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Ramirez-Lopez A.,Autonomous University of Mexico City | Ramirez-Lopez A.,Technological Autonomous Institute of Mexico | Ramirez-Lopez A.,Mexican Institute of Petroleum | Romero-Romo M.A.,Technological Autonomous Institute of Mexico | And 4 more authors.
International Journal of Minerals, Metallurgy and Materials | Year: 2012

Computational models are developed to create grain structures using mathematical algorithms based on the chaos theory such as cellular automaton, geometrical models, fractals, and stochastic methods. Because of the chaotic nature of grain structures, some of the most popular routines are based on the Monte Carlo method, statistical distributions, and random walk methods, which can be easily programmed and included in nested loops. Nevertheless, grain structures are not well defined as the results of computational errors and numerical inconsistencies on mathematical methods. Due to the finite definition of numbers or the numerical restrictions during the simulation of solidification, damaged images appear on the screen. These images must be repaired to obtain a good measurement of grain geometrical properties. Some mathematical algorithms were developed to repair, measure, and characterize grain structures obtained from cellular automata in the present work. An appropriate measurement of grain size and the corrected identification of interfaces and length are very important topics in materials science because they are the representation and validation of mathematical models with real samples. As a result, the developed algorithms are tested and proved to be appropriate and efficient to eliminate the errors and characterize the grain structures. © 2012 University of Science and Technology Beijing and Springer-Verlag Berlin Heidelberg.


Ramirez-Lopez A.,Metropolitan Autonomous University | Ramirez-Lopez A.,Technological Autonomous Institute of Mexico | Ramirez-Lopez A.,Mexican Institute of Petroleum | Palomar-Pardave M.,Metropolitan Autonomous University | And 4 more authors.
International Journal of Minerals, Metallurgy and Materials | Year: 2012

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, including 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 generate 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 routines 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 function 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 computational array and displayed on the computer screen placing color pixels on the corresponding position forming an image as is done in cellular automaton. © 2012 University of Science and Technology Beijing and Springer-Verlag Berlin Heidelberg.

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