Computational methods for a phase field model of grain growth kinetics

COMPUTATIONAL METHODS FOR A PHASE-FIELD MODEL OF GRAIN GROWTH KINETICS BIPIN KUMAR (M.Sc., IIT Kanpur, India) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE Acknowledgments I warmly express heartfelt gratitude and sincere appreciations to my supervisor Dr. Lin Ping for his guidance full with patient, stimulating ideas and invaluable advice throughout my study period in NUS. Without his supervisions, even the very first step was not possible. It has been my pleasure being his research student. I would like to express my sincere thanks to Dr. B.S.V. Patnaik for all his help and guidance. Special thanks are very much due to Dr. Shashi Bhushan (DNV), Venkateswarlu, Zacharry Harrish, Dr. Ram Singh Rana (A*Star) and Dr. Sanjiv Yadav (Chemistry). On my mind, the proximity and moral support, I had from them, has put indelible print of my memorable stay in Singapore. I would next, like to thank to my officemates Jinghui, David, Chen Yidi, Wu Lei, and Xu Ying for their assistance. Talks held with them will always remain in my thoughts. Discussion held with Shapeev Alexander are duly acknowledged. I would like to express my sincere feelings to my house mates Yogesh, Raju, Mohan, Ved and my friends Sunil, Kaushal Pandey, Jan Frode Stene, Sulakshana, Khaing Mawoo Toee and Mrs Desai. Finally, I would like to thank my parents and siblings. Without their continuous encouragement and support, nothing would have been possible for me. i ii Last but the most important feelings for Ms Preeti, who has accepted to knot with me forever. I feel lack of words to acknowledge Him who has provided me with unlimited potential but with limited capabilities. I bow my head to Him!! Contents 1 Introduction 1.1 1.2 1 Motivation and Objectives: . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Motivation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Organization of this thesis: . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Material Science and Simulations 5 2.1 What are Materials? . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Role of Computational Materials Science: . . . . . . . . . . . . . . . . 6 2.3 Relevance of Simulating Microstructural Evolutions: . . . . . . . . . . 7 2.4 Phase Field Model: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 2.4.1 What is an Interface? . . . . . . . . . . . . . . . . . . . . . . 10 2.4.2 Sharp Interface: . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.3 Diffuse Interface: . . . . . . . . . . . . . . . . . . . . . . . . . 11 Application of Phase Field Methods: . . . . . . . . . . . . . . . . . . 12 3 Theory and Model Description 3.1 14 Local Free Energy: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 iii iv CONTENTS 3.2 Need for advanced numerical approaches: . . . . . . . . . . . . . . . . 19 4 Computational Algorithms 21 4.1 Finite Difference Approach (Explicit): . . . . . . . . . . . . . . . . . 22 4.2 Finite Difference Approach (Semi Implicit): 4.3 Operator Splitting Method: . . . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . . 24 4.3.1 Exact Solution of 3rd Degree Polynomial Equation: . . . . . . 27 4.3.2 Discussion of Solution: . . . . . . . . . . . . . . . . . . . . . . 27 4.4 Active Parameter Tracking Algorithm: . . . . . . . . . . . . . . . . . 31 4.5 Finite Element Approach: . . . . . . . . . . . . . . . . . . . . . . . . 33 4.6 On the Implementation of Periodic Boundary Conditions: . . . . . . . 38 4.7 On Solving Large-Scale System of Equations: . . . . . . . . . . . . . . 39 4.7.1 Format for Sparse Matrix: . . . . . . . . . . . . . . . . . . . . 41 4.7.2 Sparse Matrix-Vector Multiplication in CSR Format: . . . . . 42 5 Results and Discussion 43 5.1 Finite Difference Method (Explicit): . . . . . . . . . . . . . . . . . . . 43 5.2 FD with Operator Splitting Method: . . . . . . . . . . . . . . . . . . 48 5.2.1 Operator Splitting: Case 1 . . . . . . . . . . . . . . . . . . . . 50 5.2.2 Operator Splitting: Case 2 . . . . . . . . . . . . . . . . . . . . 52 5.2.3 Operator Splitting: Case 3 . . . . . . . . . . . . . . . . . . . . 52 5.3 AIA-PT Method: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.4 Result Comparison and Discussion: . . . . . . . . . . . . . . . . . . . 54 5.4.1 Comparison of Result with other Researcher’s Results: . . . . 56 6 Conclusions and Suggestions for Future Work 65 CONTENTS v 6.1 Conclusions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2 Suggestions for Future Work: . . . . . . . . . . . . . . . . . . . . . . 66 Abstract Computational Materials science is fast catching up as an attractive and complementary approach to designing novel materials. An apriori understanding of micro structures and its linkage to material properties is possible through the use of workable models. In this thesis, we explore a variety of computational algorithms for solving partial differential equations that govern the kinetics of grain growth. To start with, we employ the emerging phase-field approach to model the micro structural evolutions. To start with, we have to design the appropriate energy functional. Then, solve the Allen-Cahn equations or the Complex Ginzhburg Landau equations (CGLE). Following computational schemes are devised for the phase-field equations: (i) Solve the equations by Finite difference method with a simple explicit time marching scheme (This is the most popular approach) (ii) Solve the Allen-Cahn equations using operator splitting method, where the equation is divided into a Poisson part and a cubic equation. (Note that, the cubic equation has an exact solution). vi CONTENTS vii (iii) A simple observation that only the interfaces have more than one active phase-field variables at every grid point was successfully exploited by devising an active and implicitly adaptive parameter tracking (AIA-PT) approach. (iv) Solve the equations by Combing the two ideas of operator splitting and AIA-PT. List of Figures 2.1 Demonstration of multi-scale modeling for the design of radiation resistant material : Ghoniem et al [26] . . . . . . . . . . . . . . . . . 8 2.2 Typical image of grains [27] . . . . . . . . . . . . . . . . . . . . . . . 9 3.1 Representation of Orientation field variable φ . . . . . . . . . . . . . 14 4.1 Discretization of Laplacian . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 CSR format of a sparse matrix 5.1 Temporal evolution of microstructure for FDM(explicit) for Q=48 . . 45 5.2 Temporal evolution of microstructure for FDM(explicit) for Q=60. . . 46 5.3 Temporal evolution of microstructure for FDM (explicit) scheme for . . . . . . . . . . . . . . . . . . . . . 42 Q=36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.4 The average grain area and total time taken as function of time for FDM (explicit). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.5 Temporal evolution of microstructure for FDM (explicit) scheme for Q=48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.6 The average grain area and total time taken as function of time for FDM (explicit). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 viii ix LIST OF FIGURES 5.7 Temporal evolution of microstructure for FDM (explicit) scheme for Q=48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.8 Temporal evolution of microstructure for operator splitting case 1 for Q=36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.9 The average grain area and total time taken as function of time for operator splitting case 1. . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.10 Temporal evolution of microstructure for operator splitting case 1 for Q=48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.11 The average grain area and total time taken as function of time for FDM (explicit). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.12 Temporal evolution of microstructure for operator splitting case 2 for Q=36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.13 The average grain area and total time taken as function of time for operator splitting case 2. . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.14 Temporal evolution of microstructure for operator splitting case 3 for Q=36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.15 The average grain area and total time taken as function of time for operator splitting case 3. . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.16 The average grain area and total time taken as function of time for AIA-PT method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.17 Comparison of total time taken by FDM (Explicit),different operator splitting method and AIA-PI method for 10000 time steps . . . . . . 61 5.18 Comparison of average grain area as function of time for FDM (explicit) case and operator splitting method case 1. . . . . . . . . . . . 