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Copyright 2005 ASM International. This paper was published in Journal of Phase Equilibria and Diffusion, Vol. 26, Issue 4, pp and is made available as an electronic reprint with the permission
Copyright 2005 ASM International. This paper was published in Journal of Phase Equilibria and Diffusion, Vol. 26, Issue 4, pp and is made available as an electronic reprint with the permission of ASM International. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplications of any material in this paper for a fee or for commercial purposes, or modification of the content of this paper are prohibited. JPEDAV (2005) 26: DOI: / X /$19.00 ASM International Modeling the Recrystallization Process Using Inverse Cellular Automata and Genetic Algorithms: Studies Using Differential Evolution Tushar D. Rane, Rinku Dewri, Sudipto Ghosh, Kishalay Mitra, and N. Chakraborti (Submitted July 5, 2004; in revised form April 15, 2005) Basic and Applied Research: Section I An inverse modeling approach was taken up in this work to model the process of recrystallization using cellular automata (CA). Using this method after formulating a CA model of recrystallization, differential evolution (DE), a real-coded variant of genetic algorithms, was used to search for the value of nucleation rate, providing an acceptable matching between the theoretical and experimentally observed values of fraction-recrystallized (X). Initially, the inverse modeling was attempted with a simple CA strategy, in which each of the CA cells had an equal probability of becoming nucleated. DE searched for the value of the nucleation rate yielding the best results for single-crystal iron at 550 C. A good match could not be simultaneously achieved this way for the early stages of recrystallization as well as for the later stages. To overcome this difficulty, the CA grid was divided into two zones, having lower and higher probabilities of nucleation. This resulted in good correspondence between the predicted and experimental values of X for the entire duration of recrystallization. The introduction of a distribution in the probability of nucleation made the model even closer to the actual process, in which the probability of nucleation is often nonuniform due to nonuniformity in dislocation density as well as the presence of grain/interface boundaries. 1. Introduction Thermomechanical processing of a material is carried out to tailor its microstructure and texture, which, in turn, helps to achieve the desired properties. Recrystallization is one of the mechanisms by which the microstructure and texture are altered during thermomechanical processing. It occurs by the nucleation and growth of dislocation-free grains within a region of high dislocation density. Although studied by numerous researchers worldwide, there are still a number of issues related to the process of recrystallization that have defied precise analysis. For example, a reasonably accurate experimental determination of the nucleation rate and its mesoscopic distribution is not possible as the sizes of the critical nuclei are too small, and, although growing, they tend to merge with the other growing nuclei as well. The complexities associated with a theoretical evaluation of this parameter are also enormous. Thus, in a large number of recrystallization models that exist in the literature, the value of the nucleation rate, in most cases, has been arbitrarily chosen. [1-16] In this study, we have used an evolutionary computing approach to circumvent the problem. At the core of our Tushar D. Rane, the Department of Mechanical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai , India; Rinku Dewri, the Department of Mathematics and Computing, and Sudipto Ghosh and N. Chakraborti, the Department of Metallurgical and Materials Engineering, Indian Institute of Technology, Kharagpur , India; and Kishalay Mitra, Manufacturing Industrial Practice, Tata Consultancy Service, 54B Hadapsar Industrial Estate, Pune , India. Contact simulation runs a cellular automata (CA) scheme that is further augmented through the application of genetic algorithms (GAs), which provide the CA model with values of the model parameters, and also monitor its output. Genetic algorithms are biologically inspired computing techniques that come in many different forms. [17,18] What we have used here is known as differential evolution (DE). [19,20] Very limited work has been carried out utilizing GA within the CA framework. [21] It has not been done, so far, for modeling recrystallization or any other process involving a microstructural transformation. For further clarity about the adopted methodology, brief outlines of both CA and DE are provided below. 