Rank-deficiency for problem of degenerate scale and fictitious frequencies

• Main Authors: Jheng-Lin Yang, 楊政霖
• Other Authors: Chen, Jeng-Tzong
• Format: Others
• Language: zh-TW
• Published: 2017
• Online Access: http://ndltd.ncl.edu.tw/handle/08668688750500575876

碩士 === 國立臺灣海洋大學 === 機械與機電工程學系 === 105 === In this thesis, we focus on the two rank-deficiency problems. One is motivated by the incompleteness of single-layer potential approach for the interior problem with a degenerate-scale domain and the exterior problem with bounded potential at infinity. The other is rank-deficiency problem due to fictitious frequencies in the boundary element method(BEM). For the first issue, we revisit the method of fundamental solutions(MFS). Although the MFS is an easy method to implement, it is not complete for solving not only the interior 2D problem in case of a degenerate scale but also the exterior problem with bounded potential at infinity for any scale. Following Fichera's idea for the boundary integral equation, we add a free constant and an extra constraint to the traditional MFS. The reason why the free constant and extra constraints are both required is clearly explained by using the degenerate kernel for the closed-form fundamental solution. Since the range of the single-layer integral operator lacks the constant term in the case of a degenerate scale for a two dimensional problem, we add a constant to provide a complete base. Due to the rank deficiency of the influence matrix in the case of a degenerate scale, we can promote the rank by simultaneously introducing a constant term and adding an extra constraint to enrich the MFS. For an exterior problem, the fundamental solution does not contain a constant field in the degenerate kernel expression. To satisfy the bounded potential at infinity, the sum of all source strengths must be zero. The formulation of the enriched MFS can solve not only the degenerate-scale problem for the interior problem but also the exterior problem with bounded potential at infinity. Three examples, a circular domain, an infinite domain with two circular holes and an eccentric annulus were demonstrated to see the validity of the enriched MFS. For the problem of fictitious frequencies, a self-regularization approach in the linear algebraic system of the BEM is proposed by using the singular value decomposition (SVD). Since there is no null-field equation in the indirect BEM, we employ the self-regularization method to promote the rank of the influence matrix. We adopt the bordered matrix by adding the left and right unitary vectors of zero eigenvalue. In addition, a free constant is introduced. Finally, one circle domain examples were demonstrated to see the validity of the self-regularization method for alleviating the fictitious frequencies in the indirect BEM. Keywords: method of fundamental solutions, Fichera's method, degenerate scale, self-regularization, singular value decomposition