62 LIST OF FIGURES x 5.19 Comparison of average grain area as function of time for FDM (explicit) case and operator splitting method case 2. . . . . . . . . . . . 63 5.20 Comparison of average grain area as function of time for FDM (explicit) case and operator splitting method case 3. . . . . . . . . . . . 63 5.21 Comparison of average grain area as function of time for FDM (explicit) case and all three operator splitting method. . . . . . . . . . . 64 Chapter 1 Introduction Materials as a whole can be classified as either amorphous (glossy) or crystalline. For the latter the atoms are situated in a repeating (or) periodic array over large distances, meaning a long range order exists. If all such unit cells interlock in the same way and have the same orientation, it is called a single crystal. Single crystals exist in nature, but very difficult to grow (Example: Diamond). For a given piece of material, if we find a collection of many small crystals or grains, it is called polycrystalline material, whose crystallographic orientation varies from grain to grain. Modeling such materials with a view to understand their properties is still a great challenge. Phase-field models developed in mid 90’s is a fertile ground for both Mathematicians and Physicists alike. In this thesis an attempt is made to model the microstructural evolution and kinetics of such materials. 1 CHAPTER 1. INTRODUCTION 1.1 2 Motivation and Objectives: There is an extensive literature on the use of phase-field models. These models have continuously varying field variables ψ(x) that lend themselves well to PDE’s. These models are often used as a way to approach difficult problems in interfacial dynamics such as phase-separation or diffusion limited aggregation. For instance, Cahn-Hilliard equation describes phase separation from a uniform binary mixed state to that of spatially separated multi-phase structure. 1.1.1 Motivation: The simulation and modeling route enables Mathematicians to contribute to the understanding of material processes and to achieve desirable properties and gain insights. Several unanswered, non-trivial questions exist in the field of material mechanics, which are amenable to Mathematical modeling, albeit with simplifications. Even an iota (ǫ or δ ) progress would greatly aid the material technologists and the world at large in our quest towards discovering novel materials. Grain growth is one such process, whose kinetics can be described by suitable Mathematical models. Rich amount of literature exists on these coarsening processes, which is a good ground for probabilistic and/or deterministic models such as Monte Carlo and cellular automaton. However, from Mathematical point of view, employing a set of partial differential equations is highly desirable. For polycrystalline settings, where a large number of orientations are likely, the governing coupled partial differential equations at every grid point could lead to simulations that are highly memory and CPU intensive. Assuming N ×N grid points in 2-D settings and Q order parameters (or) phase-field variables, in the system, at least N ×N ×Q coupled equations need CHAPTER 1. INTRODUCTION 3 to be solved at every grid point. This limitation has indeed attracted our attention. 1.1.2 Objectives: After a thorough and careful study of the existing literature, we find that the phasefield models have been applied to a variety of materials processing applications. However, the modelers have not paid much attention regarding numerical methods that can be employed. To this end, we have set the following Questions in our pursuit (i) Is it possible to employ efficient, stable numerical algorithms, that are superior and less CPU and memory intensive. (ii) Is it possible to compare the performance of existing algorithms vis-a-vis the proposed methods (iii) Is it possible to view the kinetics of grain growth merely as evolution of interfaces and drastically reduce the number of phase-field variables to be solved at every grid point. 1.2 Organization of this thesis: This thesis is organized as follows. Chapter 2 starts with a rudimentary discussion of what constitutes a material, the issues of interest in computational materials science and the relevance of the present simulations etc. In chapter 3 we briefly introduce the theory behind the phase-field models of interest to Grain Growth kinetics and the formation of mathematical models. A repository of numerical methods which CHAPTER 1. INTRODUCTION 4 have been employed in solving these partial differential equations is explained in chapter 4. In particular, two elegant ideas have been pursued: (i) the application of operator splitting method, (ii) active and implicitly adaptive parameter tracking (AIA-PT) approach. Results and discussion and comparison of different computational schemes is presented in chapter 5. The thesis ends with a conclusion and possible future extensions in chapter 6. Chapter 2 Material Science and Simulations 2.1 What are Materials? The Mathematical models that help us in understanding materials have far reaching ramification in design and development of novel materials. Materials not only directly impact us, but advanced materials play a crucial, and enabling role underlying virtually all technologies. A simple and operative definition of a material is to view it as a particular arrangement of atoms. In materials design and control there are three main concerns (i) the atoms to be arranged, (ii) kind of arrangement, (iii) how to arrange these atoms. The last among the three, is more of a processing question and can be understood by investigating the microstructural evolution at various scales. A variety of tools used at the human scale (say, macro scale), depend on materials for which some 5 CHAPTER 2. MATERIAL SCIENCE AND SIMULATIONS 6 other length and time scales are critical. All these scales can be broadly classified as - micro, meso and macro. Mesoscale is in between the scales of engineering and atomistic science [1]. This is a convenient scale in materials design, since the material properties and behavior is dominated by these mesoscale structures. Simulations at these scales enable us to investigate the temporal evolutions of grain microstructures. The various material properties (e.g mechanical properties) are thoroughly influenced by the grain topology and its evolution. Such internal topological evolutions have been continuously possible mainly due to the following facts : (i) different lattices (ii) recrystallization (iii) transition of phases. Furthermore, such evolutions are influenced by the connectivity of grains, which is a key microstructural feature. All these different ways of looking at the materials enable the scientists and engineers, to synthesize and process the materials, and control material structures at the molecular, nano, meso, and macro scales. [2]. 2.2 Role of Computational Materials Science: Simulations through Mathematical modeling have played a constructive role in the development of new materials, in enhancing our fundamental understanding of material behavior. Indeed, computational Materials science is an exciting synergy between materials and computing power, each feeding back into the developments of the other in a tightly coupled fashion. For instance the CPU time is expanding at an exponential rate (thanks to Moor’s law), which is only possible by delving deep into the world of materials and technology. The time scales and length scales at which one can probe through simulation is getting more and more realistic and CHAPTER 2. MATERIAL SCIENCE AND SIMULATIONS 7 close to predictable. The advances in understanding material microstructures can bring valuable insights in the field of material science [3]. Thus, simulation and modeling of material processes through micro, meso and macroscopic modeling is gaining fast acceptance as a cost effective and complementary tool to experimentation. However, achieving relevant time scales and bridging the gap among different scales is a real challenge and is in the realm of multiscale materials modeling. The following simulation modes are prerequisite in building capability for the reliable and accurate prediction of phenomena and properties in a wide range of materials [1]. • discovering novel relations and paradigms for complex behavior, • benchmarking test forms of the mathematical models, • direct calculation of parameter input for mathematical models, • validation of models by comparison the result with experiments, and • predicting the complexities involved in materials processes and phenomenon. Combining these five modes, simulation becomes a powerful, revolutionary tool to accelerate conceptual advances. 