Numerical analysis of acoustic modes using the linear least squares method of fundamental solutions

Abstract

The method of fundamental solutions (MFS) has been proved to be an accurate and efficient meshless numerical method to solve acoustic eigenproblems. Traditionally, the technique of the singular value decomposition (SVD) is employed to obtain the corresponding contours of acoustical modes after the eigenvalues are solved. However, it is found that the mode shapes are sensitive to the source locations of the MFS. In this paper, we try to derive a robust meshless numerical scheme to obtain the contours of acoustical modes based on the linear least squares method of fundamental solutions (LSMFS) by specifying an additional normalized dual boundary condition. The failure for determining the mode shapes by specifying a normalized data at boundary locations near or on the nodes are examined. Moreover, it is demonstrated that the mode shapes of degenerate eigenmodes can be distinguished by specifying the boundary data at different boundary points. Furthermore, a normalization procedure is introduced for degenerate eigenmodes. Three numerical experiments with regular and irregular boundaries are carried out to validate the proposed method. Mode shapes obtained by the linear LSMFS are in good agreement with the analytical solutions and also the results obtained by the finite element method. In addition, the robustness and accuracy of the eigenvalues obtained with respect to different locations of source points by the linear LSMFS in conjunction with direct determinant search method are also revisited.