Chapter 9: Discrete, vorticity-preserving, and stable simplicial fluids

Visual accuracy, low computational cost, and numerical stability are foremost goals in computer animation. An important ingredient in achieving these goals is the conservation of fundamental motion invariants. For example, rigid or deformable body simulation have benefited greatly from conservation of linear and angular momenta. In the case of fluids, however, none of the current techniques focuses on conserving invariants, and consequently, they often introduce a visually disturbing numerical diffusion of vorticity. Visually just as important is the resolution of complex simulation domains. Doing so with regular (even if adaptive) grid techniques can be computationally delicate. In this chapter, we propose a novel technique for the simulation of fluid flows. It is designed to respect the defining differential properties, i.e., the conservation of circulation along arbitrary loops as they are transported by the flow. Consequently, our method offers several new and desirable properties: (1) arbitrary simplicial meshes (triangles in 2D, tetrahedra in 3D) can be used to define the fluid domain; (2) the computations are efficient due to discrete operators with small support; (3) the method is stable for arbitrarily large time steps; and (4) it preserves a discrete circulation avoiding numerical diffusion of vorticity. The underlying ideas are easy to incorporate in current approaches to fluid simulation and should thus prove valuable in many applications.