CAREER: Geometric Approaches to Simulation

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Physical simulations and visual computing are pivotal in exploring physical sciences and technological innovations. Despite increasing computing power over the years, fundamental numerical challenges still await solutions. This project addresses two such challenges: (A) Simulating vortex-dominated fluids; (B) Solving partial differential equations on infinite domains. These computational problems frequently occur when one computes physical systems in large environments. The research aims to exploit geometry to tackle these problems at a fundamental level. A mathematical focus on geometry affords nuanced formulations for fluid dynamics systems that can be more resilient to numerical error. It also offers systematic approaches for identifying the invariants under transformation. Project outcomes are expected to simulate turbulent flows at microscopic scales and remove infinite-domain challenges through suitable transformations, while bringing mathematical ideas in geometry to a larger audience. A diversity-supporting education and outreach plan will likely result in closer communication between different fluid dynamics communities such as computer graphics and the climate sciences.Many physical simulations take place in a flat Euclidean space. Therefore, a deeper set of geometric tools, usually motivated by non-Euclidean geometry, has yet to draw much attention in developing numerical solvers. This project explores geometric ideas expected to solve fundamental challenges in simulation. One challenge is to simulate fluids at a high Reynolds number, which is central to computer animation, aerodynamics, climate science, etc. The dominating phenomena are intricate vortex dynamics and turbulence often lost during simulation. One thrust of the project aims to develop an implicit flow representation that faithfully encodes the vorticity, in contrast to the traditional formulation using a velocity field. Such an implicit representation can resolve Kolmogorov microscales at low cost. The other thrust, an Erlangen program for simulation, explores transformations of the domains and the variables such that a partial differential equation (PDE) is left invariant. When a transformation can turn an infinite domain into a finite domain, finite-domain PDE solvers are automatically infinite-domain PDE solvers after being wrapped by the transformation. This method would fundamentally solve numerical challenges involving infinite domains. Preliminary results suggest that many important PDE systems, including wave equations, signed distance fields, optimal transport problems, and minimal surface problems admit such a hidden symmetry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.