# Multi-Resolution SPH Simulation of a Laser Powder Bed Fusion Additive Manufacturing Process
## 4 Application: Laser Powder Bed Fusion
### Parameter Study and Some General Observations  
The presented spatial adaptivity algorithm helps achieve a more realistic LPBF simulation with less computational effort (see Figure 10). However, another incentive to introduce particle refinement into the current SPH LPBF models was to minimize their computational cost and allow for parameter studies in a reasonable amount of time. For instance, analyzing the sensitivity of melt pool dimensions to process parameters is a useful consideration that will assist in optimizing fabricated part quality. Therefore, the effect of the track energy on the melt geometry using the efficient SPH code is investigated in this section. According to the term used in [58], the track energy is defined here as \(\bar{E}=P_{L}/v_{s}\) with respect to the laser power and scan speed. Since SPH particles are incapable of representing the interface explicitly [54], a post-processing analysis needs to be performed to accurately extract the melt geometry. The \(\alpha\)-shape algorithm of [PERSON] et al. [59] was chosen for this purpose. Figure 11 illustrates the calculation steps of the melt geometry as an \(\alpha\)-shape, depicted in green. The procedure is showcased for a laser power of 100 W scanning the LPBF track at the speed of 1 m/s. Snapshots of the temperature and material phase are shown, where the state color-bar indicates the parameter \(s\) defined in Equation (22). The state scale is linear, starting from the liquid state represented by red, down to the solid state shown in blue. The  
Figure 10: Effect of spatial discretization on SPH results. Particle refinement leads to a more accurate melt surface in LPBF simulation. The blue particles inside the squares represent the finest resolution in each case. Figure 9: Initial configuration and boundary conditions of the LPBF simulation. A Gaussian laser beam is scanning the track from left to right, shown in orange. The size (i.e., mass) of each SPH particle inside the refinement frame is 1/4 of the unrefined particles in blue. SPH particles with temperatures equal or greater than the melt temperature are extracted, shown in solid grey on the right top frame. The \(\alpha\)-shape algorithm is then applied to these melted particles to find an explicit definition of the melt pool interface. Halfway through the scanning process, the melt pool seems to reach a steady state. SPH frames are thus taken from this time instant: after the laser has scanned half of the powder bed. It is clear from the state of SPH particles in Figure 11 that the laser beam (in orange) creates a highly dynamic melt pool underneath itself, fusing the powder bed along the way. A very similar behavior and zonal distinction were also observed by previous studies in FEM [12] and (single-resolution) SPH [28]. In order to set up the parameter study, a total of 16 test cases was simulated by varying the laser power from 50 to 200 W at different scan speeds of 1, 1.5, 2, and 2.5 m/s. The finest and coarsest discretization sizes in these models were, respectively, 3 \(\upmu\)m and 6 \(\upmu\)m. In Figure 12, the total melt depth \(d_{m}\) is plotted in grey as a surface function of the laser power and scan speed. The blue stems in this plot indicate the 16 simulation data points. As expected, the maximum depth is found when using the maximum power (200 W) at the minimum scan speed (1 m/s). Moreover, a second-degree polynomial equation of \(d_{m}\) expresses the best fit to the available track energies. The red curve in Figure 12 represents this graph alongside the SPH data-sets shown in black squares. For a laser power of 200 W at the scan speed of 2 m/s, Figure 13 shows the melt dimensions after the powder bed is scanned and fully solidified. As a verification that the present SPH results are correct, the simulated LPBF track is plotted next to the experimental monograph of [12] for a very similar case. The numerical calculations are represented at six locations sufficiently far from the two edges to average the melt shape.