Novel Finite Element Methods for Wave Propagation Modeling

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Title: Novel Finite Element Methods for Wave Propagation Modeling
Author: Yue, Bin
Advisors: Murthy N. Guddati, Committee Chair
Fuh-Gwo Yuan, Committee Member
G. Kumar Mahinthakumar, Committee Member
Vernon C. Matzen, Committee Member
M. Shamimur Rahman, Committee Member
Abstract: The phenomenon of wave propagation is encountered in various engineering problems related to earthquake engineering, nondestructive evaluation and acoustics. Due to the complex material and geometrical features, many of these wave propagation problems are modeled using numerical methods such as the finite element method. Most numerical methods, due to their approximate nature, incur errors in the solution. In the context of wave propagation, these errors can be classified as amplitude and dispersion errors. Of these, dispersion error tends to have more severe effect on the accuracy due to its accumulative nature. Although it is possible to reduce the dispersion error by mesh refinement, such refinement imposes unrealistic computational cost even for medium-sized problems. In light of this, researchers have long sought efficient methods that reduce the dispersion error without any mesh refinement, but such efforts have only been partially successful. This dissertation develops efficient finite element methods for simulation of time-harmonic as well as transient wave propagation. For time harmonic waves, most existing dispersion reducing methods are limited to square meshes and homogeneous acoustic media. This dissertation develops two novel finite element methods that are applicable to unstructured meshes, as well as to heterogeneous media. They are the Local mesh-dependent augmented Galerkin finite element methods and the modified integration rules. Compared with existing methods, the proposed methods have higher convergence rate while maintaining low computational cost. When applied to elastic waves, the modified integration rules can reduce the dispersion error for either longitudinal or transverse wave, but not both. In the context of transient wave propagation, the spatial error of dispersion is coupled with temporal error resulting from time discretization. This dissertation focuses on reducing these errors by utilizing the modified integration rules and a modified time integration scheme. All the existing methods have second order convergence rates, while that of the proposed method has fourth order convergence. Numerical examples are utilized to illustrate the accuracy of the proposed method.
Date: 2004-10-12
Degree: PhD
Discipline: Civil Engineering

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