Macquarie University
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Neumann Problem for Helmholtz Equation in Two-Dimensional Open Domains with Applications

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posted on 2023-03-02, 02:48 authored by Turker Topal

The research described in the thesis is devoted to a rigorous solution of the Neumann boundary value problem for the Helmholtz equation in two-dimensional open arbitrary domains, and its application to the diverse problems of practical acoustics and electro-engineering. The Method of Analytical Regularisation is used to transform the initial surface integral equation in the form of a double layer potential to well-conditioned coupled infinite systems of linear algebraic equations, thus addressing the hyper-singular kernel arising from the normal derivative of the double layer potential. The compactness of the matrix operators provides the fast convergence of the truncated versions of the infinite systems, numerical solution of which may be obtained with any predetermined accuracy. Furthermore, complex eigenvalues of natural complex oscillations for various structures have been analysed extensively, using the spectral parameter (frequency) for which the matrix operator has zero valued determinant. A root finding algorithm is developed to improve the accuracy of the calculations as well as the computation time and performance. The developed method uses the spectral map of truncated system condition number for initial approximations. There are no restrictions imposed on boundaries of domains (except requirements on smoothness of the bounding contours), frequency range and size of entries (slits). To take advantage of all these merits of the solution, an object-oriented software for its numerical implementation has been developed: bounding contours of slotted arbitrary cylinders are generated by interpolation functions and Sobolev’s approximation, providing a smooth parametrisation. The numerical results cover spectral studies and resonance excitation of the slotted cylinders. Spectral studies of classical (closed) waveguides, ridged waveguides, and magnetron-type resonator cavities reveal an excellent agreement with the results available in literature. Complex eigenvalues associated with slotted arbitrary cylinders are obtained for the first time. Classical cylinders (circular, elliptic and rectangular) as well as their corrugated and modified shapes (single and paired acoustic resonators with attached necks, polygonal and sinusoidally corrugated cylinders) are investigated. Finally, wave scattering from slotted circular, elliptic, rectangular cavities, duct and bent duct cavities, parabolic reflector antennas (with and without flanges), corner reflectors, finite sinusoidally profiled gratings, and complex shapes such as an airplane and a submarine are studied. The MAR approach allows calculation of radar cross sections, scattering patterns and field distributions for a frequency range from a few to a half-thousand wavelengths, covering the low-frequency region (Rayleigh scattering), the resonance regime (diffraction), and the high-frequency regions.


Table of Contents

1. Introduction -- 2. The Neumann Boundary Value Problem for the Helmholtz Equation and the MAR -- 3. 2D Helmholtz Resonators -- 4. Spectral Studies of Slotted Cavities -- 5. Resonance Wave Scattering and Radiation from Slotted Cylinders -- 6. A Selective Overview of Advanced Applications of the MAR -- 7. Conclusions

Awarding Institution

Macquarie University

Degree Type

Thesis PhD

Department, Centre or School

Department of Mathematics and Statistics

Year of Award


Principal Supervisor

Elena Vynogradova


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183 pages