JEST-M, Vol. 2, Issue 1, 2013 - MVJ College of Engineering

34 JEST-M, Vol. 2, Issue 1, 2013 

Micro-Structure Simulation using Cellular Automata: 

A Case Study 

Sireesha Kraleti 1 

Asst. Professor, IEM Department, MVJ College of 

Engineering, Bangalore-67, India 

siree.reesha@gmail.com 

Abstract—Dynamic re-crystallization governs the plastic flow 

behavior and the final microstructure of many crystalline 

materials during thermo-mechanical processing. Understanding 

the re-crystallization process is the key to linking dislocation 

activities at the microscopic scale to mechanical properties at the 

macroscopic scale. A modeling methodology coupling 

fundamental metallurgical principles with the cellular automaton 

(CA) technique is used to simulate the dynamic re-crystallization 

process. Cellular automata (CA) are a class of spatially and 

temporally discrete, deterministic mathematical systems 

characterized by local interaction and an inherently parallel 

form of evolution. Microstructure simulation predicts the 

nucleation and the growth kinetics of dynamically re-crystallized 

grains. Hence it can simulate different stages of micro-structural 

evolution during thermo-mechanical processing. This paper 

illustrates the simulated micro structure result of at selected 

regions on the thermo-mechanically deformed component. 

Index Terms—Cellular Automata, Microstructure Simulation, 

Dynamic Re-crystallization. (key words) 

I. INTRODUCTION 

Cellular automata (CA) are, fundamentally, the 

simplest mathematical representations of a much broader class 

of complex systems where, for the moment, "complex system” 

means any dynamical system that consists of more nonlinearly 

interacting parts. As such, CA have proven to be extremely 

useful idealizations of the dynamical behavior of many real 

complex systems, including physical fluids, neural networks, 

molecular dynamical systems, natural ecologies, military 

command and control networks, and the economy, among 

many others. Because of their underlying simplicity, CA are 

also powerful conceptual engines with which to study general 

pattern formation. They have already provided critical insights 

into the self-organization of chemical reaction-diffusion 

systems, crystal growth, pattern formation on seashells, and 

phase-transition-like phenomena in vehicular traffic flow, to 

name but a few examples. On a more practical side, CA may 

provide the basis for extremely powerful encryption 

algorithms, a subject about which there has recently been 

much heated debate. So, the cellular automata technique is 

used for the simulation of the micro-structure in the flow 

forming process. 

Cellular automata (often termed CA) are an 

idealization of a physical system in which space and time are 

Ravi Pullepudi 2 

Asst. Systems Engineer, TCS Engineering and Industrial 

Services, GR Tech. Park, Bangalore-67, India. 

ravipullepudi@gmail.com 

discrete, and the physical quantities take only a finite set of 

values. Although cellular automata have been reinvented 

several times (often under different names), the concept of a 

cellular automaton dates back from the late 1940s. During the 

following fifty years of existence, cellular automata have 

been developed and used in many different fields. 

Cellular automata (CA) are a class of spatially and 

temporally discrete, deterministic mathematical systems 

characterized by local interaction and an inherently parallel 

form of evolution. First introduced by von Neumann in the 

early 1950s to act as simple models of biological selfreproduction, 

CA are prototypical models for complex 

systems and processes consisting of a large number of 

identical, simple, locally interacting components. The study of 

these systems has generated great interest over the years 

because of their ability to generate a rich spectrum of very 

complex patterns of behavior out of sets of relatively simple 

underlying rules. Moreover, they appear to capture many 

essential features of complex self-organizing cooperative 

behavior observed in real systems. 

II. TERMINOLOGY 

Discrete lattice of cells: the system substrate consists 

of a one-, two- or three dimensional lattices of cells. 

Homogeneity: all cells are equivalent. 

Discrete states: each cell takes on one of a finite 

number of possible discrete states. 

Local interactions: each cell interacts only with cells 

that are in its local neighbourhood, discrete 

dynamics: at each discrete unit time, each cell 

updates its current state according to a transition rule 

taking into account the states of cells in its 

neighbourhood. 

III. WHY TO STUDY CA? [1] 

There are at least four partially overlapping 

motivations for studying CA, which we order according to 

increasing levels of theoretical significance 

As powerful computation engines 

As discrete dynamical system simulators 

Sireesha Kraleti,Ravi Pullepudi


As conceptual vehicles for studying pattern formation 

and complexity 

As original models of fundamental physics 

A. CA as Powerful Computation Engines 

СA allow very efficient parallel computational 

implementations to be made of lattice models in physics and 

thus for a detailed analysis of many concurrent dynamical 

processes in nature. Indeed, dedicated hardware represents one 

of the most promising practical applications of CA modeling. 

