Taylor particle-in-cell transfer and kernel correction for material point method

Abstract

The material point method (MPM) has been extensively used to solve problems involving large displacements and deformations. The standard MPM formulation, which adopts fluid-implicit-particle (FLIP) transfer, is less dissipative but also less stable than particle-in-cell (PIC) transfer. The affine PIC (APIC) transfer introduces the affine velocity and was developed to realize stable simulations while overcoming dissipations of the angular momentum in PIC. This paper presents Taylor-PIC (TPIC) transfer, which is a type of APIC transfer that combines the affine velocity based on the first-order Taylor series approximation and PIC transfer. TPIC is simple (only the particle velocity gradient is required) and inherits the key advantages of the original APIC, such as less dissipation and stability. Although TPIC does not conserve angular momentum, in contrast to APIC, the velocity gradient is preserved in the transfer between particles and the grid. This velocity gradient contains the angular information in its skew-symmetric component, which allows TPIC to adequately describe angular motion, similar to APIC.Furthermore, the MPM can cause stress oscillations near boundaries when boundary conditions are explicitly imposed, e.g., when the boundary grid velocity (or momentum) is set to zero. When affine-type transfers are used, this instability is exacerbated, and simulations can easily fail. Therefore, we propose a kernel correction method based on the weighted least squares for particles near boundaries. The proposed TPIC transfer and kernel correction are validated through five types of simulations, each using a different material—linear elastic, von Mises, Newtonian fluid, and Drucker–Prager. In the simulations, incorrect results due to stress oscillations are observed even for small deformations. Applying the corrected kernels successfully removes these spurious oscillations, and the results are consistent with the analytical solutions. Moreover, the numerical results confirm the accuracy and robustness of the proposed TPIC transfer.