Simulation of metal solidification using a cellular automaton
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I.
INTRODUCTION
C E L L U L A R automata (also known as tesselation automata [u) were introduced by Von Neumann in the early 1960s as a technique for modeling physical phenomena. 12l In this approach, the physical system of interest is divided into elements, and each element is allowed to interact iteratively with its neighbors according to some set of rules. The elements are usually mapped to the graphics display of a computer, and the development of the model system is viewed as time passes. The lattice over which the cells of the cellular automaton (CA) are distributed is usually in position space, with the iteration of the CA corresponding to the evolution of the system with time. Square or cubic cells are the most common, but other variations are possible. 13l One-dimensional CAs have been comparatively thoroughly studied from a theoretical point of view, t2] but two- or three-dimensional CAs appear to be of greater relevance in the modeling of natural phenomena. A literature search revealed that CAs have been used by biologists for simulating the growth and spread of living organisms, and in the earth sciences for modeling fractures in the Earth's crust, the flow of lava, transport of fluids through porous rocks, earthquakes, chemical reactions on the surface of mineral particles, the erosion of land forms, and the urbanization of communities. Other areas of application include astronomy and astrophysics, nonlinear optics, and chemistry. There also appears to be a body of literature devoted to the purely theoretical study of CAs. In materials science, CAs have been applied to the recrystallization of alloys, t4.51 the growth of martensite, t6,7,8l dislocation interactions,t91 inhomogeneous slip in metals, [1~ and, recently, solidification, m-~41 Other examples of applications for CAs are listed by Wolfram [:l and Pickover. m This article presents an example of a CA suitable for simulating the solidification process. The effects of varying selected parameters are examined, and the resuits are compared to the situation for real systems. Because CAs may be unfamiliar to some readers, the experimental technique is described in some detail. M.B. CORTIE, Assistant Director, is with the Physical Metallurgy Division, Mintek, Randburg 2194, South Africa. Manuscript submitted March 4, 1993. METALLURGICAL TRANSACTIONS B
II.
BACKGROUND
A. The Solidification of Metals In the conventional approach, solidification and many other natural phenomena are modeled using differential equations. For example, the variation of temperature with time in a volume of liquid metal may be estimated from integration of the unsteady heat conduction equation over time and space:
OT --
Ot
qs = V(a.
VT) + - -
p.c~
[1]
where p is density, cp is the specific heat capacity, qs is the latent heat of solidification, t is time, ce is thermal diffusivity, and T is temperature./15] (Equation [1] assumes that p and cp are constant.) However, this approach generally yields an approximation of the position of the solidification front with time
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