# Multi-Resolution SPH Simulation of a Laser Powder Bed Fusion Additive Manufacturing Process
## 2 Computational Framework
### Mechanical Model  
To model the dynamics of the melt pool region in LPBF, it is usually assumed that the liquid is incompressible and the liquid pool is in a laminar flow regime. As a result of this simplification, the Navier-Stokes equations for mass and momentum conservation in a Lagrangian frame arrive at:  
\[\frac{d\rho}{dt} =-\rho\
abla\cdot\underline{u} \tag{13}\] \[\rho\,\frac{d\underline{v}}{dt} =-\
abla p\underline{l}+\mu\
abla^{2}\underline{u}+\rho\underline{ g}+\underline{b} \tag{14}\]  
where \(p\) is the pressure, \(\underline{l}\) the unity tensor, \(\mu\) the dynamic (shear) viscosity, \(\underline{g}\) the gravitational acceleration, and \(\underline{b}\) any other volumetric body forces. To compute the spatial discretization of these equations, the \(\delta\)-SPH scheme proposed by [PERSON] et al. [42] is employed. Consequently, Equations (13) and (14) become:  
\[\big{\langle}\frac{d\rho_{i}}{dt}\big{\rangle}_{i} \approx\rho_{i}\sum_{j}(\underline{v}_{ij}+\underline{\Upsilon}_ {ij})\cdot\
abla W_{ij}\,V_{j} \tag{15}\] \[\big{\langle}\frac{d\underline{v}_{i}}{dt}\big{\rangle}_{i} \approx-\sum_{j}\frac{1}{\rho_{i}}\left(\frac{p_{j}}{\Gamma_{i}}+ \frac{p_{i}}{\Gamma_{j}}\right)\
abla W_{ij}\,V_{j}+\sum_{j}\frac{1}{\rho_{i} }\,\left(\frac{2\mu_{i}\mu_{j}}{\mu_{i}+\mu_{j}}\right)\,\pi_{ij}\
abla W_{ij} \,V_{j}+\underline{g}+\frac{1}{\rho_{i}}b_{i} \tag{16}\]  
where \(\pi_{ij}\) was defined via Equation (9), \(\Gamma_{i}=\sum_{j}V_{j}W_{ij}\) is a renormalization factor widely used for free surface problems (e.g., in [43]), and \(\underline{\Upsilon}_{ij}\) is the \(\delta\)-SPH diffusion term given by [42]:  
\[\underline{\Upsilon}_{ij}=-2\delta\,h\,c_{0}\left[\left(\rho_{j}-\rho_{i} \right)\frac{(\underline{r}_{i}-\underline{r}_{j})}{|\underline{r}_{i}- \underline{r}_{j}|^{2}}\right] \tag{17}\]  
in which \(\delta\) denotes the \(\delta\)-SPH smoothing parameter, \(c_{0}\) is the sound speed, and \(\bar{\rho}=\rho-\rho_{0}\). Very often in weakly compressible SPH frameworks (see in [44; 45; 46; 47], for example), an equation of state is enforced to approximately satisfy the incompressibility condition. [PERSON] et al. [28] proposed a modified equation of state that accounts for thermal expansion, thus well-suited to the LPBF application. This temperature-dependent equation of state computes the pressure \(p\) from:  
\[p=c_{0}^{2}\,(\rho-\rho_{0}(T)) \tag{18}\]  
where \(\rho_{0}\) is the reference density expressed by:  
Figure 2: Illustration of the volumetric heat source modeling approach and the laser intensity distribution.  
\[\hat{\rho}_{0}(T)=\rho_{0,T_{r}}\bigg{[}1+\alpha_{T}\bigg{(}1-\frac{T}{T_{r}}\bigg{)} \bigg{]} \tag{19}\]  
in which \(T_{r}\) is refereed to as a thermal reference value for the temperature field, \(\rho_{0,T_{r}}\) a thermal reference value for the reference density field, and \(\alpha_{T}\) the volumetric thermal expansion coefficient. To complete the SPH momentum Equation (16), surface tension forces are exerted to surface particles as a traction boundary condition. In this paper, the SPH-based surface tension model employs the continuum surface force (CSF) scheme of [PERSON] et al. (2018) and follows the algorithm proposed by [PERSON] et al. (2018). In a nutshell, the surface force term (without recoil pressure) can be computed from:  
\[\underline{E}_{s}=-\sigma\kappa\underline{\underline{u}}+\underbrace{\frac{d \sigma}{dT}\bigg{[}\
abla T-\bigg{(}\
abla T\cdot\frac{\
abla c}{|\
abla c|} \bigg{)}\frac{\
abla c}{|\
abla c|}\bigg{]}|\
abla c|}_{\text{Marangoni term (tangential)}} \tag{20}\]  
where \(\sigma\) is the surface tension coefficient, \(\kappa\) is the surface curvature, \(\underline{\underline{u}}\) the surface normal vector, and \(c\) the color function suggested by [PERSON] et al. (2018) to track the interface location. The term \(d\sigma/dT\) is sometimes referred to as the thermo-capillary coefficient, too. The gradient operators in Equation (20) are replaced by the same normalized SPH formulation used for the pressure term in Equation (16).