Behind the scenes of most movie special effects are computers crunching intense mathematical equations. And some of the most complex types of animation equations describe fluid motion: anything from lava flow to an explosion to the rise and disappearance of smoke rings. But many times, the equations available to animators aren’t good enough to accurately represent and control fluids, says Mathieu Desbrun, professor of computer science at the California Institute of Technology, in Pasadena. In order to make fluid animation look good enough, he says, some animators opt to draw it by hand–a time-consuming process.

But Desbrun’s research could make fluids flow more nicely on screen. He and his team are developing an entirely new approach to fluid motion, based on new mathematics called discrete differential geometry, that use equations designed specifically to be solved by computers rather than people. Ultimately, he says, they have the potential to cut the cost and time of making a piece of animation. “Now that we’re using computers, it’s a whole new ball game,” he says.

Before computers, Desbrun explains, mathematicians and physicists developed equations for the motion of objects such as solids and fluids, and many of them were solvable by hand. Over the past few decades, it became clear that computers could be used to solve many of the more difficult equations, so computer scientists and mathematicians took the known set of equations and tried to modify them for the task. They reworked the equations explaining the physical rules, effectively breaking them into hundreds of numerous chunks so that the digital brain of a computer, which is good at working on a lot of these chunks at one time, could solve them.

While much successful work has been done using this approach, says Desbrun, these equations still only approximate motion, and they tend to produce unnaturally flowing liquids. For instance, in the case of a whirlpool, over time the traditional approach introduces errors into the motion, producing artificial viscosity: the visual result is a swirling whirlpool that slows down for no obvious reason. An animator must step in to modify the frames to make sure that the liquid keeps moving the right way.

Desbrun’s approach is to write new equations that are based on physical properties that aren’t expressed in the traditional equations. For instance, traditional equations include information about the velocity of a liquid, and this is used to approximate, or provide an inexact description of, a liquid’s motion if it starts to swirl around. But Desbrun’s equations bypass simple velocity and instead describe the swirling motion exactly, and in a way that computers can easily crunch. “Instead of just approximating them, we can capture the dynamics faithfully,” he says. “And we show it pays off visually.”

In a paper published this month in the journal *Transactions on Graphics*, Desbrun and his team describe the approach they take to model swirling fluids around and within solid objects such as a snow globe. The traditional approach would approximate the velocity of the liquid at various points in space and time and use this to approximate its motion along a circular path. But Desbrun’s equations model the actual circulation of the liquid, as if it were a property as fundamental as velocity.

To simulate the circulation of liquid, the researchers must capture the fundamental property of that circulation, called flux. Flux, or the amount of liquid that moves through a space at any given time, is captured by breaking the whirlpool into tiny pieces and determining the flow at each piece. These values are folded into the motion equation, allowing the liquid to flow more accurately.

So far, says Desbrun, the results are promising. This approach “has been shown to provide good statistical predictability … ensuring high visual quality.”

The research could be significant for the computer-graphics community, says Eva Kanso, professor of aerospace and mechanical engineering at the University of Southern California, in Los Angeles, who models fluids computationally. “Traditionally,” she says, “the trend was to use fast computation that is similar to reality but not based on real physics. It’s a big step for the computer-graphics community to look at physical laws and try to simulate them, especially now with a big demand on more-realistic animation.”

James O’Brien, professor of computer science at the University of California, Berkeley, says that if the traditional computational method and Desbrun’s method were to go head-to-head, there wouldn’t be much difference in the amount of time it takes to render an animation. However, he says, “the real point is getting better-looking results for the same amount of effort.”

Right now, says Desbrun, his new computation approach isn’t ready for prime time in the software found at animation studios, but colleagues at Columbia University are exploring the option. “We haven’t pushed our research to the point where we could help movie companies to add more control over the way fluid flows,” he says. But, he adds, if the equations are used in software, artists could, with the click of a button, easily modify special effects and animation far more accurately than they can today.

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