Development and applications of hypersingular boundary integral equations for three-dimensional acoustics and elastodynamics
Liu, Yijun
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https://hdl.handle.net/2142/21553
Description
Title
Development and applications of hypersingular boundary integral equations for three-dimensional acoustics and elastodynamics
Author(s)
Liu, Yijun
Issue Date
1992
Doctoral Committee Chair(s)
Rizzo, Frank J.
Department of Study
Mechanical Science and Engineering
Discipline
Theoretical and Applied Mechanics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Applied Mechanics
Mathematics
Physics, Acoustics
Language
eng
Abstract
The boundary integral equation/boundary element method (BIE/BEM) has emerged as a powerful alternative tool to other numerical methods for many problems in engineering. The hypersingular BIE's, which are derivatives of conventional BIE's, are indispensable for the analyses of many problems in mechanics by BIE/BEM, such as wave scattering, crack problems, plate bending, thin body and thin inclusion problems, for which the conventional BIE's are insufficient or fail. However, the application of hypersingular BIE's had been very limited because of the difficulty in dealing with the hypersingular integrals involved. In this thesis, the hypersingular BIE's for 3-D acoustic and elastic wave problems are presented in weakly-singular forms. For this purpose, three integral identities for the fundamental solutions of both potential and elastostatic problems are established and employed. These weakly-singular forms of the hypersingular BIE's can be handled easily and no special quadratures are needed in the numerical computation. The composite BIE formulations, which use a linear combination of the conventional and hypersingular BIE's, are applied to overcome the fictitious eigenfrequency difficulty (nonunique solutions) existing in the conventional BIE formulations of exterior acoustic and elastic wave problems. Overhauser $C\sp1$ continuous boundary elements, which satisfy the smoothness requirement of the hypersingular BIE's, are implemented for these composite BIE formulations and compared with the traditional $C\sp0$ conforming quadratic and non-conforming quadratic elements. Numerical examples of scattering in both acoustic and elastic media clearly demonstrate the effectiveness and efficiency of the developed formulations.
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