Control and optimization problems in fluid mechanics

提出的研究将涉及一系列的主题，将理论和计算流体力学与现代控制和优化理论联系起来。我们的目标是开发和验证一套先进的计算方法，以解决该领域的一些基本问题。本课题所建立的理论基础将为今后开展更多的应用研究提供基础。在提议的工作的一部分，我们将重新审视以连续涡量分布为特征的无粘和不可压缩流动的基于涡的模型，如涡斑和涡片。这些模型对于理解水动力推进的机制非常重要，例如，对于鱼的运动。我们的研究将为这些经典模型增加新的价值，为解决基于这些模型制定的控制和优化问题提供一套工具。在本文的另一部分中，我们将把控制和优化方法扩展到涉及多物理场效应的流动现象，并以界面和接触线为特征。这种流作为模型在许多重要的工业应用中无处不在。在数学上，所有这些现象都被描述为“自由边界问题”，其中必须确定域的形状作为解决方案的一部分。这一事实对这类系统的反问题的数学公式具有深远的影响。因此，人们需要开发特殊的方法，将经典的应用数学方法与一些新技术相结合。此外，在本研究中，我们还将寻求对经典流体力学中有关过障流的一些突出问题作出贡献。这笔拨款将使Protas博士能够培养几名研究生进行跨学科研究，这些研究以应用数学为基础，也涉及科学计算、物理和工程科学的元素。

The proposed research will address a range of topics that bridge the theoretical and computational fluid mechanics with the modern theory of control and optimization. Our goal is to develop and validate a suite of advanced computational methods for a number of fundamental problems in this field. The theoretical foundations established by this project will provide the basis for more applied investigations to be pursued in the future. In one part of the proposed effort we will revisit vortex-based models for inviscid and incompressible flows characterized by continuous vorticity distributions, such as vortex patches and vortex sheets. These models are relevant for understanding the mechanisms of hydrodynamic propulsion important, for instance, for fish-like locomotion. Our research will add new value to these classical models by providing a suite of tools for solution of control and optimization problems formulated based on such models. In another part of the proposed effort we will extend the methods of control and optimization to flow phenomena involving multiphysics effects and featuring interfaces and contact lines. Such flows are ubiquitous as models in a number of important industrial applications. Mathematically, all of these phenomena are described as 'free-boundary problems' in which the shape of the domain must be determined as a part of the solution. This fact has far-reaching consequences for the mathematical formulation of inverse problems for such systems. As a result, one is required to develop special approaches combining classical methods of applied mathematics with a number of novel techniques. In addition, in this investigation we will also seek to make contributions to some outstanding problems in classical hydrodynamics concerning flows past obstacles. The funding provided by this grant will allow Dr. Protas to train several graduate students in interdisciplinary research rooted in applied mathematics and involving also elements of scientific computing, physics and engineering science.