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Application of the asynchronal cellular automata method in the heat conductivity problems investigation

L. Shumilyak, V. Zhikharevich, S. Ostapov
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Description: It is well known that many physical properties of crystalline materials obtained by the method of directed crystallization are determined by the distribution of impurity in the melt and its ability to accumulate in the form of separate grains, cells etc. This occurs due to concentrated overcooling and leads to deterioration of the mechanical, electrical and physical properties of the material. It is one of the reasons for their fragility. A series of experiments is needed to investigate the optimal conditions for the growth of semiconductor materials with required properties. The time required is not always available, and the labor and material resources cost is rather high. Therefore, in recent years great attention is paid to the development of the technology of process simulation. The tasks of heterogeneous dynamical systems numerical simulation are very relevant today, since they allow observing the evolutionary patterns of such systems in real-time, especially when it comes to problems with nonlinear parameters of materials, complex boundary and initial conditions, phase transitions with moving limits, etc. In the vast majority of such cases, it is almost impossible to obtain analytical solutions, and classical based on difference schemes solution numerical methods may be unstable. The classical model of physical processes is based on differential equations. But practical application of it does not allow us to receive acceptable results. In real practical tasks it is often used in the simplest cases with a number of limitations and assumptions. In this regard, alternative approaches are increasingly popular in recent years. Widely used imitation or agent models, where each agent can be attributed their own rules of conduct. The method of cellular automata (СA) is one of such simulation approach. It provides not only a description of the physical properties of the material but also can provide for changes at the micro-level. In particular, heat transfer processes are naturally approximated by continuous models of cellular automata. Just creation of a qualitative model of the process, on the basis of computational experiments, allows to predict the properties of the resulting material. Cellular automata are the most effectively used to describe the behavior of a system the collective behavior of which is determined by the local behavior of its constituent elements, when the system is highly heterogeneous, and averaging of variables throughout the system can hardly reflect its status adequately as a whole. Therefore, while modeling the melting process, accompanied by the first order phase transition, we chose the cellular automata technique. The work is devoted to the use of cellular automaton model for the study of some basic physical processes. The main approaches and the general methodology of the development of cellular automata models are considered on the example of heat transfer processes modeling. It is shown that these models can become an alternative to the use of classical differential equations. It is proved that the model in the form of a system of cellular automata equations is a very convenient tool for nonlinear heat transfer problems studying and can describe a rather complicated system behavior, despite the simplicity of its description. The typical problems of the thermal conductivity theory and their solving by the cellular automata method are considered. The analysis of accuracy of calculations by the CA-method was carried out. A comparison of the computational speed for the Stefan problem with the help of the CA-method and known net methods has been made.


Keywords: phase transition, cellular automata, thermal conductivity

Reference:
Shumyliak, L.M., Zhykharevych, V.V. and Ostapov, S.E. (2018), “Doslidzhennia metodu asynkhronnykh klitynnykh avtomativ pry zastosuvanni v zadachakh teploprovidnosti” [Application of the asynchronal cellular automata method in the heat conductivity problems investigation], Information Processing Systems, Vol. 1(152), pp. 74-79. https://doi.org/10.30748/soi.2018.152.11.