A Nonlinear Vertex-based Model for Animation of Two-dimensional Dry Foam

Foam is the natural phenomenon of bubbles that arise due to nucleation of gas in liquids. The current state of art in Computer Graphics rarely includes foam effects on large scales. In this paper we introduce a vertexbased, quasi-static equilibrium model from the field of Computational Physics as a new paradigm for foam effects. Dynamic processes like gas diffusion and bubble collapse are added prior equilibration. Animationwise the numerical model is well behaved and stable and can converge even if the foam is locally ill-defined. A novel contribution is the Ghost-Bubble method that allows foam simulations with free dynamic boundary conditions. The presented model is interesting and well suited for 2D graphics applications like video games and procedural or animated textures. 1 FOAM ANIMATION IN COMPUTER GRAPHICS Many methods to simulate liquid fluids have been presented in Computer Graphics (Stam, 1999; Losasso et al., 2004; Selle et al., 2005; Losasso et al., 2006) with a common focus on realism. We believe that the next step in fluids dynamics lies within the subject of liquid foams, or froths. The scientific work on foam dynamics in Computer Graphics is sparse. In our opinion the most promising result to animate beer froth still has issues with the motion and behavior of the foam (Cleary et al., 2007) but the still frames from the foam simulations seem very convincing. We argue that the foam animation problems in Computer Graphics are caused by the fact that foams are treated as fluids and not as real foams. We believe that the theoretic foundation of foam dynamics must be obtained from Computational Physics. In this paper we revisit a vertex-based foam model from the field of Computational Physics and derive a mathematical model with a discretization that is suitable for the purpose of Computer Graphics. Our focus is directed towards two-dimensional foams. This dimensional restriction is due to that physics literature commonly only agrees on the processes of two-dimensional foams and how they are behaving whereas three-dimensional foams are not yet completely understood. Two-dimensional foams also exist in the real three-dimensional world, e.g. a liquid foam constrained between two glass plates or the single layer of foam resting on a surface. Additionally, in this paper we only focus on the behavior of dry foams as wet foams are not as accurate described by current physical models (Weaire et al., 2003). We handle the dynamics and internal forces of the dry foam along with external interactive contributions, e.g. topological changes, gas diffusion, and bubble collapses, but omit external dynamic forces and collision handling. However, the method is based on a Lagrangian representation and as such additional external couplings and interactions can be handled at vertex level. This paper introduces the Ghost-Bubble method which is a novel contribution for the two-dimensional vertex-based foam model. The Ghost-Bubble method allows dry foam simulations with free dynamic surfaces and finite boundaries. We derive a nonlinear Newton method for the dry foam model and our discrete model can converge even if the foam is locally ill-defined. The model has less than 1% total error even though we use first order finite difference approximations. The presented physics-based dry foam model is interesting and well suited for twodimensional graphics applications like video games and procedural or animated textures.