Analytical derivation and numerical experiments of degenerate scale for an ellipse in BEM

Chen, Jeng‐Tzong; Lee, Ying‐Te; Kuo, Shyh‐Rong; Chen, Yiwei

doi:10.1016/j.enganabound.2012.03.002

Cited by 22 publications

(13 citation statements)

References 23 publications

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“…For the second kind NFM, although the description is briefly given in many papers of Chen with his group [7][8][9][10][11][12], but no numerical computation is provided so far. Hence, there arise a few questions.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Neumann problems of Laplace׳s equation in circular domains with circular holes by methods of field equations

Lee¹,

Li²,

Huang³

et al. 2015

Engineering Analysis with Boundary Elements

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Section: Introductionmentioning

confidence: 98%

“…The Fourier series are also used to approximate the known or unknown Dirichlet and Neumann boundary conditions, and numerous papers have been published for different physical problems. For Laplace's equation we only cite [7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Neumann problems of Laplace׳s equation in circular domains with circular holes by methods of field equations

Lee¹,

Li²,

Huang³

et al. 2015

Engineering Analysis with Boundary Elements

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“…From the viewpoint of mathematics, there are three ways to understand the occurring mechanism of degenerate scale. One is the non-uniqueness solution in the BIEM or BEM [18,19]. Another is the unit logarithmic capacity corresponding to the conformal mapping in the theory of complex variables [20,21].…”

Section: Introductionmentioning

confidence: 98%

Revisit of a degenerate scale: A semi-circular disc

Chen

Kuo

Kao

et al. 2015

Journal of Computational and Applied Mathematics

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a b s t r a c tBoundary element method (BEM) has been employed in engineering analysis since 1956, it has been widely applied in the engineering. However, the BEM/BIEM may result in an ill-conditioned system in some special situations, such as the degenerate scale. The degenerate scale also relates to the logarithmic capacity in the modern potential theory. In this paper, three indexes to detect the degenerate scale and five regularization techniques to circumvent the degenerate scale are reviewed and a new self-regularization technique by using the bordered matrix is proposed. Both the analytical study and the BEM implementation are addressed. For the analytical study, we employ the Riemann conformal mapping of complex variables to derive the unit logarithmic capacity. The degenerate scale can be analytically derived by using the conformal mapping as well as numerical detection by using the BEM. In the theoretical aspect, we prove that unit logarithmic capacity in the Riemann conformal mapping results in a degenerate scale. We revisit the Fredholm alternative theorem by using the singular value decomposition (SVD, the discrete system) and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. Besides, a zero index by using the free constant in Fichera's approach is also proposed to examine the degenerate scale. According to the relation between the SVD structure and Fichera's technique, we numerically provide a new self-regularization method in the matrix level. Finally, a semi-circular case and a special-shape case are designed to demonstrate the validity of six regularization techniques.

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“…In Ang and Kang [1], complex boundary elements are studied. For circular and elliptic domains, the NFM is developed by Chen's group, see [8][9][10][11][12][13]. Recently, the explicit collocation equations and semi-analytic solutions are derived in Li et al [25], and the conservative schemes are developed in Lee et al [19].…”

Section: Introductionmentioning

confidence: 99%

Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

Zhang

Wei

et al. 2015

Engineering Analysis with Boundary Elements

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Analytical derivation and numerical experiments of degenerate scale for an ellipse in BEM

Cited by 22 publications

References 23 publications

Neumann problems of Laplace׳s equation in circular domains with circular holes by methods of field equations

Neumann problems of Laplace׳s equation in circular domains with circular holes by methods of field equations

Revisit of a degenerate scale: A semi-circular disc

Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

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