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The study uses the Phase-field method to simulate the bubble behavior of liquid-solid interface in the solidification.The flow use the mass equation,momentum equation,and energy equation for simulating the variation of bubble. This pattern considers that three-phase of solid, liquid,and gas coexist with the different density and viscosity coefficient,and the external force considers surface tension and the gravity force. In addition,the mass transfer also can¡¦t neglect around interface. The result discuss the production of pore when the velocity of solidification is fast,but bubble leaves when the velocity of solidification is slow.

No description available.

Ferroelectric ceramics are broadly used in applications including actuators, sensors and information storage. An understanding of the microstructual evolution and domain dynamics is vital for predicting the performance and reliability of such devices. The underlying mechanism responsible for ferroelectric constitutive response is ferroelectric domain wall motion, domain switching and the interactions of domain walls with other material defects. In this work, a combined theoretical and numerical modeling framework is developed to investigate the nucleation and growth of domains in a single crystal of ferroelectric material. The phase-field approach, applying the material electrical polarization as the order parameter, is used as the theoretical modeling framework to allow for a detailed accounting of the electromechanical processes. The finite element method is used for the numerical solution technique. In order to obtain a better understanding of the energetics of fracture within the phase-field setting, the J-integral is modified to include the energies associated with the order parameter. Also, the J- integral is applied to determine the crack-tip energy release rate for common sets of electromechanical crack-face boundary conditions. The calculations confirm that only true equilibrium states exhibit path-independence of J, and that domain structures near crack tips may be responsible for allowing positive energy release rate during purely electrical loading. The small deformation assumption is prevalent in the phase-field modeling approach, and is used in the previously described calculations. The analysis of large deformations will introduce the concept of Maxwell stresses, which are assumed to be higher order effects that can be neglected in the small deformation theory. However, in order to investigate the material response of soft dielectric elastomers undergoing large mechanical deformation and electric field, which are employed in electrically driven actuator devices, manipulators and energy harvesters, a finite deformation theory is incorporated in the phase-field model. To describe the material free energy, compressible Neo-Hookean and Gent models are used. The Jaumann rate of the polarization is used as the objective polarization rate to make the description of the dissipation frame indifferent. To illustrate the theory, electromechanical instabilities in composite materials with different inclusions will be studied using the finite element methods. / text

Phase-field models are emerging as a promising strategy to simulate interfacial phenomena. Rather than tracking interfaces explicitly as done in sharp interface descriptions, these models use a diffuse order parameter to monitor interfaces implicitly. This implicit description, as well as solid physical and mathematical footings, allow phase-field models to overcome problems found by predecessors. Nonetheless, the method has significant drawbacks. The phase-field framework relies on the solution of high-order, nonlinear partial differential equations. Solving these equations entails a considerable computational cost, so finding efficient strategies to handle them is important. Also, standard discretization strategies can many times lead to incorrect solutions. This happens because, for numerical solutions to phase-field equations to be valid, physical conditions such as mass conservation and free energy monotonicity need to be guaranteed. In this work, we focus on the development of thermodynamically consistent algorithms for time integration of phase-field models. The first part of this thesis focuses on an energy-stable numerical strategy developed for the phase-field crystal equation. This model was put forward to model microstructure evolution. The algorithm developed conserves, guarantees energy stability and is second order accurate in time. The second part of the thesis presents two numerical schemes that generalize literature regarding energy-stable methods for conserved and non-conserved phase-field models. The time discretization strategies can conserve mass if needed, are energy-stable, and second order accurate in time. We also develop an adaptive time-stepping strategy, which can be applied to any second-order accurate scheme. This time-adaptive strategy relies on a backward approximation to give an accurate error estimator. The spatial discretization, in both parts, relies on a mixed finite element formulation and isogeometric analysis. The codes are available online and implemented in PetIGA, a high-performance isogeometric analysis framework.

No description available.