2.3 Relevance of Simulating Microstructural Evolutions: All engineering materials contain certain type of microstructure, virtually in every material processing phase. The microstructural evolutions are common in many fields including biology, hydrodynamics, chemical reactions, and phase transformations. CHAPTER 2. MATERIAL SCIENCE AND SIMULATIONS 8 Figure 2.1: Demonstration of multi-scale modeling for the design of radiation resistant material : Ghoniem et al [26] Microstructure is a broader term, which refers to spatial distribution of structural features which can be phases of different compositions and/or crystal structures, or grains of different orientations, domains of different structural variants, domains of different electrical and magnetic polarization, and structural defects. The length scales of these structures is typically in the range of nanometers to a few tens of microns. Microstructural evolution takes place to reduce the total free energy that may include the bulk chemical free energy, interfacial energy, elastic strain energy, magnetic energy, electrostatic energy, and/or under applied external fields such as applied stress, electrical, temperature, and magnetic fields[25] . Indeed, the simulation of microstructures is a hunt for key mechanisms associated with that scale. The current trend in materials modeling is to feed on to higher scales as depicted in Fig.2.1. In Fig.2.2, a typical micrograph from [27] depicts material microstruc- CHAPTER 2. MATERIAL SCIENCE AND SIMULATIONS 9 Figure 2.2: Typical image of grains [27] ture, where a few grains can be visually identified. The capability of predicting equilibrium and non-equilibrium phase transformation phenomena at a microstructural scale is among the most challenging topics in materials science. Therefore, it is highly desirable that we are able to understand the kinetics of microstructural evolutions. 2.4 Phase Field Model: A phase field implicitly describes a microstructure, both the compositional/ structural domains and the interfaces, as a whole by using a set of field variables. The field variables are continuous across the interfacial regions and hence the interfaces in a phase-field model are diffuse. There are two types of field variables, conserved and non-conserved. Conserved variables have to satisfy the local conservation condition. Interfaces between phases is central to understanding the elegant features of phase-field models. CHAPTER 2. MATERIAL SCIENCE AND SIMULATIONS 2.4.1 10 What is an Interface? An interface refers to a boundary between two different phases. From a mathematical point of view these problems constitute a class of so-called moving boundary problems. Usually analytical treatment of these type of problems is very restricted. Thus, it seems natural to search for adequate numerical treatment. In numerical treatment the key part is to simulate the effect of the boundary by treating it explicitly, i.e. with no smearing of information at the interface resulting in numerical diffusion. For moving boundaries, techniques for such an explicit treatment include block-structured domain decomposition, overset meshes or unstructured boundaryconforming curvilinear grids to discretize the domain are employed, in which the computations are performed on the fixed grid while at the same time the interface is tracked explicitly as independent curve. The phase boundaries are more than the interfaces known from materials science. Talking of interface in a broader sense one could include evolving boundaries from, e.g. combustion, image processing, computer vision, control theory, seismology and computer aided design, as well. 2.4.2 Sharp Interface: The interface between two distinct domains is infinetly thin. In the sense of numerics, if a grid is assumed to have been superimposed on the domain, at every grid point either we have phase 1 or phase 2 but not a combination of these two. This is precisely the reason for the popularity of the diffuse interface models. The boundary condition typically yields a normal velocity at which the interface is moving. This is called sharp interface approach. CHAPTER 2. MATERIAL SCIENCE AND SIMULATIONS 2.4.3 11 Diffuse Interface: In the case of diffuse interface any density of an extensive quantity (e.g. the mass density, number density, orientation etc.), between two co-existing phases varies smoothly from its value in one phase to its values in the other. Essentially the diffuse interface is connected to such an additional order parameter. Clearly such models have advance numerical treatment as well as understanding of interfacial growth phenomena. The Phase field method to model interfacial growth is to understand it as a numerical technique which helps to overcome the necessity of solving for the precise location of the interfacial surface explicitly in each time step of numerical simulation. This can be achieved by the introduction of one or several additional phase field variables. They are the key elements of the resulting phase-field modeling approach for studying systems out of equilibrium. In such an approach the phase-field variables are continuous fields which are functions of x, and time t. They are introduced to describe the different relevant phases. Typically these fields vary slowly in bulk regions and rapidly, on length scales of the order of the correlation length ξ, near interface. ξ is also a measure for the finite thickness of the interface. The free energy functional F determine the phase behavior. Together with equations of motion this yields a complete description of the evolution of the system. For example, in a binary alloy the local concentration or sub-lattice concentration can be described by such fields. With the contribution of Cahn and collaborators, the phase-field models are more than just trick to overcome numerical difficulties. Rather they are rigorously derived based on the variational principles of irreversible thermodynamics. In another CHAPTER 2. MATERIAL SCIENCE AND SIMULATIONS 12 approach, a given sharp interface formulation of the growth problem is the correct description of the physics under consideration. On the basis of this assumption, a phase-field model can be justified by simply showing that it is asymptotic to the correct sharp interface description, i.e. the latter arises as the sharp interface limit of the phase-field model when the interface width is taken to be zero. Therefore employed in this way phase-field models do not seem to be of much help to elucidate the physics of the interfacial region beyond what is captured within the sharp interface model equations. One can assume a phase-field model to be thermodynamically consistent and to describe a physical situation, for which an established sharp interface formulation exists, as well, certainly, in the sharp interface limit the phase-field model should correspond precisely to that sharp interface formulation. 2.5 Application of Phase Field Methods: A variety of material processing applications have been comfortably handled by Phase-field models. The application of phase-field method have been focused on the three major materials processes: solidification, solid-state phase transformation, and grain growth and coarsening. Examples of existing phase-field applications are: Solidification • Pure liquid. • Pure liquid with fluid flow. • Binary alloys. • Multicomponent alloys. • Nonisothermal solidification. Solid-State Phase Transformations • Spinodal phase separation. CHAPTER 2. MATERIAL SCIENCE AND SIMULATIONS 13 • Precipitation of cubic ordered intermetallic precipitates from a disordered matrix. • Cubic-tetragonal transformations. • Hexagonal to orthorhombic transformations. • Ferroelectric transformations. • Phase transformations under an applied stress. • Martensitic transformation in single and polycrystals Coarsening and Grain Growth • Coarsening. • Grain growth in single-phase solid . • Grain growth in two-phase solid. • Anisotropic grain growth. Chapter 3 Theory and Model Description Basic to phase field theories are continuous field variables φ1 (x, t), φ2 (x, t), . . . , φQ (x, t) (referred to as order parameters) which are functions of material points x and time t. In the context of polycrystalline materials, the order parameters represent the volume fraction of grains of a particular orientation. A schematics microstructure represented by the orientation fields in 2-D is shown as in fig (3.1). Figure 3.1: Representation of Orientation field variable φ 14 15 CHAPTER 3. THEORY AND MODEL DESCRIPTION The order parameters φi are non-conserved quantities therefore their local evolution rates are linearly proportional to the variational derivative of the total free energy with respect to local order parameter. Thus the evolution of the order parameters is specified by the time dependent Ginzburg-Landau equations for each of the Q order parameters δF ∂φi = −Li , ∂t δφi i = 1, 2, . . . , Q (3.1) where Li are mobility coefficients, t is time and F is the total free energy functional. For this study the free energy functional is taken to be of the form Q F= f (φ1 , φ2 , . . . , φQ ) + V i=1 κi |∇φi |2 2 dV (3.2) where κi are positive constants for an isotropic gradient energy and f is a local free energy density. The origin of grain boundary energy comes from the gradient energy terms, (∇φi )2 , which are non-zero only around the grain boundaries. 