Li, Tongtong
(2021)
Mixed Formulations for Fluid-poroelastic Structure Interaction.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
This is the latest version of this item.
Abstract
This thesis focuses on the development of mixed finite element methods for the coupled problem arising in the interaction between free fluid flow and flow in a deformable poroelastic medium. We adopt the Stokes or the Navier-Stokes equations to model the free fluid region, and the Biot system to describe the poroelastic medium. On the interface, mass conservation, balance of stresses and the slip with friction conditions are imposed via the Lagrange multiplier method.
We first develop a new mixed elasticity formulation for the Stokes-Biot problem. We establish the existence and uniqueness of a solution for the continuous weak formulation and perform stability and error analyses for the semi-discrete continuous-in-time mixed finite element approximation. We present numerical experiments that verify the theoretical results and illustrate the robustness of the method with respect to the physical parameters.
We then extend the previous results for the Stokes-Biot problem by considering dual-mixed formulations in both the fluid and structure regions. Well-posedness and stabil- ity results are established for the continuous weak formulation, as well as a semi-discrete continuous-in-time formulation with non-matching grids. In addition, we develop a new multipoint stress-flux mixed finite element method by involving the vertex quadrature rule. Well-posedness and error analysis with corresponding rates of convergences for the fully- discrete scheme are complemented by several numerical experiments.
Next, we propose an augmented fully mixed formulation for the coupled quasi-static Navier-Stokes – Biot model by introducing a ”nonlinear-pseudostress” tensor linking the pseudostress tensor with the convective term in the Navier-Stokes equations and augment- ing the variational formulation with suitable Galerkin redundant terms. We show well- posedness, derive stability and error analysis results for the associated mixed finite element approximation and conduct several numerical experiments.
Finally, we derive a fully mixed formulation with weakly symmetric stresses for the Navier-Stokes – Biot model. We develop an extension of the multipoint stress-flux mixed finite element method that allows for local elimination of the fluid and poroelastic stresses, vorticity, and rotation, resulting in a positive definite finite volume scheme. A numerical convergence study is presented for the fully discrete scheme.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| Date: |
8 October 2021 |
| Date Type: |
Publication |
| Defense Date: |
9 July 2021 |
| Approval Date: |
8 October 2021 |
| Submission Date: |
12 July 2021 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
250 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
numerical analysis, mixed finite element methods, FPSI, Stokes-Biot model, Navier-Stokes -- Biot model, multipoint stress-flux, augmented formulation, finite volume method |
| Date Deposited: |
08 Oct 2021 19:19 |
| Last Modified: |
08 Oct 2021 19:19 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/41623 |
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Mixed Formulations for Fluid-poroelastic Structure Interaction. (deposited 08 Oct 2021 19:19)
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