Numerical Methods for Phase-Field Model and its Applications to Multi-Phase Flow

Abstract

Numerical Methods for Phase-field Model and its Applications to Multi-phase Flow Numerical methods for solving partial differential equations (PDEs) were used first by John von Neumann in the mid of 1940s as an effective tool. Since then, numerical computing has become the most versatile tool to experiments and complement theory. In principle, all mathematical equations can be easily solved numerically in comparison to the theoretical approach. They are low cost, high efficiency and no danger when compared with experimental approach. The driving force of numerical methods comes from practical application in all branches of science, engineering and other disciplines. The starting point of computational methods is a mathematical model, the form and origin of which depends on the particular field of study. There exist many important physical and biological processes in nature that can be represented by mathematical models. However, a physical and mathematical interpretation of the models and their numerical solutions is always a vital part of the computational science. Although, analytical solutions are difficult to obtain for many complex phenomena governed by nonlinear PDEs. However, with the rapid development in computational techniques and availability of high-speed computers, there is a continuously widening scope of nonlinear problems that can be solved numerically. Appropriate numerical algorithms, in particular those for solving time-dependent nonlinear PDEs are in heart of many of advanced scientific computations and software development. Moving free boundary problems are present in nature and many areas of physical and biological sciences. Examples include impact of a droplet on a solid surface, image segmentation, surface waves, jet breakup, realistic interfaces in animation movies, crystal and tumor growth and many others where the simulation of moving interfaces plays a key role in the problem to be solved. In dealing with moving boundary problems, an important consideration is how to model the moving boundary or interfacial surface on which the boundary conditions are imposed. Mathematical models adopted both in analytical and numerical studies for variety of free boundary problems are classified into two types, sharp interface and diffuse interface models. Sharp interface models like (level set method ) assumes that the interface has zero thickness. However, in phase transition, the existence of transition zone introduced an idea of diffuse interface by Gibbs, which allow the interface to have finite thickness. A type of diffuse interface model with particular interest is phase-field based model by the introduction of an order-variable representing the interface. In such approach, the phase-field variable is continuous as a function of space and time. Nevertheless, the partial differential equations describing the two-phase flow are highly nonlinear and numerical simulation is often necessary to solve them. This is why phasefield methods are numerically attractive with no tracking of interface explicitly but can be obtained as a part of the solution processes. This work deals with the development of numerical methods for solving phase-field models with some real world applications. There are several discretization method like finite difference method, finite volume method and finite element method etc. We have adapted the conforming finite element method for spatial discretization and have used different diagonally implicit schemes for time discretization. The performance of the proposed numerical algorithms in term of their accuracy and CPU time are demonstrated. The comparisons with analytical, experimental and numerical results are also provided for validation and verification of the computed results. The numerical simulations were carried out using DUNE-PDELab, which is a software tool for solving partial differential equations. This thesis is organized as follows: In chapter 1, an introduction of the phase-field models with some fundamental aspects and applications are given. Chapter 2 provides numerical methods with some discretization techniques and their modifications. Chapter 3 is based on the development of an efficient time stepping scheme for solving of 2ndorder nonlinear Allen-Cahn equation. Error estimates at different degrees of freedom are also provided using available exact solution. In chapter 4, the method developed in chapter 2 is further extended to solve 4th-order Cahn-Hillard equation with variable mobility. In chapter 5, a system of Allen-Cahn equation coupled with heat equation is solved with its application to dendritic crystal growth phenomena. Chapter 6 is based on the mathematical modeling and its numerical simulation of tumor hypoxia targeting in cancer treatment. Chapter 7 concludes this work with some future research directions.