3.1 Local Free Energy: In the phase-field model one of the key component is local free-energy density function. The main difference among different phase field models lies in the construction of the energy function f as a function of field variables. In many of phase-field models, particularly in solidification modeling a double-well form function is used as f (φ) = 4∆f 1 1 − φ2 + φ4 2 4 (3.3) where φ is a field variable. The free-energy function has a doubly degenerate minima represented by φ = −1 and φ = +1. For example, in the case of solidification, 16 CHAPTER 3. THEORY AND MODEL DESCRIPTION φ = −1 and φ = +1 represent the liquid and solid states, respectively. ∆f is the potential height between the two states with the minimum free energy. If φ represents a conserved composition field, the two minima represent the two equilibrium phases with different compositions and ∆f is the driving force for the transformation of a single homogeneous phase (φ = 0) to a hetrogenous mixture of two phases represented by φ = −1 and φ = +1. For some processes, it may be more desirable to have two minima of free energy located at φ = 0 and φ = 1, then following function f (φ) = 4(∆f )φ2 (1 − φ2 ) (3.4) can be employed[25]. Another free energy function that has sometimes been employed [25] in phasefield models is the double-obstacle potential f (φ) = ∆f (1 − φ2 ) + I(φ) (3.5) where I(φ) = ∞ for |φ| > 1 and 0 otherwise. On the similar way Chen et. al.[30], we choose a specific form of the free energy function as Q f (φ1 , φ2 , . . . , φQ ) = i=1 1 1 − αφ2i + βφ4i + γ 2 4 Q Q φ2i φ2j (3.6) i=1 j=i such that the energy is independent of the orientation of the grains. In the above equation φi are order parameters and α, β, and γ are the phenomenological parameters. It can be shown that the above simple free-energy model gives a second-order phase transition whereas crystallization is always first order. This energy function also satisfy the main requirement, “the function should provide large numbers of potential wells with equal well depth located at (φ1 , φ2 , · · · , φQ ) = (1, 0, · · · , 0),(0, 1, · · · , 0), · · · · · · (0, 0, · · · , 1) respectively” as described in [32]. 17 CHAPTER 3. THEORY AND MODEL DESCRIPTION If we assume α = β = 1, then the first summation from right hand side of equation (3.6) becomes Q 1 1 − φ2i + φ4i 2 4 f (φ) = i=1 and then δf =0 δφi ⇒ −φi + φ3i = 0 ⇒ φi (φ2i − 1) = 0 i.e. φi = 0, 1 and -1. It shows that each term in the first summation in the right hand side is a doublewell potential with well located at φi = −1 and φi = 1. However, the first summation alone cannot satisfy the requirement [32] since it will have total number of 2Q minima located at the positions where each φi is either equal to -1 or 1, such as (φ1 , φ2 , · · · , φQ ) = (1, 1, · · · , 1). Therefore, the cross terms were added to equation (3.6). By choosing proper value of γ, the potential equation can fulfill that requirement. One can rewrite the equation (3.6), for choosing a proper γ, as follows Q α f (φ1 , φ2 , · · · , φQ ) = − 2 φ2i i=1 β + 4 Q Q φ4i Q φ2i φ2j +γ i=1 j=i i=1 on simplification it can be written as α f (φ1 , φ2 , · · · , φQ ) = − 2 Q φ2i i=1 β + 4 2 Q φ2i Q Q φ2i φ2j +γ i=1 j=i i=1 β − 2 Q Q φ2i φ2j i=1 j=i which on rearranging terms yields α f (φ1 , φ2 , · · · , φQ ) = − 2 Q φ2i i=1 β + 4 2 Q φ2i i=1 β + (γ − ) 2 Q Q φ2i φ2j , i=1 j=i (3.7) 18 CHAPTER 3. THEORY AND MODEL DESCRIPTION Case 1: γ = β/2 In this case equation (3.7) becomes α f =− 2 2 i=1 2 Q β φ2i + 4 φ2i i=1 Therefore δf =0 δφi Q φ2i φi = 0 ⇒ −αφi + β i Q ⇒ φi −α + β φ2i =0 i=1 Thus, for γ = β2 , f has infinitely many degenerate minima for Q > 1, located at the loci described by Q φ2i = i=1 α =1 β For Q=2, these loci form a circle, for Q=3 they form a sphere etc. Case 2 : γ < β/2 In this case f will have 2Q minima with each φi being either equal to 1 or -1. Case3 : γ > β/2 For this case f has 2Q minima located at (φ1 , φ2 , · · · , φQ ) =(1, 0, · · · , 0),(0, 1, · · · , 0), · · · · · · (0, 0, · · · , 1). From these three cases it can be concluded that in order to fulfill the main requirement by function f , γ has to be greater than β/2. In modeling grain growth, each of the 2Q minima represents a specific crystallographic orientation of grains. Using (3.2) and (3.6) in the evolution equations (3.1), we obtain the governing equations for grain growth ∂φi = −Li ∂t Q −αφi + βφ3i + 2γφi j=i φ2j − κi ∇2 φi , i = 1, 2, . . . , Q (3.8) CHAPTER 3. THEORY AND MODEL DESCRIPTION 19 which are a set of Q coupled, non-linear, parabolic equations. In the diffuse interface description, the free energy of an inhomogeneous system, such as microstructure depends on the gradients of the field variables. α in the equation is the gradient energy coefficient, which characterizes the energy penalty due to the field inhomogeneties at the interfaces i.e. the interfacial energy contribution to the total free energy. For a given free energy model and a given set of gradient energy coefficients, the specific interfacial energy can be calculated for an equilibrium interface. It is important to realize that the integral of the gradient energy term only counts part of the interfacial energy. 3.2 Need for advanced numerical approaches: In real materials, there are infinetly many grain orientations that are likely i.e. Q = ∞. However for the purpose of computer simulations, the number of field variables (ordered parameters) has to be a finite quantity. But there are two drawbacks in using a limited number of order parameters. Since each order parameter represents a particular grain orientation, intermediate values of orientations are excluded. This may be viewed as restricting the likely grain orientations to a finite discrete set of angles (with loss of rotational invariance of the free energy) or each order parameter as representing a wider range of orientations. In the context of solidification of a polycrystalline material, this prevents the potential nucleation of a new randomly oriented grain. Obviously, a larger number of order parameters is required for a more continuous description of frequency distribution of grains of different orientations. Furthermore, another unfortunate consequence of limiting the number of order parameters is the coalescence of grains during grain growth. Coalescence CHAPTER 3. THEORY AND MODEL DESCRIPTION 20 is the situation in which two grains which have the same order parameters, come into contact and instantaneously form a single large grain. This leads to incorrect growth rates and unphysical grain shapes. The likelihood of coalescence involving a given grain varies with the probability p(Q) that at least one of its second-nearest neighboring grains shares the same orientation and is given by [43] p(Q) ≃ 1 − 1 − 1 Q Z (3.9) where Z represents the average number of second nearest-neighbor grains in the microstructure. In the limit of large Q, p(Q) is approximately equal to Z/Q. In 2D simulations, an average grain has about 6 sides and hence, there will be 12 nearest neighbor grains [44]. With Q = 48, the probability of grain coalescence is about p(48) = 0.223. Approximately 110 order parameters are needed to effectively suppress p(Q) to below 10% [33]. The problem is even more acute in 3D with each grain having about 14 sides on average (Z ≈ 28), resulting in a probability of coalescence of 0.36 [44]. Therefore, more than 200 order parameters would have to be employed at each grid point in order to keep p(Q) below 13% in 3D [33]. There have been some attempts to avoid this effect when using smaller Q values, via a dynamic reorientation algorithm [33]. The quest for a multi-order parameter models with the framework to handle an unlimited number of phase field variables would be seen as a great asset by modelers. Precisely we propose such an approach which can handle theoretically ∞ phase-field variables. Chapter 4 Computational Algorithms In this chapter, we explain some of the computational algorithms used in implementing the phase-field models discussed above. One of the simple approaches is the one by Chen et al [25, 31, 32] is briefly described and then other significant approaches for solving the PDE are discussed. We start with the popular explicit method of solving the governing equations and elaborate other significant approaches which have been attempted in solving the equations (3.8). In this thesis, the following approaches have been attempted: (1) Explicit approach using FD (2) Semi-implicit approach using FD (3) Operator splitting method (OSM) (4) Active and implicitly adaptive parameter tracking (AIA-PT) method (5) Finite element approach. 