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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/32126

Title: Analytical derivation and numerical experiments of degenerate scales for an ellipse in BEM
Authors: Chen J.T.;Lee Y.T.;Kuo S.R.;Chen Y.W.
Contributors: NTOU:Department of Harbor and River Engineering
Keywords: Degenerate scale;Degenerate kernel;Logarithmic capacity;Conformal mapping
Date: 2011-11
Issue Date: 2012-06-15T07:29:45Z
Publisher: The Second Workshop on Boundary Element Method and Boundary Integral Equation Method
Abstract: Degenerate scale of an ellipse is studied by using the dual boundary element method (BEM), degenerate kernel and unit logarithmic capacity. Degenerate scale stems from either the nonuniqueness of logarithmic kernel in the BIE or the conformal radius of unit logarithmic capacity in the complex variable. Numerical evidence of degenerate scale in BEM is given. Analytical formula for the degenerate scale can be derived not only from the conformal mapping in conjunction with unit logarithmic capacity, but also can be derived by using the degenerate kernel. Eigenvalues and eigenfunctions for the weakly singular integral operator in the elliptical domain are both derived by using the degenerate kernel. It is found that zero eigenvalue results in the degenerate scale. Based on the dual BEM, the rank-deficiency (mathematical) mode due to the degenerate scale is imbedded in the left unitary vector for weakly singular and strongly singular integral operators. On the other hand, we obtain the common right unitary vector of a rigid body (physical) mode in the influence matrices of strongly singular and hypersingular operators after using the singular value decomposition. Null field for the exterior domain and interior nonzero fields are analytically derived and numerically verified in case of the normal scale while the interior null field and nonzero exterior field are obtained for the homogeneous Dirichlet problem in case of the degenerate scale. No failure CHEEF point is confirmed in the nonzero exterior field to overcome the degenerate-scale problem. To deal with the nonuniqueness-solution problem, the constraint of boundary flux equilibrium instead of rigid body term, CHEEF and hypersingular BIE, is added to promote the rank of influence matrices to be full rank. Both analytical and numerical results agree well in the demonstrative example of an ellipse.
Relation: 36(9), pp.1397–1405
URI: http://ntour.ntou.edu.tw/handle/987654321/32126
Appears in Collections:[河海工程學系] 演講及研討會

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