux (CHF) when compared to the base fluid (HFE 7200). A pool boiling model was developed using the phase field method available in COMSOL. Although these simulations are computationally expensive, they provide an alternate solution to evaluate several candidate fluids generated using the CAMD approach.

Dans ce travail de thèse, nous nous sommes intéressés à la simulation de l’évolution de la microstructure d’un polymère. Plus précisément, nous avons étudié le changement de la morphologie du polyéthylène téréphthalate (PET) sous l’effet de différents mécanismes. Ces simulations sont réalisées par la méthode des champs de phase. Il s’agit d’une méthode basée sur l’équation de Cahn-Hilliard ou l’équation de Ginzburg-Landau. Elle utilise un paramètre d’ordre pour décrire l’état du matériau, des variables thermodynamiques et cinématiques. Ainsi on peut décrire l’évolution d’une microstructure sans suivre l’interface et ainsi reproduire l’évolution de la structure cristalline sphérolitique qui apparait lors d’une cristallisation induite par la température. Dans le cadre d’un changement de morphologie induit par la température, le calcul par champ de phase a été simulé par la méthode de différences finies et la méthode d’éléments finis. Le coefficient cinétique a été identifié à partir de données expérimentales de la littérature. En introduisant un modèle du champ de phases multiples (the MPF model) on a aussi simulé l’évolution de plusieurs sphérolites et gérer la jonction lorsque deux sphérolites se rencontrent. La croissance et la jonction des sphérolite a été modélisée par la méthode d’éléments finis : elle reproduit parfaitement l’évolution expérimentale de cristallisation isotherme d’un polymère. En comparant ces résultats avec le modèle macroscopique d’Avrami, une évaluation de la constante d'Avrami, K(T), a été discutée en fonction des fluctuations des conditions initiales (positions et taille des germes).Dans le cadre de la cristallisation induite par la déformation mécanique, nous avons couplé le champ de phase aux équations de la mécanique pour un comportement viscoélastique différent pour chaque phase. L’influence, sur la cristallisation et l’orientation, de la déformation, de la vitesse de sollicitation, du contraste entre les phases sont étudiées et comparées qualitativement aux observations expérimentales. Il s’agit d’une étude préliminaire qui devra être poursuivie et affinée afin de prédire une morphologie plus réaliste / In this thesis work, we are interested in simulating the evolution of the microstructure of a polymer. In particular, we have studied in the morphology change of polyethylene terephthalate (PET) under different mechanisms. These simulations carried out by the phase field simulation. This method based on the Cahn-Hilliard equation or the Ginzburg-Landau equation. It uses an order parameter to describe the state of material, thermodynamic and kinetic variables. Thus we can describe the microstructure evolution without tracking the interface (which would require complex remeshing) and reproduce the evolution of the crystalline structure within the polymers, for example the growth of spherulites which appear during crystallization induced by temperature. Within the scope the morphology changing by the temperature, the evolution of phase field simulation is performed by the finite difference method and the finite element method. The kinetic coefficient is adjusted in order to fit the experiment data in of the literature. We introduce the multiphase field model (the MPF model) in order to simulate the evolution of several spherulites and to describe the junction of spherulites. The growth and junction of spherulites have been modeled by the finite element method and nicely reproduced in comparing the experimental evolution of isothermal crystallization of a polymer. Comparing these results with the Avrami macroscopic model, an evaluation of the Avrami constant, K (T), was discussed according to the fluctuations of the initial conditions (positions and size of the germs).In the following part, we study the crystallization induced by mechanical deformation. We are interested in the viscoelastic model to simulate the induced crystallization of PET in plane stress. The phase field model coupled to mechanics will be presented. Different viscoelastic behaviors have been considered for each phase. The influence on crystallization and orientation of the deformation, the stress velocity and the contrast between the phases are studied and compared qualitatively with the experimental observations. This is a preliminary study that will have to be continued in order to predict a more realistic morphology

In this thesis, a robust, reliable and efficient isogeometric analysis framework is presented that allows for an adaptive spatial discretization of non-linear and time-dependent multi-field problems. In detail, B\'ezier extraction of truncated hierarchical B-splines is proposed that allows for a strict element viewpoint, and in this way, for the application of standard finite element procedures. Furthermore, local mesh refinement and coarsening strategies are introduced to generate graded meshes that meet given minimum quality requirements. The different strategies are classified in two groups and compared in the adaptive isogeometric analysis of two- and three-dimensional, singular and non-singular problems of elasticity and the Poisson equation. Since a large class of boundary value problems is non-linear or time-dependent in nature and requires incremental solution schemes, projection and transfer operators are needed to transfer all state variables to the new locally refined or coarsened mesh. For field variables, two novel projection methods are proposed and compared to existing global and semi-local versions. For internal variables, two different transfer operators are discussed and compared in numerical examples. The developed analysis framework is than combined with the phase-field method. Numerous phase-field models are discussed including the simulation of structural evolution processes to verify the stability and efficiency of the whole adaptive framework and to compare the projection and transfer operators for the state variables. Furthermore, the phase-field method is used to develop an unified modelling approach for weak and strong discontinuities in solid mechanics as they arise in the numerical analysis of heterogeneous materials due to rapidly changing mechanical properties at material interfaces or due to propagation of cracks if a specific failure load is exceeded. To avoid the time consuming mesh generation, a diffuse representation of the material interface is proposed by introducing a static phase-field. The material in the resulting transition region is recomputed by a homogenization of the adjacent material parameters. The extension of this approach by a phase-field model for crack propagation that also accounts for interface failure allows for the computation of brittle fracture in heterogeneous materials using non-conforming meshes.

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