## 1 Introduction  
Multi-crystalline solidification of silicon is a low-cost way of producing photovoltaic cells although it is less efficient than single crystalline silicon [1]. The solidification process is an important step in manufacturing components. During solidification, the shape of the solid-liquid interface plays an important role in the formation of defects and the selection of grain. For silicon, the Jackson factor is greater than two [2]. Therefore, the interface tends to form facets. The formation mechanism of a faceted crystal-melt interface was investigated by in situ observation [3; 4; 5]. It has been shown that when a wavy perturbation is introduced into a planar crystal-melt interface, the perturbation would result in zigzag facets. [PERSON] et al. [6; 7] extended phase field simulation of facet formation from two to three dimensions for directional solidification of the silicon thin film. [PERSON] et al. [8] investigated the growth of a twin-related silicon dendrite through a novel phase field model. [PERSON] et al. [9] applied a phase field model to compute the evolution of two grains during the solidification of silicon. However, phase field cannot calculate such a large-scale microstructure growth as the transition from faceted front to equiaxed dendrites in the process of the solidification of multi-crystalline silicon ingot, yet cellular automaton (CA) method is an alternative. In this paper, therefore, we use cellular automaton (CA) method to calculate the facet growth of silicon. The growth behavior of columnar to equiaxed transition (CET) of multi-crystalline silicon is an important phenomenon that affects the final microstructure of a multi-crystalline silicon ingot. X-ray examination have shown that a transition from planar front to equiaxed growth is likely to occur in case of faceted interfaces, observed in multi-crystalline silicon ingots [10]. A few models and mechanisms have been proposed to understand the columnar to equiaxed transition (CET). [PERSON] [11] developed a three-dimensional cellular automaton-finite element (CA-FE) model to simulate the microstructural evolution of multi-crystalline silicon ingots. In addition, this model can represent the occurrence of equiaxed grains observed ahead of a planar faceted interface due to carbon segregation during solidification. Although cellular automaton-finite element (CA-FE) model can calculate the columnar to equiaxed transition (CET) process, it is difficult to calculate the flow field, and the model is also very complicated. However, cellular automaton-lattice Boltzmann (CA-LB) method can be used to calculate the complex melt convection, especially the fluid flow between the gaps of different grains which is difficult for finite element (FE) to compute and the model is simple. The cellular automaton-lattice Boltzmann (CA-LB) method has been widely used in the field of simulated crystal growth [12; 13; 14]. In general, macroscopic simulation can improve the production process, yet it cannot simulate crystal growth mechanism accurately. Microscopic simulation has a solid theoretical basis, it is subjected to the efficiency of computation and cannot simulate large-scale of crystal growth. In mesoscopic scale, the cellular automaton combining lattice Boltzmann method is a good means to simulate crystal growth. [PERSON] et al. [15] set up a three-dimensional cellular automaton (CA) model to simulate the microstructure change of iron carbon alloy in the solidification process. This model took into account the solid-liquid interface curvature, surface energy and dendrites anisotropy. Meanwhile, it analyzed the grain and dendrite growth under different undercooling and thermal conductivity parameter. [PERSON] et al. [16] solved the solid-liquid phase change problems by the lattice Boltzmann (LB) method and developed a new approach to treat the latent heat source term by modifying the equilibrium distribution function for the temperature.