A Hybrid Reproducing Kernel Particle Method for Three-Dimensional Advection-Diffusion Problems

In this paper, a hybrid reproducing kernel particle method (HRKPM) for three-dimensional (3D) advection-diffusion problems is presented. The governing equation of the advection-diffusion problem includes the second derivative of the field function to space coordinates, the first derivative of the field function to space coordinates and time, so it is necessary to discretize the time domain after discretizing the space domain. By introducing the idea of dimension splitting, a 3D advection-diffusion problem can be transformed into a series of related two-dimensional (2D) ones in the dimension splitting direction. Then, the discrete equations of these 2D problems are established by using the RKPM, and these discrete equations are coupled by using the difference method. Finally, by using the difference method to discretize the time domain, the formula of the HRKPM for solving 3D advection-diffusion problem is obtained. Numerical results show that the HRKPM has higher computational efficiency than the RKPM when solving 3D advection-diffusion problems.

Penetration modeling of ultra‐high performance concrete using multiscale meshfree methods

Terminal ballistics of concrete is of extreme importance to the military and civil communities. Over the past few decades, ultra‐high performance concrete (UHPC) has been developed for various applications in the design of protective structures because UHPC has an enhanced ballistic resistance over conventional strength concrete. Developing predictive numerical models of UHPC subjected to penetration is critical in understanding the material's enhanced performance. This study employs the advanced fundamental concrete (AFC) model, and it runs inside the reproducing kernel particle method (RKPM)‐based code known as the nonlinear meshfree analysis program (NMAP). NMAP is advantageous for modeling impact and penetration problems that exhibit extreme deformation and material fragmentation. A comprehensive experimental study was conducted to characterize the UHPC. The investigation consisted of fracture toughness testing, the utilization of nondestructive microcomputed tomography analysis, and projectile penetration shots on the UHPC targets. To improve the accuracy of the model, a new scaled damage evolution law (SDEL) is employed within the microcrack informed damage model. During the homogenized macroscopic calculation, the corresponding microscopic cell needs to be dimensionally equivalent to the mesh dimension when the partial differential equation becomes ill posed and strain softening ensues. Results of numerical investigations will be compared with results of penetration experiments.

A semi-Lagrangian reproducing kernel particle method with particle-based shock algorithm for explosive welding simulation

The explosive welding process is an extreme-deformation problem that involves shock waves, large plastic deformation, and fragmentation around the collision point, which are extremely challenging features to model for the traditional mesh-based methods. In this work, a particle-based Godunov shock algorithm under a semi-Lagrangian reproducing kernel particle method (SL-RKPM) is introduced into the volumetric strain energy to accurately embed the key shock physics in the absence of a mesh or grid, which is shown to also ensure the conservation of linear momentum. For kernel stability, a deformation-dependent anisotropic kernel support update algorithm is proposed, which is shown to capture excessive plastic flow and material separation. A quasi-conforming nodal integration is adopted to avoid the need of updating conforming cells which is tedious in extreme deformations. It is shown that the proposed formulation effectively captures shocks, jet formation, and smooth-to-wavy interface morphology transition with good agreement with experimental results.