**1. Introduction**

During the solidification process of the alloy, the fine grains formed on the surface of the mold will fall off, and the solidified dendrite arms will also be remelted and fractured, resulting in a large number of free equiaxed dendrites in the liquid phase area, which will move under the action of natural convection and gravity [1]. The movement and falling process of a large number of group grains not only has an important influence on the formation of positive segregation at the top of ingot, A-type and V-type segregation [2], but also is the main reason for the formation of triangular cone-shaped negative segregation at the bottom of large ingot [3,4].Therefore, it is of great significance to add the calculation of grain movement process into the numerical model of ingot macrosegregation to improve the prediction accuracy.

At present, the phase field method is mostly used to simulate grain movement in the world. In 2008, Do-Quang M et al. [5] used the phase field-virtual domain method to simulate the growth and movement of single dendrite under the action of gravity. In 2012, karagadde and Bhattacharya [6] used the enthalpy method (EF) to calculate the growth of dendrites, the volume of fluid method (VOF) to calculate the movement behavior of dendrites, and the immersion boundary method (IBM) to deal with the solid-liquid interface. The simulation results show that the multi dendrite growth pattern is significantly different from that of the single dendrite. In 2013, Medvedev et al. [7] used Phase Field-Lattice Boltzmann method (LBM) coupling model to calculate the dendrite growth and movement behavior of aluminum copper alloy under the action of shear flow and pipe flow. In 2015,

Rojas and Takaki [8] used a PF-LBM model to simulate the growth and movement of dendrites under shear flow, and analyzed the effect of solution flow on the growth and movement of dendrites. In the same year, Takaki et al. [9] added GPU technology to the program code, which greatly improved the simulation efficiency and scale, and used the technology to simulate the settlement behavior of single dendrite in the gravity field. In 2017, Qi et al. [10] proposed a new phase-field model incorporating dendrite-melt two-phase flow, and modified the boundary layer of growth kinetics equation, so that it can better reflect the relationship between the growth rate of dendrite tip and the flow direction of fluid. In 2018, Takaki et al. [11] established a new phase field model to simulate the growth and movement of multi-dendrite, and coupled this model with lattice Boltzmann to simulate the growth, movement, collision, and growth behavior after bonding of multi-dendrite.

However, the grid of phase field method is very small and the amount of calculation is huge, which greatly limits the number of equiaxed dendrites. It is impossible to simulate the solidification process of ingots with a large number of equiaxed and columnar dendrites [12]. The Cellular automaton (CA) method has a small amount of calculation and a fast calculation speed, and is undoubtedly more suitable for calculating multi-dendritic motion behavior. Currently, only the work of Liu et al. [13] used the CA method to calculate the moving dendrite. He can only simulate the settlement of a single dendrite, and the dendrite cannot rotate, obviously this is different from the actual situation. This paper improves the calculation accuracy of the concentration field and simulates the movement of multiple dendrites.

In addition, the temperature, flow, and solute fields need to be calculated when simulating dendrite motion in the melt. The LBM method developed in recent years can effectively calculate the passage process of dendrites in the melt, so it has been widely used. This paper uses the LBM method to calculate three fields. The dendrite in the melt must interact with the melt during the movement. This article uses the Ladd method to deal with this effect, because the Ladd method uniformly processes the solid phase and the liquid phase, so as to avoid the mass and momentum loss caused by the solid phase node covering the liquid phase node in the process of dendrite movement. However, the difficulty in dealing with the solid-liquid interaction lies in the calculation of the solute field. The solute diffusion coefficient of the two phases is quite different, therefore it cannot directly treat the two phases as a whole to deal with the boundary. Therefore, this paper presents a method to deal with the solute field of multi grain movement, which realizes the calculation of solute field in the real sense of dendrite movement.
