## 2 SLM Process Modeling in Moving Coordinate System
### Material Properties  
* Thermal conductivity In this study, we used AISI 316L stainless steel as our material. Two states of AISI 316L, the powder state and the solidified state, were considered in this simulation. [PERSON] et al. (2010) proposed a model to determine the effective thermal conductivity of packed powder beds. They found that the effective thermal conductivity of the powder bed is much smaller than that of bulk material. Therefore, the effective thermal conductivity \(k\) including phase change in this study is defined as Equation (9) \[ \begin{array}{l}\text{Powder zone:}\\ k(T)=\left\{\begin{array}{cc}\ \ \ \ \left(\frac{\widetilde{p}\widetilde{n}}{ \widetilde{x}}\widetilde{x}\right)\text{-}k_{solid}\ \text{(}T\ \text{)}\ if\ T<T_{m}-\Delta T\\ k_{liquid}\ \text{(}T\ \ \ \ \ \ if\ T>T_{m}+\Delta T\end{array}\right.\\ \text{Solidified zone and substrate:}\\ k(T)=\left\{\begin{array}{cc}k_{solid}\ \text{(}T\ \ \ \ if\ T<T_{m}-\Delta T\\ k_{liquid}\ \text{(}T\ \ \ \ if\ T>T_{m}+\Delta T\end{array}\right.\end{array} \right.\end{array}\] (9) where \(k_{solid}\) is the thermal conductivity of the corresponding bulk material, \(\widetilde{p}\) denotes the packing density of the powder layer, \(\widetilde{n}\) is the average coordination number (note that the coordination number is the number of contact points between a particle and all its surrounding particles), and \(\widetilde{x}\) is the contact size ratio. \(T_{m}\) is the melting temperature, and the phase change is assumed to occur in the temperature between \(T_{m}-\Delta T\) and \(T_{m}+\Delta T\). * DensitySimilarly, the powder bed can be considered to be a mixture of solid state and gas state. Therefore, the effective density \(\rho\) including phase change in this study is expressed as Equation (10)  
\[ \begin{array}{l}\text{Powder zone:}\\ \rho(T)=\left\{\begin{array}{ll}(1-\varnothing)\cdotp_{solid}(T)+\varnothing \cdotp_{gas}(T)&if\ T<T_{m}-\Delta T\\ \rho_{liquid}(T)&if\ T>T_{m}+\Delta T\end{array}\right\}\\ \text{Solidified zone and substrate:}\\ \rho(T)=\left\{\begin{array}{ll}\rho_{solid}(T)&if\ T<T_{m}-\Delta T\\ \rho_{liquid}(T)&if\ T>T_{m}+\Delta T\end{array}\right.\end{array}  \tag{10}\]  
where \(\rho_{powder}\) is the effective density of the powder bed, \(\rho_{solid}\) is the density of the corresponding bulk material, \(\rho_{gas}=1.2\) kg/m\({}^{3}\) is the density of gas, and \(\varnothing\) is the porosity, the ratio of the volume occupied by the gas to the total volume of the medium. * Heat capacity  
The effective heat capacity c including phase change in this study can be described as Equation (11)  
\[ \begin{array}{l}\text{Powder zone:}\\ c(T)=\left\{\begin{array}{ll}\frac{(1-\varnothing)\cdotp_{solid}c_{solid}+ \varnothing\cdotp_{gas}c_{gas}}{p_{powder}}&if\ T<T_{m}-\Delta T\\ \\ c_{liquid}(T)&if\ T>T_{m}+\Delta T\end{array}\right.\\ \text{Solidified zone and substrate:}\\ c(T)=\left\{\begin{array}{ll}c_{solid}(T)&if\ T<T_{m}-\Delta T\\ c_{liquid}(T)&if\ T>T_{m}+\Delta T\end{array}\right.\end{array}  \tag{11}\]  
where \(c_{solid}\) and \(c_{gas}=1000\) J/kgK are the heat capacities of the corresponding bulk material and air, respectively. Figure 3 illustrates the data of temperature-dependent material properties of AISI 316L stainless steel. It shows that the effective thermal conductivity and density of the powder bed is much smaller than that of bulk material; however, the effective heat capacity of the powder bed is almost equal to that of the corresponding bulk material.