Collaborative Research: Computational Methods for Optimal Transport via Fluid Flows

合作研究：流体流动优化传输的计算方法

基本信息

批准号：
2111315
负责人：
Yangwen Zhang
金额：
$ 8.65万
依托单位：
University of Delaware
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111315&HistoricalAwards=false
关键词：
Collaborative Research Computational Methods Optimal

项目摘要

Transport and mixing in fluids is a topic of fundamental interest in engineering and natural sciences, with broad applications ranging from industrial and chemical mixing on small and large scales, to preventing the spreading of pollutants in geophysical flows. This project focuses on computational methods for control of optimal transport and mixing of some quantity of interest in fluid flows. The question of what fluid flow maximizes mixing rate, slows it down, or even steers a quantity of interest toward a desired target distribution draws great attention from a broad range of scientists and engineers in the area of complex dynamical systems. The goal of this project is to place these problems within a flexible computational framework, and to develop a solution strategy based on optimal control tools, data compression strategies, and methods to reduce the complexity of the mathematical models. This project will also help the training and development of graduate students across different disciplines to conduct collaborative research in optimal transport and mixing, flow control, and computational methods for solving these problems.The project is concerned with the development and analysis of numerical methods for optimal control for mixing in fluid flows. More precisely, the transport equation is used to describe the non-dissipative scalar field advected by the incompressible Stokes and Navier-Stokes flows. The research aims at achieving optimal mixing via an active control of the flow velocity and constructing efficient numerical schemes for solving this problem. Various control designs will be investigated to steer the fluid flows. Sparsity of the optimal boundary control will be promoted via a non-smooth penalty term in the objective functional. This essentially leads to a highly challenging nonlinear non-smooth control problem for a coupled parabolic and hyperbolic system, or a semi-dissipative system. The project will establish a novel and rigorous mathematical framework and also new accurate and efficient computational techniques for these difficult optimal control problems. Compatible discretization methods for coupled flow and transport will be employed to discretize the controlled system and implement the optimal control designs numerically. Numerical schemes for the highly complicated optimality system will be constructed and analyzed in a systematic fashion. New incremental data compression techniques will be utilized to avoid storing extremely large solution data sets in the iterative solvers, and new model order reduction techniques specifically designed for the optimal mixing problem will be developed to increase efficiency. The synthesis of optimal control and numerical approximation will enable the study of similar phenomena arising in many other complex and real-world flow dynamics.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.

流体中的运输和混合是一个对工程和自然科学的基本兴趣的话题，其广泛应用从小规模和大尺度上的工业和化学混合到防止地球物理流中污染物的扩散。该项目着重于控制流体流中一定数量兴趣的最佳运输和混合的计算方法。哪种流体流动流动的问题使混合速率最大化，减慢速度，甚至引导了一定数量的兴趣对所需的目标分布，这引起了复杂动力学系统领域中广泛的科学家和工程师的极大关注。该项目的目的是将这些问题放置在灵活的计算框架中，并基于最佳控制工具，数据压缩策略以及降低数学模型复杂性的方法制定解决方案策略。该项目还将帮助对不同学科的研究生进行培训和发展，以进行最佳运输和混合，流量控制和解决这些问题的计算方法的协作研究。该项目涉及对流体流中混合的最佳控制的数值方法的开发和分析。更确切地说，传输方程用于描述不可压缩的Stokes和Navier-Stokes流动所推荐的非隔离标量场。该研究旨在通过积极控制流速和构建有效的数值方案来实现最佳混合，以解决此问题。将研究各种控制设计以引导流体流动。最佳边界控制的稀疏性将通过目标功能中的非平滑惩罚项来促进。从本质上讲，这导致了一个高度挑战性的非线性非平滑控制问题，对于耦合的抛物线和双曲线系统或半脱轴性系统。该项目将建立一个新颖而严格的数学框架，并为这些困难的最佳控制问题提供新的准确有效的计算技术。耦合流和运输的兼容离散方法将用于离散受控系统并以数值实现最佳控制设计。高度复杂的最优系统的数值方案将以系统的方式构建和分析。新的增量数据压缩技术将被用来避免在迭代求解器中存储极大的解决方案数据集，并且将开发专门为最佳混合问题设计的新模型订单减少技术以提高效率。最佳控制和数值近似的综合将使在许多其他复杂和现实世界中产生的类似现象的研究能够研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的审查标准来评估的支持。