Null-field boundary integral equations approach for solving Helmholtz problems of free vibration and water wave containing circular and elliptical boundaries

• Main Authors: Jia-Wei Lee, 李家瑋
• Other Authors: Jeng-Tzong Chen
• Format: Others
• Language: en_US
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/23613276688349156796

碩士 === 國立臺灣海洋大學 === 河海工程學系 === 98 === Following the success of five advantages of the null-field boundary integral equation method (BIEM), the Helmholtz problems containing elliptical boundaries are solved by using the null-field BIEM in conjunction degenerate kernels and eigenfunction expansion in this thesis. Not only problems of interior free vibration and exterior water wave problems are considered. To fully utilize the property of ellipse for solving the Helmholtz problems, the elliptic coordinates and the associated Mathieu functions are adopted. The closed-form fundamental solution is expressed in terms of the degenerate kernel in the elliptic coordinates. Besides, the boundary densities are expanded by using the eigenfunction expansion. A Jacobian term may exist in the degenerate kernel, boundary density or boundary contour integration and they can cancel each other out. By scaling the boundary flux using a Jacobian term, the orthogonal relations can be reserved in the boundary contour integral and contour integration along elliptical boundaries can be analytically determined. By this way, the unknown coefficients can be easily determined through a linear algebraic system after matching boundary conditions. This approach is one kind of semi-analytical methods since errors only occur from the truncation of the number of the eigenfunction expansion terms in the real implementation. Although spurious eigenvalues of interior eigenproblem as well as fictitious frequencies for the exterior problem for elliptical boundaries may appear in the BIEM, it is interesting to find that those of them happen to be zeros of the modified Mathieu functions of the first kind or their derivatives. The appearances of spurious eigenvalues and fictitious frequencies are effectively suppressed by using three alternatives including the Combined Helmholtz Interior integral Equation Formulation (CHIEF) method, Burton & Miller approach and the singular value decomposition (SVD) updating technique. Besides, the near-trapped mode for an array of four elliptical cylinders is also observed. To avoid the addition theorem by translating the Bessel and Mathieu functions, the adaptive observer system is employed to solve the Helmholtz problems containing circular and elliptical boundaries at the same time in a semi-analytical manner. Finally, a general-purpose program was developed for solving eigenproblems or water wave problems containing arbitrary number, different sizes and various locations of circular and elliptical boundaries.