Stable Reproducing Kernel Particle Method for Studying Munitions Penetration into Geo-Materials

Mechanics of penetration of projectiles into the geo-material has long been of interest in the military and industry and remains an active research area because of the complex processes and deformations during penetration, including material flow, cavity formation, and fracture. These response characteristics present significant challenges to continuum mesh-based methods due to the severe distortion of the elements. However, particle-based methods such as the Reproduced Kernel Particle Method (RKPM) can avoid the above issue by developing the approximation function without the requirement of elements. Particularly, the semi-Lagrangian RK, which combines the advantages of being the Lagrangian description and utilizing the Eulerian kernels, performs effectively in extreme deformation problems. However, the standard semi Lagrangian framework suffers from the complexity of the convective effect arising from the Eulerian kernel. Full discrete semi-Lagrangian equations with consistent temporal integration lead to a set of nonlinear equations and thus necessitates the use of a nonlinear algorithm, which reduces accuracy while increasing computational cost. This work addresses the aforementioned issues by proposing a concurrent semi-Lagrangian approximation for the time-dependent variable. The proposed approximation eliminates the convective component from the time-dependent variable by interpolating the RK shape function with the corresponding generalized nodal coefficients. The proposed approximation avoids the need to compute the convective component, simplifying the semi-discrete equation. Overall, the concurrent semi-Lagrangian RK performs more effectively than the existing formulation. Furthermore, the temporal stability of the proposed formulation has been examined using a closed form eigenvalue estimation for the system under large deformations. A comprehensive study of the eigenvalue and critical time step is conducted to determine the effective parameters for the stability of the system. The estimated analytical eigenvalue and the critical time step were found to be consistent with the numerical estimate for several benchmark problems. Furthermore, the domain integration scheme in the semi-Lagrangian RK framework remains a challenge because it either necessitates a conforming integration cell, which is computationally expensive, or the surface integral, which is challenging to acquire. In this work, a stable nonconforming integration called the Midpoint Integration Method (MPIM) has been developed based on an extension of modified Simpson’s rule to address the domain-integration issue. The method is derived in such a way that the evaluation point is solely dependent on the nodal point, with no requirement of the conforming subdomain. The resultant method is free of conforming subdomains and surface integrals and can be applied directly to different meshfree formulations. To reduce the overall computational cost of integration, a variant of the MPIM scheme has been developed. An eigenvalue and numerical analyses are performed to investigate the stability, convergence, and effectiveness of the proposed method in the large deformation problems. Overall, MPIM shows superior performance when compared with other nodal integration schemes. In addition, simulations for the high-impact speed penetration into the geo-materials using the proposed stable RK method have been performed. For the purpose of verifying the numerical solution, a variety of different experimental setups, projectile types, geo-materials, range of impact velocities and angles are taken into consideration. Several parameters such as penetration depth, exit velocity, exit angle, penetration length, and exit time are numerically evaluated and verified with the experimental data. Overall, the numerical results show stable solution even for high-speed impact problems and agrees well with the experimental data.