Collaborative Research: Variational Structure Preserving Methods for Incompressible Flows: Discretization, Analysis, and Parallel Solvers

该项目的目标是开发改进的模型和数值方法，推进不可压缩流体流动模拟的最新技术。关键思想是在计算机算法中更好地执行几何和物理定律，为所开发的模型和方法建立坚实的数学框架，以及设计允许在超级计算机上有效实施的算法。虽然流体流动的模拟是广泛的工程应用中的关键子任务，但当前的工具和技术通常是不可靠的，并且对于最先进的方法来说，即使使用数千个处理器，也需要花费数周或数月（或可能永远无法完成精确的解决方案）来执行汽车周围，通过核反应堆或飞机部件周围的流动模拟。该项目旨在开发数学预测模型和数值方法，以比最先进的方法更有效的方式提供更准确的解决方案。PI将通过构建比现代方法更好地遵循几何结构和物理守恒定律的模型和方法，构建不可压缩流模拟的有效方法。关键部分是：i）构建新的方法，这些方法是有效的，可以正确地解释涡度动力学，能量，螺旋度和质量守恒-这将需要开发有效的边界条件和重要的分析，以建立一个坚实的数学框架; ii）为这些方法开发更有效的代数求解器，可以在数千个处理器上使用; iii）与合作者一起对基准问题以及应用问题进行大规模测试。更广泛的影响包括：i）开发模拟高速不可压缩流的有效方法，这将改善环境工程，心血管模拟以及大气和海洋科学中广泛应用的设计过程; ii）通过参与研究培训研究生和本科生;以及iii）开发大规模并行代码，作为交易II库的一部分公开提供。

The goal of this project is to develop improved models and numerical methods that advance the state of the art for incompressible fluid flow simulation. The key ideas are the better enforcement of geometric and physical laws in the computer algorithms, building a solid mathematical framework for the models and methods developed, and the devising of algorithms that will allow for efficient implementation on supercomputers. Although the simulation of fluid flow is a critical subtask in a wide spectrum of engineering applications, current tools and techniques are often unreliable and it is not uncommon for state of the art methods to take weeks or months (or possibly never finish with an accurate solution), even with thousands of processors, to perform simulations of flows around a car, through a nuclear reactor, or around part of an airplane. The project aims to develop mathematical prediction models and numerical methods that will provide more accurate solutions in a more efficient manner than state-of-the-art methods.The PIs will construct efficient methods for incompressible flow simulation by constructing models and methods that better adhere to geometric structure and physical conservation laws than modern methods. The key components are to i) construct novel methods that are efficient and can correctly account for vorticity dynamics, and energy, helicity, and mass conservation -- this will require development of efficient boundary conditions and significant analysis to build a solid mathematical framework; ii) develop more efficient algebraic solvers for these methods that can be used on thousands of processors; iii) large scale testing on benchmark problems as well as on application problems with collaborators. Broader impacts include i) developing efficient methods for simulating high speed incompressible flows, which will improve the design process for a wide spectrum of applications in environmental engineering, in cardiovascular simulations, and in atmosphere and ocean sciences; ii) training graduate and undergraduate students through research involvement; and iii) developing large scale, parallel codes to be made publicly available as part of the deal.II library.