Control Theory, Qualitative Analysis, and Approximation of Coupled Structure-Flow Interaction Systems

耦合结构-流相互作用系统的控制理论、定性分析和逼近

• 批准号: 2348312
• 负责人: Pelin Guven Geredeli
• 金额: $ 20万
• 依托单位: Clemson University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-10-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348312&HistoricalAwards=false
• 关键词: Control; Theory; Qualitative; Analysis; Approximation

The project aims to develop qualitative and quantitative analysis of certain fluid structure interaction (FSI) partial differential equation (PDE) systems. Such coupled PDE systems mathematically describe various biological phenomena, such as interactions of blood flow within encapsulating blood vessel wall structures. In particular, mammalian blood vascular walls, being composed of viscoelastic materials, undergo large deformations due to the hemodynamic forces generated during the blood transport process. This interaction between arterial walls and incompressible blood flow is mathematically realized by a composite (multilayered) FSI PDE precisely of the type under consideration in this project. In addition, this project will focus on those FSI PDE models, generally composed of elastic dynamics coupled to compressible or incompressible Stokes or Navier Stokes fluid flows, that are known to describe a variety of phenomena seen in civil engineering: for instance, the interaction of fluid or gas flows with displacing elastic membranes, and the aerodynamics of structures such as bridges and tall buildings. For such FSI dynamics, the focus of the project research will be the development of a continuous and numerical approximation theory, relevant to cases when the fluid flow PDE component manifests the Navier-Stokes nonlinearity, as well as when nonlinearities emanate from the plate PDE component. This project will also provide research training opportunities for both undergraduate and graduate students. Moreover, to promote study in the STEM fields, this project will include K-12 outreach activities.This research entails the development of novel methodologies to address issues of existence, uniqueness, longtime behavior, and numerical approximation of solutions to multilayered FSI systems. Part of the project research aims to: (I.i) establish novel mathematical methodologies to determine the existence and uniqueness of solutions to those FSI that consist of multilayered elastic equations coupled to incompressible fluid flows; (I.ii) ascertain the qualitative behavior of such solutions, including the possibility of obtaining optimal rates of rational decay, as time evolves. Such results in (I.i) and (I.ii) could provide qualitative insight concerning the incidence and pathology of those aneurysms caused by arterial wall deformations during the mammalian blood transportation process. Moreover, the project research aims to: (II.i) provide intrinsically novel mixed variational formulations to obtain solutions of coupled (compressible and incompressible) Navier Stokes-fully nonlinear Kirchoff plate FSI systems which describe certain phenomena in civil engineering; (II.ii) construct a mathematical control theory relative to the boundary control of said FSI PDE systems, in the physically relevant case that boundary control is active in the plate component. This project anticipates that the mixed variational approaches to wellposedness noted in (II.i) will give rise to implementable numerical approximation schemes for the solutions of multilayered FSI systems, with faster convergence rates than those in the existing literature, and with less computational cost. Moreover, the boundary controllability project work noted in (II.ii) is consonant with certain fluid mechanical applications; namely, the intent of the boundary control law is to induce a mixing of the fluid velocity within its 3D chamber, to ultimately attain a nonchaotic or quiescent flow state.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.

该项目的目的是发展定性和定量分析的某些流体结构相互作用（FSI）的偏微分方程（PDE）系统。这种耦合PDE系统在数学上描述了各种生物现象，例如包封血管壁结构内的血流相互作用。特别地，由粘弹性材料组成的哺乳动物血管壁由于在血液输送过程中产生的血液动力学力而经历大的变形。动脉壁和不可压缩的血流之间的这种相互作用在数学上是由复合材料（多层）FSI PDE精确地在这个项目中考虑的类型来实现的。此外，本项目将重点关注那些FSI PDE模型，通常由弹性动力学耦合到可压缩或不可压缩的斯托克斯或Navier Stokes流体流动组成，这些模型已知用于描述土木工程中的各种现象：例如，流体或气体流动与位移弹性膜的相互作用，以及桥梁和高层建筑等结构的空气动力学。对于这样的流固耦合动力学，该项目研究的重点将是一个连续的和数值近似理论的发展，相关的情况下，当流体流动PDE组件表现出的Navier-Stokes非线性，以及当非线性源自板PDE组件。该项目还将为本科生和研究生提供研究培训机会。此外，为了促进STEM领域的研究，本项目将包括K-12的推广活动。该研究需要开发新的方法来解决多层FSI系统解的存在性、唯一性、长期行为和数值近似等问题。项目研究的一部分旨在：（I.i）建立新的数学方法，以确定由多层弹性方程耦合到不可压缩流体流动组成的FSI解的存在性和唯一性;（I.ii）确定这些解的定性行为，包括随着时间的推移获得最佳合理衰减率的可能性。（I.i）和（I.ii）中的这些结果可以提供关于哺乳动物血液运输过程中动脉壁变形引起的动脉瘤的发病率和病理学的定性见解。此外，项目研究旨在：（II.i）提供本质上新颖的混合变分公式，以获得耦合的解描述土木工程中某些现象的（可压缩和不可压缩）Navier Stokes-完全非线性Kirchoff板流固耦合系统（II.ii）构造与所述FSI PDE系统的边界控制相关的数学控制理论，在物理上相关的情况下，边界控制在板部件中是有效的。该项目预计，（II.i）中提到的适定性的混合变分方法将为多层FSI系统的解提供可实现的数值逼近方案，其收敛速度比现有文献中的更快，并且计算成本更低。此外，边界可控性项目工作中指出，（II.ii）与某些流体机械应用一致;即，边界控制律的目的是在其3D腔室内引起流体速度的混合，该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准。