Date of Award
5-2025
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
Committee Chair/Advisor
Dr. Hyesuk Lee
Committee Member
Dr. Margaret Wiecek
Committee Member
Dr. Timo Heister
Committee Member
Dr. Leo Rebholz
Abstract
We consider two primary areas of physical application in this work: fluid interaction systems with either linear elastic structures or with poroelastic structures, and thin film polymers, where the majority of the work focuses on the fluid-structure interaction systems.
In the first chapter, we present a strongly coupled partitioned method for fluid structure interaction (FSI) problems based on a monolithic formulation of the system which employs a Lagrange multiplier (LM). We prove that both the semi-discrete and fully discrete formulations are well-posed. To derive the partitioned scheme, a Schur complement equation, which implicitly expresses the Lagrange multiplier and the fluid pressure in terms of the fluid velocity and structural displacement, is constructed based on the monolithic FSI system. Solving the Schur complement system at each time step allows for the decoupling of the fluid and structure subproblems, making the method non-iterative between subdomains. We investigate bounds for the condition number of the Schur complement matrix and present initial numerical results to demonstrate the performance of our approach, which attains the expected convergence rates.
Next, we consider the employment of a projection-based reduced order model (ROM) on one or both subdomains. The inclusion of ROMs in the non-iterative partitioned scheme provides a more robust framework and offers a cheaper alternative to the full order model in terms of computational time and the size of the linear systems. Utilizing the supremizer enrichment technique, we offer detailed investigations into the performance of our method with respect to the use of supremizers and with respect to the basis sizes of the reduced order variables. Results indicate that the ROM-ROM coupled formulation yields results that agree well with the full order solution in a shorter computational time and with a large reduction in the size of the algebraic systems.
We then move from considering a linear elastic structure to a poroelastic one, in which a fluid flow modeled by Darcy's law saturates an elastic structural skeleton. Similarly to the linear elastic case, we develop a monolithic formulation of the system that uses three LMs to enforce boundary conditions. We show well-posedness, stability, and convergence results for this formulation and then develop a non-iterative partitioned method based on the monolithic system for its solution. We propose a preconditioner for the Schur complement system at the center of this partitioned scheme and demonstrate the numerical performance of the method over a range of parameters. Implementing ROMs in this context, we conduct reproductive and predictive studies with respect to the physical parameters. As in the linear elastic case, our method proves to be robust, and the computational gains provided by the inclusion of the ROMs improves the viability of the scheme.
Lastly, the self-healing process of thin film polymers is modeled numerically under two different temperature regimes. We then consider optimization problems which provide insight into the behavior and structure of these self-healing polymers.
Recommended Citation
de Castro, Amy, "Domain Decomposition for Coupled Systems of Fluid-Structure Interaction and Numerical Modeling for Thin Film Polymers" (2025). All Dissertations. 3940.
https://open.clemson.edu/all_dissertations/3940
Author ORCID Identifier
https://orcid.org/0009-0003-7544-4829