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

Iowa State University, Ames IA

Investigators

Abstract

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.