Electronic Theses and Dissertations

Date

2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

Roberto Triggiani

Committee Member

Hongqiu Chen Chen

Committee Member

Irena Irena Lasiecka (Chair)

Committee Member

Ben McCarty

Committee Member

Roberto Triggiani (advisor)

Abstract

The class of fluid-structure interaction models in the physical dimension d = 2,3 has been the object of intense mathematical studies in the past several years. In the case of static interface, justified to be appropriate for small, rapid oscillations of the structure, the typical configuration considers a structure – modeled by a d-dimensional wave-type equation –surrounded by a fluid –modeled by the d-dimensional dynamic Stokes equation involving pressure. Coupling takes place at the common interface of the two media where the individual equations evolve and is given by differential operators. Thus, the overall system provides a physical illustration of hyperbolic- parabolic coupling. The present dissertation represents a key distinctive novelty over past literature whereby the structure equation includes a (strong) Kelvin-Voigt damping. Two specific cases of structure are analyzed, both with physically significant interface conditions: (1) a wave with Kelvin-Voigt damping (d = 2, 3), and (2) a plate with Kelvin-Voigt damping (d = 2). In both cases, this dissertation provides the following main results: (i) analyticity of the s.c. contraction semigroup (on a natural physical space), describing the overall coupled dynamics; (ii) a detailed (and surprising) spectral analysis of its generator; (iii) uniform stability of such semigroup, either on the entire space or, when this is not true, on a natural subspace of codimension one, obtained by factoring out the one dimensional eigenspace corresponding to the zero eigenvalue. The final chapter includes the original work done during a summer 2021 NSF-MSGI Internship at the Oak Ridge National Laboratory, under the sponsorship of senior scientist Dr. J. Nutaro. It introduces a novel quantized state integration method that is analogous to the explicit, second order Runge-Kutta technique. For a single differential equation over an interval where the solution is monotonic, such method exhibits an error proportional to the square of the integration quantum. It is a surprising feature which solves in the positive a long-standing open problem. A theoretical proof is given, illustrated by a numerical example.

Comments

Data is provided by the student.

Library Comment

Dissertation or thesis originally submitted to ProQuest.

Notes

Open Access

Download

Share

COinS