Fluid Simulation in Bases of Laplacian Eigenfunctions
Tyler de Witt
Full Text PDF
M. Sc. Thesis, 2010
Department of Computer Science
University of Toronto
We present a novel method for the simulation of incompressible fluids. In contrast to
existing grid based and particle methods, we choose a spatial representation of vorticity in a
basis of Laplacian eigenfunctions. In this thesis, we show that unique properties of this basis
make it useful for computer graphics applications. Particularly, the Navier-Stokes equations
reduce to a compact form that is elegant and practical, permitting time integration schemes
that operate directly in the reduced space of basis coefficients. These time integration
schemes are efficient and energy preserving. For a number of useful geometries, our basis
functions are analytic. We extend our method to work on simplicial meshes through the use
of discrete exterior calculus.
Images and Videos
A minimum of three basis functions is needed to observe interesting
dynamic behavior. This animation shows a vortex moving clockwise around the domain as it is
carried by the flow of a larger vortex. This confirms what is described mathematically by the
Helmholtz version of the Euler equations: the vorticity is advected by the flow.
Vorticity advected clockwise by larger vortex
Top left, bottom left: fluid simulation on 16x16 MAC grid with linear interpolation of velocity field. Grid artifacts are most noticeable at the centres of vorticies where the
curvature is high. Top right, bottom right: fluid simulation using analytic basis functions for
velocity field. The velocity field remains smooth and accurate in the areas of high curvature.
Mesh free simulation
viscosity is simulated by decaying the energy in basis fields with a time constant proportional to their
eigenvalue. Accounting for viscosity in our model is trivial and does not require precompu-
tation. It can be controlled at run time by changing a single parameter.
These smoke simulations were rendered through volumetric ray tracing of
density fields created by radial basis functions at advected particle positions. Not counting
rendering costs, in both of these examples each simulation step took approximately 10ms,
after a precomputation of less than 10 minutes.
Smoke in bust model: Ray traced
Smoke in head model: Ray traced
Some basis fields exhibit symmetry in one or more
spatial directions. The dynamics amongst these basis fields will produce fluid motion that
remains symmetric. By initializing an inviscid flow with a few of these basis functions, the
resulting dynamics will change perpetually, but always retain an element of symmetry.
We use this property along with a few liberal adjustments to design a performance based artistic fluid
simulation. For example, one modification we employ is to clamp particle positions to
around the boundary of the fluid simulation domain. Particles will tend to
accumulate into dense layers, which are continually accumulated and lifted off, creating a
pleasing visualization of the vortices in the field.
Performance based symmetric fluid animation.