## 3 Computational Methods  
Substantial advances in computing resources and numerical modeling capabilities require the implementation of complex numerical approximations to govern analytical models of penetration mechanics. Numerical approaches have many benefits over analytical methods and empirical methods based on experimental tests. The numerical methods are most suitable to model the frictional effect, influence of inhomogeneity on projectile instability, and target erosion during the penetration, which is not possible to incorporate in analytical methods [75]. The numerical simulation could offer the detailed description of the entire penetration process from all aspects and are cost effective. However, the accuracy of numerical solution depends on analytical models of inherent behavior and associated numerical approximations of penetration problem. This necessitates the development of in-situ experimental techniques to not only improve our observed understanding, but also to validate the simulated solutions. The experimental methods or even analytical methods quantify projectile velocity, stress, final penetration depth into the target, and the final target/projectile deformation [76]. Of these, only the stress may be measured with any temporal resolution given normally used methods, and none yield any information about target material behavior during the process beyond general bulk results. The numerical methods can predict the particle response along with the bulk response of the target. The numerical models can simulate the \"non-uniform distribution of grains, material strength including strain hardening with time and temperature-independent properties\" [77], mesoscale interactions, spall, phase change, void collapse, grain failure, and mode of contact building on the basic Hertzian philosophy, exploring different descriptions of penetration [77; 78]. As a continuum modeling of the phase of projectile penetration in the soil target, numerical methods readily analyze complex target and projectile geometries, i.e., multi-layered materials, obliquity, by treating the soil media as a single component bulk material, where the removal of soil porosity is controlled by an additional constitutive relationship such as the P-\(\alpha\)[79], P-\(\lambda\)[80], or P-\(\varepsilon\)[81] model. [PERSON] et al. [82] and [PERSON] et al. [83] discusses many computational simulations of penetration mechanism and works with utilizing computer software to model penetration and other related behavior of soil media. Numerical modeling such as the use of constitutive granular media models are defined as a continuum, and particle methods are used to model granular media at the mesoscale. \"Continuum constitutive models\" can be computationally efficient compared to mesoscale particle models. However, experimental tests are typically needed to calibrate parameters of the constituent model. Mesoscale models are usually computationally expensive using particle methods. Owing to major developments in computational powers in recent years, mesoscale models are increasingly being used as an alternative to experimental tests. The past investigators used three types of numerical methods to perform penetration analysis of missiles into geological targets, namely \"Finite Differences method (FDM), Finite Elements method (FEM), and Discrete Elements method (DEM)\". The application of Finite Element Method to simulate the penetration in soil media is due to [PERSON] et al. [84] and [PERSON] et al. [85]. The Finite element simulation of ball drop test along with experimental study for clay material is presented by [PERSON] [86]. The impact of the steel ball with the clay was monitored with a high-speed camera to study steel ball-clay interaction. The simulations showed that linear yield stress curve model was not adequate to model the penetration dynamics of clay and showed too-heavy oscillations after impact in the simulations with this model. Figure 6: Two-stage penetration model.