Affine particle in cell method for MAC grids and fluid simulation

Abstract

We present a new technique for transferring momentum and velocity between particles and MAC grids based on the Affine-Particle-In-Cell (APIC) framework [1, 2] previously developed for co-located grids. APIC represents particle velocities as locally affine, rather than locally constant as in traditional PIC. These previous APIC schemes were designed primarily as an improvement on Particle-in-Cell (PIC) transfers, which tend to be heavily dissipative, and as an alternative to Fluid Implicit Particle (FLIP) transfers, which tend to be noisy. The original APIC paper [1] proposed APIC-style transfers for MAC grids, based on a limit for multilinear interpolation. We extend these to the case of smooth basis functions and show that the proposed transfers satisfy all of the original APIC properties. In particular, we achieve conservation of angular momentum across our transfers, something which eluded [1]. Early indications in [1] suggested that APIC might be suitable for simulating high Reynolds fluids due to favorable retention of vortices, but these properties were not studied further and the question was left unresolved. One significant drawback of APIC relative to FLIP is that energy is dissipated even when ∆t = 0. We use two dimensional Fourier analysis to investigate dissipation in this important limit. We investigate dissipation and vortex retention numerically in the general case to quantify the effectiveness of APIC compared with PIC, FLIP, and XPIC.

Effective time step restrictions for explicit MPM simulation

“…We note that other collision treatments also exist, such as an extension for cutting [HFG * 18], a penalty collision treatment for implicit integration [DS19], and a special treatment for sand [DBD16]. For domain boundaries, reflection boundary conditions for MPM simulation of fluids were introduced in [DSS19], and give the basis for our reflection boundary conditions. Unlike existing level set treatments for MPM, the reflection boundary conditions prevent particles from penetrating domain boundaries.…”

“…We propose a novel method for applying collisions or boundary conditions to an MPM simulation at domain walls based on the reflection boundary conditions of [DSS19]. The motivation for [DSS19] is as follows: when touching a mirror, the mirror itself applies the same forces to your hand that a physical reflection of your hand would. Thus, boundary conditions can be enforced by running the simulation as though a reflected copy of each particle existed on the other side of the domain walls.…”

“…Reflection boundary conditions were used in [DSS19] to enforce fluid boundary conditions at domain walls, including no-slip, free surface, and free slip conditions. Our method builds on [DSS19] with three improvements: (1) whereas [DSS19] modified mass and momentum, we modify velocity directly; this conserves mass and (where appropriate) momentum, (2) [DSS19] applied mass and momentum modification at various stages in their algorithm, whereas we require only a single velocity correction step at the end of the grid evolution, and (3) [DSS19] did not support friction or separation; we extend our formulation to these cases. Unlike [SSC * 13], where particles can seep into objects by a grid cell or so, our method stops particles at domain walls.…”

“…Modified mass, momentum, and forces. For motivation and completeness, we briefly review the reflection boundary conditions presented in [DSS19]. In that work, the boundary conditions are enforced by solving a modified grid system, constructed by transferring data from both the regular particles p as well as a virtual, reflected copy r(p) (which is never actually constructed).…”

“…Additionally, we introduce a novel reflection boundary condition for MPM based on [DSS19], reformulating their method to conserve mass and momentum, and extending it to handle Coulomb friction and separation.…”