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We present the first model ever, that describes explicitly ice crystal growth in a sugar solution during freezing. This 2-D model uses the phase field method, supplemented with realistic, and predictive theories on the thermodynamics and (diffusion) kinetics of this food system. We have to make use of a novel type of phase field to obtain realistic, micron-sized ice crystals, and exclusion of sugar from the crystalline phase. Via simulation of a single ice crystal, we identify important time scales governing the growth. These times scales are also important for the coarsening of the ice morphology in freezing systems with multiple ice crystals. These simulations show that the average ice crystal size is governed by the freezing rate via a power law, similar to an empirical relation from literatures, which is deduced from experiment. The presented model is viewed as a good basis for even more realistic simulations of crystal growth in food.

Spinodal decomposition was studied during aging of Fe-Cr alloys by means of the numerical solution of the linear and nonlinear Cahn-Hilliard differential partial equations using the explicit finite difference method. Results of the numerical simulation permitted to describe appropriately the mechanism, morphology and kinetics of phase decomposition during the isothermal aging of these alloys. The growth kinetics of phase decomposition was observed to occur very slowly during the early stages of aging and it increased considerably as the aging progressed. The nonlinear equation was observed to be more suitable for describing the early stages of spinodal decomposition than the linear one. La descomposición espinodal se estudió durante el envejecido de aleaciones Fe-Cr mediante la solución numérica de las ecuaciones diferenciales parciales lineal y no linear de Cahn y Hilliard usando el método de diferencias finito explícito. Los resultados de la simulación numérica permitieron describir apropiadamente el mecanismo, morfología y cinética de la descomposición de fases durante el envejecido isotérmico de estas aleaciones. La cinética de crecimiento de la descomposición de fases ocurrió muy lentamente durante las primeras etapas de envejecido, y se incrementó considerablemente con el tiempo de envejecido. La ecuación no lineal parece ser más apropiada para describir las primeras etapas de la descomposición espinodal que la ecuación lineal.

The Mg–Zn–Y alloys show a good mechanical strength which can be achieved with the precipitation hardening by intermediate phases (X, W and I phase) in Mg solid solution (α phase). However, an accurate control of the microstructure formation is required in order to obtain good mechanical properties. In this study, experimental observations of microstructures of the Mg–Zn–Y system have been performed. Then we have focused on developing CALPHAD (CALculation of PHAse Diagrams) thermodynamic database to obtain the Gibbs free energy to draw phase diagram of the system and to understand the precipitation behavior of the intermediate phases. In order to understand the formation of microstructures, we have performed simulations of solidification of the alloy with use of multi-phase-field method. At the beginning the solidification process has been calculated for a large area, then the zoomed in region of the lamellar structures of the α phase and the W phase have been analyzed. Resulting optimum lamellar spacing reproduce experimental one well.

This paper proposes a new topology optimization method, which can adjust the geometrical complexity of optimal configurations, using the level set method and incorporating a fictitious interface energy derived from the phase field method. First, a topology optimization problem is formulated based on the level set method, and the method of regularizing the optimization problem by introducing fictitious interface energy is explained. Next, the reaction–diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed, which uses the finite element method to solve the equilibrium equations and the reaction–diffusion equation when updating the level set function. Finally, several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method.

The boundary heat flow has important significance for the microstructures of directional solidified binary alloy. Interface evolution of the directional solidified microstructure with different boundary heat flow was discussed. In this study, only one interface was allowed to have heat flow, and Neumann boundary conditions were imposed at the other three interfaces. From the calculated results, it was found that different boundary heat flows will result in different microstructures. When the boundary heat flow equals to 20 W·cm-2, the growth of longitudinal side branches is accelerated and the growth of transverse side branches is restrained, and meanwhile, there is dendritic remelting in the calculation domain. When the boundary heat flow equals to 40 W·cm-2, the growths of the transverse and longitudinal side branches compete with each other, and when the boundary heat flow equals to 100-200 W·cm-2, the growth of transverse side branches dominates absolutely. The temperature field of dendritic growth was analyzed and the relation between boundary heat flow and temperature field was also investigated.

This paper discusses a structural optimization method that optimizes shape and topology based on the phase field method. The proposed method has the same functional capabilities as a structural optimization method based on the level set method incorporating perimeter control functions. The advantage of the method is the simplicity of computation, since extra operations such as re-initialization of functions are not required. Structural shapes are represented by the phase field function defined in the design domain, and optimization of this function is performed by solving a time-dependent reaction diffusion equation. The artificial double well potential function used in the equation is derived from sensitivity analysis. The proposed method is applied to two-dimensional linear elastic and vibration optimization problems such as the minimum compliance problem, a compliant mechanism design problem and the eigenfrequency maximization problem. The numerical examples provided illustrate the convergence of the various objective functions and the effect that perimeter control has on the optimal configurations.

application/pdf, Article, 日本機械学会論文集C編 . 2009, 75 (758), p.2633-2641

We discuss in this video phase-field approximations of the Willmore functional and the associated $${\mathrm L}^{2}$$L2-flow. After recollecting known results on the approximation of the Willmore energy and its $${\mathrm L}^{1}$$L1 relaxation, we derive the expression of the flows associated with various approximations, and we show their behavior by formal arguments based on matched asymptotic expansions. We introduce an accurate numerical scheme, whose local convergence is proved, to describe with more details the behavior of two flows, the classical and the flow associated with an approximation model due to Mugnai. We propose a series of numerical simulations in 2D and 3D to illustrate their behavior in both smooth and singular situations.

