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This textbook provides a fast-track pathway to numerical implementation of phase-field modeling—a relatively new paradigm that has become the method of choice for modeling and simulation of microstructure evolution in materials. It serves as a cookbook for the phase-field method by presenting a collection of codes that act as foundations and templates for developing other models with more complexity. Programming Phase-Field Modeling uses the Matlab/Octave programming package, simpler and more compact than other high-level programming languages, providing ease of use to the widest audience. Particular attention is devoted to the computational efficiency and clarity during development of the codes, which allows the reader to easily make the connection between the mathematical formulism and the numerical implementation of phase-field models. The background materials provided in each case study also provide a forum for undergraduate level modeling-simulations courses as part of their curriculum.

### 1. An Overview of the Phase-Field Method and Its Formalisms

Abstract
A microstructure can be described as the spatial arrangement of the phases and possible defects that have different compositional and/or structural character (for example, the regions composed of different crystal structures and having different chemical compositions, grains of different orientations, domains of different structural variants, and domains of different electric or magnetic polarizations). The size, shape, volume fraction, and spatial arrangement of these microstructural features determine the overall properties of any type of multiphase and/or multicomponent materials.
S. Bulent Biner

### 2. Introduction to Numerical Solution of Partial Differential Equations

Abstract
Many of the fundamental theories of physics and engineering, including the phase-field models, are expressed by means of systems of partial differential equations, PDEs. A PDE is an equation which contains partial derivatives, such as
$$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial {x}^2}$$
in which u is regarded as function of length x and time t. There is no real unified theory for PDEs. They exhibit their own characteristics to express the underlying physical phenomena as accurately as possible. Since PDEs can be hardly solved analytically, their solutions relay on the numerical approaches. A brief of summary of the numerical techniques involving their spatial and temporal discretization is given below. These techniques will be applied to solving the equations of the various phase-field models throughout the book and their detailed descriptions and implementations are given in relevant chapters. There are numerous textbooks also available on the subjects, of which some of them are listed in the references.
S. Bulent Biner

### Chapter 3. Preliminaries About the Codes

Abstract
The Matlab/Octave programming language was chosen for the codes presented in the book. In the development of the codes, it is assumed that the reader has some degree of experience in computer programming. There are differences in syntax between Matlab/Octave programming and the other traditional programming languages such as Fortran, C, and C++. The readers who are not familiar with Matlab/Octave programming may find useful to consult the extensive documentations that are provided at their websites. In addition, there are plenty of books and online tutorials available, of which some of them are listed in reference section.
S. Bulent Biner

### Chapter 4. Solving Phase-Field Models with Finite Difference Algorithms

Abstract
Finite difference algorithms offer a more direct approach to the numerical solution of partial differential equations than any other method. Finite difference algorithms are based on the replacement of each derivative by a difference quotient. Finite difference algorithms are simple to code, economic to compute, and easy to parallelize for the distributed computing environments. However, they also have disadvantages in terms of accuracy and imposing complex boundary conditions. For better understanding of the method, the solution of one-dimensional transient heat conduction is given as an example together with the source code in this section.
S. Bulent Biner

### 5. Solving Phase-Field Models with Fourier Spectral Methods

Abstract
As described in Chap. 2, finite difference and finite element methods have local character and the unknown functions are interpreted by usually low-order polynomials over small sub-domains. In contrast, spectral methods make use of global representation, usually with high-order polynomials or Fourier series. The rate of convergence of spectral approximations depends only on the smoothness of the solution. They achieve much higher accuracy with much smaller number of sampling points in comparison to other two methods. This fact is known in the literature as “spectral accuracy.” The spectral methods most often are successful with domains in periodic nature, which is the case in most of the phase-field modeling simulations. Again, the application of the Fourier spectral method will be demonstrated to the solution of one-dimensional transient heat conduction in this section. This source code, solving this simple problem given below, forms the foundation of the algorithms that will be developed in this chapter.
S. Bulent Biner

### 6. Solving Phase-Field Equations with Finite Elements

Abstract
The finite element method, FEM, and sometimes also called finite element analysis, FEA, was originally developed in the aircraft industry in 1960s [1, 2]. Therefore, it is a very mature algorithm and widely used in engineering and science as a general numerical approach for the solution of PDEs subject to known boundary and initial conditions. The use of piecewise continuous functions over subregions of domain to approximate the unknown function was first introduced by Courant . This approach was later formalized [4, 5] and term finite elements for these subregions was introduced by Clough . Therefore, similar to finite difference technique the FEM is also local in nature. However, FEM has superior and unique characteristics to describe very complex geometries and boundaries of domains. There are plenty of textbook and online materials covering both theoretical and practical aspects of FEM or FEA and some of them are listed in the reference section.
S. Bulent Biner

### 7. Phase-Field Crystal Modeling of Material Behavior

Abstract
The phase-field crystal, PFC, method introduced by Elder and coworkers [1–3], can be viewed as multiscale simulation algorithm that bridges the classical molecular dynamics, MD, simulations and the phase-field methods covered in the previous chapters. PFC method introduces an order parameter defined as the local-time-averaged atomic number density which is able to produce periodicity of crystal lattices. In the model, any perturbation or lattice defects result in an increase in the free energy, thus enabling to obtain the information which has been only possible by the atomistic simulations previously. In addition, PFC method produces various atomistic events in much larger spatial and temporal dimensions that are not easily accessible with current MD simulation techniques. Therefore, PFC method has emerged as an attractive simulation approach.
S. Bulent Biner

### 8. Concluding Remarks

Abstract
Any numerical algorithm can be characterized in terms of its properties, which are accuracy, flexibility to handle many different problems, robustness, and computational efficiency in terms of amount of required coding and computational resources memory, storage and processor. Very often, it is difficult to achieve all of these properties in one particular algorithm. The aim of this book was that the reader, after hands on experience, can understand the fine details and strengths and weaknesses of each given solution’s methods sufficiently well and can adopt the suitable algorithm needed for their phase-field models. It is hoped that this objective was achieved by facilitating such comparative study with a collection of algorithms and case studies.
S. Bulent Biner

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S. Bulent Biner