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）部分微分方程（PDE）系统进行定性和定量分析。这种耦合的PDE系统数学描述了各种生物学现象，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传递过程中产生的血液动力学而导致较大的变形。动脉壁与不可压缩血流之间的这种相互作用是通过该项目中所考虑的类型的复合（多层）FSI PDE来数学上实现的。 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.对于此类FSI动力学，项目研究的重点将是发展连续和数值近似理论的发展，这与流体流量PDE组件表现出Navier-Stokes非线性的情况以及非线性从平板PDE组件中散发出来的情况有关。该项目还将为本科生和研究生提供研究培训机会。此外，为了促进在STEM领域的研究，该项目将包括K-12外展活动。这项研究需要开发新的方法，以解决生存，唯一性，长期行为以及对多层FSI系统解决方案的数值近似。项目研究的一部分旨在：（i.i）建立新的数学方法论，以确定解决方案的存在和独特性的FSI，由多层弹性方程组成的FSI与不可压缩的流体流相连；（i.ii）确定此类解决方案的定性行为，包括随着时间的变化而获得最佳理性衰减速率的可能性。（i.i）和（i.ii）中的这些结果可以提供有关在哺乳动物血液运输过程中由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异配方，以获取耦合（可压缩且不可压缩的）Navier Stokes的溶液 - 非常非线性的Kirchoff Plate FSI FSI系统，描述了民用工程中某些现象；（ii.ii）在与上述FSI PDE系统的边界控制相关的数学控制理论中，在物理上相关的情况下，边界控制在板组件中处于活动状态。该项目预计（II.I）中指出的良好性的混合变化方法将导致多层FSI系统解决方案的可实现数值近似方案，其收敛速度比现有文献中的融合速率更快，并且计算成本较低。此外，（II.II）中指出的边界可控项目工作与某些流体机械应用相吻合；也就是说，边界控制法的目的是诱导其3D室内流体速度的混合，以最终达到非偶然或静止的流量状态。该奖项反映了NSF的法定任务，并认为通过基金会的知识率和更广泛的影响，可以通过评估来进行评估，以审查Criteria。