# Benchmarking Numerical Methods for Impact and Cratering Applications
## 1 Introduction  
Impact cratering is a phenomenon that happens across a wide range of spatial scales, from large geological scales which encompass examples such as asteroid impacts to very small scales such as microparticle impacts. Particularly on the largest scales, experimental data become limited as kilometer scales cannot be reached in the laboratory environment, and many impact craters of interest are extraterrestrial. Thus, models operative at larger spatial scales become critical for studying and understanding these impact craters. Consequently, validation and verification of these models is important for the generation of reliable results. In lieu of experimental data, code-to-code comparisons can be informative for understanding differences in modeling approaches and the overall effect on results. Impact cratering is a complex process that can be divided into three stages: contact and compression; excavation; crater modification [1, 2]. The initial stage, contact and compression, occurs as the impactor strikes the target, transferring momentum and resulting in a shock wave [1, 2, 3]. During this stage, target and impactor materials behave as fluids, with hydrodynamics governing the motion of material [1, 2, 3]. The second stage, excavation, occurs as material is ejected from the forming crater. This ejected material may be in the form of solid particles or in the form of vapor [1, 2]. During the final stage, crater modification, strength and/or gravity dominate the crater formation, depending on the cratering regime of the impact [1, 2]. As excavation ends and modification begins, material begins to settle into the crater floor, and other processes such as slumping and the overturned flap occur [1, 2]. For this final stage especially, more accurate modeling of solid materials, including material strength and damage, is essential to better understanding impact crater formation through computational models. This work presents a code-to-code comparison between a hydrodynamics code (hydrocode) and a finite-discrete element method (FDEM) code. The hydrocode approach is a more traditional approach for addressing impact cratering problems [4; 5; 6; 7; 8; 9], while the FDEM approach has been traditionally used for brittle fracture in geomaterials [10; 11; 12]. These approaches are operative on similar spatial scales, but the underlying formulation, particularly in how each approach accounts for damage, is very different. Hydrocodes solve the equations for conservation of mass, momentum, and energy, along with constitutive relations describing the response of different materials. Hydrocode approaches have been under development for decades [13; 14; 15] and thus have many attractive features for addressing impact cratering problems. One such feature is the ability of these models to employ Eulerian, Lagranian, or a combination of these two descriptions of the equations of motion [15; 16]. In Eulerian codes, material is allowed to freely move or 'flow' through a spatially fixed mesh [15]. Such methods have advantages because simulations are not limited by the consequences of excessive grid or element deformation, though they are not without drawbacks. For example, tracking the interfaces between materials presents a challenge in the form of mixed material cells, in which multiple materials exist in a single cell. In this case, material properties are often averaged [15; 16]. Conversely, in Lagrangian approaches, the mesh deforms with the material, making the tracking of interfaces between materials trivial. Thus, this approach has advantages particularly for problems considering solid materials. However, these formulations are subject to stability issues and/or inaccuracies when elements experience significant deformation, and mesh tangling can result [13; 15]. In order to capitalize on the advantages of both Lagrangian and Eulerian approaches, hybrid methods, or arbitrary Lagrangian-Eulerian (ALE) methods, have emerged [16].