## 2 Finite Element Modeling
### FE Models and Meshes  
Figures 3 and 4 show the FE meshes for SLM AlSi10 Mg and SLM Inconel 718, respectively. Mesh refinement only appears along the trajectory of the heat source, and the minimum element size in the substrate is \(5(length)\times 5(width)\times 9(depth)\)\(\upmu\)m.  
Figure 5 shows the material properties of AlSi10 Mg and Inconel 718. Since the experiments of [PERSON] et al. (2018) and [PERSON] et al. (2018) are used as references, therefore, the thicknesses of the powder layer are 20 \(\upmu\)m for AlSi10 Mg and 40 \(\upmu\)m for Inconel 718. The apparent heat capacity method is employed for Inconel 718 to incorporate the latent heat (L = 210 kj \(\ast\) kg\({}^{-1}\)) due to the phase changing between solid and liquid ([PERSON] et al., 2018). The substrate material properties of Al2024 used in SLM AlSi10 Mg are exactly the same as [PERSON] et al. (2018). For the sake of convergence, the thermal conductivity of powder varies linearly from 20 degrees to temperature fusion. The thermal properties of powder at ambient temperature (20 degrees) are calculated from the technique presented by [PERSON] et al. (2018), and the effective properties of powder are described by the following equations:  
\[\rho_{powder}=(1-\Phi)\rho_{compact} \tag{3}\]  
Figure 4: FE mesh (450,133 nodes/430,570 3D elements) for SLM Inconel 718: left view (**a**); front view (**b**); top view (**c**); global view (**d**). Figure 3: FE mesh (286,196 nodes/271,034 3D elements) for SLM AlSi10 Mg: left view (**a**); front view (**b**); top view (**c**); global view (**d**). \[K_{\textit{powder}}=\left\{ \begin{array}{ll}K_{\textit{compact}}(1-\Phi)^{n}& \text{if $T=20$ degrees}\\ \dfrac{K_{\textit{compact}}-K_{\textit{compact}}(1-\Phi)^{n}}{T_{\textit{ melting}}-20}*(T-20)&\text{otherwise}\\ K_{\textit{compact}}&\text{if $T>T_{\textit{melting}}$}\end{array} \right. \tag{4}\]  
where \(\Phi\) is the porosity of powder. In this study, values of \(\Phi=0.25\) and \(n=4\) are taken [11]. Since the pure heat transfer model has neglected fluid flow in melt pool and multi-physics problems related to the keyhole (see Figure 6), which leads to an under-estimation of molten pool depth and over-estimation of molten pool width, in the literature, the anisotropic thermal conductivity technique has been reported to be useful to improve the precision of molten pool dimensions. This technique has been used to simulate deep penetration by multiplying thermal conductivity by an enhancement factor [33; 38; 39], as shown in Equation (5). \[ \begin{array}{l}\text{K}_{\text{xx}}=\lambda_{\text{xx}}\cdot\text{k}\\ \text{K}_{\text{yy}}=\lambda_{\text{yy}}\cdot\text{k}\\ \text{K}_{\text{zz}}=\lambda_{\text{zz}}\cdot\text{k}\end{array}  \tag{5}\]  
where \(\text{K}_{\text{xx}},\text{K}_{\text{yy}},\text{K}_{\text{zz}}\) represent the thermal conductivity in each direction. \(\lambda_{\text{xx}}\), and \(\lambda_{\text{yy}}\) and \(\lambda_{\text{zz}}\) are the temperature-dependant enhancement factors of k (presented in Table 5). Figure 5: The temperature-dependent material properties used in this study. The depending temperature (\(T_{c}\)) and the numerical value of (\(M\)) should be estimated and validated from experiments. The enhancement factors are defined according to the following equation:  
\[ \begin{array}{l}\lambda_{\text{xx}}=\lambda_{\text{yy}}=\lambda_{\text{zz}}=1,\text{ T}_{1}<T_{c}\\ \lambda_{\text{xx}}=\lambda_{\text{yy}}=1,\quad\lambda_{\text{zz}}=M,\quad \text{ T}_{1}\geq T_{c}\end{array}  \tag{6}\]  
where \(T_{c}\) is greater than temperature fusion and \(M\) is generally set to 2-30 [40]. The melt pool depth is supposed to be in the \(z\) direction. [PERSON] et al. [33] reported that values of \(T_{c}=750\) degrees and \(M=15\) give a good correlation with experiments in SLM AlSi10 Mg; therefore, the same values are used in the present study. However, the enhanced factors are calculated by comparing the predicted values with the experimental results, which is quite complex and time consuming.