Control and optimization problems in fluid mechanics

拟议的研究将解决一系列的主题，桥梁的理论和计算流体力学与控制和优化的现代理论。我们的目标是开发和验证一套先进的计算方法，在这个领域的一些基本问题。本项目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。在所提出的努力的一部分，我们将重温基于涡的模型，其特点是连续的涡度分布，如涡补丁和涡面的无粘和不可压缩流。这些模型是相关的理解重要的水动力推进机制，例如，鱼一样的运动。我们的研究将增加新的价值，这些经典的模型，提供了一套工具，用于解决控制和优化问题的基础上制定这样的模型。在另一部分的建议的努力，我们将扩展的控制和优化的方法，涉及多物理场效应的流动现象，并具有接口和接触线。这种流动作为模型在许多重要的工业应用中是普遍存在的。在数学上，所有这些现象都被描述为“自由边界问题”，其中域的形状必须作为解的一部分来确定。这一事实具有深远的影响的数学公式的反问题，这样的系统。因此，人们需要开发特殊的方法结合经典的应用数学方法与一些新的技术。此外，在这项研究中，我们也将寻求作出贡献的经典流体力学中的一些悬而未决的问题，有关流动的障碍。这笔赠款提供的资金将使Protas博士能够培训几名研究生进行跨学科研究，这些研究植根于应用数学，并涉及科学计算，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.