Materials Science Forum, Volume 475-479, Phase Field Method (PFM) is hybridized with Cluster Variation Method (CVM) to investigate the ordering dynamics of L10-disorder transition at atomistic and microstructural scales simultaneously. For this, coarse graining operation is attempted on the inhomogeneous free energy functional of CVM. The resultant gradient energy coefficient is found out to be dependent on temperature and order parameters, which is in marked contrast to a conventional PFM formalism. Electronic structure total energy calculations for Fe-Pd system are incorporated to the hybridized scheme and the first principles calculation of microstructural evolution process is attempted.

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 thesis focuses on phase-field methods for the approximation of motion by mean curvature. In particular, we propose new models to take into account some constraints of volume, anisotropic surface tension and contact angle at the boundaries of the evolution area.; Cette thèse s'intéresse aux méthodes de champs de phase pour l'approximation de mouvements par courbure moyenne. En particulier, nous proposons de nouveaux modèles pour prendre en compte des contraintes de volumes, des tensions de surfaces anisotropes et des angles de contact au bord du domaine d'évolution.

This thesis focuses on phase-field methods for the approximation of motion by mean curvature. In particular, we propose new models to take into account some constraints of volume, anisotropic surface tension and contact angle at the boundaries of the evolution area., Cette thèse s'intéresse aux méthodes de champs de phase pour l'approximation de mouvements par courbure moyenne. En particulier, nous proposons de nouveaux modèles pour prendre en compte des contraintes de volumes, des tensions de surfaces anisotropes et des angles de contact au bord du domaine d'évolution.

The estimation of optical flow fields from image sequences is incorporated in a Mumford-Shah approach for image denoising and edge detection. Possibly noisy image sequences are considered as input and a piecewise smooth image intensity, a piecewise smooth motion field, and a joint discontinuity set are obtained as minimizers of the functional. The method simultaneously detects image edges and motion field discontinuities in a rigorous and robust way. It is able to handle information on motion that is concentrated on edges. Inherent to it is a natural multiscale approximation that is closely related to the phase. eld approximation for edge detection by Ambrosio and Tortorelli. We present an implementation for two-dimensional image sequences with finite elements in space and time. This leads to three linear systems of equations, which have to be solved in a suitable iterative minimization procedure. Numerical results and different applications underline the robustness of the approach presented.

In this paper we present a critical comparison of the suitability of several numerical methods, level set, moving grid and phase eld model, to address two well known Stefan problems in phase transformation studies: melting of a pure phase and diusional solid state phase transformations in a binary system. Similarity solutions are applied to verify the numerical results. The comparison shows that the type of phase transformation considered determines the convenience of the numerical techniques. Finally, it is shown both numerically and analytically that the solid-solid phase transformation is a limiting case of the solid-liquid transformation.

In this paper we present a critical comparison of the suitability of several numerical methods, level set, moving grid and phase eld model, to address two well known Stefan problems in phase transformation studies: melting of a pure phase and diusional solid state phase transformations in a binary system. Similarity solutions are applied to verify the numerical results. The comparison shows that the type of phase transformation considered determines the convenience of the numerical techniques. Finally, it is shown both numerically and analytically that the solid-solid phase transformation is a limiting case of the solid-liquid transformation.

Al-Zn-Si alloy coatings are widely used for the corrosion protection of steel sheets. In addition to the favorable corrosion properties, the Al-43wt%Zn-1.6wt%Si coatings present several features which are of metallurgical interest. The first one is their surface appearance dominated by the typical spangle morphology exhibiting dull and shiny areas on the coating surface. Further on, the dendrite tips do apparently not grow along the 〈100〉 crystallographic directions that are typical for fcc-metals, but rather along directions in between 〈100〉 and 〈110〉. In addition, continuous variations of the crystallographic orientation, up to 35°, are observed within individual grains of Al-Zn-Si coatings [Sémoroz1 01]. The objective of the present study is to establish a better understanding of the microstructure development which is responsible for the formation of both the characteristic spangle pattern and the important variations of crystallographic orientation within the grains. In order to elucidate these questions, the present study includes three main axes, (i) a detailed microstructure characterization of industrially solidified samples, (ii) modeling work which encompasses microstructure modeling by the phase field method and a geometrical model, as well as the determination of the solid-liquid interfacial energy anisotropy by an inverse method, and (iii) re-solidification experiments aimed at studying the behavior of Al-Zn-Si layers under modified solidification conditions. The results show that the dendrite network spreads quickly in the coating layer at a temperature between 530 to 535°C. During growth, the dendrite tips are separated from the confining boundaries by a thin, solute-rich liquid film. It was found that the preferred dendrite growth directions are in between 〈100〉 and 〈110〉, 28.5° from 〈100〉. Further on, a mathematical expression for the interfacial energy anisotropy of the considered alloy has been determined. The combination of the geometrical model and surface topography measurements allowed concluding that the spangle pattern is due to preferential dendrite growth along one of the two boundaries confining the melt layer. In addition, the new experimental evidence forced to discard the mechanisms previously proposed for the formation of intragranular crystallographic misorientations. The experimental findings acquired during this study indicate that the solidification shrinkage occurring in the area of the grain envelope is the driving force for the formation of the observed intragranular misorientations. The solidification shrinkage leads to the development of tensile stresses in the oxide film covering the coating while it solidifies. These stresses apply on the dendrite network and lead to plastic deformation in the tip area of the growing dendrite arms.