Momentum mapping format

Material point method discrete format

The Momentum Mapping Format is a key technique in the Material Point Method (MPM) for transferring physical quantities such as momentum, mass, and stress between a material point and a background grid. The Material Point Method (MPM) is a numerical technique using a mixed Eulerian-Lagrangian description. It discretises the computational domain with material points and employs a background grid to solve the momentum equations. Proposed by Sulsky et al. in 1994, MPM has since been expanded to various fields such as computational solid dynamics. Currently, MPM features several momentum mapping schemes, with the four main ones being PIC (Particle-in-cell), FLIP (Fluid-Implicit Particle), hybrid format, and APIC (Affine Particle-in-Cell). Understanding these schemes in-depth is crucial for the further development of MPM.

Background

MPM represents materials as collections of material points (or particles). Unlike other particle methods such as SPH (Smoothed-particle hydrodynamics) and DEM (Discrete element method), MPM also uses a background grid to solve the momentum equations arising from particle interactions. MPM can be categorized as a mixed particle/grid method or a mixed Lagrangian-Eulerian method. By combining the strengths of both frameworks, MPM aims to be the most effective numerical solver for large deformation problems. It has been further developed and applied to various challenging problems such as high-speed impact (Huang et al., 2011), landslides (Fern et al., 2019), saturated porous media (He et al., 2024), and fluid-structure interaction (Li et al., 2022).

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.

Affine Particle In Cell Method

In the PIC scheme, particle velocities during the Grid-to-Particle (G2P) substep are directly overwritten by extrapolating the nodal velocities to the particles themselves:

$V_{p}^{n+1}=\sum _{I}^{}S_{Ip}^{n}V_{I}^{n+1}$

In the FLIP scheme, the material point velocities are updated by interpolating the velocity increments of the grid nodes over the current time step: