Large-Scale Liquid Simulation on Adaptive Hexahedral Grids

Abstract

Abstract-Regular grids are attractive for numerical fluid simulations because they give rise to efficient computational kernels. However, for simulating high resolution effects in complicated domains they are only of limited suitability due to memory constraints. In this paper we present a method for liquid simulation on an adaptive octree grid using a hexahedral finite element discretization, which reduces memory requirements by coarsening the elements in the interior of the liquid body. To impose free surface boundary conditions with second order accuracy, we incorporate a particular class of Nitsche methods enforcing the Dirichlet boundary conditions for the pressure in a variational sense. We then show how to construct a multigrid hierarchy from the adaptive octree grid, so that a time efficient geometric multigrid solver can be used. To improve solver convergence, we propose a special treatment of liquid boundaries via composite finite elements at coarser scales. We demonstrate the effectiveness of our method for liquid simulations that would require hundreds of millions of simulation elements in a non-adaptive regime.

A stream function solver for liquid simulations

“…Thus, a large part of work has focused on improving it, e.g. with fast iterative solvers [Zhu et al 2010;Ferstl et al 2014] and by introducing flexible ways to couple multiple grids [English et al 2013]. A variety of different discretizations have been proposed over the years, many of them using tetrahedral meshes [Klingner et al 2006;Batty et al 2010], Elcott and colleagues [2007] proposed a discretization based on generic simplices while focusing on preserving circulations in the flow.…”

Power diagrams and sparse paged grids for high resolution adaptive liquids

“…later improved the discretization of the pressure projection to recover second order accuracy while preserving symmetry, although it is unclear how to incorporate irregular boundary conditions near T-junctions because the necessary modifications to the Poisson stencil are mutually incompatible; we overcome this limitation to allow refinement and boundaries to seamlessly co-exist without loss of accuracy. Ferstl et al [2014] proposed a finite element method (FEM) that subdivides octree surface cells cut by surface geometry to yield a conforming mesh, applying Nitsche's method for the free surface and stabilization to minimize checkerboarding. Like Losasso's method, it requires boundaries to be uniformly resolved.…”

“…The price for this choice was the loss of orthogonality between the stored velocity components at the T-junction cell faces and the corresponding discrete pressure gradients across those faces. This approximation reduces the accuracy of the pressure to first order, and gives rise to non-physical parasitic currents at T-junctions even for hydrostatic scenarios, as discussed by various authors [Ando et al 2013;Chentanez et al 2007;Ferstl et al 2014;. subsequently proposed a modification that raises the accuracy of the pressure field to second order, but allows for adaptivity only in regions completely interior to the fluid.…”

A Dimension‐reduced Pressure Solver for Liquid Simulations

“…Ferstl et al [FWD13] showed that regular grid multigrid schemes do not guarantee convergence in general without significantly more complicated data-structures. In contrast, our method does not share these problems, because our dimension-reduction strategy is guaranteed to converge and our surface-aware basis satisfies Dirichlet boundary conditions exactly.…”

“…(5) also can be regarded as a Galerkin-based coarsening scheme that is used for certain classes of multigrid methods [FWD13]. In this context our approach represents a modified 2-level scheme with a zero initial guess, and a boundary-aware prolongation/interpolation operator, described in the next section.…”