2. About Cellular Automata CA [22 26] have been compared with a synthetic model of the universe in which the physical laws are expressed in terms of simple local rules in a discrete space-time structure. The efficacy of CA techniques lies in the fact that they can predict the behavior of a system at the macroscopic level when the rules governing the system at a microscopic level are known. They have already been used for modeling a large number of physical systems belonging to diverse disciplines like biology, chemistry, geology, and biochemistry. [22-26] In material science, the CA techniques are initially applied to problems related to solidification processes. [27,28] The methodology has been extended to phase transformation in subsequent studies, and recrystallization is one area that has been investigated in the process. [10-16] As indicated before, CA deal with a discrete dynamic system, [22-26] the behavior of which is completely specified Journal of Phase Equilibria and Diffusion Vol. 26 No Section I: Basic and Applied Research in terms of some local rules. To be more specific, CA deal with an array of cells the evolution of which is characterized by three features: the state of the cells; their neighborhood; and the rules for their transition. To construct any CA to simulate a specific problem, one needs to make a number of choices, the first being the selection of specific lattice geometry. Cellular automata require this lattice to be regular. In fact, it can be a linear (one-dimensional), triangular, square, hexagonal (two-dimensional [2-D]), or cubic (threedimensional) array of cells. Once the lattice, L, is decided upon, we choose a neighborhood in which cells can interact. The neighborhood of a cell is described as a set of cells surrounding it. It is further elaborated in Fig. 1. For the nearest neighbors on a square lattice, the neighborhood, which is called the von-neumann neighborhood of radius r, is described by: N i,j = k,l L : k i + l j r Another common neighborhood is the Moore neighborhood of radius r, defined as: Fig. 1 Examples of different neighborhoods N i,j = k,l L : k i r and l j r In the formal definition of CA, one usually requires the lattice to be infinite in all dimensions. For the considerations of computability and complexity, this is reasonable and necessary. However, it is difficult to simulate a truly infinite lattice on a computer, and therefore, one needs to prescribe some boundary conditions. Furthermore, the system that one wants to simulate may also require certain natural boundaries. Periodic, reflexive and fixed value boundaries are some of the options that are available in this scenario. This is elaborated in Fig. 2. Along with the boundary conditions, the initial conditions are also required to start a CA simulation. In most cases the initial condition significantly influences the subsequent evolution. The initial conditions can be very specially constructed or, depending upon the nature of the problem, they can be randomly generated as well. One important consideration in the generation of initial conditions is that many CA rules conserve some quantities like the total number of particles, the total momentum, or energy and so on. Often some spurious quantities (i.e., quantities that are not conserved in the system to be modeled) can be conserved as well. In generating the initial conditions, care must be taken that the intended values of the conserved quantities are reached, especially when the random initial conditions are used, and that the spurious conserved quantities do not produce any undesired effects. According to the definition of CA, each cell is a finite automaton, and therefore the set of states has a finite size. In addition, this set is usually kept fairly small because it simplifies the specification of rules, whereas a reason to use a large number of states can arise in some situations for better approximating a continuous system. A common construction of CAs is required to have several variables in each cell that must be stored in the state. In this case, we construct the state set S as the cross product of the sets for each variable. Formally, the state set is described as S=UXV, where U Fig. 2 Different types of boundary condition and V are the set of values that the variables u and v can take. The size of the set S is S U V. The most important aspect of CA is the transition rule or transition function. This is what affects the evolution most, and it depends on the lattice geometry, the neighborhood, and the state set. The transition rule determines how the state of a cell can change depending on the state of its neighbors. Rules can be directly specified by writing down the outcome of each possible configuration of states in the neighborhood. They can also be probabilistically specified where the outcome can be one of many states with associated probabilities. Formally, a transition rule can be expressed as: T : S n XS s P where S and n are the state set and the number of cells in the neighborhood. s P is a state from S, and it can occur with probability P. For direct specification of the rules, there will be only one state s S with P 1, and zero for the rest, for each possible configuration of the neighborhood. However, for probabilistic transition a positive value can be associated with one or more states in S. In fact, CA can be considered an idealization of a physical system in which the space and time are discrete, and the physical quantities take only a finite set of values. In many real systems, the time evolutions of physical quantities are often governed by nonlinear partial differential equations. Owing to such nonlinearities, the solutions of these systems can often be very complex. Cellular automata provide an alternative approach for studying the behavior of such dynamic systems, subjecting them to an easier analysis by virtue of their inherent simplicity. 