The Least Squares Trefftz Method and the Method of External Source for the Eigenfrequencies of Waveguides

Hui-Ming Fang ； Chia-Ming Fan ； Yan-Cheng Liu ； Sung-Shan Hsiao

《Journal of Marine Science and Technology》 21卷6期 (2013/12) Pp. 703-710

https://doi.org/10.6119/JMST-013-0626-1

並列摘要

In this study, the least squares Trefftz method (LSTM) is adopted for analyzing the eigenfrequencies problems governed by homogeneous Helmholtz equations. The Trefftz method, one kind of boundary-type meshless collocation methods, does not need mesh generation and numerical quadrature. Since the system of linear algebraic equations obtained by Trefftz method is highly ill-conditioned, the least squares method is adopted to stabilize the numerical scheme in this study. In the eigenproblem, the response amplitudes from an external source are used to determine the resonant frequencies. By adding an external source, the homogeneous boundary condition becomes inhomogeneous. Then we can employ the LSTM to easily solve this problem. In this paper, the LSTM and the method of external source are used to solve this eigenfrequencies problems governed by Helmholtz equations. Several numerical examples are provided to verify the accuracy and the simplicity of the proposed numerical scheme.

並列關鍵字

least squares Trefftz method ； eigenfrequencies problem ； Helmholtz equation ； boundary-type meshless method

參考文獻

Abou-Dina, M.S., “Implementation of Trefftz method for the solution of some elliptic boundary value problems,” Applied Mathematics and Computation, Vol. 127, pp. 125-147 (2002).

Cai, Z.J., “Best estimates of RBF-based meshless Galerkin methods for Dirichlet problem,” Applied Mathematics and Computation, Vol. 215, pp. 2149-2153 (2009).

Chen, J.T., Liu, L.W., and Hong, H.K., “Spurious and true eigensolu tions of Helmholtz BIEs and BEMs for a multiply connected problem,” Proceedings of the Royal Society A, Vol. 459, pp. 1897-1924 (2003).

Chen, J.T., Lee, Y.T., and Shieh, S.C., “Revisit of two classical elasticity problems by using the Trefftz method,” Engineering Analysis with Boundary Elements, Vol. 33, pp. 890-895 (2009).

Chen, J.T., Lee, Y.T., Yu, S.R., and Shieh, S.C., “Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image con cept,” Engineering Analysis with Boundary Elements, Vol. 33, pp. 678-688 (2009).