Constrained Lagrangian and Hamiltonian Mechanics in Fluid-Body Interactions: Analytical Modeling and Computational Methods

该项目提出了固体与涡旋流体相互作用的理论和计算建模的新视角，阐明了拉格朗日和哈密顿对这种相互作用的描述中可积和不可积约束的基本存在。该项目侧重于四个不同的主题：(1)陀螺升力与非完整约束力的形式等价；(2)流动附着或停滞条件作为完整或非完整约束的数学处理；(3)将具有离散涡分布的自由体相互作用的哈密顿结构分解为约束和无约束分量；(4)实现了保持固体表面涡脱落与相应表面力发展之间平衡原理的变分积分方法。在自然界和工程世界中，涉及流体与固体结构的动态相互作用的问题比比皆是。对这种相互作用的深入了解有望改善从机器人车辆到能量收集设备等系统的设计，但支撑这种相互作用的微妙物理学对传统的数学建模方法提出了挑战。本项目将探索流体-结构相互作用建模与其他机械现象建模之间的基本相似之处，目的是阐明普遍存在的物理原理和新的分析技术。

This project advances a novel perspective on the theoretical and computational modeling of interactions among solid bodies and vortical fluids, illuminating the fundamental presence of integrable and nonintegrable constraints in Lagrangian and Hamiltonian descriptions of such interactions. The project focuses on four distinct topics: (1) the formal equivalence of gyroscopic lift forces to nonholonomic forces of constraint, (2) the mathematical treatment of conditions of flow attachment or stagnation as holonomic or nonholonomic constraints, (3) the resolution of the Hamiltonian structure underpinning interactions of free bodies with discrete vortex distributions into constrained and unconstrained components, and (4) the realization of variational integration methods which preserve balance principles between vortex shedding from solid surfaces and the corresponding development of surface forces.Problems involving the dynamic interaction of fluids with solid structures abound in the natural and engineered worlds. A deeper understanding of such interactions promises to improve the design of systems ranging from robotic vehicles to energy-harvesting devices, but the subtle physics underpinning such interactions presents challenges to traditional mathematical modeling methods. This project will explore fundamental parallels between the modeling of fluid-structure interactions and the modeling of other mechanical phenomena, with the goal of illuminating pervasive physical principles and novel analytical techniques.