Innovations in computer graphics and geometry processing are revolutionizing the way we simulate physical phenomena like fire and flames, crucial for creating breathtaking visual effects in both video games and movies, as well as aiding in the fabrication of intricate designs through 3D printing.
At the core of these advancements lie mathematical constructs known as partial differential equations (PDEs), pivotal for modeling these natural processes. Notably, the category of second-order parabolic PDEs elucidates how phenomena can achieve smoothness over time, with the heat equation being the quintessential example that illustrates how heat disperses across a surface or within a volume proportionately as time unfolds.
While geometry processing researchers have developed numerous algorithms to wrestle with these equations on curved surfaces, many solutions typically cater only to linear problems or singular PDEs. However, a groundbreaking approach by MIT’s Computer Science and Artificial Intelligence Laboratory (CSAIL) researchers addresses a broader spectrum of potentially nonlinear challenges.
In a recent paper published in the Transactions on Graphics and showcased at the SIGGRAPH conference, the team reveals an algorithm capable of tackling various nonlinear parabolic PDEs on triangle meshes. This is achieved by deconstructing the problem into three simpler equations that can be resolved with the existing tools familiar to graphics experts. This novel framework aims to enhance our understanding of shapes and accurately model complex dynamical processes.
Lead author Leticia Mattos Da Silva SM ’23, a PhD candidate at MIT focusing on electrical engineering and computer science, asserts, “We present a structured approach: if you aim to solve a second-order parabolic PDE numerically, simply adhere to these three sequential steps. Each step involves resolving a more straightforward issue using established geometry processing tools, ultimately providing a solution to the intricate second-order parabolic PDE.”
To achieve this, Da Silva and her collaborators employed Strang splitting, a technique enabling researchers in geometry processing to compartmentalize the PDE into manageable components they can efficiently solve.
The first step requires advancing a solution in time by addressing the heat equation, a fundamental equation that illustrates how heat emanates from a source and spreads across a given shape. Imagine using a blowtorch on a metal plate; this equation precisely describes how the heat disperses from that focal point. This stage can be executed effortlessly with linear algebra techniques.
Next, if the parabolic PDE reveals additional nonlinear dynamics beyond just heat spread, the second step of the algorithm involves accounting for these nonlinear aspects by solving a Hamilton-Jacobi (HJ) equation, which classifies as a first-order nonlinear PDE.
Although generic HJ equations can pose significant challenges, Mattos Da Silva and her team demonstrated that applying their splitting method yields an HJ equation solvable through convex optimization algorithms. This technique is a common tool among geometry processing researchers, easily integrating with their existing reliable software. In the final phase, the algorithm leverages the heat equation once again to push forward the solution of the more complicated second-order parabolic PDE.
This framework holds promise for efficiently simulating fire and flames among various applications. “Simulating flames involves a vast pipeline, of which a PDE solver is a critical component,” comments Mattos Da Silva. Within these frameworks lies the pivotal G-equation, a nonlinear parabolic PDE modeling flame front propagation, which the researchers’ framework can adeptly resolve.
Furthermore, the team’s algorithm can tackle the diffusion equation within the logarithmic domain, where it transitions into a nonlinear context. Senior author Justin Solomon, an associate professor of EECS and head of the CSAIL Geometric Data Processing Group, has previously developed cutting-edge techniques for optimal transport requiring the logarithm of diffusion results. Da Silva’s framework enhances computational reliability by processing diffusion directly in the logarithmic realm, allowing for a more stable methodology to ascertain a geometric average across distributions on surface meshes, as illustrated by a koala model.
While this framework predominantly addresses nonlinear challenges, it remains effective for linear PDEs as well. For example, it successfully resolves the Fokker-Planck equation, wherein heat disperses linearly alongside additional terms drifting concurrently. A practical application of this approach modeled how swirling motions evolve across a triangulated sphere, yielding results reminiscent of captivating latte art.
The researchers emphasize that their project is merely a stepping stone toward confronting the challenges posed by nonlinearity in various PDEs relevant to graphics and geometry processing. Although their current focus is on static surfaces, aspirations exist to extend their efforts into dynamic surfaces. Additionally, while their framework adeptly manages single parabolic PDE tasks, tackling problems involving coupled parabolic PDEs—common in biological and chemical systems—remains a future goal.
Mattos Da Silva and Solomon collaborated with Oded Stein, an assistant professor at USC’s Viterbi School of Engineering, on this paper. Their innovative work received support from various prestigious entities, including an MIT Schwarzman College of Computing Fellowship funded by Google, MathWorks Fellowship, the Swiss National Science Foundation, U.S. Army Research Office, U.S. Air Force Office of Scientific Research, U.S. National Science Foundation, MIT-IBM Watson AI Lab, and Google Research, among others.
Photo credit & article inspired by: Massachusetts Institute of Technology
