The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates

International Journal for Numerical Methods in Engineering, 77(2) (2009), 177–194
http://dx.doi.org/10.1002/nme.2404

In this paper we study the application of the method of fundamental solutions (MFS) to the numerical calculation of the eigenvalues and eigenfunctions for the 2D bilaplacian in simply connected plates. This problem was considered in Kang and Lee (J. Sound Vib. 2001; 242(1):9–16) using wave-type functions and in Chen et al. (Eng. Anal. Boundary Elem. 2004; 28:535–545) using radial basis functions for circular and rectangular domains. The MFS is a mesh-free method that was already applied to the calculation of the eigenvalues and eigenfunctions associated with the Laplace operator (cf. Appl. Math. Lett. 2001; 14(7):837–842; Eng. Anal. Boundary Elem. 2005; 29(2):166–174; Comput. Mater. Continua 2005; 2(4):251–266). The application of this method to the bilaplace operator was already considered in Chen and Lee (ECCOMAS Thematic Conference on Meshless Methods, Lisbon, 2005) for multiply connected domains, but only for simple shapes. Here we apply an algorithm for the choice of point-sources, as in Alves and Antunes (Comput. Mater. Continua 2005; 2(4):251–266), which leads to very good numerical results for simply connected domains. A main part of this paper is devoted to the numerical analysis of the method, presenting a density result that justifies the application of the MFS to the eigenvalue biharmonic equation for clamped plate problems. We also present a bound for the eigenvalues approximation error, which leads to an a posteriori convergence estimate