Algorithms, Data-driven Methods and Analysis in Fluid Dynamics and Fluid-Structure Interactions
Restricted (Penn State Only)
- Author:
- Luo, Yushuang
- Graduate Program:
- Mathematics (PHD)
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 03, 2022
- Committee Members:
- Corina Drapaca, Outside Unit & Field Member
John Harlim, Major Field Member
Xiantao Li, Co-Chair & Dissertation Advisor
Alexei Novikov, Professor in Charge/Director of Graduate Studies
Helge Jenssen, Major Field Member
Wenrui Hao, Co-Chair & Dissertation Advisor - Keywords:
- Fluid-structure Interactions
Reduced-order Modeling
Data-driven Modeling
Compressible Euler System
Self-similar Riemann Problems
Radial Solutions - Abstract:
- For the first part, we study numerical methods for fluid-structure interactions (FSI). FSI problems are often too complex to solve analytically. On the other hand, numerically solving the whole system can be computaionally expensive. Our work focuses on stability-preserving reduced-order modeling techniques. A projection-based reduced-order modeling method is proposed and applied to the immersed boundary method (IBM) for biofluid systems. The reduced-order model (ROM) are derived from projecting the full-order model (FOM) on selected subspaces such that incompressibility and the Lyapunov stability are both preserved. We also address the practical issue of efficiently computing the reduced-order model using an interpolation technique. Next, a data-driven modeling approach for more general dynamics problem with latent variables is introduced without knowledge of the FOM. The data-driven model includes artificial latent variables in the state space, in addition to observed variables. We present a model framework where the stability of the coupled dynamics can be easily enforced. The model is implemented by recurrent cells and trained using back propagation through time. For both the projection-based method and the data-driven method, benchmark examples from order reductions are used to demonstrate the efficiency, robustness, and stability. Classic FSI problems are experimented to illustrate the accuracy and predictive capability of the proposed approaches. For the second part, we study the compressible Euler system for gas dynamics. We construct self-similar solutions to Riemann problems for the 1-dimensional isothermal Euler system. Such self-similar solutions always contain exactly two shock waves, necessarily generated at time $0_+$ and move apart along straight lines. We also provide physical interpretation of the solution structure, describing the behavior of the solution in the emerging wedge between the shock waves. We then move on to the 3-dimensional linearized Euler system. Radial solutions are used to construct examples of BV instability and $L^\infty$ blowup. Global existence of a class of radial solutions is shown using an argument based on scaling of the dependent variables, with variation estimates.