## 2 Finite Element Modeling
### Heat Source Modeling  
Considering the interactions between the powder and laser, the laser penetration depth \(\eta\) is generally equal to the thickness of the powder layer. Therefore, the 3D Gaussian heat source is represented as  
\[ \begin{array}{l}\text{q}(\text{x},\text{y},\text{z})=\left\{\begin{array}{ ll}\frac{2\lambda P}{\pi r_{0}^{2}\eta}\exp\left[-2\frac{r_{0}^{2}}{r_{0}^{2}} \right]\exp\left(-\frac{|z|}{\eta}\right);&\text{if $z\leq\eta$ and $r\leq r_{0}$}\\ 0;&\text{otherwise}\end{array}\right.\end{array}  \tag{7}\]  
where \(r_{0}\), \(\lambda\), \(P\), \(r\), and \(z\) are, respectively, the laser beam radius, energy absorptivity, laser power, the distance between the beam center and the point, and the distance from the top surface. In case of deep penetration, since the governing Equation (1) neglects the fluid flow within the melt pool and the effects of the keyhole (see Figure 6), the melt pool size simulated with the Gaussian heat source described by Equation (7) will therefore underestimate the penetration and over-estimate the width. Figure 6: Thermal effects neglected by FEM: fluid flow (**a**); multiple reflections due to keyhole (**b**). We can start with a comprehensive analysis related to these neglected effects. 1. The fluid flow within the melting pool (Figure 6a) can lead to a more homogeneous temperature distribution by transporting the received energy with the mass flow. Therefore, the anisotropic thermal conductivity technique is well suited to consider the transport effect. In case of high energy input, higher-temperature fluid from the surface heated by a laser is carried toward the bottom of the melt pool [41], which results in a deep-penetration region in the central melt pool. Thus, we propose to increase the parameter \(\eta\) to simulate the penetration. Since \(\eta\) evolves at each time step, \(\eta_{t}\) is now used in Equation (8). 2. The effect of multiple reflections on the keyhole (Figure 6b) has been studied by [PERSON] et al. [42]. This effect changes the energy distribution in depth, and the maximal power is no longer present at the top surface, which can lead to deeper penetration. One should note that this effect cannot be represented by imposing an enhancement factor for conductivity. The energy distribution described by Equation (7) should be modified by replacing \(\exp\left(-\frac{|z|}{\eta}\right)\) by \(\exp\left(-\frac{|z-z_{0}|}{\eta_{t}}\right)\), where \(z_{0}\) represents the position of maximal power in the depth direction. This parameter is not a fixed value due to the instability of the keyhole, and it is impossible to measure this value from experiments at the moment. Therefore, we propose \(z_{0}\), which is proportional to \(\eta_{t}\), for simplicity. To conclude, the thermal effects related to Figure 6a can be compensated by increasing the parameter of \(\eta_{t}\), while \(\eta_{t}\) cannot be superior to the melt pool depth. The \(\eta_{t}\) is calculated in an explicit manner, where \(\eta_{t}\) is updated based on the average temperature of controlled volume (see Figure 7) calculated from instant \(t-dt\). The effect of multiple reflection (Figure 6b) can be represented by employing \(\exp\left(-\frac{|z-z_{0}|}{\eta_{t}}\right)\) in Equation (8), and the melt pool depth will be equally enhanced. In order to ensure a constant energy input as well as consider the variation of \(\eta_{t}\) and \(z_{0}\), Equation (7) is transformed into the following form:  
\[\mathrm{q}(\mathrm{x},\mathrm{y},\mathrm{z})=\left\{ \begin{array}{ll}\frac{ 2N\lambda\rho}{\pi r_{0}2^{\eta_{t}}}\exp\left[-2\frac{r_{0}^{2}}{r_{0}^{2}} \right]\exp\left(-\frac{|z-z_{0}|}{\eta_{t}}\right)&\text{; if z not greater than }\eta\\ 0&\text{; otherwise}\end{array} \right. \tag{8}\]  
where \(N\) is recalculated at the beginning of each time step to ensure a constant energy input, and \(\eta_{t}\) is also recomputed at beginning of each time step according to the average temperature calculated from selected nodes.