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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

基本信息

批准号：
2206200
负责人：
Pelin Guven Geredeli
金额：
$ 20万
依托单位：
Iowa State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-15 至 2023-10-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206200&HistoricalAwards=false
关键词：
Control Theory Qualitative Analysis Approximation

项目摘要

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.

该项目旨在对某些流体结构相互作用（FSI）偏微分方程（PDE）系统进行定性和定量分析。这种耦合偏微分方程系统在数学上描述了各种生物现象，例如封装血管壁结构内血流的相互作用。特别是，由粘弹性材料组成的哺乳动物血管壁由于血液输送过程中产生的血流动力而发生较大变形。动脉壁和不可压缩血流之间的这种相互作用是通过复合（多层）FSI PDE 在数学上实现的，正是本项目所考虑的类型。此外，该项目将重点关注那些 FSI PDE 模型，这些模型通常由与可压缩或不可压缩斯托克斯或纳维斯托克斯流体流耦合的弹性动力学组成，已知这些模型可描述土木工程中的各种现象：例如，流体或气体流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。对于此类 FSI 动力学，该项目研究的重点将是开发连续数值逼近理论，该理论与流体流动 PDE 分量表现出纳维-斯托克斯非线性以及非线性源自板 PDE 分量的情况相关。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，该项目将包括 K-12 的推广活动。该研究需要开发新颖的方法来解决多层 FSI 系统解的存在性、唯一性、长期行为和数值近似等问题。该项目研究的一部分旨在： (I.i) 建立新颖的数学方法，以确定由与不可压缩流体流动耦合的多层弹性方程组成的 FSI 解的存在性和唯一性； (I.ii) 确定此类解决方案的定性行为，包括随着时间的推移获得最佳理性衰减率的可能性。 (I.i) 和 (I.ii) 中的这些结果可以提供有关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，该项目研究的目的是：（II.i）提供本质上新颖的混合变分公式，以获得描述土木工程中某些现象的耦合（可压缩和不可压缩）纳维斯托克斯-完全非线性基尔霍夫板 FSI 系统的解； (II.ii)在边界控制在板组件中有效的物理相关情况下，构建与所述FSI PDE系统的边界控制相关的数学控制理论。该项目预计 (II.i) 中提到的适定性混合变分方法将为多层 FSI 系统的解决方案带来可实施的数值逼近方案，其收敛速度比现有文献中的收敛速度更快，并且计算成本更低。此外，（II.ii）中提到的边界可控性项目工作与某些流体机械应用相一致；也就是说，边界控制法的目的是诱导 3D 室内的流体速度混合，最终达到非混沌或静态流动状态。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。