IPAB Seminar - 29/08/2024

Title: Structure-preserving fluid simulation

Abstract: The Incompressible Euler Equation that governs fluids with low viscosity exhibits rich phenomena such as intricate vortex interaction at different scales.  For decades researchers in computational fluid dynamics and computer graphics have been looking for computational methods to capture these phenomena.  Popular methods based on particle-in-cell (e.g. Fluid Implicit Particle FLIP) heavily depends on interpolations and approximations, and thus little could be said about structure preservation.  Discrete differential geometric fluid simulation methods relies on a grid of fixed resolution, making it hard to capture small scale phenomena.  We describe a new approach to discretize the Euler equation.  We start with a Hamiltonian formulation of the incompressible Euler Equations: a Hamiltonian flow using the Lie-Poisson bracket on the dual space of the Lie algebra to the volume-preserving diffeomorphism group.  Then, using a mimetic interpolation from isogeometric analysis, we construct a modified Hamiltonian system that governs our discrete Euler flow, still using the same Lie-Poisson structure.  The resulting discretization, when paired with a geometric time integration scheme, is energy and coadjoint orbit preserving.  Moreover it is similar to a FLIP method.  The method enjoys multiple additional exactness properties, and is demonstrated numerically with outstanding stability, energy, and Casimir preservation.  The method produces small scale phenomena even at low grid resolutions.  With further topological analysis in the Lie-Poisson framework, we also point out a discrepancy in many existing vorticity methods, and explain a simple method to resolve the problem.

Bio: Albert Chern is an assistant professor in Computer Science and Engineering at UC San Diego.  He received his PhD at Caltech advised by Prof. Peter Schröder in 2017, followed by a postdoctoral position at TU Berlin working with Prof. Ulrich Pinkall before the position at UC San Diego in 2020.  Chern's research interest lies in differential geometry and its application to computational mathematics, fluid dynamics, and computer graphics.  He is a recipient of the NSF CAREER Award.