## 1 Introduction  
Multi-crystalline silicon (mc-Si) is a very useful material owing to its excellent integrated properties [1]. Compared with other solar energy materials, mc-Si grown via directional solidification is of higher quality and cheaper than others in the large market of solar cell products [2; 3]. In past years, numerical simulation for silicon crystal growth has been carried out by many researchers. It is well known that the grain structure of mc-Si ingot has a strong effect on the conversion efficiency of solar cells because the grain boundaries are the locations of lattice defects which interact with minority carriers [4; 5]. [PERSON] et al. designed an insulation partition in a seeded directional solidification (DS) furnace and the result shows that it can reduce the total heating power consumption as well as improve the efficiency of solar cells [6]. [PERSON] et al. designed different moving partition blocks to investigate the solidification front shape and thermal stress of quasi-single-crystal silicon ingots [7]. [PERSON] et al. studied the heat and mass transfer and solute segregation during single-crystal growth by a numerical method [8]. [PERSON] et al. have investigated the grain structure in an mc-Si ingot to predict the evolution of grain structures [9]. [PERSON] et al. have investigated the facet formation mechanism and grain competition during the silicon crystal growth via adaptive phase field modeling, which presented the morphological evolution and thermal field near the interface as well as the comparison with experimental results [10]. Their work focuses on the improvement of macro-processes or the study of micro-mechanisms. Anisotropy, faceting, and twining are important features in the crystal growth of many materials, and they cannot be overlooked when studying the microscopic growth mechanism of silicon. The phase-field method is a good method for the study of crystal micro-growth, but it is important to calculate the efficiency of computation and cannot simulate large-scale crystal growth. Macroscopic simulation of crystal growth is an important method for improving the growth process, but this method ignores some crystal growth mechanisms, such as anisotropy and faceting, etc. which result in the physical mechanism of crystal growth not being accurately studied. Mesoscale simulation combining the cellular automation and finite difference or finite element methods, which is widely used in metal casting to simulate crystal growth and crystal grain structures, efficiently bridges the gap between micro- and macroscopic scale method. [PERSON] and [PERSON] were the first ones to apply the cellular automaton (CA) method to predict solidification grain structures [11]. The material was considered to be a group of cells, and the motion of each cell was calculated based on the behavior of their neighboring environment. The CA method is not only simple; it is also a powerful tool to achieve high level of complexity and diversity. It has been used to simulate the dendritic growth in the solidification process as well as the grain boundary motions in polycrystalline materials [12; 13]. [PERSON] et al. developed a 3-D cellular automaton finite difference (3-D CAFD) model to predict the solidification grain structure of Al-3 wt%Si binary alloy [14]. Their research results can be of considerable help for controlling grain structure and segregation in complex casting products. [PERSON] et al. used the Cellular Automata Finite Element (CAFE) modeling technique to simulate silicon grain growth. They ignored the effect of anisotropy on silicon crystal growth in the calculation model, and compared the simulation results with real ingot grain structures [15]. The simulation results were in good agreement with the reality. The CA is a mesoscale method, which neglects some details in microscopic simulation method and has important guiding significance for macroscopic simulation.