Energy and eigenmode analyses in coupled fluid-structure problems using immersed methods
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Authors
Shen, Chen
Issue Date
2025-08
Type
Electronic thesis
Thesis
Thesis
Language
en_US
Keywords
Mechanical engineering
Alternative Title
Abstract
Fluid-structure interaction (FSI) simulations often suffer from spurious energy gain or loss and numerical instabilities when the fluid and solid domains are discretized on non-matching meshes. Motivated by the need for robust and energy-consistent coupling schemes in non-boundary-fitted numerical methods, this work presents a comprehensive theoretical and computational framework for analyzing energy behavior and eigenmode in FSI problems using the modified Immersed Finite Element Method (mIFEM). The first part of this work develops a rigorous energy analysis of the mIFEM formulation, deriving both semi-discrete and fully-discrete energy balance that quantify the interplay among kinetic, potential, and numerical energy across non-conforming meshes. A penalty-based interface enforcement strategy is introduced, shown to be dimensionally consistent, and capable of reducing interface velocity errors to solver tolerance with a single tunable parameter, without injecting spurious energy. To quantify transfer errors arising on overlapping meshes, a general bound is established, and a variational projection strategy is proposed. The study compares energy transfer schemes that including nodal interpolation and variational projection to identify sources of conservation error. Numerical verifications confirm the accuracy of the derived energy balance and demonstrate the fidelity of energy exchange in both uncoupled and coupled configurations. In the second part, the first eigenmode analysis of mIFEM is performed for a simplified coupled viscous-elastic system. A coupled eigen-system is derived for a simplified two-dimensional configuration, rigorously enforcing both traction and kinematic continuity. An adaptive argument principle algorithm is implemented to detect unstable roots; grid convergence studies show that it reliably detects all roots while remaining insensitive to contour placement. Numerical studies demonstrate that the mIFEM formulation, in the idealized high penalty limit, shows no unstable eigenmodesfor a broad range of density ratios, thereby providing the theoretical foundation for future eigen analyses of discretized systems. Together, the energy analysis and eigen-system developed in this work furnish a diagnostic suite for assessing algorithmic correctness, guide the design of strictly conservative transfer operators, and open a pathway toward provably stable, high-fidelity simulations of complex FSI phenomena using mIFEM.
Description
August2025
School of Engineering
School of Engineering
Full Citation
Publisher
Rensselaer Polytechnic Institute, Troy, NY