Analytical and numerical investigation for true and spurious eigensolutions of an elliptical membrane using the real-part dual BIEM/BEM

Abstract

The paper performs analytical and numerical investigation of the true and spurious eigensolutions of an elliptical membrane using the real-part boundary integral equation method (BIEM) following the successful work on a circular case by using the dual boundary element method (BEM) (Kuo et al. in Int. J. Numer. Methods Eng. 48:1401–1422, 2000). We extend to the elliptical case in this paper. To analytically study the eigenproblems of an elliptical membrane, the elliptical coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the degenerate kernel by using the elliptical coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the degenerate kernel, boundary density and boundary contour integration but they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integral. It is interesting to find that the BIEM using the real or the imaginary-part kernel to deal with an elliptical membrane yields spurious eigensolutions. This finding agrees with those corresponding to the circular case. The spurious eigenvalues in the real-part BIEM are found to be the zeros of the mth-order (even or odd) modified Mathieu functions of the second kind or their derivatives. To verify this finding, the BEM is implemented. Furthermore, the commercial finite-element code ABAQUS is also utilized to provide eigensolutions for comparisons. It is found that good agreement is obtained.