With the help of such hardware, many heretofore intractable 

but technologically important problems such as fluid flow near 

and around airplane wings, are becoming computationally 

accessible for the first time. 

B. CA as Discrete Dynamical System Simulators 

CA allows systematic investigation of complex 

phenomena by embodying any number of desirable physical 

properties. Reversible CA, for example, can be used as 

laboratories for studying the relationship between microscopic 

rules and macroscopic behavior- exact computability ensuring 

that the memory of the initial state is retained exactly for 

arbitrarily long periods of time. Discrete models of turbulence 

are particularly impressive in that they clearly show that very 

simple finite dynamical implementations of local conservation 

laws (defined so that the discrete system is computationally 

universal) are capable of exactly reproducing continuum 

system behavior on the micro scale. 

C. CA as Conceptual Vehicles for Exploring Pattern 

Formation 

CA can be treated as abstract discrete dynamical 

systems embodying intrinsically interesting, and potentially 

novel, behavioral features. The central motivation is to 

abstract the general principles governing self-organizing 

structure formation. Related interests include formal 

classification of the dynamical behavior of complex systems; a 

better understanding of the relationship between the dynamics 

of continuous and discrete systems; and the quantification of 

complexity-as a system property. All of these questions are 

particularly accessible through studying the generic behavior 

of СA systems. 

D. CA as Original Models of Fundamental Physics 

CA allows studies of radically new discrete 

dynamical approaches to microscopic physics, exploring the 

possibility that nature locally and digitally processes its own 

future states. This paper is devoted to a prolonged discussion 

of such potentially ground breaking models of physics. Using 

the fact that computationally universal systems are capable of 

arbitrarily complicated behavior (in the sense that they can 

mimic any computation performed by a conventional 

computer), the idea is to construct fundamentally discrete field 

theories to compete with existing continuous models. The 

emphasis in this class of models is emphatically not to 

construct a lattice-gauge-like theory; rather, in the same way 

as lattice gas CA successfully reproduce continuous fluid flow 

despite never having heard of the Navier-Stokes equations, so 

the hope is to abstract a set of microphysical laws that 

reproduce known behavior on the macro scale. 

IV. REQUIREMENTS OF CELLULAR AUTOMATA [2] 

A Cellular Automation generally requires 

A regular lattice of cells covering a portion of a d- 

dimensional space. 

A set ф (r', t) = { ф 1(r', t), ф 2(r', t), ф 3(r', t)................ 

ф m(r', t)} of Boolean variables attached to each site r 

of the lattice and giving the local state r’ of each cell 

at the time t = 0, 1, 2,.... 

A rule R = {R 1 , R 2 ..... R m } which specifies the time 

evolution of the state's ф (r',t) in the following way as 

shown in Eq. (4.1). 

(r ', t 1) R ( (r ', t), (r ' 1, t), (r ' 2, t),.... (r ' q, t)) 

j 

j 

Where in Eq. (1) r' k designate the cells 

belonging to a given neighborhood of cell r'. In the above 

definition, the rule R is identical for all sites and is applied 

simultaneously to each of them, leading to a synchronous 

dynamics. It is important to notice that the rule is 

homogeneous, that is it cannot depend explicitly on the cell 

position r. However, spatial (or even temporal) in 

homogeneities can be introduced by having some 

(r ') systematically 1 in some given locations of the lattice 

j 

to mark particular cells for which a different rule applies. 

Boundary cells are a typical example of spatial in 

homogeneity. Similarly, it is easy to alternate between two 

rules by having a bit which is 1 at even time steps and 0 at odd 

time steps. In our definition, the new state at time t + 1 is only 

a function of the previous state at time t. It is sometimes 

necessary to have a longer memory and introduce a 

dependence on the states at time t - l, t - 2,..., t -k. Such a 

situation is already included in the definition if one keeps a 

copy of the previous states in the current state. 

