# Benchmarking Numerical Methods for Impact and Cratering Applications
## 1 Introduction
ALE methods use a mix of Lagrangian and Eulerian schemes in which the solution from a distorted Lagrangian mesh is remapped to a spatially fixed Eulerian mesh. In addition, adaptive mesh refinement (AMR) techniques have been developed to provide dynamic mesh resolution in regions where material interfaces or shock fronts exist [17]. While hydrocodes can capture a wide range of multiphysics, they typically do not discretely resolve many features important to material deformation. Examples include cracks, fragmentation, or grains/particles within polycrystalline or granular microstructures. While such limitations may seem like a disadvantage from a physical standpoint, from the computational standpoint, this type of homogenization often makes this formulation more computationally efficient than other approaches, which allows for the ability to study very large systems in reasonable amounts of times. FDEMs operate on similar spatial scales as hydrocodes. FDEMs were introduced by [PERSON] in the late 1980s/early 1990s for the simulation of \"transient dynamics on fragmenting solids\" [18; 19]. As the name indicates, FDEM merges the advancements achieved by both the Finite-Element Method (FEM) and the Discrete-Element Method (DEM). From the start, FDEM's FEM portion was designed to be able to handle finite displacements, finite rotations, and finite strain-based deformation of the solid [20; 21]. The FEM approach was then combined with contact detection, contact interaction, and objective discrete crack initiation and crack propagation solutions that were inherited from the DEM [20; 22]. FDEM provides a framework for representing the transition from continuum to discontinuous (i.e., fracture) of solid materials when they are subjected to loads that exceed the material's strength. This transition is allowed to occur at the boundaries of the finite elements. The first implementations of FDEMs were mainly focused on the simulation of fracture and fragmentation processes for cementitious materials [23]. These models used assumptions that the material inside the finite elements behaves elastically and that the localization process (fractures) was limited to the boundaries of the finite elements. More recently, a growing need for the simulation of problems where the aforementioned assumptions were no longer valid because of the large extent of permanent deformation mechanisms has led to the incorporation of plasticity models inside of the finite elements [24]. Traditional FDEM-based simulations involve the resolution of fracture, fragmentation, and contact problems. A typical solution for contact is via the implementation of penalty methods [25], which are computationally robust but require the use of relatively high values of contact stiffness (i.e., penalty coefficients), which in turn require smaller time step sizes. Combined with the fact that contact problems also require a spatial search for contacts every certain number of time steps [26; 27], the result is that FDEM is more computationally expensive than traditional continuum-based models. Another drawback of the FDEM, because it is a Lagrangian-based method, is the time-step size restrictions imposed by highly deformed elements [13; 15]. This type of situation usually occurs in areas located close to impacts or around high strain sources, such as explosions. As mentioned previously, this work is focused on a code-to-code comparison of a hydrocode and an FDEM code, with the goal of benchmarking both numerical methods for impact cratering simulations and also evaluating the effects that the different numerical methods have on the overall solutions. In this work, hydrocode simulations use the ALE code Free LAGrange (FLAG), and FDEM simulations use the Hybrid Optimization Software Suite (HOSS). Both codes are developed and maintained by Los Alamos National Laboratory [28; 29; 30; 31; 32; 33]. The following section contains brief summaries of each approach. Section 3 describes the test problem and model setups for both FLAG and HOSS simulations.