21 22 CHAPTER 4. COMPUTATIONAL ALGORITHMS 4.1 Finite Difference Approach (Explicit): This is a simple and straightforward approach in which the all updates at a given grid point for all the variables is based on the known values of the solution system. It implies, the system of equations need not be solved at every iteration. Although it appears as a significant advantage, explicit schemes cause a restriction on the time step (In this case, Laplacian term limits the value of ∆t). The semi discretization equation (3.8) in temporal direction gives ηin+1 − ηin = Li αi ηin − Li βi (ηin )3 − 2 γLi ηi ∆t Q ηjn j=i 2 + Li κi ∇2 ηi (4.1) Here, ηi represents the value of any of field-variable φJ , J = 1, 2, · · · , Q, as described in the previous chapter, in our 2-D grid system. The Lapalcian term in equation (4.1) can be discretized as 1 ∇ ηi = (∆x)2 n 4 2 l=1 (ηl − ηi ) Although in a numerical sense, a second order accuracy is acceptable for these simulations, it would not taken into account all the neighboring grid points. To this end a Monte Carlo styled 4th order accuracy is preferred by Fan et al [32]. Note that the weights are different at grid points which are along the diagonals. 1 1 ∇ ηi = 2 (∆x) 2 2 4 4 1 (ηl − ηi ) + (ηm − ηi ) 4 m=1 l=1 n where ∆x is the grid size, l represents the set of first nearest neighbors of i and m is the set of set of second neighbors of i as shown in Fig. 4.1). In this way all 23 CHAPTER 4. COMPUTATIONAL ALGORITHMS Figure 4.1: Discretization of Laplacian eight neighbors can be included along with node i for the discretization. The two neighbor model approximation, for the Laplacian, has been shown to improve the numerical stability over the one-neighbor model. Using this in equation (4.1), we get Q ηin+1 = ηin + ∆tLi αi ηin − ∆tLi βi (ηin )3 ∆tLi κi 1 (∆x)2 2 − 2.∆tγLi ηi 4 l=1 (ηjn )2 + j=i 4 (ηl − ηi ) + 1 (ηm − ηi ) 4 m=1 on simplification it can be written as ηin+1 = ηin + ∆tLi αi ηin − ∆tLi βi (ηin )3 − 2.∆tγLi ηi + ∆tLi κi 2(∆x)2 4 l=1 (ηl − ηi ) + ∆tLi κi 4(∆x)2 4 m=1 (ηm − ηi ) 24 CHAPTER 4. COMPUTATIONAL ALGORITHMS Q ηin+1 = ∆tLi αi ηin − 2γ ηin Li ∆t j=i (ηjn )2 − 3.0 ∗ Li κi ∆t Li κi ∆t + 0.5 ∗ (∆x)2 (∆x)2 4 ηln l=1 4 +0.25 ∗ Li κi ∆t ( η n ) + ηin − ∆tLi βi ηi3 (4.2) (∆x)2 m=1 m In the above equation the summation over l and m can be expanded as follows (coordinates are according to Fig.4.1), 4 ηln+1 = [ηi−1,j,k + ηi+1,j,k + ηi,j−1,k + ηi,j+1,k ]n+1 l=1 4 n+1 ηm = [ηi−1,j−1,k + ηi+1,j−1,k + ηi+1,j+1,k + ηi+1,j−1,k ]n+1 m=1 4.2 Finite Difference Approach (Semi Implicit): Semi-implicit methods enable the use of higher time steps (∆t) in each iteration. However, one has to solve a system of equations since some of the non-linear terms are treated implicitly. Equation (3.8) can also be approximated as ηin+1 − ηin = Li αi ηin+1 − Li βi (ηin )3 − 2 γLi ηin+1 ∆t Q ηjn j=i 2 + Li κi ∇2 ηi (4.3) To discretize the Laplacian term in the semi discrete problem (4.3), we use 5th order finite difference scheme same as in the previous section ∇2 ηin+1 1 1 = 2 (∆x) 2 4 4 1 (ηl − ηi ) + (ηm − ηi ) 4 m=1 l=1 n+1 (4.4) 25 CHAPTER 4. COMPUTATIONAL ALGORITHMS Therefore by using equation(4.4) in the equation (4.3), we get Q ηin+1 = ∆t + Li αi ηin+1 ∆t Li κi (∆x)2 − ∆t 1 2 Li βi (ηin )3 4 − 2∆t γLi ηin+1 4 l=1 (ηl − ηi ) + 1 (ηm − ηi ) 4 m=1 (ηjn )2 j=i n+1 + ηin which on simplification yields Q ηin+1 + = ∆t Li αi ηin+1 − ∆t 3∆tLi κi n+1 0.5 ∗ ∆tLi κi η − (∆x)2 i (∆x)2 Li βi (ηin )3 4 l=1 ηln+1 − − 2∆t γLi ηin+1 0.25 ∗ ∆tLi κi (∆x)2 (ηjn )2 j=i 4 n+1 ηm + ηin m=1 after rearranging the terms, we get Q ηin+1 1.0 − Li αi ∆t + 2γ Li ∆t (ηjn )2 j=i Li κi ∆t 0.5 ∗ Li κi ∆t + 3.0 ∗ − 2 (∆x) (∆x)2 4 ηln+1 l=1 4 −0.25 ∗ Li κi ∆t ( η n+1 ) = ηin − ∆tLi βi ηi3(4.5) (∆x)2 m=1 m Equation (4.5) represents a system of equations A x = B. There are Q phase field variables at every grid point, which mimics as Q possible orientations of grain at a given point. At each layer there are N ∗ N grid points. Thus the total number of grid points are N 2 × Q. Any grid point is denoted by η[i][j][k], i=1 to N, j= 1 to N and k = 1 to Q. Therefore equation(4.5) is written, in terms of i, j and k, as Q 1.0 − Li αi ∆t + 2γ Li ∆t j=i (ηjn )2 + 3.0 ∗ Li κi ∆t 0.5 ∗ Li κi ∆t ηi,j,k − 2 (∆x) (∆x)2 4 −0.25 ∗ 4 l=1 ηln+1 |i,j,k Li κi ∆t n 3 ( η n+1 )|i,j,k = ηi,j,k − ∆tLi βi ηi,j,k (∆x)2 m=1 m 26 CHAPTER 4. COMPUTATIONAL ALGORITHMS As the number of grid points N 2 and number of field variables Q increases, the size of matrix (A) to be solved will be too large. For such system of equations iterative solvers are the best option. Therefore this system of equations was solved by Krylov Subspace Methods viz. CGM,GMRES and Steepest Descent Method, which are described in the coming section 4.7. 4.3 Operator Splitting Method: In this proposed method, we take note of the fact that by splitting the terms of the equation(3.8), an analytical solution can be applied, as it forms forms a 3rd degree polynomial [4]. The cubic equation can be solved with an exact solution and the solution procedure is as below : Step 1: Discretization in temporal direction n+1/2 ηi − ηin n+1/2 = Li κi ∇2 ηi ∆t (4.6) Step 2: Formulation of 3rd degree polynomial equation Q n+1/2 ηin+1 − ηi ∆t = Li αi ηin+1 − Li βi (ηin+1 )3 −2γ Li ηin+1 (ηjn+1 )2 (4.7) j=i Equation (4.7) can be rearranged into a polynomial of degree 3 as , 1 (ηin+1 )3 + Li βi ∆t Q 1.0 − Li αi ∆t + 2 γ Li ∆t j=i n+ 1 ηj 2 2 n+ 1 ηi 2 n+ 1 η 2 =0 − i Li βi ∆t (4.8) 27 CHAPTER 4. COMPUTATIONAL ALGORITHMS Equation (4.8) has the exact solution. Hence, we only need to solve linear equation in the step 1. The nonlinear part can be dealt with simply from the closed form solution (4.8). 4.3.1 Exact Solution of 3rd Degree Polynomial Equation: The exact solution of 3rd degree polynomial can be found by Cardan’s Formula [49]. According to this formula, equation (4.8) has the solutions (ηin+1 )1 = √ 3 (ηin+1 )2 = ω √ 3 (ηin+1 )3 = ω 2 where ω = −1+ i 2 √ 3 , A = − 2q − q2 4 + A+ B A + ω2 √ 3 p3 27 √ 3 A+ω , √ 3 B √ 3 B B = − 2q + and 1 p= Li βi ∆t Q 1.0 − Li αi ∆t + 2 γ Li ∆t and q2 4 + n+ 21 p3 27 2 ηj j=i n+ 1 η 2 q=− i Li βi ∆t 4.3.2 Discussion of Solution: The above solution has two forms namely algebraic and trigonometric. Algebraic Solution Here we note that p and q are real numbers. Now consider the function D = 4p3 + 27q 2 . CHAPTER 4. COMPUTATIONAL ALGORITHMS 28 Case 1: D is positive : The square root of q2 4 + p3 27 A and B will be real, and by = √ 3 D 108 will be real and we will take it positive. Then, √ A we mean the real cube root of A. Also then 3 B will be the real cube root of B. Hence equation (4.8) has a real root (ηin+1 )1 = √ 3 A+ √ 3 B but two other (complex roots), √ 3 A + ω2 B √ √ √ √ 3 √ 3A− 3B A+ 3B =− +i 3 2 2 √ √ 3 3 (ηin+1 )3 = ω 2 A + ω B √ √ √ √ 3 √ 3A− 3B A+ 3B =− −i 3 2 2 (ηin+1 )2 = ω √ 3 will be imaginary conjugates since A and B are not equal. Case 2 : D is negative: In this case the square root of q2 4 + p3 27 =i D − 108 is purely imaginary and both numbers q A=− +i 2 −D 108 q B =− −i 2 −D 108 are imaginary, so that the roots of equation (4.8) are expressed through the cube roots of imaginary numbers, and yet all three of them are real. For this let √ 3 A=a+bi CHAPTER 4. COMPUTATIONAL ALGORITHMS 29 be one of the the cube roots of A. Since B is conjugate to A, the number a − bi will √ be one of the cube roots of B, and it must take equal to 3 B. Thus √ 3 A = a + b i, √ 3 B =a−bi and from the Cardan’s rule the roots will be (ηin+1 )1 = 2 a (ηin+1 )2 = (a + b i)ω + (a − b i)ω 2 √ = −a − b 3 and (ηin+1 )3 = (a + b i)ω 2 + (a − b i)ω √ = −a + b 3 are real and unequal. But for the computational purpose this solution is not suitable because sometime one have to calculate the cube root of the imaginary part. This case is called irreducible case. There is another way to deal with irreducible case i.e. when D is negative. Trigonometric Solution: In spite of algebraic difficulties inherent in the irreducible case, it is possible to present the roots in a form suitable for numerical computation by extracting the the cube root of q A = − +i 2 − p3 q2 − 4 27 trigometrically. The square of modulus of A is 2 ρ = q − 2 2 q − 2 2 p3 − 27 CHAPTER 4. COMPUTATIONAL ALGORITHMS −p 3 = 30 3 hence 3/2 −p ρ = 3 √ p −p = √ 27 The argument of A can be determined either by its cosine or by its tangent √ 27 q √ , cos φ = 2 p −p √ − −D √ tan φ = q 27 on the condition that φ is taken in the first or second quadrant according as q is negative or positive. Having found ρ and φ, we can take √ 3 A = √ 3 ρ cos φ φ + i sin 3 3 = −p 3 cos φ φ + i sin 3 3 B = −p 3 cos φ φ − i sin 3 3 and √ 3 Then, since ω = cos120o + i sin 120o , the roots of equation will be given by −p φ cos 3 3 φ −p cos 60o − 3 3 (ηin+1 )1 = 2 (ηin+1 )2 = 2 (ηin+1 )3 = 2 −p φ cos 60o + 3 3 (4.9) (4.10) (4.11) 31 CHAPTER 4. COMPUTATIONAL ALGORITHMS 4.4 Active Parameter Tracking Algorithm: In this section, we illustrate the proposed active and implicitly adaptive parameter tracking (AIA-PT) algorithm. As mentioned earlier, at every grid point only a few phase field variables are non-zero and contribute to the evolution of grains via the migration of the interfaces. At each point O(x, t), the set of ordered pairs P(x, t) = {(i, ηi ) : |ηi (x, t)| > ε} (4.12) contains the list of active phase field variables with magnitude greater than ε (chosen to be a small positive threshold value)1 . In order to account for the possibility of a grain entering or leaving a point from neighboring points, we consider the active parameter list: Q(x, t) which is the union of the sets P at the two levels of nearest neighbors 1 Q(x, t) = p,q=−1 P(x + p∆e1 + q∆e2 , t) (4.13) The number of active parameters at a point is Qa (x, t) = |Q(x, t)| where |S| denotes the cardinality of set S. With this identification of the active phase field variables at a point, only Qa (x, t) governing equations (3.8) are required to be solved   ∂ηi  = −Li −αηi + βηi3 + 2γηi ∂t Qa j=1 ξj =i  ηξ2j − κi ∇2 ηi  , ∀ (i, ηi )     (ξj , ηξj )(j=1,...,Qa )  ∈ Q(x, t). (4.14) Specifically, the algorithm consists of the following steps repeated for each iteration (for each grid point in the domain): 1 We use the notation S(x, t) to emphasize that the set S is to be constructed for each point x and at each time step t. CHAPTER 4. COMPUTATIONAL ALGORITHMS 32 1. The list of all active phase field variables from the first and second nearest neighbors Q is assembled from stored values of P. 2. Qa governing equations (4.14) are solved for all (i, ηi ) ∈ Q. The Laplacian term is evaluated from neighboring active parameter sets P(x + p∆e1 + q∆e2 , t), p, q = −1, 0, 1. 3. A new set P(x, t + ∆t) is assembled from order parameters satisfying |ηi (x, t + ∆t)| > ε obtained from (2) and stored. In more generic terms the steps (1) and (3) are akin to the mesh refinement/derefinement strategy often used by computational physicists in a variety of other settings (shocks, vortex structures etc). However, in the present context they serve the purpose of implicitly tracking the grain boundaries using minimal data storage. As the grain boundary moves away from a given lattice point, the phase field variable representing that grain is automatically not tracked any more, since it is no longer part of the active set of any of the neighbors. We find that there is usually only one effective grain orientation the grid points away from grain boundaries, while two or more variables are likely at the interface between adjoining grains. As can be imagined, at initial times of the simulation when triple or higher junctions are possible for small grains, a larger number of active order parameters exist at several points. With increasing grain size, there remain smaller numbers of active order parameters at each point. We found that the number of effective grain orientations never crossed 10 in our 2D simulations. It can be easily seen that increasing Q has no effect on the CPU time or memory requirements since the number of active order parameters at each point remains unchanged. This provides significant savings in CPU time and memory. Furthermore, when the CHAPTER 4. COMPUTATIONAL ALGORITHMS 33 number of active order parameters is one at each grid point and for all its neighbors (which is fairly typical inside a grain), a simple test condition allows us to determine if a given grid point is interior to a grain or not. If the grid point is completely interior to a grain, we do not solve the corresponding evolution equation for the order parameter at that point. At later times in the simulations when the grain boundary area is a small fraction relative to the overall domain, an additional computational savings is thus obtained. We end our description of the APT algorithm with a remark on the development of the initial microstrucure from a liquid state. Chen and co-workers start with ηi , i = 1, 2, . . . , Q taking small random values at each grid point. We perform simulations using two methods: In the first approach, we perform simulations using the standard algorithm for a few hundred time steps using as large a Q as computationally feasible till grains form. Then we reassign unique order parameters to all the grains and switch to the APT algorithm. Alternatively, we start by assigning small random values to a single unique order parameter at each grid point, with the aim of simulating coalescence free grain growth from the initial time t = 0. 4.5 Finite Element Approach: In this section weak formulation for the governing equation is presented. Weak Formulation: Multiplying equation (4.6) by test function N(x,y), we get ηin+1 − ηin · N = ∆t · N ∂ 2 ηi ∂ 2 ηi + ∂x2 ∂y 2 34 CHAPTER 4. COMPUTATIONAL ALGORITHMS By integrating both side, we get ηin+1 − ηin ·N dΩ = ∆t· ∂ 2 ηi ∂ 2 ηi + ∂x2 ∂y 2 N· dΩ (4.15) Ω Now consider the identities N· ∂ ∂x ∂ηi ∂x = ∂ ∂ηi ∂N ∂ηi N· − ∂x ∂x ∂x ∂x N· ∂ ∂y ∂ηi ∂y = ∂ ∂ηi ∂N ∂ηi N· − ∂y ∂y ∂x ∂y Using these in the equation (4.15), we get ηin+1 − ηin ·N = ∆t. ∂ ∂x Ω Ω N ∂ηi ∂x + ∂ ∂y N ∂ηi ∂y − ∂N ∂ηi ∂N ∂ηi − ∂x ∂x ∂x ∂x (4.16) Applying the divergence theorem gives ηin+1 − ηin ·N = ∆t − ∂N ∂ηi ∂N ∂ηi − dΩ ∂x ∂x ∂y ∂y Ω Ω +∆t N ∂ηi ∂x ηx + N ∂ηi ∂y ηy ds (4.17) Γ The equation (4.17) is called the weak form of original equation. Finite Element Approximation: Consider a finite dimensional space and take the approximation of ηe , where ’e’ represents the element in number in the space, as NE e ηje Nje η = j=1 dΩ 35 CHAPTER 4. COMPUTATIONAL ALGORITHMS Substitute this in equation (4.17), we get NE l=1 NE Ωe ηle (n+1) − ηle (n) N dΩe = l=1 Ωe N. +∆t ∂N ∂Nle ∂x ∂x − ∆t ∂N ∂Nle ∂y ∂y ηle − ∂Nle e e η ηx + N ∂x l ηle dΩe ∂Nle e e (4.18) η ηy ds ∂y l Γe Consider the triangular element (i.e. N E=3). Now apply Galerkin Method, i.e., take the test function, N (x, y) as the shape function Nie (x, y), then the equation (4.18) can be expressed as 3 l=1 Ωe ηle (n+1) − ηle (n) N dΩe = ∆t × Ωe 3 Nie + l=1 Γe ∂Nie ∂Nle ∂x ∂x − ηle − ∂Nie ∂Nle ∂y ∂y ηle dΩe ∂Nle e e ∂N e ηl ηx + Nie l ηle ηye dS i = 1, 2, 3 (4.19) ∂x ∂y The last term in the above equation will be taken care automatically therefore we have 3 l=1 Ωe ηln+1 Nle Ni dΩe = ∆t × Ωe − ∂Nie ∂Nle ∂x ∂x ηle − ∂Nie ∂Nle ∂y ∂y ηle dΩe 3 ηle Nle Nie i = 1, 2, 3 (4.20) + l=1 Γe Consider right hand side of equation (4.20) 36 CHAPTER 4. COMPUTATIONAL ALGORITHMS (RHS)ei = ∆t × Ωe ∂Nie ∂Nle ∂x ∂x − ∂Nie ∂Nle ∂y ∂y ηle − ηle dΩe 3 ηle Nle Nie i = 1, 2, 3 + (4.21) l=1 Γe Thus using equation(4.21), the equation(4.20) can written in the matrix form as       2 N N N N N     2 1 3 1 1    (RHS)e1  η1e     Ωe Ωe  Ωe                                            2 e e N N N N N = (4.22)  1 2 3 2  η2   (RHS)2  2  Ωe  Ωe Ωe                                              e  e     2 η (RHS) N1 N3 N2 N3 N3 3 3 Ωe Ωe Ωe The integrals can be calculated by using formula, β Nlα Nm Nnγ dx dy = α!β!γ! (2∆) (α + β + γ + 2)! (4.23) Ωe and Ωe ∇Nl ∇Nm dx dy = bl bm + c l c m , 4∆ (4.24) where b l = y m − yn and c l = x n − xm Therefore Nle Nle dx dy = Ωe 2∆e ∆e = 4! 12 (4.25) 37 CHAPTER 4. COMPUTATIONAL ALGORITHMS The right hand side part of equation(4.21) can be calculated as follows (RHS)ei = ∆t × Ωe − ∂Nle ∂Nle ∂x ∂x ηle − ∂Nie ∂Nle ∂y ∂y ηle dΩe 3 ηl Nle Nie i = 1, 2, 3 + l=1 Γe which is equivalent to  (RHS)ei = ∆t  3 l=1 Ωe   (∇Ni .