# Research Advances in Large Deformation Analysis and Applications of the Material Point Method
## 6 Application in Impact Penetration  
The MPM is not only capable of effectively simulating the damage process of target plate materials under impact but also accurately assessing the degree of material damage [79]. With the advantages of easy boundary condition application, no tensile instability issues, and high efficiency in handling large deformation problems, it is particularly well-suited for solving complex problems involving extreme deformations, such as impact penetration. [PERSON] and [PERSON] [80] utilized the MPM to obtain numerical solutions for Taylor problems such as the impact penetration of a steel ball into a rigid wall, which matched experimental results and confirmed the significant advantage of the MPM in evolving complex damage behavior at high speeds. [PERSON] [81] studied the impact penetration of steel shot at different impact velocities into thin boron carbide and Al-6061 plates. [PERSON] et al. [82] used the GIMP to simulate the penetration of shaped charge jets into aluminum to enhance numerical stability. However, the inherent no-slip contact of the standard MPM cannot solve penetration problems. [PERSON] et al. [16] proposed two novel contact algorithms and introduced them into the MPM. By simulating large deformation scenarios such as the impact penetration of steel balls, as depicted in Figure 15, it can be observed that the target shape parameter h/D approaches the experimental result with the decrease in the cell size and particle space. This demonstrated that the simulation results obtained with the proposed contact algorithms are closer to the experimental results than those obtained with the standard MPM. To enhance the accuracy of solving penetration problems with smooth pressure and stress fields, [PERSON] et al. [83] proposed a B-spline MPM (cBSMPM) based on the Lagrange multiplier method, which employs B-splines to define the background grid, as shown in Figure 16.  
Figure 15: Final deformed shape of the target plate: (**a**) experimental graph; (**b**) obtained by the first contact method; (**c**) obtained by the second contact method [16]. The corresponding basis function for computational grid node \(I\) is  
\[N_{i,j}^{pq}(\xi,\eta)=N_{i}^{p}(\xi)N_{j}^{q}(\eta) \tag{17}\]  
where \(N_{i}^{p}(\xi)\) is the B-spline basis function of polynomial degree \(p\) for the open uniform knot vector \(N=\xi_{0},\xi_{1},\dots,\xi_{n+p}\), and \(N_{j}^{q}(\eta)\) is the B-spline basis function of polynomial degree \(q\) for the open uniform knot vector \(M=\eta_{0},\eta_{1},\dots,\eta_{n+q}\). Therefore, the higher the degree of the basis functions, the higher the solution accuracy, as shown in Figure 17. [PERSON] et al. demonstrated through the study of high-speed tungsten projectile penetration into thick plates and other impact penetration examples that increasing the degree of B-spline basis functions can improve the solution accuracy of smooth stress/pressure fields for impact and penetration problems. For penetration problems involving rigid materials, only certain areas experience significant deformation, and the material deformation is minimal at the initial moment. To investigate the damage effects of a projectile on ballistic gelatin, [PERSON] et al. [84] introduced a contact transition criterion in the adaptive finite element MPM to convert contacting gelatin finite elements into MPM particles, with the coupling between the steel ball and gelatin being achieved on the MPM background grid. They successfully simulated the physical process of a steel ball penetrating ballistic gelatin at a high speed. At the beginning of the calculation, both the steel ball and gelatin are finite element bodies (as shown in Figure 18). The results indicate that the deformation of the ball is limited, while a local area of the gelatin experiences significant deformation, and the remaining areas of the gelatin have limited deformation.