Restoring the missing vorticity in advection-projection fluid solvers

Abstract

Most visual effects fluid solvers use a time-splitting approach where velocity is first advected in the flow, then projected to be incompressible with pressure. Even if a highly accurate advection scheme is used, the self-advection step typically transfers some kinetic energy from divergence-free modes into divergent modes, which are then projected out by pressure, losing energy noticeably for large time steps. Instead of taking smaller time steps or using significantly more complex time integration, we propose a new scheme called IVOCK (Integrated Vorticity of Convective Kinematics) which cheaply captures much of what is lost in self-advection by identifying it as a violation of the vorticity equation. We measure vorticity on the grid before and after advection, taking into account vortex stretching, and use a cheap multigrid V-cycle approximation to a vector potential whose curl will correct the vorticity error. IVOCK works independently of the advection scheme (we present examples with various semi-Lagrangian methods and FLIP), works independently of how boundary conditions are applied (it just corrects error in advection, leaving pressure etc. to take care of boundaries and other forces), and other solver parameters (we provide smoke, fire, and water examples). For 10 ∼ 25% extra computation time per step much larger steps can be used, while producing detailed vorticial structures and convincing turbulence that are lost without correction. Table 1: Algorithm abbreviations used through out this paper. IVOCK The computational routine (Alg.1) correcting vorticity for advection SF Classic Stable Fluids advection [Stam 1999] SF-IVOCK IVOCK with SF advection SL3 Semi-Lagrangian with RK3 path tracing and clamped cubic interpolation BFECC Kim et al.'s scheme [2005], with extrema clamping([Selle et al. 2008]) BFECC-IVOCK IVOCK with BFECC advection MC Selle et al.'s MacCormack method [2008] MC-IVOCK IVOCK with MacCormack FLIP Zhu and Bridson's incompressible variant of FLIP [2005] FLIP-IVOCK FLIP advection of velocity and density, SL3 for vorticity in IVOCK.

An explicit structure‐preserving numerical scheme for EPDiff

“…An initial set of vortices, each localized along a curve (vortex filament) or a hypersurface (vortex sheet), interact in a non‐local manner as they are advected by the sum of their individual induced velocity fields. In the case of an inviscid and incompressible fluid (the Euler equation regime), this motion is described by the vorticity equation (see e.g., [ZBG15]), which can be written as…”

Model-reduced variational fluid simulation

“…and Fiume 1993], wavelet ], empirical mode decomposition [Gao et al 2013], subgrid turbulence [Schechter and Bridson 2008;Narain et al 2008], curl correction [Zhang et al 2015], or through enforcing Lagrangian coherent structure [Yuan et al 2011] can be done straightforwardly. We demonstrate the efficiency of our resulting integrator through a number of examples in 2D, 3D, and curved 2D domains, as well as its versatility by pointing out how to extend its use to magnetohydrodynamics, subgrid scale models, and other fluid equations.…”

“…Stam [1999] introduced semi-Lagrangian advection and a sparse Poisson solver which brought much improved efficiency and stability. However, these improvements came at the cost of significant dissipation-a common issue that one can partially mitigate via vorticity confinement [Steinhoff and Underhill 1994], reinjection of vorticity with particles [Selle et al 2005], or curl correction [Zhang et al 2015]. Significantly less dissipative time integrators were also proposed through semi-Lagrangian advection of vorticity [Elcott et al 2007], or even energy-preserving methods [Mullen et al 2009].…”

Fast Fluid Simulations with Sparse Volumes on the GPU

“…Recently, Affine Particle‐In‐Cell (APIC) [JSS*15] were introduced to stably remove the dissipation problems of PIC, providing exact conservation of angular momentum during particle‐to‐grid transfers. The Integrated Vorticity of Convective Kinematics (IVOCK) [ZBG15] method was introduced for approximately restoring the dissipated vorticity during advection, independent of the advection method.…”