Analytical investigation for true and spurious eigensolutions of multiply-connected membranes containing elliptical boundaries using the dual BIEM

Abstract

a b s t r a c tIt is well known that the boundary element method may induce spurious eigenvalues while solving eigenvalue problems. The finding that spurious eigenvalues depend on the geometry of inner boundary and the approach utilized has been revealed analytically and numerically in the literature. However, all the related efforts were focused on eigenproblems involving circular boundaries. On the other hand, the extension to elliptical boundaries seems not straightforward and lacks of attention. Accordingly, this paper performs an analytical investigation of spurious eigenvalues for a confocal elliptical membrane using boundary integral equation methods (BIEM) in conjunction with separable kernels and eigenfunction expansion. To analytically study this eigenproblem, the elliptic coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the separable kernel by using the elliptic coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the separable kernel, boundary density and boundary contour integration and they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integration. In this way, a similar finding about the mechanism of spurious eigenvalues is found and agrees with those corresponding to the annular case. To verify this finding, the boundary element method and the commercial finite-element code ABAQUS are also utilized to provide eigensolutions, respectively, for comparisons. Good agreement is observed from comparisons. Based on the adaptive observer system, the present approach can deal with eigenproblems containing circular and elliptical boundaries at the same time in a semi-analytical manner. By using the BIEM, it is found that spurious eigenvalues are the zeros of the modified Mathieu functions which depend on the inner elliptical boundary and the integral formulation. Finally, several methods including the CHIEF method, the SVD updating technique and the Burton & Miller method are employed to filter out the spurious eigenvalues, respectively. In addition, the efficiency of the CHIEF method is better than those of the SVD updating technique and the Burton & Miller approach, since not only hypersingularity is avoided but also computation effort is saved.

SH-wave scattering by a semi-elliptical hill using a null-field boundary integral equation method and a hybrid method

“…Mathematically speaking, degenerate kernel is a finite‐rank approximation of the closed‐form fundamental solution. Successful applications to vibration (Chen et al . 2011a) and water wave problems (Lee & Chen 2010a; Chen & Lee 2011) have also been done.…”

An Efficient Automatic Mesh Generator With Parabolic Arcs in Julia for Computation of TE and TM Modes for Waveguides

“…The external boundary radius is R = 2, while the major and minor semi-lengths of the hole are a = 1, b = 0. 5, respectively, which is the ellipse, as described in [9] displayed in Fig. 3.…”

“…Here, the consistency of the numerical solution to the true eigenfrequencies is validated with the suggested methodology for different mesh sizes for a discretization of the domain using 9679, 2473, 630, 169 elements for linear, quadratic, cubic, and quintic order, respectively. The numerical outcome reported for the present case is contrasted with the conclusions described in [9] and is tabulated in Table I, which lists the initial nine possible eigenfrequencies. A good agreement was made.…”

“…Numerical methods are very much essential for analysis of propagation of these waves as there is no single direct empirical approach available at present. During recent years, many techniques such as the finite element method (FEM) [6]- [8] or the boundary element method (BEM) [9], [10] have been employed for numerical eigen analysis. Along with these, several other methods for analyzing waveguides have also been proposed in recent years such as for obtaining cut-off wavenumbers of TE and TM modes, a surface integral equation approach was examined with the most common types of waveguides by Swaminathan et al [11].…”

Natural mode analysis of an acoustic cavity with multiple elliptical boundaries by using the collocation multipole method

“…Chen et al [5][6] applied the null-field boundary integral equation method (BIEM) to solve the eigenproblems with circular boundaries. Recently, Chen et al extend the BIEM to deal with torsion problems containing multiple elliptical inclusions [13] and eigenproblems of a confocal elliptical membrane [14], performing an analytical investigation of spurious eigenvalues and providing several remedies to suppress them. It is well known that the BIEM (or BEM) belongs to the boundary-type method which can reduce the dimension of the original problem by one.…”