*3.1. Dendritic Nucleus Model*

During the solidification process of the molten pool, the interface between the molten pool of liquid and the solid substrate is the nucleation surface affiliated with the grains' growth in the molten pool. A heterogeneous nucleation method was used to simulate the solidification evolution process of the grains in the molten pool. A quasi-continuous nucleation model was established based on the Gaussian distribution function. The term *dn/d*(Δ*T*) was used to describe the variation of the grains' nucleation density, and the total density of nuclei *n*(Δ*T*) formed at a certain undercooling Δ*T* was given as [21]:

$$m(\Delta T) = \int\_0^{\Delta T} \frac{d\mathbf{n}}{d(\Delta T)} d(\Delta T) \tag{8}$$

The change rate of nucleation density varies with the supercooling degree satisfied with Gaussian distribution. It can be calculated by the following expression:

$$\frac{dn}{d(\Delta T)} = \frac{n\_{\text{max}}}{\sqrt{2\pi}\Delta T} \exp\left[-\frac{1}{2}\left(\frac{\Delta T - \Delta T\_{\text{N}}}{\Delta T\_{\text{or}}}\right)^{2}\right] \tag{9}$$

where *n*max is the maximum nucleation density and Δ*T*<sup>σ</sup> is the standard deviation of undercooling. In the calculation process, the nucleation point location was randomly chosen.
