CAREER: Geometric Approaches to Simulation

物理模拟和视觉计算是探索物理科学和技术创新的关键。尽管多年来计算能力不断增强，但根本性的数字挑战仍有待解决。该项目解决了两个这样的挑战：(A)模拟以涡流为主的流体；(B)求解无穷大区域上的偏微分方程组。当人们在大型环境中计算物理系统时，这些计算问题经常发生。这项研究的目的是利用几何学在根本层面上解决这些问题。对几何学的数学关注为流体动力学系统提供了微妙的公式，可以更好地抵御数值误差。它还提供了识别变换下的不变量的系统方法。预计项目成果将在微观尺度上模拟湍流，并通过适当的变换消除无限领域的挑战，同时将几何中的数学思想带给更多的受众。支持多样性的教育和推广计划可能会导致不同流体动力学社区之间更紧密的沟通，如计算机图形学和气候科学。许多物理模拟在平坦的欧几里德空间进行。因此，在开发数值求解器时，通常由非欧几里德几何驱动的更深层次的几何工具集尚未引起太多关注。这个项目探索几何思想，期望解决模拟中的基本挑战。一个挑战是在高雷诺数下模拟流体，这是计算机动画、空气动力学、气候科学等的核心。主要现象是复杂的涡动力学和湍流在模拟过程中经常丢失。该项目的一个主要目的是开发一种隐式流动表示法，与使用速度场的传统公式形成对比，该隐式流表示法准确地编码了涡度。这种隐式表示法能以较低的成本分辨柯尔莫戈洛夫微尺度。另一个推力是用于模拟的Erlangen程序，它探索区域和变量的转换，从而使偏微分方程(PDE)保持不变。当变换可以将无限域转化为有限域时，有限域PDE求解器被变换包裹后自动成为无限域PDE求解器。这种方法将从根本上解决涉及无限域的数值挑战。初步结果表明，许多重要的PDE系统，包括波动方程、带符号距离场、最优传输问题和最小表面问题，都承认这种隐藏的对称性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.