Date of Award

Summer 2009

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

X. Sheldon Wang

Second Advisor

Eliza Zoi-Heleni Michalopoulou

Third Advisor

Michael Siegel

Fourth Advisor

David James Horntrop

Fifth Advisor

Hongya Ge

Abstract

In this work, mixed finite element formulations are introduced for acoustoelastic fluid- structure interaction (FSI) systems. For acoustic fluid, in addition to displacement- pressure (u/p) mixed formulation, a three-field formulation, namely, displacement-pressure-vorticity moment formulation (u - p -Λ) is employed to eliminate some zero frequencies. This formulation is introduced in order to compute the coupled frequencies without the contamination of nonphysical spurious non-zero frequencies. Furthermore, gravitational forces are introduced to include the coupled sloshing mode. In addition, u/p mixed formulation is the first time employed in solid. The numerical examples will demonstrate that the mixed formulations are capable of predicted coupled frequencies and mode shapes even if primary slosh, structural, and acoustic modes are within separate frequency ranges. That is to say, the mixed finite element formulations are used to deal with fluid and solid monolithically. In numerical analysis, boundary conditions, wetted surface, and skew systems are considered in order to obtain the symmetrical, nonsingular mass and stiffness matrices. An implicit time integration scheme, the Newmark method, is employed in the transient analysis. Appropriate finite elements corresponding to the mixed finite element formulations are selected based on the inf-sup condition, which is the fundamental solvability and stability condition of finite element methods. In addition, the inf-sup values of the FSI system using a sequence of three meshes are evaluated in order to identify and confirm that the 'locking' effect does not occur. The numerical examples in this work will also show that by imposing external forces near different coupled frequencies, predominant slosh, structural, and acoustic motions can be triggered in the FSI systems. Further, it is discussed that the frequency range on which energy mainly focuses can be evaluated with Fast Fourier Transform, if the system is activated by single-frequency excitations.

In the second part, fluid-structure interaction systems with both immersed flexible structures and free surfaces are employed to study the traditional mode superposition methods and singular value decomposition (SVD) based model reduction methods, e.g., principal component analysis (PCA). The numerical results confirm that SVD-based model reduction methods are reliable by comparing the Rayleigh-Ritz quotients obtained by the principal singular vector and the natural frequencies of the system. If an initial excitation is loaded on a nodal points on the free surface or the structure, the corresponding natural frequency by the transient data of the first few time snapshots can be captured. Excellent agreements are confirmed between the original transient solutions and the data reconstructed with a few dominant principal components. The figures of energy are also plotted in order to verify the realization of this objective, which is recovering the transient data with a few principal components without losing dominant characteristics. The numerical results further demonstrate that different time steps lead to distinct mode shapes of the FSI system, if a combined eigenmode is given as the initial displacement. This is because the natural frequency of sloshing, structural, and acoustic modes are separated. Therefore, the errors between original transient data and recovered results vs different time steps are compared in order to find the appropriate time step and further capture all the eigenmodes. Finally, the coarse-grained system is employed to study the long-time behavior of the FSI system based on model reduction methods. The extrapolation results in coarse temporal scale can be obtained based on dominant principal components provided by PCA. The data at some time instances in fine temporal scale can be neglected. The numerical results show excellent agreement for some generic initial conditions.

Included in

Mathematics Commons

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