# Benchmarking Numerical Methods for Impact and Cratering Applications
## 4 Results and Discussion
### Benchmarking against Other Hydrocodes

Table 2: Computational resources required for the 3D impact simulations. The pressure decay from FLAG and HOSS obtained for several impact cases, including 5 km/s normal and oblique impacts and 20 km/s normal and oblique impacts, are compared against other hydrocodes. Unless otherwise noted, all subsequent comparisons are for 3D simulations. To compare the pressure decay for 3D normal and oblique impact angles, pressure values are recorded along vertical, diagonal (45\({}^{\circ}\) from vertical), and horizontal lines extending outward from the impact point. These lines are shown schematically as dashed lines in Figure 1.  
In the case of the normal impacts, the results can also be compared to the analytical solutions in Table 3, shown as a constant dotted line in Figure 9. This figure, which includes digitized results from [PERSON] et al. [5], shows that all of the codes tested produce pressure values at the impact point that underpredict the analytical solution. The degree to which the codes underpredict the analytical solution varies highly in the 5 km/s impact velocity case. In the 20 km/s impact velocity case, the codes in general come very close to the analytical prediction, although most predictions are just under the analytical value. In this case, HOSS shows the best match to the analytic value close to the point of impact. However, it also shows some subsequent oscillations before the shock decay regime, whereas hydrocode simulations resulted in a slightly lower maximum pressure but a smoother decay curve. It is worth noting that for the 5 km/s normal impact case, the 2D FLAG simulations slightly overpredict the analytic value (see Table 3), while the 3D simulations underpredict the analytic value (see Figure 9a). This dimensional difference can be attributed to to the increased number of directions in which shocks can travel and dissipate in 2D versus 3D as well as the 1D calculation of the analytic solution. Figures 10-12 show the shock decay computed with FLAG and HOSS for oblique impact scenarios in comparison to the data presented by [PERSON] et al. [5]. Overall, the results from the HOSS and FLAG simulations have similar shock pressure decay to \begin{table}
\begin{tabular}{c c c c c} \hline
**Maximum Pressure** & **1D Analytic Solution** & **[PERSON] et al. [5] Mean \({}^{\star}\)** & **FLAG** & **HOSS** \\ \hline
5 km/s\({}_{\blacksquare}\)point of impact & 58.725 GPa & – & 59.58 GPa & 52.66 GPa \\ Relative Error & – & – & 1.46 \% & – 10.33 \% \\
5 km/s\({}_{\blacksquare}\)200 m into target & 58.725 GPa & 40.4 GPa & 55.77 GPa & 54.85 GPa \\ Relative Error & – & –31.2 \% & –5.03 \% & –6.59 \% \\
20 km/s\({}_{\blacksquare}\)point of impact & 506.25 GPa & – & 492.63 GPa & 438.96 GPa \\ Relative Error & – & –269 \% & –13.29 \% \\
20 km/s\({}_{\blacksquare}\)685 m into target & 506.25 GPa & 379.0 GPa & 407.99 GPa & 393.24 GPa \\ Relative Error & – & –25.14 \% & –19.41 \% & –22.32 \% \\ \hline \end{tabular} \({}^{\star}\) Note that the relative errors shown for these data were calculated with respect to the analytical values shown in this table, which show slight variations to relative error presented in the original paper. This difference comes from the different reference densities of Al-6061 and Al-2024 [3].  
\end{table} 
Table 3: Maximum pressures from 2D simulations in FLAG and HOSS compared to the theoretical maximum. These values are also compared to the mean of the values reported in [PERSON] et al. [5] by various codes. The relative error is reported with respect to the 1D analytical solution. Figure 9: 3D normal impact, vertical tracers from FLAG and HOSS simulations, plotted with data from [PERSON] et al. [5] for (**a**) 5 km/s impact and (**b**) 20 km/s impact. the codes tested by [PERSON] et al. [5]. The FDEM simulations exhibit a fair amount of variability in the region directly below the impact point. Farther from the impact point (roughly 1 km), the pressures calculated by both HOSS and FLAG exhibit stable, power-law decay with distance.