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International Journal of Minerals, Metallurgy and Materials

Volume 19, Number 8, Aug 2012, Page 699

DOI: 10.1007/s12613-012-0616-0

Corresponding author: A. Ramírez-López E-mail: adaramil@yahoo.com.mx

A cellular automata model for simulating grain structures with straight and

hyperbolic interfaces

A. Ramírez-López

1, 2, 3)

, M. Palomar-Pardavé

, D. Muñoz-Negrón

, C. Duran-Valencia

S. López-Ramirez

, and G. Soto-Cortés

1) Materials and Energy Department, Metropolitan Autonomous University, Mexico

2) Department of Industrial Engineering, Technological Autonomous Institute of Mexico, Mexico

3) Department of Molecular Engineering, Mexican Institute of Petroleum, Mexico

(Received: 11 May 2011; revised: 4 July 2011; accepted: 11 July 2011)

Abstract: A description of a mathematical algorithm for simulating grain structures with straight and hyperbolic interfaces is shown. The

presence of straight and hyperbolic interfaces in many grain structures of metallic materials is due to different solidification conditions, in-

cluding different solidification speeds, growth directions, and delaying on the nucleation times of each nucleated node. Grain growth is a

complex problem to be simulated; therefore, computational methods based on the chaos theory have been developed for this purpose.

Straight and hyperbolic interfaces are between columnar and equiaxed grain structures or in transition zones. The algorithm developed in this

work involves random distributions of temperature to assign preferential probabilities to each node of the simulated sample for nucleation

according to previously defined boundary conditions. Moreover, more than one single nucleation process can be established in order to gen-

erate hyperbolic interfaces between the grains. The appearance of new nucleated nodes is declared in sequences with a particular number of

nucleated nodes and a number of steps for execution. This input information influences directly on the final grain structure (grain size and

distribution). Preferential growth directions are also established to obtain equiaxed and columnar grains. The simulation is done using rou-

tines for nucleation and growth nested inside the main function. Here, random numbers are generated to place the coordinates of each new

nucleated node at each nucleation sequence according to a solidification probability. Nucleation and growth routines are executed as a func-

tion of nodal availability in order to know if a node will be part of a grain. Finally, this information is saved in a two-dimensional computa-

tional array and displayed on the computer screen placing color pixels on the corresponding position forming an image as is done in cellular

automaton.

Keywords: grain growth; interfaces; grain size and shape; computational methods; algorithms; cellular automata; computer simulation

1. Introduction

Simulation of grain structures is a very important topic in

materials science. Metallic pieces are frequently processed

and formed by foundry, laminated, rolling, etc. During manu-

facturing and processing, different grain sizes and mor-

phologies are obtained according to particular solidification

conditions and thermal treatments. Fractals, stochastic

methods, cellular automata, models based on Monte Carlo,

random walk, and the chaos theory have been employed to

simulate the anisotropic features of grain structures and the

phenomena related such as recrystallization and grain de-

formation during processing.

Many authors [1-15] have developed mathematical mod-

els to simulate grain growth during solidification of metallic

materials (metals and alloys). The beginners [2] used basic

geometrical models due to the limited computational capaci-

ties. Nevertheless, the development of new mathematical

methods, more efficient algorithms, and the increment in

computational data speed and storage capacities have been

possible to solve complex problems that involve no-regular

geometries and random processes using Monte Carlo [1, 2, 4,

7] and cellular automaton [5, 8, 11-15]

methods in problems

related to materials science. Some of these authors [1, 4-8,

700 Int. J. Miner. Metall. Mater., Vol.19, No.8, Aug 2012

10-12] have developed models for solidification processes

as a function of solid and liquid fractions for different metals.

Others [1-4, 6, 9, 12-15] have been working in models to

simulate dendrite growth, which is the basic structure in

primary metallic products obtained after foundry.

In a previous work, Ramírez et al. [15] explained the de-

velopment of a basic computational algorithm to create

grain structures as a function of an initial stochastic tem-

perature distribution during the solidification of a squared

metallic sample. Heterogeneous distributions of temperature

provide nodes with particular probabilities to be nucleated.

The results are zones with big and small grains. The assig-

nation of preferential growth directions gives samples with

different grain sizes. Nevertheless, in the original algorithm,

there was only one single nucleation execution.

The grain structures formed in metals are heterogeneous

and the grain morphology can also be very different due to

the factors involved in the manufacturing process. Grains

with equiaxed and columnar morphologies, with different

sizes and distributions, can be simulated as shown in Figs.

1(a) and 1(b), respectively, using the original algorithm de-

veloped [15]. Nevertheless, the interfaces between the grains

(grain boundaries) are always straight (straight lines for

two-dimensional (2D) models and flat planes for three-di-

mensional (3D) models) as shown in better detail in the

close-ups, which show the boundaries between these. This

work describes an improved algorithm to generate more

complex 2D grain structures. The new algorithm includes

multiple nucleation sequences to generate different grain

sizes and hyperbolic interfaces between them.

Fig. 1. Different grain structures with straight interfaces: (a) equiaxed grains; (b) columnar grains.

Straight lines are presented in grain boundaries during the

evolution of the simulation as shown in Figs. 1(a) and 1(b).

These are formed because all the nodes are nucleated in only

one single sequence and because all of them have the same

growth rate. Nevertheless, this does not happen during real

solidification.

Fig. 1(a) shows an equiaxed grain structure. This kind

resulted after the metal is foundry and cast. Grain size dis-

tribution is frequently a function of the nucleation node dis-

tribution in the sample and the growth procedure. Neverthe-

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