V. NEIGHBOURHOODS (NH) 

A cellular automata rule is local, by definition. The 

updating of a given cell requires one to know only the state of 

the cells in its vicinity. The spatial region in which a cell 

needs to search is called the neighborhood. In principle, there 

(1) 



is no restriction on the size of the neighborhood, except that it 

is the same for all cells. However, in practice, it is often made 

up of adjacent cells only. If the neighborhood is too large, the 

complexity of the rule may be unacceptable as the complexity 

usually grows exponentially fast with the number of cells in 

the neighborhood (NH). 

For two-dimensional cellular automata, two 

neighborhoods are often considered, the von Neumann 

neighborhood, which consists of a central cell (the one which 

is to be updated) and its four geographical neighbors north, 

west, south and east as shown in Fig. 1. The Moore 

neighborhood contains, in addition, second nearest neighbors 

north-east, north-west, south-east and south-west, that is a 

total of nine cells as shown in Fig. 2 

Fig. 1 Von Neumann NH 

Where in Figs.1 and 2, the shaded region indicates 

the central cell which is updated according to the state of the 

cells located within the domain marked with the bold line. 

Another useful neighborhood is the so-called 

Margolus neighborhood which allows a partitioning of space 

and a reduction of rule complexity. The space is divided into 

adjacent blocks of two-by-two cells. The rule is sensitive to 

the location within this so-called Margolus block, namely 

upper-left, upper-right, lower-left and lower-right. The way 

the lattice is partitioned changes as the rule is iterated. It 

alternates between an odd and an even partition, as shown in 

Fig. 3. As a result, information can propagate outside the 

boundaries of the blocks, as evolution takes place. The key 

idea of the Margolus neighborhood is that when updating 

occurs, the cells within a block evolve only according to the 

state of that block and do not immediately depend on what is 

in the adjacent blocks. To understand this property, consider 

the difference with the situation of the Moore neighborhood 

shown in Fig. 4.3 After iteration, the cell located on the west 

of the shaded cell will evolve relatively to its own Moore 

neighborhood. Therefore, the cells within a given Moore 

neighborhood evolve according to the cells located in a wider 

region. The purpose of space partitioning is to prevent these 

"long distance" effects from occurring. 

In Fig.3, the lattice is partitioned in 2 x 2 blocks. Two 

partitioning are possible - odd and even partitions, as shown 

by the blocks delimited by the solid and broken lines, 

respectively. The neighborhood alternates between these two 

situations, at even and odd time steps. Within a block, the cells 

are distinguishable and are labeled ul, ur, 11 and lr for upperleft, 

upper-right, lower-left and lower-right. The cell labeled lr 

in the figure will be become ul, at the next iteration, with the 

alternative partitioning. 

VI. BOUNDARY CONDITIONS 

Fig. 2 Moore NH 

In practice, when simulating a given cellular 

automata rule, one cannot deal with an infinite lattice. The 

system must be finite and have boundaries. Clearly, a site 

belonging to the lattice boundary does not have the same 

neighborhood as other internal sites. In order to define the 

behavior of these sites, a different evolution rule can be 

considered, which sees the appropriate neighborhood. This 

means that the information of being, or not, at a boundary is 

coded at the site and, depending on this information, a 

different rule is selected. Following this approach, it is also 

possible to define several types of boundaries, all with 

different behavior. Instead of having a different rule at the 

limits of the system, another possibility is to extend the 

neighborhood for the sites at the boundary. 

Fig. 3 Margolus NH 



pseudorandom bit which is 1 with probability 1/2. This 

mechanism can mimic a probabilistic cellular automaton. 

VIII. CASE-STUDY- THERMO-MECHANICALLY DEFORMED 

COMPONENT MICROSTRUCTURE SIMULATION 

Fig. 4 Types of boundary conditions 

The various possible types of boundary conditions 

obtained by extending the neighborhood for one dimensional 

lattice are as illustrated in Fig.4. The shaded block in Fig. 4 

represents a virtual cell which is added at the extremity of the 

lattice (left extremity, here) to complete the neighborhood. A 

very common solution is to assume periodic (or cyclic) 

boundary conditions, that is one supposes that the lattice is 

embedded in a torus-like topology. In the case of a twodimensional 

lattice, this means that the left and right sides are 

connected, and so are the upper and lower sides. We assume 

that the lattice is augmented by a set of virtual cells beyond its 

limits. A fixed boundary is defined so that the neighborhood is 

completed with cells having a pre-assigned value. An 

adiabatic boundary condition (or zero-gradient) is obtained by 

duplicating the value of the site to the extra virtual cells. A 

reflecting boundary amounts to copying the value of the other 

neighbor in the virtual cell. The nature of the system which is 

modeled will dictate the type of boundary conditions that 

should be used in each case. 