La méthode d'élimination sur le terrain est devenu ces dernières années la méthode de choix pour modéliser la formation des motifs de la microstructure lors de la solidification. Pour monocristaux, accord quantitatif avec des expériences et des solutions analytiques ont été obtenues. La modélisation des polycristaux, qui sont composées de nombreux grains d'une même phase thermodynamique, mais différentes orientations du réseau cristallin, est beaucoup moins avancée. Deux types de modèles ont été proposés: les modèles multi-phase-champ d'utiliser un champ de phase pour chaque grain, et les modèles d'orientation-champ d'utiliser un petit nombre de domaines, mais ont des termes non analytiques dans leur énergie libre fonctionnel. Ce travail examine les divers aspects de la phase de modélisation du champ de polycristaux et est divisé en trois parties. Dans la première, une nouvelle possibilité de décrire l'orientation locale est explorée, en utilisant un paramètre d'ordre tensoriel qui représente automatiquement la symétrie locale du système. Cette approche est testée en phase de développement d'un modèle de champ pour la transition de phase nématique-isotrope dans les cristaux liquides. Le modèle est appliqué pour simuler la solidification directionnelle''''d'un cristal liquide. L'effet du couplage entre l'orientation et la forme nématique interface est étudiée. Les résultats de simulation pour la stabilité d'une interface plane en bon accord avec une analyse de stabilité généralisée, qui tient compte d'une condition nouvelle d'ancrage à l'interface: l'orientation à l'interface nématique est le résultat de l'interaction entre la déformation en vrac et l'anisotropie d'interface. La forme et la stabilité des cellules bien développé est également influencée par cet effet. Numériquement, l'utilisation d'un paramètre d'ordre tensoriel simplifie le traitement des symétries dans le système de manière significative, tandis que les équations de mouvements deviennent beaucoup plus compliquées. Dans la deuxième partie, les joints de grains sont étudiés sur une échelle plus petite longueur, en utilisant un modèle de cristal phase de terrain, où les propriétés élastiques et des dislocations apparaissent naturellement. Avec ce modèle, l'ordre local dans les interfaces est examiné et la stabilité des films liquides entre deux grains solides est étudiée ci-dessous le point de fusion. Cette situation peut être décrite par un potentiel d'interaction entre les deux interfaces solide-liquide, qui est extraite numériquement. Les résultats sont comparés avec un modèle phénoménologique qui se trouve à tenir pour les joints de grains à forte inclinaison, où les dislocations se chevauchent. Pour les joints de grains à faible angle, autour de préfusion dislocation ainsi qu'une brisure de symétrie (paires de dislocations forme) est observée. En conséquence, le potentiel d'interaction devient nonmonotonous, et se compose d'une attraction à longue portée et une répulsion à courte portée. Dans la troisième partie, un nouveau modèle de phase sur le terrain est développé en utilisant une variable d'angle pour décrire l'orientation cristalline. Contrairement aux modèles déjà existants, l'énergie libre est construit sans un terme proportionnel au module du gradient du champ de l'orientation. Au lieu de cela, le gradient de la norme au carré est utilisé, mais il est couplé à la phase du champ avec une fonction de couplage singulier. Diverses simulations référence sont réalisés afin de tester le modèle. Il se trouve qu'elle présente plusieurs artefacts tels que la rotation et le mouvement du grain parasite interface, mais ces effets sont extrêmement petites, telles que le modèle donne des résultats satisfaisants que si la surfusion est très faible. Enfin, les problèmes observés sont analysés et des moyens d'obtenir une meilleure description de la dynamique de l'angle de champ sont discutées.

Diffusion controlled phase transformations are studied bymeans of the phase-field method. Morphological evolution ofdendrites, grains and Widmanst\"atten plates is modeled andsimulated. Growth of dendrites into highly supersaturated liquids ismodeled for binary alloy solidification. Phase-field equationsthat involve both temperature and solute redistribution areformulated. It is demonstrated that while at low undercoolingheat diffusion does not affect the growth of dendrites, i.e.solidification is nearly isothermal, at high cooling rates thesupersaturation is replaced by the thermal undercooling as thedriving force for growth. In experiments many crystals with different orientationsnucleate. The growth of randomly oriented dendrites, theirsubsequent impingement ant formation of grain boundaries arestudied in two dimensions using the FEM on adaptive grids. The structure of dendrites is determined by growthconditions and physical parameters of the solidifying material.Effects of the undercooling and anisotropic surface energy onthe crystal morphology are investigated. Transition betweenseaweeds, doublons and dendrites solidifying out of puresubstance is studied and compared to experimental data. Two-and three-dimensional simulations are performed in parallel onadaptive and uniform meshes. A phase-field method based on the Gibbs energy functional isformulated for ferrite to austenite phase transformation inFe-C. In combination with the solute drag model, transitionbetween diffusion controlled and massive transformations as afunction of C concentration and temperature is established byperforming a large number of one dimensional calculations withreal physical parameters. In two dimensions, growth ofWidmanstaetten plates is governed by the highly anisotropicsurface energy. It is found that the plate tip can beapproximated as sharp, in agreement with experiments. Keywords:heat and solute diffusion, solidification,solid-solid phase transformation, microstructure, crystalgrowth, dendrite, grain boundary, Widmanstaetten plate,phase-field, adaptive mesh generation, FEM. / <p>NR 20140805</p>

