A cellular automata model for simulating grain structures with straight and hyperbolic interfaces

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.