312 Journal of Phase Equilibria and Diffusion Vol. 26 No Basic and Applied Research: Section I 3. Basic Differential Evolution The most common forms of GAs involve a binary encoding of the problem variables. An individual is formed by concatenation of the variables in their binary form. A randomly generated set of individuals constitutes the initial population containing a number of trial solutions. This population now evolves from generation to generation through some repeated, well-devised applications of the selection, crossover, and mutation operators, each devised analogously to their biological counterparts. Following a Darwinian framework, the individuals with higher fitness (i.e., the relatively better solutions) continue to emerge, survive, and reproduce until their ultimate convergence. Although highly robust and ubiquitous, the binary GAs are not devoid of problems. As discussed elsewhere, [18] they often require storage of very large binary arrays, in addition to repeated variable mapping between the binary and real space. Furthermore, in a situation known as Hamming cliff, [29,30] they become highly sluggish and often unsuitable for an essentially real variable problem. In DE, such difficulties are easily bypassed by doing away with binary representation altogether. A population of real-coded solution vector is allowed to evolve in DE, and both the crossover and mutation operations are conducted using a realcoded procedure. The crucial idea behind DE is to generate new trial variable vectors using direction information from the existing vectors. [19,20] The method is initiated by adding a weighted difference between two existing vectors to a third member, which essentially constitutes its mutation process. Through a specially designed crossover operation, the mutated vector is now recombined with a fourth member of the population to yield an offspring, and following a greedy strategy, the child replaces the parent if, and only if, it obtains a better fitness. DE starts by randomly generating a population of realcoded solution vectors. For example, in the case of a fourvariable problem, X i, [x 1,x 2,x 3,x 4 ], will denote a typical solution vector in which the components x 1,x 2,x 3, and x 4 are the real-coded representation of the problem variables at a particular generation. A generation count G is maintained to keep track of the number of cycles completed during the evolutionary process. The initial population corresponds to generation G 0. Each of the members is then assigned a fitness value that represents how good the solution is in the problem domain. The concept of fitness here is essentially the same as that in the other forms of GAs. Next, for each vector X i,g, the corresponding mutated vectors are generated using three other randomly picked individuals such that: initial generations, the population remains random, and, consequently, the differential term (X r2,g X r3,g ) remains large, resulting in a larger amount of mutation. As the population moves toward convergence, the system requires only a small amount of mutation, which is automatically achieved in DE because the differential term also tends to become small in such cases. Furthermore, by adjusting the parameter F, one can easily extend or contract the search domain. These are two additional advantageous features of DE that are normally not available in the binary coded GAs. A specially designed crossover operator now takes over to introduce diversity into the new parameter vector V. It involves probabilistically replacing a few elements of the vector with original values from X i,g, to form the final trial vector U: u j = y j, ifr j P cr = x i, ifr j P cr where x j belongs to X i,g, r j is a random number, P cr is the crossover probability, and u i and v i denote the typical components of the vectors U and V. Additionally, it is always ensured that U is able to retain at least one component of the mutated vector V. A fitness value is computed for this trial vector as well. A comparison of the fitness values is now made between the parent X i,g and the child U, and the one having better fitness is passed on to the next generation as X i,g+1. This completes one generation of the evolutionary process, and the whole process is repeated with this newly generated population. The same procedure continues until the solution vectors converge. The basic operations involved in DE are shown schematically in Fig. 3 for an arbitrary variable problem. 