Keywords: heat conduction, cellular automata.

Cellular automata (CA) are dynamic systems which are discrete in space and time, operating on a regular multi-dimensional grid. In a CA model, each cell may be in one of a predetermined number of states. As the simulation progresses step by step, the state of a particular cell depends on its state in the previous period and the state of its immediate neighbours according to a simple rule applied to all cells on the grid. The development of CA can be traced back to 1940s when Von Neumann et al. [1] studied biological reproduction and crystal growth. Up till today, CA has been successfully applied to various complex systems, such as theoretical biology [2], computability theory [3], fluid dynamics [4] and evacuation simulation [5].

This paper proposes a new numerical approach to calculate steady-state thermal conduction in anisotropic solids based on the concept of cellular automata. This approach is developed without the consideration of governing partial differential equation for heat transfer. Generally, a CA model requires that geometrical and time domain are discretized into cells and intervals, respectively. To analyze heat conduction in solids, the basic concept of CA is extended in this manuscript. In the spatial domain concerned, a specific local rule is established for a number of random nodes, instead of a regular grid of cells. Meanwhile, since only steady-state heat conduction is studied, a virtual time domain is generated and divided into many constant intervals to match the features of CA.

Both two and three-dimensional examples show that the proposed approach has great potential in analyzing heat transfer in a space domain. As a mesh free numerical tool, this approach is very convenient to use as compared with other numerical methods such as finite element method. Moreover, this approach performs using a local rule and this makes the development of this approach very simple since it avoids complicated mathematical derivation.

1

V. Neumann, A. Burks, "Theory of self-reproduction automata", University of Illinois Press, Urbana, 1966.

2

G.B. Ermentrout, L. Edlestein-Keshet, "Cellular automata approaches to biological modelling", Journal of Theoretical Biology, 160, 97-133, 1993. doi:10.1006/jtbi.1993.1007

3

Th. Buchholz, M. Kutrib, "On time computability of functions in one-way cellular automata", Acta Inform, 35, 329-352, 1998. doi:10.1007/s002360050123

4

S. Wolfram, "Cellular automata fluids: Basic theory", Journal of statistical physics, 45, 475-525, 1986. doi:10.1007/BF01021083

5

W. Yuan, K. Tan, "An evacuation model using cellular automata", Physica A, 384, 549-566, 2007. doi:10.1016/j.physa.2007.05.055

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