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CAREER: Geometric Approaches to Simulation

职业：模拟的几何方法

基本信息

批准号：
2239062
负责人：
Albert Chern
金额：
$ 67.95万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-04-01 至 2028-03-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2239062&HistoricalAwards=false
关键词：
CAREER Geometric Approaches Simulation

项目摘要

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.

物理模拟和可视化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提高，但基本的数值挑战仍有待解决。这个项目解决了两个这样的挑战：（A）模拟涡流占主导地位的流体;（B）解决无限域上的偏微分方程。这些计算问题经常发生在大型环境中计算物理系统时。这项研究旨在利用几何学在基础层面上解决这些问题。对几何形状的数学关注为流体动力学系统提供了细致入微的公式，可以更好地适应数值误差。它还提供了系统的方法来确定变换下的不变量。项目成果预计将在微观尺度上模拟湍流，并通过适当的转换消除无限域的挑战，同时将几何中的数学思想带给更多的观众。一个支持多样性的教育和推广计划可能会导致不同的流体动力学社区，如计算机图形学和气候科学之间的更密切的沟通。许多物理模拟发生在一个平坦的欧几里得空间。因此，一套更深层次的几何工具，通常由非欧几里德几何的动机，尚未引起人们的注意，在发展数值求解器。该项目探讨了几何思想，有望解决模拟中的基本挑战。一个挑战是模拟高雷诺数的流体，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流动力学和湍流，在模拟过程中经常丢失。该项目的一个目标是开发一种隐式流表示法，与使用速度场的传统公式相反，该隐式流表示法忠实地编码涡量。这样的隐式表示可以以低成本解决柯尔莫哥洛夫微尺度。另一个推力，一个埃尔兰根程序的模拟，探讨变换的域和变量，使偏微分方程（PDE）是左不变的。当变换可以将无限域转换为有限域时，有限域PDE求解器在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，许多重要的偏微分方程系统，包括波动方程，签署的距离场，最佳运输问题，和最小的表面问题承认这样一个隐藏的symmetrical.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

期刊论文数量（5）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Sparse Stress Structures from Optimal Geometric Measures

最佳几何测量的稀疏应力结构

DOI：
10.1145/3610548.3618193
发表时间：
2023
期刊：
SA '23: SIGGRAPH Asia 2023 Conference Papers
影响因子：
0
作者：
Rowe, Dylan;Chern, Albert
通讯作者：
Chern, Albert

Fluid Cohomology

流体上同调

DOI：
10.1145/3592402
发表时间：
2023
期刊：
ACM Transactions on Graphics
影响因子：
6.2
作者：
Yin, Hang;Nabizadeh, Mohammad Sina;Wu, Baichuan;Wang, Stephanie;Chern, Albert
通讯作者：
Chern, Albert

Exterior Calculus in Graphics: Course Notes for a SIGGRAPH 2023 Course