4. Modeling the Recrystallization Process Models of recrystallization can be divided into two broad classes: analytical and probabilistic. Analytical models are based on Johnson-Mehl-Avrami-Kolmogorov relationship. V = X rl,g + F. X r2,g X r3,g where each one of X r1,g,x r2,g, and X r3,g are randomly selected, but mutually different, vectors from the population. F is a real and constant factor that controls the amplification of the differential variation. The term F(X r2,g X r3,g ) determines the extent of the mutation of the vector X r1,g. The mutation here is self-adaptive in nature. For the Fig. 3 Mutation and crossover operations in DE Journal of Phase Equilibria and Diffusion Vol. 26 No Section I: Basic and Applied Research Monte-Carlo and CA models are probabilistic approaches. A CA model of recrystallization has certain advantages over other models. One of the important advantages of a CA simulation is that it can be directly related to the evolution of microstructure and features like the spatial distribution of a second phase can be demonstrated convincingly. Cellular automata models have been successfully used in predicting the variation of fraction-transformed (X) as a function of time during recrystallization. [10-16] The rate of nucleation is one of the parameters in the CA models. However, as indicated before, a reasonably accurate estimation of the rate of nucleation is not possible because of several constraints. Thus, an evolutionary inverse CA modeling was undertaken in this study to estimate the nucleation rate and the mesoscopic distribution of recrystallization probability on the basis of the available experimental data concerning the variation of X with time. The objective of inverse modeling is to estimate the input parameter settings for which the model prediction closely matches the corresponding experimental results. Therefore, it is essentially a problem of optimization in which we search for the parameter settings that would result in a minimum deviation of the model prediction. Thus, in this case, the set of parameters constitutes the variable vector used in the optimization scheme. Because the search space for the CA parameters is quite wide, the need for a heuristic approach was automatically felt. The method of DE was used to find the right choice of the parameter vector that would minimize an error function. On an industrial scale, the parameters obtained by inverse calculation can be used for the prediction of X and also the spatial distribution of the recrystallized zones as a function of time. This basic methodology is further elaborated below. 4.1 The Implementation of Cellular Automata In the present work, the CA represent a small portion of the 2-D cross section of the material undergoing recrystallization, and it is composed of a 2-D array of cells. The 2-D array consists of a total of 10 6 cells to represent an area of 0.25 cm 2. Thus, cell, the length (or breadth) of each cell, is equal to 5 m, providing enough accuracy to the model. Each of the squares can be in one of the two states: recrystallized or un-recrystallized. Initially (i.e., at time 0), all of the cells are in an un-recrystallized state. In each subsequent time step, few cells are transformed into the recrystallized state from their un-recrystallized state. The CA model predicts the distribution of the recrystallized region at different time steps by adopting the following strategy. Transformation of an unrecrystallized cell to a recrystallized one, with the increment in the time step, occurs in two different ways in our model. Few cells are randomly chosen and transformed into recrystallized cells, if they are found to be unrecrystallized. Physically, this simulates the process of nucleation. Thus, each unrecrystallized cell has a probability of nucleation, which is given by the number of nucleation events for each time step increment (n r ) divided by the total number of cells. The second way in which the transformation of the cells occurs is by the growth of the recrystallized region in the Fig. 4 CA, which simulates the growth phenomenon of the recrystallized zone in a real material. During this process, the unrecrystallized cells within a neighborhood of recrystallized cells transit to the recrystallized state. Because the neighborhood can be of many different types, its proper description warrants further clarification. In the present work, each cell in the recrystallization CA model is assumed to be associated with an extended Moore neighborhood, as shown in Fig. 4. The extended Moore neighborhood consists of 24 cells around the central cell as shown in Fig. 4. This neighborhood was chosen to incorporate the slowing down of the growth process, as has been done by Hesselbarth and Gobel. [10] This will be explained later. For defining the neighborhood of the boundary cells, a periodic boundary condit
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