The Material Point Method (MPM) community has developed several momentum mapping schemes, among which PIC, FLIP, the hybrid scheme, and APIC are the most common. The FLIP scheme is widely used for dynamic problems due to its energy conservation properties, although it can introduce numerical noise and instability (Bardenhagen, 2002), potentially leading to computational failure. Conversely, the PIC scheme is known for numerical stability and is advantageous for static problems, but it suffers from significant numerical dissipation (Brackbill et al., 1988), which is unacceptable for strongly dynamic responses. Nairn et al. combined FLIP and PIC linearly (Nairn, 2015) to create a hybrid scheme, adjusting the proportion of each component based on empirical rather than theoretical analysis. Hammerquist and Nairn (2017) introduced an improved scheme called XPIC-m (for eXtended Particle-In-Cell of order m), which addresses the excessive filtering and numerical diffusion of PIC while suppressing the noise caused by the nonlinear space in FLIP used in MPM. XPIC-1 (eXtended Particle-In-Cell of order 1) is equivalent to the standard PIC method. Jiang et al. (2017, 2015) introduced the Affine Particle In Cell (APIC) method, where particle velocities are represented locally affine, preserving linear and angular momentum during the transfer process. This significantly reduces numerical dissipation and avoids the velocity noise and instability seen in FLIP. Fu et al. (2017) introduced generalized local functions into the APIC method, proposing the Polynomial Particle In Cell (PolyPIC) method. PolyPIC views G2P (Grid-to-Particle) transfer as a projection of the particle's local grid velocity, preserving linear and angular momentum, thereby improving energy and vorticity retention compared to the original APIC. Additionally, PolyPIC retains the filtering properties of APIC and PIC, providing robustness against noise.