■ Abstract The phase-field method has recently emerged as a powerful computational approach to modeling and predicting mesoscale morphological and microstructure evolution in materials. It describes a microstructure using a set of conserved and nonconserved field variables that are continuous across the interfacial regions. The temporal and spatial evolution of the field variables is governed by the Cahn-Hilliard nonlinear diffusion equation and the Allen-Cahn relaxation equation. With the fundamental thermodynamic and kinetic information as the input, the phase-field method is able to predict the evolution of arbitrary morphologies and complex microstructures without explicitly tracking the positions of interfaces. This paper briefly reviews the recent advances in developing phase-field models for various materials processes including solidification, solid-state structural phase transformations, grain growth and coarsening, domain evolution in thin films, pattern formation on surfaces, dislocation microstructures, crack propagation, and electromigration.

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Phase-Field Models for Microstructure Evolution

Isogeometric analysis of the Cahn–Hilliard phase-field model

In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.

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Finite element methods for surface PDEs

In this paper, we review the recent development of phase-field models and their numerical methods for multi-component fluid flows with interfacial phenomena. The models consist of a Navier-Stokes system coupled with a multi-component Cahn-Hilliard system through a phase-field dependent surface tension force, variable density and viscosity, and the advection term. The classical infinitely thin boundary of separation between two immiscible fluids is replaced by a transition region of a small but finite width, across which the composition of the mixture changes continuously. A constant level set of the phase-field is used to capture the interface between two immiscible fluids. Phase-field methods are capable of computing topological changes such as splitting and merging, and thus have been applied successfully to multi-component fluid flows involving large interface deformations. Practical applications are provided to illustrate the usefulness of using a phase-field method. Computational results of various experiments show the accuracy and effectiveness of phase-field models.

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Phase-Field Models for Multi-Component Fluid Flows

This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

Computation of geometric partial differential equations and mean curvature flow

A mathematical analysis is carried out for the Cahn–Hilliard equation where the free energy takes the form of a double well potential function with infinite walls. Existence and uniqueness are proved for a weak formulation of the problem which possesses a Lyapunov functional. Regularity results are presented for the weak formulation, and consideration is given to the asymptotic behaviour as the time becomes infinite. An investigation of the associated stationary problem is undertaken proving the existence of a nontrivial stationary solution and further regularity results for any stationary solution. Stationary solutions are constructed in one and two dimensions; a formula for the number of stationary solutions in one dimension is derived. It is then natural to study the asymptotic behaviour as the phenomenological parameter λ→0, the main result being that the interface between the two phases has minimal area.

The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis

We consider a fully practical finite element approximation of the Cahn--Hilliard equation with degenerate mobility 
$$ \textstyle \frac{\partial u}{\partial t}= \del .(\,b(u)\, \del (-\gamma\lap u+\Psi'(u))) , $$ where $b(\cdot)\geq 0$ is a diffusional mobility and $\Psi(\cdot)$ is a homogeneous free energy. In addition to showing well posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore, an iterative scheme for solving the resulting nonlinear discrete system is analyzed. We also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy. Finally, some numerical experiments are presented.

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Finite Element Approximation of the Cahn--Hilliard Equation with Degenerate Mobility

We consider the Cahn-Hilliard equation with a logarithmic free energy and non-degenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally some numerical experiments are presented.

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Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility

The use of parabolic double obstacles problems for approximating curvature dependent phase boundary motion is reviewed. It is shown that such problems arise naturally in multi-component diffusion with capillarity. Formal matched asymptotic expansions are employed to show that phase field models with order parameter solving an obstacle problem approximate curvature dependent phase boundary motion. Numerical simulations of surfaces evolving according to their mean curvature are presented.

Curvature Dependent Phase Boundary Motion and Parabolic Double Obstacle Problems

We consider a fully discrete finite element approximation of the nonlinear cross-diffusion population model: Find ui, the population of the ith species, i=1 and 2, such that * where j≠i and gi(u1,u2):=(μi−γii ui−γij uj) ui. In the above, the given data is as follows: v is an environmental potential, ci ∈ ℝ, ai ∈ ℝ are diffusion coefficients, bi ∈ ℝ are transport coefficients, μi ∈ ℝ are the intrinsic growth rates, and γii ∈ ℝ are intra-specific, whereas γij, i≠j,  ∈ ℝ are interspecific competition coefficients. In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d≤3. Finally some numerical experiments in one space dimension are presented.

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James F. Blowey

Papers

The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis

Finite Element Approximation of the Cahn--Hilliard Equation with Degenerate Mobility

Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility

Curvature Dependent Phase Boundary Motion and Parabolic Double Obstacle Problems

Finite element approximation of a nonlinear cross-diffusion population model