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Solving Phase-Field Equations with Finite Elements

6.1

Introduction

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 [3]. This approach was later formalized [4, 5] and term finite elements for these subregions was introduced by Clough [6]. 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, [7–11]. We will start this section by first introducing the isoparametric representation of domain and its numerical integration. Then, the strong and weak forms of FEM formulation will be demonstrated for simple transient heat transfer problem, since all the case studies given in this chapter utilize the weak form of phase-field equations in their FEM implementation. Then, a working FEM code for linear elasticity will be

developed by using the formulism based on the principals of virtual work. This code will form the backbone for solving different phase-field models with FEM given in this chapter. Therefore, the reader is urged to review this code and associated functions, even if they are seasoned FEM programmers. There are also dedicated books, such as [12–14], just covering Matlab/ Octave implementation of FEM. It is possible to code a FEM with 50 lines or less with Matlab/ Octave as demonstrated in [15–17]. However, to bring the clarity to the computational aspects and its underlying fundamentals of the algorithm, we will proceed with more traditional implementation of FEM by following the procedures and the notation given in [18–20].

6.2

Isoparametric Representation and Numerical Integration

The main concept in the development isoparametric formulation is that the geometry of an element is defined using its nodal coordinates and the same shape functions which are used to interpolate the nodal unknowns (i.e., displacements, temperatures, chemical concentrations). In isoparametric formulism, it is convenient to express the shape functions in terms of a nondimensional, localized coordinate system (ζ and η) in which they vary from
1 to 1 over an element. The adaptation of this local

# Springer International Publishing Switzerland 2017 S.B. Biner, Programming Phase-Field Modeling, DOI 10.1007/978-3-319-41196-5_6

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6 Solving Phase-Field Equations with Finite Elements h

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Fig. 6.1 Global and local definition of isopa