## 1 Introduction  
Predicting the penetration depth of the projectile is one of the essential aspects of the geometric design of the projectile [1]. In particular, projectiles are expensive, preparation and experiments of projectiles take so much time and cost. Additionally, the effects of design variables are challenging to study accurately due to measuring errors and experiment problems. Recent developments in computational capabilities and techniques have made computational simulation very efficient in many ways [2, 3]. Simulation considering deformation of the projectile and the target costs too much computational time. Also, the deformation and fracture characteristics of the target and the contact problem between the target and the projectile are required [4]. Thus, a theoretical and empirical based model for penetration has been developed for rapid prediction. [PERSON] [5] developed an empirical-based simple analytic equation for predicting the penetration depth of the projectile. [PERSON]'s equation was derived based on the energy method and some empirical variables. The advantage of [PERSON]'s equation is that the calculation time is short, and the use is simple. However, in the calculation, the deformation of the projectile is not calculated. It is difficult to apply to various environments. Another method considering lateral loading was developed based on [PERSON]'s equation [6]. The calculation was completed with small time increments. The reaction forces were calculated from [PERSON]'s equation and applied to the projectile with an explicit method. However, it was not possible to consider the change of the reaction force with respect to deformation of the projectile. The cavity pressure method [7; 8; 9] is an analytical method that calculates the reaction forces resulting from targets with respect to the expansion speed of the cavity. Normal pressure on the projectile nose is approximated by the results from the radial stress on the cavity surface. [PERSON] and [PERSON] [7] developed penetration equations for soil targets with Mohr-Columb and Tresca yield criteria. According to the yield criteria, the boundary condition of the elastic-plastic interface is different. [PERSON] and [PERSON] [8] developed a spherical cavity-expansion penetration model for concrete targets for an incompressible and compressible material. [PERSON] and [PERSON] [10] derived a spherical cavity pressure distribution of the aluminum plate, considering compressible, strain hardening, and strain-rate dependence. The integration of the projectile surface with the cavity pressure presents axial reaction forces. From [PERSON]'s law, the acceleration and the penetration depth of the projectile were calculated. However, the cavity pressure method cannot consider the deformation of the projectile. When the deformation of the projectile is extensive (i.e., aluminum target), the cavity pressure method is not proper. There have been some studies regarding the effects of the projectile geometries. [PERSON] and [PERSON] [11] studied the effect of a non-deformable projectile having different geometrical characteristics on the penetration with the cavity expansion method. A general geometry function was introduced to define the geometrical characteristics of the projectile. When deformation of the projectile is very small, the prediction of the penetration using the cavity-expansion method is appropriate. [PERSON] et al. [12] studied the penetration and perforation of the fiber-reinforced plastic (FRP) laminate with rigid projectiles. The finite cavity method [13] is a method that links the advantages of finite element analysis with the advantages of analytical methods. Cavity pressure calculates the pressure distribution at this point when the point moves at a certain speed and calculates the reaction forces of the projectile. Finite element analysis was applied to predict the deformation and trajectory of the projectile. [PERSON] et al. [14] and [PERSON] et al.