On the ill-posed behavior and its remedy of the Trefftz type boundary element method

• Main Authors: Ru-feng Liu, 劉如峯
• Other Authors: Weichung Yeih
• Format: Others
• Language: en_US
• Published: 2005
• Online Access: http://ndltd.ncl.edu.tw/handle/85092751929704562428

博士 === 國立臺灣海洋大學 === 河海工程學系 === 93 === This dissertation proposes the Trefftz boundary-type methods to solve partial differential equations, such as Laplace equation, Helmholtz equation and Poisson equation, for a two-dimensional domain. Such methods can avoid the difficulties of singular or hypersingular integration in the traditional BEM. It is found that the ill-posed nature, i.e., the numerical instability, exists in the solver due to the regular formulation of the Trefftz method. The Riemann- Lebesgue lemma, which simply describes why the Fredholm integral equation of the first kind will encounter numerical instability, is adopted to explain the inherent ill-posed behavior theoretically. It also has been proved that the proposed indirect Trefftz methods have the same mathematical structure as the direct Trefftz methods. Due to its indirect nature, the present approach can represent field quantities within its own mathematical formulations. How to choose an appropriate set of basis functions when the origin is placed inside or outside the domain is critically evaluated along with this scope. Based on the argument, it is further explained that for a multiply connected domain of genus one, to place the origin inside the hole is the only selection. Based on the argument, only M-Trefftz and F-Trefftz methods can deal with a multiply connected domain with holes more than one. In order to deal with the numerical instability, a combined use of the truncated singular value decomposition method and the L-curve concept has been suggested as an effective remedy for the Laplace equation. An automatic searching algorithm for finding the corner point of L-curve is developed. It is proved that there exits no spurious eigensolution in our formulation for the eigenproblem, i.e., solving the Helmholtz equation for a finite domain. To overcome the numerical instability, both the generalized singular value decomposition and Tikhonov’s regularization method are used to cope with. For solving the Helmholtz equation in unbounded domain, only lowing the order of basis functions can eliminate the ill-posed behavior and the correct results of spectrum and field quantities are obtained. In order to solve Poisson equation, we used the radial basis function to find the particular solution first and then to solve the resulting Laplace equation by means of the Trefftz method. The truncated singular value decomposition method and the L-curve concept have been suggested to deal with the numerical instability. Numerical results show that the current approach can successfully deal with various problems including the Poisson equation of , for which the iteration process should be adopted. As for treating the Helmholtz equation as a Poisson equation at a wave number near the eigenvalue, only when the boundary excitation matches the modal shape of the corresponding eigenvalue, no convergence for the iteration process is reported. Such a characteristic can support the engineer to determine the response of structure when the wave number is close to the eigenvalue because that one solves it under the same condition by using the Helmholtz integral equation may result in numerical overflow due to the inversion of a nearly singular matrix. Several numerical examples are demonstrated to show the validity of the proposed approach. It is concluded that to become a robust and universal solver the Trefftz methods require treatment on the numerical instability, which is the main contribution of this dissertation.