# Multi-Resolution SPH Simulation of a Laser Powder Bed Fusion Additive Manufacturing Process
## 3 Validation
### Weld Pool Experiment: Validation of Multiphysics Modeling  
The laser spot welding experiments of [PERSON] et al. [55] are taken as reference to validate the fully coupled thermal-mechanical-material model. Specifically, the test case pertains \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Density** & **Dynamic Viscosity** & **Surface Tension Coefficient** \\ \hline Symbol & \(\rho\) & \(\mu\) & \(\sigma\) \\ Unit & kg/m\({}^{3}\) & Pa-s & N/m \\ \hline Value & 1000 & 0.001 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of water used in the SPH simulation. Figure 5: Deformation of a square droplet to a circle under the effect of surface tension. Figure 6: Convergence of the SPH results to the analytical solution in a liquid droplet test. to a 304 stainless steel block welded by a laser pulse of power 1967 W, beam radius of 0.428 mm, and pulse duration of 3 ms. The workpiece is 1.3 mm by 1.3 mm and 0.4 mm height with material properties listed in Table 2. The convective heat transfer coefficient and emissivity factor are taken here as \(h_{c}=20\) W/(m\({}^{2}\):K) and \(\epsilon=0.4\), respectively. Expressed by [PERSON] et al. [28], the reference density is considered herein to be a function of temperature:  
\[\rho_{0}(T)= \begin{cases}8020-0.50(T-298)&\text{solid}\\ 7332-6.03(T-1673)&\text{transition}\\ 6840-0.70(T-1773)&\text{liquid}\end{cases}  \tag{25}\]  
To account for the thermo-capillary convection (i.e., Marangoni effect) at the melt-pool interface, the surface tension coefficient is also taken to be temperature-dependent. A common choice for \(\sigma(T)\) in this application was suggested by [PERSON] et al. [56] as:  
\[\sigma(T)=\sigma_{0}-A(T-T_{m})-R\,T\,\Gamma_{s}\,\log\left[1+k_{l}\,a_{s}\, \exp\left(-\frac{\Delta H_{0}}{R\,T}\right)\right] \tag{26}\]  
in which \(\sigma_{0}=1.943\) N/m, \(A=0.0005\) N/(m\(\cdot\)K), \(R=8.3145\) J/(mol\(\cdot\)K), \(\Gamma_{s}=1.3\times 10^{-8}\) (kg\(\cdot\)mol)/m\({}^{2}\), \(k_{l}=0.0032\), and \(\Delta H_{0}=-166\times 10^{3}\) J/(kg\(\cdot\)mol). Together with the thermal gradient, this temperature-dependent coefficient is plugged in Equation (20) to approximate the tangential Marangoni surface tension force. A 3D SPH model of this benchmark is created by discretizing the block with approximately 25k particles. This setup leads to a discretization size of \(dx=0.045\) mm and a time step in the order of \(10^{-7}\) s. As for the thermal boundary conditions, a constant temperature of \(T_{0}=293\) K is imposed on the bottom of the substrate (i.e., Dirichlet boundary condition). Furthermore, heat loss is included via radiation and convection to air through all open surfaces (i.e., Neumann boundary condition). Figure 7 and Table 3 provide the simulation results of the weld pool geometry at \(t=3\) ms. We measured the radius and depth of the weld pool by a line distance calculator in ParaView. According to the data in Figure 7, the predicted weld pool dimensions are compared against the results from two other references, both experimental and computational, in Table 3. The comparison shows that the present values of weld pool dimension agree well with those of [PERSON] et al. [55] and [PERSON] et al. [28]. Also, the geometry of the weld pool simulated by SPH is similar to that of the physical experiment, clearly seen through the side-by-side comparison in Figure 7. Even at this fairly low resolution, the 3D SPH model predicts the depth of the weld pool by an error of 4% and its radius by 10%.