Computer graphics and geometry processing research provide the tools needed to simulate physical phenomena like fire and flames, aiding the creation of visual effects in video games and flicks in addition to the fabrication of complex geometric shapes using tools like 3D printing.
Under the hood, mathematical problems called partial differential equations (PDEs) model these natural processes. Amongst the various PDEs utilized in physics and computer graphics, a category called second-order parabolic PDEs explain how phenomena can change into smooth over time. Essentially the most famous example on this class is the warmth equation, which predicts how heat diffuses along a surface or in a volume over time.
Researchers in geometry processing have designed quite a few algorithms to unravel these problems on curved surfaces, but their methods often apply only to linear problems or to a single PDE. A more general approach by researchers from MIT’s Computer Science and Artificial Intelligence Laboratory (CSAIL) tackles a general class of those potentially nonlinear problems.
In a paper recently published within the Transactions on Graphics journal and presented on the SIGGRAPH conference, they describe an algorithm that solves different nonlinear parabolic PDEs on triangle meshes by splitting them into three simpler equations that could be solved with techniques graphics researchers have already got of their software toolkit. This framework may also help higher analyze shapes and model complex dynamical processes.
“We offer a recipe: If you wish to numerically solve a second-order parabolic PDE, you’ll be able to follow a set of three steps,” says lead writer Leticia Mattos Da Silva SM ’23, an MIT PhD student in electrical engineering and computer science (EECS) and CSAIL affiliate. “For every of the steps on this approach, you’re solving a less complicated problem using simpler tools from geometry processing, but at the top, you get an answer to the tougher second-order parabolic PDE.”
To perform this, Da Silva and her coauthors used Strang splitting, a method that permits geometry processing researchers to interrupt the PDE down into problems they know the way to solve efficiently.
First, their algorithm advances an answer forward in time by solving the warmth equation (also called the “diffusion equation”), which models how heat from a source spreads over a shape. Picture using a blow torch to warm up a metal plate — this equation describes how heat from that spot would diffuse over it. This step could be accomplished easily with linear algebra.
Now, imagine that the parabolic PDE has additional nonlinear behaviors that will not be described by the spread of warmth. That is where the second step of the algorithm is available in: it accounts for the nonlinear piece by solving a Hamilton-Jacobi (HJ) equation, a first-order nonlinear PDE.
While generic HJ equations could be hard to unravel, Mattos Da Silva and coauthors prove that their splitting method applied to many necessary PDEs yields an HJ equation that could be solved via convex optimization algorithms. Convex optimization is a typical tool for which researchers in geometry processing have already got efficient and reliable software. In the ultimate step, the algorithm advances an answer forward in time using the warmth equation again to advance the more complex second-order parabolic PDE forward in time.
Amongst other applications, the framework could help simulate fire and flames more efficiently. “There’s an enormous pipeline that creates a video with flames being simulated, but at the guts of it’s a PDE solver,” says Mattos Da Silva. For these pipelines, a necessary step is solving the G-equation, a nonlinear parabolic PDE that models the front propagation of the flame and could be solved using the researchers’ framework.
The team’s algorithm may solve the diffusion equation within the logarithmic domain, where it becomes nonlinear. Senior writer Justin Solomon, associate professor of EECS and leader of the CSAIL Geometric Data Processing Group, previously developed a state-of-the-art technique for optimal transport that requires taking the logarithm of the results of heat diffusion. Mattos Da Silva’s framework provided more reliable computations by doing diffusion directly within the logarithmic domain. This enabled a more stable method to, for instance, find a geometrical notion of average amongst distributions on surface meshes like a model of a koala.
Although their framework focuses on general, nonlinear problems, it could possibly even be used to unravel linear PDE. As an example, the strategy solves the Fokker-Planck equation, where heat diffuses in a linear way, but there are additional terms that drift in the identical direction heat is spreading. In a simple application, the approach modeled how swirls would evolve over the surface of a triangulated sphere. The result resembles purple-and-brown latte art.
The researchers note that this project is a place to begin for tackling the nonlinearity in other PDEs that appear in graphics and geometry processing head-on. For instance, they focused on static surfaces but would love to use their work to moving ones, too. Furthermore, their framework solves problems involving a single parabolic PDE, however the team would also prefer to tackle problems involving coupled parabolic PDE. All these problems arise in biology and chemistry, where the equation describing the evolution of every agent in a combination, for instance, is linked to the others’ equations.
Mattos Da Silva and Solomon wrote the paper with Oded Stein, assistant professor on the University of Southern California’s Viterbi School of Engineering. Their work was supported, partly, by an MIT Schwarzman College of Computing Fellowship funded by Google, a MathWorks Fellowship, the Swiss National Science Foundation, the U.S. Army Research Office, the U.S. Air Force Office of Scientific Research, the U.S. National Science Foundation, MIT-IBM Watson AI Lab, the Toyota-CSAIL Joint Research Center, Adobe Systems, and Google Research.
