A Cut‐Cell Geometric Multigrid Poisson Solver for Fluid Simulation

Abstract

Figure 1: Three frames from a simulation of smoke flowing past a sphere on a 256 × 512 × 256 grid. AbstractWe present a novel multigrid scheme based on a cut-cell formulation on regular staggered grids which generates compatible systems of linear equations on all levels of the multigrid hierarchy. This geometrically motivated formulation is derived from a finite volume approach and exhibits an improved rate of convergence compared to previous methods. Existing fluid solvers with voxelized domains can directly benefit from this approach by only modifying the representation of the non-fluid domain. The necessary building blocks are fully parallelizable and can therefore benefit from multi-and many-core architectures.

Sawtooth cycle revisited

“…They also solved the linear complementarity problem using a multigrid method . Weber et al proposed a cut‐cell finite volume discretization for fluid simulations to improve the convergence rate of multigrid solvers …”

A positive-definite cut-cell method for strong two-way coupling between fluids and deformable bodies

“…Such cut-cell discretizations have become increasingly common in uid animation, for example in the context of spatial adaptivity [Batty et al 2010], multigrid methods [Weber et al 2015], detailed splashes [Edwards and Bridson 2014], and thin solids [Azevedo et al 2016]. This trend can be attributed to the simplicity of cutcells (both conceptual and with respect to implementation) as well as their expressive power: uxes across small geometrical details can be mathematically captured without re ning or re-orienting the computational grid.…”

“…The approach we take is to generalize an increasingly popular family of symmetric positive-de nite cut-cell uid methods [Azevedo et al 2016;Batty et al 2007;Bridson 2015;Gibou and Min 2012;Ng et al 2009;Roble et al 2005;Weber et al 2015] to the case of strongly (or monolithically) coupled two-way interactions with deformable solids, in which solid and uid dynamics are solved simultaneously. To reconcile the Eulerian uid and Lagrangian solid domains, we construct a mutually conforming cut-cell mesh at each frame by clipping the cells of the uid grid against the deformable solid object's geometry.…”

A Multilevel SPH Solver with Unified Solid Boundary Handling

“…Figure illustrates cutaway views of fluid simulation with our coarsening scheme, at the particle, the finest grid, and the second finest grid levels. It is worth noting that we tested the cut‐cell approach [WMRSF15]. However, the convergence was not improved, unlike the case of the grid‐based simulations.…”

“…To address this problem, in the Eulerian grid‐based approach, scalable multigrid (MG) methods have been used [MST10, CM11, CM12, FWD14,WMRSF15] with the regular Cartesian grid, which can also be used to construct the hierarchy. This multilevel approach is also adopted for deformable body simulation by constructing the hierarchy from embedded grid structures [ZSTB10, MZS*11] or computing nearly optimal coarser‐level structures from finer‐levels [Mül08, WOR10].…”