∇Nl ) dxdy  ηle +  3 l=1 Ωe  Ni Nl  (4.26) Using equations (4.23), (4.24) and (4.25) in the equation(4.26) we can write   3 3 1 (RHS)ei = ∆t − (4.27) Nie Nle dxdy  ηle (bi bl + ci cl ) ηle +  4∆ e l=1 l=1 Ωe This equation can be written in the matrix form as     2 2   b1 b2 + c 1 c 2 b1 b3 + c 1 c 3   η1    b1 + c 1       −∆t e  b b +c c  2 2 (RHS)i = b + c2 b2 b3 + c 2 c 3   η2  2 1 2 1 4∆e         2 2  b1 b3 + c 1 c 3 b3 b2 + c 3 c 2 b3 + c 3 η3        η1    2 1 1       ∆e   1 2 1  η +  2  12          1 1 2 η3  Substituting this value in to the equation (4.21), we get, (4.28) CHAPTER 4. COMPUTATIONAL ALGORITHMS ∆e 12       38 n n+1       2 2     η1  2 2 1  b1 b2 + c 1 c 2 b1 b3 + c 1 c 3   η1    b1 + c 1            −∆t    2 2 = 1 2 1   η2  b2 b1 + c 2 c 1 b + c2 b2 b3 + c 2 c 3   η2  4∆e               2 2    η3  b1 b3 + c 1 c 3 b3 b2 + c 3 c 2 b3 + c 3 η3 1 1 2 n      η1    2 1 1        ∆e   + 1 2 1   η2  (4.29) 12          1 1 2 η3  Equation (4.29) is a system of equations [A]e {x} = [B]e associated to element e. After generating system of equations associated to each elements, all of them are assembled which in turn results in large system of equation [A] x = [B]. Again this system becomes larger and larger as the number of grid point increases. And so the non stationary iterative solvers (Krylov subspace solvers) have to employ to solve it. 4.6 On the Implementation of Periodic Boundary Conditions: All the above methods employ a periodic boundary condition. The relevance and implementation details are given below. In materials science it is more meaningful to simulate a domain as part of the whole. In other words, the whole is made up of smaller domains like our representative zone of interest. This view is easily implemented by using periodic boundary conditions. When the boundary conditions on all the domain boundaries is periodic, in a more abstract sense, we are unifying ’holism’ into our reductionist simulation domain. In a more day to day example, CHAPTER 4. COMPUTATIONAL ALGORITHMS 39 the bigger domain is like a loaf of bread, where all the bread pieces are the same, except at the edges. And our domain of interest is focused on studying a slice of bread, in the center assuming that the shape of bread remains the same. In a mathematical settings, on a grid of size N ×N , one can write the differencing equations by assuming that the domain repeats itself, once it crosses the boundary. This is achieved by the modulus (%) operator in C language. For example the value of ηN +1 is not available, since it is not part of the domain. By applying periodic boundary condition, on any grid point i, we can apply i % N and equate it to an appropriate value which is known from within the domain. The value of ηN +1 is η1 , η0 is ηN and so on... The same idea holds good for both 2-d and 3-d settings. 4.7 On Solving Large-Scale System of Equations: The iterative methods are commonly used to solve the large-scale linear systems. These iterative schemes are classified as stationary and non stationary methods. In stationary iterative methods the current value of a variable depends only on the immediate previous level, whereas in non stationary methods current variable values are updated based on several of previous iteration values. In the first category the methods lies are Gauss-Seidel, Jacobi, successive over relaxation (SOR) etc. Methods like conjugate gradient method (CGM), conjugate gradient squared (CGS), conjugate gradient residual (CGR), biconjugate gradient (Bi-CGM), biconjugate gradient stabilized (Bi-CGSTAB), generalized minimal residual (GMRES) etc., are based on Krylov subspace [12, 13] and are nonstationary. Krylov Subspace Methods: These methods comes under the category of nonstationary iterative methods. Unlike CHAPTER 4. COMPUTATIONAL ALGORITHMS 40 in stationary methods these methods do not have an iteration matrix. CGM and GMRES are two such methods. These methods minimize at the mth iteration some measure of error over the affine space x0 + Kk where x0 is the initial iterate and the Km is the mth Krylov subspace defied as Km = Span{r0 , Ar0 , · · · Am−1 r0 } f or k ≥ 1 where r = b − Ax rk = b − Axk are the residual vectors. Conjugate Gradient Method : It is an effective method for symmetric positive-definite system. The method proceeds by generating vector sequence of iterate (i.e., successive approximations to the solution), residuals corresponding to the iterates, and search directions used in updating the iterates and residual. In every iteration of the method, the inner products are found in order to update the scalars which are defined to make the sequence satisfy certain orthogonal conditions. Details of the iteration scheme is as follows: For given the input, A, b, ǫ (tolerance) following step has to be done: 1. Set r0 = b − Ax0 2. Do while (ρk ) 2 > ǫ ||b||2 1 CHAPTER 4. COMPUTATIONAL ALGORITHMS 41 (a) If k = 0 then p0 = 0, else βk = rkT rk T rk−1 rk−1 (b) wk = Apk (c) αk = rkT rkT pT k wk (d) xk+1 = xk + αk + αk pk (e) rk+1 = rk − αk wk (f ) ρk+1 = ||rk+1 ||22 (g) k = k + 1 (4.30) The reliability and robustness of iterative methods can be improved dramatically by the use of preconditioned techniques. In fact, pre-conditioning is critical in making iterative methods practically useful [12, 13]. 4.7.1 Format for Sparse Matrix: In the case of Semi-implicit approach and finite element approach, the matrix, A, is highly sparse and big. Therefore a special data structure is required to store them and do the matrix operations. Compressed Sparse Row (CSR), Compressed Sparse Column (CSC), index format and Modified Sparse Row (MSR) etc. are some of common sparse matrix formats; in this work CSR format [9, 10] are implemented. In this format, a matrix is represented by three arrays one is data array aval(1 : nz) stores nz nonzero entries ai,j in row major order, where within each row the elements appear in the same order as in the dense equivalent. Another is integer array col(1 : nz), which gives the column index for each element in aval and one more integer array F IR(i : n + 1) (FIR stands for first in row), gives indices to aval such CHAPTER 4. COMPUTATIONAL ALGORITHMS 42 Figure 4.2: CSR format of a sparse matrix that FIR(i) denotes the position in aval where row i starts, and F IR(i+1)−F IR(i) gives the number of nonzero entries in row i. The F IR(n + 1) is set to total number of nonzero entries plus one. For example consider a 4×4 matrix as shown in Fig.4.2. 4.7.2 Sparse Matrix-Vector Multiplication in CSR Format: For multiplying a sparse matrix to a vector a particular row and number of nonzero entries form the array a has to be identified. It can be done by using the integer array F IR which gives the pointer to a row. The total number of nonzero element in a row i can be calculated as F IR(i + 1) − 1. After identifying these, the corresponding column index of these entries need to be found. The integer array col has the one-one correspondence to array aval. The algorithm for sparse matrix vector multiplication is given in eq (4.31). f or i = 1 to n f or j = F IR(i) to F IR(i + 1) − 1 ax(i) = a(j) × x(col(j)) end end (4.31) Chapter 5 Results and Discussion In this chapter we compare standard approaches vis-a-vis the proposed computational algorithms and their performance. Some discussions related to the implementation (during coding) of these algorithms are also presented. 5.1 Finite Difference Method (Explicit): Comparison exercise is the first mandatory step for all numerical simulations. Their credibility will have a serious dent, if it is not done in a systematic and thorough fashion. For this reason, we use the same parameter values as Fan and Chen [32] κi = 2.0 Li = 1.0 α = β = γ = 1 i = 1, 2 · · · · · · Q (5.1) The spatial time step is chosen ∆t =0.25. The periodic BCs are applied along the boundaries, implies the current simulation domain forms a repeatable segment of the whole. Initial condition is specified by assigning small random values to all field variables 43 CHAPTER 5. RESULTS AND DISCUSSION 44 at every grid point. Different sizes of computational cells are chosen for comparing the size dependence of grain growth. The solution vector is essentially the phase field variables values at every grid point of the computational domain, obtained at a specific iteration. To visualize the microstructure evolution using the orientation field variables the following function is defined Q ηk2 Ψr = (5.2) k=1 The function Ψr has the value 1.0 within grains and significantly smaller value at grain boundaries. The microstructure evolution is driven by the excess free energies associated with the grain boundaries which in turn results in an increase of grain size. The simulation results on a 256 × 256 grid for 48 order parameters(Q) are in Fig.5.1. In Fig.5.2 result for Q=60 is shown. It is clear by visually comparing the these two figures that the grain growth is faster in the case of Q=48 than in case of Q=60. It reflects the fact that chances of coalescence is higher in case of Q=48 than in Q=60. Unwanted and unphysical coalescence is an artifact of the simulations. To this end, in section 5.3 we present a method which can deal with large number of Q values (theoretically ∞). The microstructure evolution using a 512 × 512 grid with 36 order parameters is shown in Fig.5.3. For this evolution Fig. 5.4 depicts the average grain area and total time taken as a function of number of time step. The average grain area is obtained by dividing the total number of the grains with the corresponding grid size employed. Fig.5.5 shows the evolution contour on 512×512 grid for Q=48 and plots for average