# Benchmarking Numerical Methods for Impact and Cratering Applications
## 3 Code Verification Problem  
The code verification simulations were based on the problem initially introduced by [PERSON] et al. in a multi-code benchmarking study [5]. The verification problem consists of a 1 km diameter sphere of aluminum (Al-6061) impacting an aluminum target at 5 km/s and 20 km/s [5]. A schematic of this test problem is shown in Figure 1. Most of the results do not consider material strength, as in the initially proposed test problem, which allows for focus on the early stages of crater formation as well as for direct comparison to previously published results. In addition to modeling the materials as strengthless, simulations of this same verification problem using models for both material strength and damage are discussed later in Section 4.4. Relevant material parameters are included in the problem setup discussions for each code.  
Figure 1: 2D schematic of the Al-Al (Al-6061) test problem. The Al target can be modeled either as a rectangular block, as in FLAG, or as a semicircle, as in HOSS. Both configurations are shown in this schematic. The insets show examples of the different meshes employed by the two methods. The dashed lines show lines in which pressure decay data were taken in 2D and 3D runs for comparison in Sections 4.1 and 4.3.  
Several simulations were completed using this test problem configuration, many of which are consistent with those performed by [PERSON] et al. [5] so that comparisons between results could be made. These simulations included a mesh resolution study in 2D, which tested mesh resolutions in the range of 5-40 cells per projectile radius (cppr). For the mesh resolution study, shock pressure decay was recorded at tracer particles along the impactor trajectory from the point of impact to 10 km into the target, shown with a dashed line in Figure 1 (labeled as the vertical tracer). 3D simulations included both normal impacts and impacts 45\({}^{\circ}\) from vertical (also referred to as the oblique impact). Shock pressure decay was recorded in 3D simulations, but a mesh resolution study was not performed because of the significant computational cost of 3D simulations. Similar to the 2D case, the shock pressure decay was measured at different tracers from the impact point, including a vertical, horizontal, and 45\({}^{\circ}\) from vertical (i.e., diagonal) tracer line, all of which are indicated with dashed lines in Figure 1. Note that for the diagonal tracers, data were taken at approximately one projectile radius (\(\sim\)0.5 km) downrange from the impact. This tracer placement is consistent with data presented previously by [PERSON] et al., which also reported diagonal tracer data originating from one projectile radius downrange [5].  
Because the target and impactor had the same material composition, the theoretical maximum pressure can be calculated for a 1D impact:  
\[P=\rho_{0}(C_{0}+SU_{P})U_{p}, \tag{3}\]  
where \(P\) is the maximum pressure, \(\rho_{0}\) is the initial density, \(C_{0}\) is the reference sound velocity (at 0 pressure), \(S\) is the linear EOS coefficient, and \(U_{p}\) is the particle velocity, equal to half of the impact velocity when target and impactor have the same material [3]. \(S\) can also be expressed in terms of shock velocity \(U_{s}\) with the following relationship: \(U_{s}=C_{0}+SU_{P}\)[3]. Substituting the values for the verification problem into Equation (3) yields a maximum pressure of 58.725 GPa for the 5 km/s impact and 506.25 GPa for the 20 km/s impact.