VII. TYPES OF CELLULAR AUTOMATA 

According to its above definition, a cellular 

automaton is deterministic. The rule R is some well-defined 

function and a given initial configuration will always evolve 

the same way. However, as we shall see later on, it may be 

very convenient for some applications to have a certain degree 

of randomness in the rule. For instance, it may be desirable 

that a rule selects one outcome among several possible states; 

with a probability P. Cellular automata whose updating rule is 

driven by external probabilities are called probabilistic cellular 

automata. On the other hand, those which strictly comply with 

the definition given above are referred to as deterministic 

cellular automata. 

In practice, the difference between probabilistic and 

deterministic cellular automata is not so important. 

Randomness enters into the rule through an extra bit which, at 

each time step, is 1 with probability p and 0 with probability 1 

- p, independently at each lattice cell. The question is then 

how to generate a random bit. The very simple deterministic 

cellular automata rules have an unpredictable behavior that is 

there is no way to know what state 

a cell will assume at a future stage, unless the evolution is 

actually performed. Such a rule can be used to produce a 

Dynamic re-crystallization (DRX) is commonly associated 

with high temperature plastic deformation of metallic 

materials with a relatively low-to-medium stacking fault 

energy. 

Dynamic re-crystallization generally has the following 

characteristics [3] 

A critical dislocation density is required for the 

nucleation of DRX 

Dynamically re-crystallized grains are equaled, and 

the mean grain size remains constant, i.e., grain 

growth does not occur during deformation 

Dynamically re-crystallized grain size is a strong 

function of the flow stress and a weak function of the 

deformation temperature. 

As the Discrete Lattice model is computationally more 

intensive than the statistical model, it is impractical to perform 

a microstructure simulation at every element / node. Rather, 

the simulation is performed at various user-selected points. 

Thus, selecting the simulation should be more closely 

analyzed with respect to microstructure evolution – either at 

known “hot-spots” in the real work pieces, for example, or at 

regions of widely different state variable values of strain, 

strain rate, or temperature. Since the FEM inputs for the 

microstructure module are presently exclusively these state 

variables, it is redundant to perform simulations in regions of 

the work piece which experience identical state variable 

histories. 

Another reason to select multiple modeling points across the 

work piece is if the initial microstructures at those locations 

are known to differ – for example, with different initial grain 

sizes, grain shapes, grain disorientations, precipitate sizes or 

volume fractions, second phase sizes or volume fractions, 

and/or dislocation density. In this case, even if the state 

variable histories are identical, the results of the 

microstructure evolution. 

Simulation may be different due to the difference in initial 

microstructure conditions. Regardless, due to the extra 

computational time of the microstructure modeling module, 

care should be taken when considering how many, and in what 

locations, the simulations should be performed. 



Fig. 1. Fig5. (a). Grain orientation 

(b).Dislocation density 

(c). Grain boundaries 

(d). Grain orientation + Grain boundaries 

Fig.5 shows the simulated micro structure result of the 

selected point 1 on the thermo-mechanically deformed 

component. 

IX. CONCLUSION: 

A CA technique coupled with metallurgical principles has 

enabled accurate simulation of micro structural evolution 

during DRX of metallic materials. This technique is used for 

simulating the microstructure of the thermo mechanically 

processed components. Microstructure simulation predicts the 

nucleation and the growth kinetics of dynamically 

recrystallised grains, orientation and dislocation density. 

REFERENCES 

[1]. Andrew Ilachinski, A text book on " Cellular Automata-A 

Discrete Universe ". World Scientific Publishing Co. Pte. 

Ltd.,2001. 

[2]. Bastien Chopard and Michel Droz, A text book on 

"Cellular Automata - Modeling of Physical Systems". 

Cambridge University press, 1998. 

[3]. R. Ding, Z.X. Guo “Modelling of dynamic recrystallization 

using an extended cellular automaton 

approach”. Computational Materials Science 23 (2002) 209– 

218. 

[4]. Ravi Pullepudi, A. Gopala Krishna “Finite Element 

Simulation of a Flow Forming Process for an Aerospace 

Component”, M.Tech Thesis, JNTU K, Kakinada, 2011. 

[5]. Google and Wikipedia.

JEST-M, Vol. 2, Issue 1, 2013 - MVJ College of Engineering

Create successful ePaper yourself

Delete template?

Save as template?