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Numerical examination for degenerate scale problem for ellipse‐shaped ring region in BIE
Author(s) -
Chen Y. Z.,
Wang Z. X.,
Lin X. Y.
Publication year - 2007
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1985
Subject(s) - mathematics , discretization , degenerate energy levels , ellipse , mathematical analysis , integral equation , algebraic equation , boundary value problem , scale (ratio) , numerical analysis , kernel (algebra) , ring (chemistry) , boundary (topology) , geometry , pure mathematics , nonlinear system , physics , chemistry , organic chemistry , quantum mechanics
This paper investigates the degenerate scale problem for an ellipse‐shaped ring region in boundary integral equation (BIE). A homogenous integral equation is introduced. The integral equation is reduced to an algebraic equation after discretization. The critical value for the degenerate scale can be obtained from the vanishing condition of a determinant. It is proved that there are two critical values for the degenerate scale, rather than one. This finding is first proposed in the paper. Two particular problems with known solutions are examined numerically. The loadings applied on the exterior boundary may result in a resultant force in the x ‐direction or in the y ‐direction. The improper numerical solutions have been found once the real size approaches the critical value. Two techniques for avoiding the improper solutions are suggested. The techniques depend on the appropriate choice of the used size or adding a constant in a kernel of the integral equation. It is proved that both techniques will give accurate numerical results. Numerical examinations for the problem are emphasized in the paper. Copyright © 2007 John Wiley & Sons, Ltd.