带圆孔圆域拉普拉斯方程Dirichlet问题的对偶零场方法

M. G. Lee, L. P. Zhang, Z. C. Li, A. Kazakov
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引用次数: 0

摘要

对偶技术在许多工程论文中得到了广泛的应用,用于处理边界元法的奇异性和病态性。在本文中,我们考虑了具有一个圆孔的圆形域的拉普拉斯方程。给出了第一类和第二类零场方法(NFM)的显式代数方程,以供应用。传统上,第一类和第二类NFM分别用于Dirichlet和Neumann问题。然而,为了绕过Dirichlet问题的退化尺度,本文将第二类和第一类NFM同时用于外边界和内边界,称为对偶NFM(DNFM)。本文探讨了其优良的稳定性和最优收敛速度。通过简单的高斯消去法或迭代法,可以很容易地得到数值解。近年来,退化尺度的研究很活跃,提出了许多去除技术,其中可能需要先进的求解方法,如截断奇异值分解(TSVD)和超定系统。相比之下,本文中的DNFM的求解方法要简单得多,并且具有很小的退化尺度导致算法奇异的风险。
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Dual null field method for Dirichlet problems of Laplace's equation in circular domains with circular holes
The dual techniques have been widely used in many engineering papers, to deal with singularity and ill-conditioning of the boundary element method (BEM). In this paper, we consider Laplace's equation with circular domains with one circular hole. The explicit algebraic equations of the rst and second kinds of the null eld method (NFM) are provided for applications. Traditionally, the rst and the second kinds of the NFM are used for the Dirichlet and the Neumann problems, respectively. To bypass the degenerate scales of Dirichlet problems, however, the second and the rst kinds of the NFM are used for the exterior and the interior boundaries, simultaneously, called the dual NFM (DNFM) in this paper. The excellent stability and the optimal convergence rates are explored in this paper. By using the simple Gaussian elimination or the iteration methods, numerical solutions can be easily obtained. Recently, the study on degenerate scales is active, many removal techniques are proposed, where the advanced solution methods may be needed, such as the truncated singular value decomposition (TSVD) and the overdetermined systems. In contrast, the solution methods of the DNFM in this paper are much simpler, with a little risk of the algorithm singularity from degenerate scales.
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CiteScore
1.00
自引率
25.00%
发文量
15
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