# Multi-Resolution SPH Simulation of a Laser Powder Bed Fusion Additive Manufacturing Process
## 2 Computational Framework
### Thermal Model  
In LPBF, the energy required for melting and fusing metallic powders together is provided by a laser. The thermal effects are, therefore, addressed by solving the heat equation for incompressible flows including a laser heat source as:  
\[\rho\,c_{p}\,\frac{dT}{dt}=\underline{\underline{r}}:\
abla\underline{u}+ \
abla\cdot(k\
abla T)+Q^{\text{laser}} \tag{6}\]  
where \(\underline{\underline{r}}\) is the shear stress tensor, \(\underline{u}\) the velocity, \(\rho\) the density, \(c_{p}\) the specific heat capacity, \(k\) the thermal conductivity, and \(Q^{\text{laser}}\) is the laser heat source term. The viscous heatingand conduction terms in Equation (6) are computed respectively by the SPH schemes of [PERSON] et al. [38], and [PERSON] and Monaghan [39], which are both suitable for multiphase applications. These formulations for particle \(i\) are given by:  
\[\langle\underline{\underline{r}}_{i}:\
abla\underline{u}_{i}\rangle \approx \sum_{j}\Biggl{(}\frac{\mu_{i}\mu_{j}}{\mu_{i}+\mu_{j}}\Biggr{)} \ \frac{m_{i}}{\rho_{j}}\ \pi_{ij}(\underline{u}_{j}-\underline{u}_{i})\cdot\
abla W_{ij} \tag{7}\] \[\langle\
abla\cdot(k_{i}\
abla T_{i})\rangle \approx \sum_{j}4\Biggl{(}\frac{k_{i}k_{j}}{k_{i}+k_{j}}\Biggr{)}\ \frac{m_{i}}{\rho_{j}}\ (T_{i}-T_{j})\ \frac{( \underline{t}_{i}-\underline{t}_{j})}{|\underline{t}_{i}-\underline{t}_{j}|^{ 2}+\eta^{2}}\cdot\
abla W_{ij} \tag{8}\]  
in which:  
\[\pi_{ij}=2(n_{D}+2)\frac{(\underline{u}_{j}-\underline{u}_{i})\cdot( \underline{r}_{j}-\underline{r}_{i})}{|\underline{r}_{j}-\underline{r}_{i}|^ {2}} \tag{9}\]  
and \(\mu\) is the viscosity, \(n_{D}\) the dimension factor, and \(\eta=0.1h\) inserted here as a small parameter for ensuring non-zero denominators. For SPH Laplacian schemes more sophisticated than Equation (8) please refer to [40].  
The heat source term \(Q^{\text{laser}}\) in Equation (6) still needs to be defined. In this paper, a volumetric heat source modeling approach based on the Beer-Lambert law is used for calculating the laser energy absorption. As such, increasing the penetration depth decreases the laser absorptivity exponentially. Figure 2 illustrates this scheme, in which the normalized Beer-Lambert law for the powder bed can be written as:  
\[I(z)=\frac{\beta\exp(-\beta z)}{[1-\exp(-\beta L)]} \tag{10}\]  
where \(L\) is the powder bed depth and \(\beta\) the extinction coefficient, taken as a constant value according to [28; 41]. The intensity distribution of the laser in the radial direction, \(I(r)\), follows a normalized Gaussian distribution:  
\[I(r)=\frac{4P_{L}}{\pi R^{2}}\exp\biggl{(}-\frac{4\ r^{2}}{R^{2}}\biggr{)} \tag{11}\]  
with \(P_{L}\) indicating the laser power and \(R\) the laser beam radius. Finally, the laser heat source appeared in Equation (6) can be calculated as \(Q^{\text{laser}}=a_{L}\ I(r,z)\) with \(a_{L}\) denoting the absorption coefficient. The implementation details of a 2D Beer-Lambert laser absorption model for SPH frameworks can be found in [28]. Figure 2 gives a graphical illustration of how this absorption model is implemented in the present code.  
To complete the solution of Equation (6), the set of boundary conditions, including the Dirichlet and Neumann boundary conditions, are imposed on one layer of surface particles. In this work, heat loss \(q_{l}\) can occur via radiation and convection through open surfaces, calculated from:  
\[q_{l}=-\Bigl{[}h_{c}\left(T_{s}-T_{\infty}\right)+\epsilon\sigma\left(T_{s}^{ 4}-T_{\infty}^{4}\right)\Bigr{]} \tag{12}\]  
where \(h_{c}\) is the convective heat transfer coefficient, \(\sigma=5.67\times 10^{-8}\) W/(m\({}^{2}\cdot\)K\({}^{4}\)) the Stefan-Boltzmann constant, \(\epsilon\) the emissivity factor, and \(T_{s}\) and \(T_{\infty}\) are the surface and background temperatures.