Operator splitting methods have been applied to nonlinear partial differential equations that involve operators of different nature. The main idea of these methods is to decompose a complex equation into simpler sub-equations, which can be solved separately. The main advantage of the operator splitting methods is that they provide a great flexibility in choosing different numerical methods, depending on the feature of each sub-problem. In this dissertation, we have developed highly accurate and efficient numerical methods for several nonlinear partial differential equations, which involve both linear and nonlinear operators. We first propose a fast explicit operator splitting method for the modified Buckley-Leverett equations which include a third-order mixed derivatives term resulting from the dynamic effects in the pressure difference between the two phases. The method splits the original equation into two equations, one with a nonlinear convective term and the other one with high-order linear terms so that appropriate numerical methods can be applied to each of the split equations: The high-order linear equation is numerically solved using a pseudo-spectral method, while the nonlinear convective equation is integrated using the Godunov-type central-upwind scheme. The spatial order of the central-upwind scheme depends on the order of the piecewise polynomial reconstruction: We test both the second-order minmod-based reconstruction and fifth-order WENO5 one to demonstrate that using higher-order spatial reconstruction leads to more accurate approximation of solutions. We then propose fast and stable explicit operator splitting methods for two phase-field models (the molecular beam epitaxy equation with slope selection and the Cahn-Hilliard equation), numerical simulations of which require long time computations. The equations are split into nonlinear and linear parts. The nonlinear part is solved using a method of lines combined with an efficient large stability domain explicit ODE solver. The linear part is solved by a pseudo-spectral method, which is based on the exact solution and thus has no stability restriction on the time step size. We have verified the numerical accuracy of the proposed methods and demonstrated their performance on extensive one- and two-dimensional numerical examples, where different solution profiles can be clearly observed and are consistent with previous analytical studies. / Zhuolin Qu

As the operating conditions of the pressurized water reactor (PWR) have been increased towards the thermal limits of the core for economics, the subcooled boiling heat transfer performance of the rod bundles under normal operating conditions has become an increasingly important design focus. Effective field models such as two-fluid model, on which most previous numerical studies in the nuclear fields have focused, cannot predict detailed phenomenon of subcooled boiling because it involves complex multiphase dynamics, such as nucleation, growth, detachment bubbles from a wall, deformation, break-up, coalescence, and condensation. It also requires numerous, additional closure relations. On the other hand, direct numerical simulations with interfacial tracking enable us to capture specific two-phase flow and do not require additional empirical closure relations. In this thesis, we simulate isothermal, two-dimensional bubble dynamics as a starting point toward direct simulation of the subcooled boiling. We adopt a lattice Boltzmann method with the phase-field model. The lattice Boltzmann method is a mesoscopic approach well-adapted to the simulation of complex fluids and is simple to implement. The phase field model can capture complex topological deformation, such as coalescence and break-up, with better numerical stability than other interfacial tracking methods like Volume of Fluid (VOF) and level set methods. We validate the present method for stationary and moving two-phase interfaces by comparing with theoretical solutions for a single static bubble in a stationary liquid and a capillary wave, respectively. In addition, the capability of the current method to simulate the coalescence of two bubbles and droplets is validated by comparing with experimental data. To see the applicability of the method to problems involving complex bubble behaviors and interactions with a high-density ratio as in subcooled boiling water, we simulate rising single and double bubbles in a viscous fluid. For a single bubble problem, the bubble shapes and terminal velocity agreed well with the experimental results for different fluid dynamic conditions. For a double bubble case, the current method can capture the interaction and dynamics of the bubbles. Thus, it is expected that this study can serve as a stepping-stone extension to convective subcooled boiling heat transfer in the nuclear reactor core.