A hyperbolic geometric flow for evolving films and foams

Abstract

Simulating the behavior of soap films and foams is a challenging task. A direct numerical simulation of films and foams via the Navier-Stokes equations is still computationally too expensive. We propose an alternative formulation inspired by geometric flow. Our model exploits the fact, according to Plateau's laws, that the steady state of a film is a union of constant mean curvature surfaces and minimal surfaces. Such surfaces are also well known as the steady state solutions of certain curvature flows. We show a link between the Navier-Stokes equations and a recent variant of mean curvature flow, called hyperbolic mean curvature flow , under the assumption of constant air pressure per enclosed region. Instead of using hyperbolic mean curvature flow as is, we propose to replace curvature by the gradient of the surface area functional. This formulation enables us to robustly handle non-manifold configurations; such junctions connecting multiple films are intractable with the traditional formulation using curvature. We also add explicit volume preservation to hyperbolic mean curvature flow, which in fact corresponds to the pressure term of the Navier-Stokes equations. Our method is simple, fast, robust, and consistent with Plateau's laws, which are all due to our reformulation of film dynamics as a geometric flow.

Keypoint-driven line drawing vectorization via PolyVector flow

“…The applications have consisted mostly of methods for surface smoothing and fairing [Desbrun et al 1999;Leng et al 2013;Taubin 1995] and/or multi-resolution processing [Guskov et al 1999;Kobbelt 2000;Kobbelt et al 1998]. It has also found use in fluid dynamics [Ishida et al 2017;Misztal et al 2012;Thürey et al 2010;Zhang et al 2012].…”

A model for soap film dynamics with evolving thickness

“…Their method enforces the incompressibility of the air phase and the preservation of circulation by using divergence-free vorticity primitives as simulation degrees of freedom. Ishida et al [2017], on the other hand, enforced an integral constraint on the air phase to preserve volume, and developed a geometric flow for surface tension computation. Their approach does not guarantee circulation preservation, but it is more computationally efficient than Da et al's N-body vortex solver.…”

“…These thickness dynamics result in a physical model for film advection, mass conservation, draining, evaporation, and surface tension ripples. We discretize the equations on a non-manifold triangle mesh surface and couple it to an existing bubble solver [Ishida et al 2017]. We use the resulting simulation framework to reproduce the aforementioned thickness-dependent natural phenomena, as illustrated in Figure 1.…”

A level‐set method for the evolution of cells and tissue during curvature‐controlled growth

“…In previous works, we have proposed a cell‐based mathematical model of tissue growth to account for the mechanistic influence of curvature on cell crowding and cell spreading in the co‐evolution of cell density and tissue interface . This mathematical model reduces to a specific type of hyperbolic curvature flow in which the normal acceleration of the interface is proportional to curvature . The hyperbolic character of this curvature flow gives rise to a rich set of interface movement patterns.…”

“…These equations are expressed in a manifestly covariant form independent of space dimension. We note that our method is also applicable to other surface‐bound dynamic processes that affect the evolution of an interface, including etching processes, active membranes, and thin films and foams …”