grain area and total time is shown in Fig. 5.6 45 CHAPTER 5. RESULTS AND DISCUSSION 250 250 200 200 150 150 100 100 50 50 50 100 150 200 250 50 100 t = 100 250 250 200 200 150 150 100 100 50 50 50 100 150 200 250 50 100 t = 3000 250 200 200 150 150 100 100 50 50 100 200 250 150 t = 7000 150 200 250 200 250 t = 5000 250 50 150 t = 1000 200 250 50 100 150 t = 10000 Figure 5.1: Temporal evolution of microstructure for FDM(explicit) for Q=48 46 CHAPTER 5. RESULTS AND DISCUSSION 250 250 200 200 150 150 100 100 50 50 50 100 t = 100 150 200 250 250 250 200 200 150 150 100 100 50 50 50 100 150 200 250 50 100 t = 1000 150 200 250 50 100 t = 5000 150 200 250 50 100 150 t = 10000 200 250 t = 3000 250 250 200 200 150 150 100 100 50 50 50 100 t = 7000 150 200 250 Figure 5.2: Temporal evolution of microstructure for FDM(explicit) for Q=60. CHAPTER 5. RESULTS AND DISCUSSION 47 Figure 5.3: Temporal evolution of microstructure for FDM (explicit) scheme for Q=36 48 CHAPTER 5. RESULTS AND DISCUSSION (b) (a) 1400 1600 Total Time (in min) Average Gain Area 1200 1800 Q = 36 1000 800 600 400 Q =36 1400 1200 1000 800 600 400 200 200 0 0 2000 4000 6000 8000 Number of Time Steps 10000 2000 4000 6000 8000 10000 12000 Number of Time Steps Figure 5.4: The average grain area and total time taken as function of time for FDM (explicit). An attempt is further made to demonstrate the evolution for higher resolution. In studies, most of the evolutions are generally shown for 512 × 512 grid. Here, for predicting grain growth with higher resolution a grid size of 1024 × 1024 is considered. The results are successfully generated. The Fig 5.7 shows its microstructure evolution as a function of time. It may, however, be noted in Fig 5.7 that this shows more clear picture of grain boundaries than as in Fig. 5.5. It is worth mention here in generating the growth shown in Fig. 5.7 the computational time taken is much higher than that of the growth shown in Fig. 5.5 and Fig. 5.2 etc. 5.2 FD with Operator Splitting Method: In the case of operator splitting method (OSM), once again validation is a key component to verify if the simulation results are accurate. To this end, we perform the simulations by bifurcating the simulations using OSM, after the nucleation 49 CHAPTER 5. RESULTS AND DISCUSSION 500 500 450 450 400 400 350 350 300 300 250 250 200 200 150 150 100 100 50 50 100 200 300 400 500 100 200 t=100 500 500 450 450 400 400 350 350 300 300 250 250 200 200 150 150 100 100 50 50 100 200 300 400 t=3000 100 500 500 450 450 400 400 350 350 300 300 250 250 200 200 150 150 100 100 50 50 100 150 200 250 300 t=7000 400 500 200 300 400 500 t=5000 500 50 300 t=1000 350 400 450 500 50 100 150 200 250 300 350 400 450 500 t=10000 Figure 5.5: Temporal evolution of microstructure for FDM (explicit) scheme for Q=48 50 CHAPTER 5. RESULTS AND DISCUSSION (a) (b) 1400 12000 Q=48 Titme Ttime (in min) Average Grain Area 1200 1000 800 600 400 8000 6000 4000 2000 200 0 0 10000 2000 4000 6000 8000 10000 12000 0 0 Number of Time Steps 2000 4000 6000 8000 10000 12000 Number of Time Steps Figure 5.6: The average grain area and total time taken as function of time for FDM (explicit). phase. This enabled a direct comparison of the results of OSM as against the explicit scheme. Note that, OSM allows a time step which is about four times the earlier scheme. the OSM is applied and the value of initial time step ∆t is changed when a well defined grain structure is formed. The choice of time step at which this method is applied depends on the new value of ∆t. For this study this method is implemented in three ways different in the value of ∆t is considered. Each case is described below: 5.2.1 Operator Splitting: Case 1 The OSM is applied at time after 2000 iteration with new value of ∆t = 4 ∗ ∆tinit , where ∆tinit is the original value of ∆t in FDM(explicit) case. Total 4000 iterations are run to complete the total 10000 iterations as of the FDM(explicit) case. For a value of Q =36, the temporal evolution of microstructure is generated using 512×512 grid. The generated growth is shown in the Fig. 5.8. The average grain area can be [...]... to that sharp interface formulation 2.5 Application of Phase Field Methods: A variety of material processing applications have been comfortably handled by Phase- field models The application of phase- field method have been focused on the three major materials processes: solidification, solid-state phase transformation, and grain growth and coarsening Examples of existing phase- field applications are:... Simulations 2.1 What are Materials? The Mathematical models that help us in understanding materials have far reaching ramification in design and development of novel materials Materials not only directly impact us, but advanced materials play a crucial, and enabling role underlying virtually all technologies A simple and operative definition of a material is to view it as a particular arrangement of atoms... as phase- separation or diffusion limited aggregation For instance, Cahn-Hilliard equation describes phase separation from a uniform binary mixed state to that of spatially separated multi -phase structure 1.1.1 Motivation: The simulation and modeling route enables Mathematicians to contribute to the understanding of material processes and to achieve desirable properties and gain insights Several unanswered,... transformations • Ferroelectric transformations • Phase transformations under an applied stress • Martensitic transformation in single and polycrystals Coarsening and Grain Growth • Coarsening • Grain growth in single -phase solid • Grain growth in two -phase solid • Anisotropic grain growth Chapter 3 Theory and Model Description Basic to phase field theories are continuous field variables φ1 (x, t), φ2 (x,...LIST OF FIGURES x 5.19 Comparison of average grain area as function of time for FDM (explicit) case and operator splitting method case 2 63 5.20 Comparison of average grain area as function of time for FDM (explicit) case and operator splitting method case 3 63 5.21 Comparison of average grain area as function of time for FDM (explicit) case and all three operator splitting... (Example: Diamond) For a given piece of material, if we find a collection of many small crystals or grains, it is called polycrystalline material, whose crystallographic orientation varies from grain to grain Modeling such materials with a view to understand their properties is still a great challenge Phase- field models developed in mid 90’s is a fertile ground for both Mathematicians and Physicists alike... coalescence of grains during grain growth Coalescence CHAPTER 3 THEORY AND MODEL DESCRIPTION 20 is the situation in which two grains which have the same order parameters, come into contact and instantaneously form a single large grain This leads to incorrect growth rates and unphysical grain shapes The likelihood of coalescence involving a given grain varies with the probability p(Q) that at least one of its... There have been some attempts to avoid this effect when using smaller Q values, via a dynamic reorientation algorithm [33] The quest for a multi-order parameter models with the framework to handle an unlimited number of phase field variables would be seen as a great asset by modelers Precisely we propose such an approach which can handle theoretically ∞ phase- field variables Chapter 4 Computational Algorithms... Typical image of grains [27] ture, where a few grains can be visually identified The capability of predicting equilibrium and non-equilibrium phase transformation phenomena at a microstructural scale is among the most challenging topics in materials science Therefore, it is highly desirable that we are able to understand the kinetics of microstructural evolutions 2.4 Phase Field Model: A phase field implicitly... Interfaces between phases is central to understanding the elegant features of phase- field models CHAPTER 2 MATERIAL SCIENCE AND SIMULATIONS 2.4.1 10 What is an Interface? An interface refers to a boundary between two different phases From a mathematical point of view these problems constitute a class of so-called moving boundary problems Usually analytical treatment of these type of problems is very ... that sharp interface formulation 2.5 Application of Phase Field Methods: A variety of material processing applications have been comfortably handled by Phase-field models The application of phase-field. .. The average grain area and total time taken as function of time for operator splitting case 60 5.16 The average grain area and total time taken as function of time for AIA-PT... equation for the order parameter at that point At later times in the simulations when the grain boundary area is a small fraction relative to the overall domain, an additional computational savings