Linkage between unit logarithmic capacity in the theory of complex variables and the degenerate scale in BEM/BIEMM

It is well known that BEM/BIEM result in degenerate scale for two-dimensional Laplace problem subjected to the Dirichlet boundary condition. In such a case, nontrivial boundary flux exist even the trivial boundary potential is given. It is proved that the unit logarithmic capacity in the Riemann conformal mapping with respect to the unit circle results in a null field for the interior domain. The logarithmic capacity is defined as the coefficient of the linear term in the Riemann conformal mapping. When the logarithmic capacity is equal to one, a trivial interior field can be obtained but exterior field is derived to be nonzero using ln function. Two mapping functions, Riemann conformal mapping for geometry and log function for physics, are both required. This matches well with the BEM result that an interior trivial field has nonzero boundary flux in case of degenerate scale. Regarding the ordinary scale, BIE results in a null field in the exterior domain owing to the Green’s third identity. It is interesting to find that ordinary and degenerate scales result in a null field in the exterior and interior domains, respectively. To demonstrate this finding, three cases of circle, ellipse and right triangle are demonstrated. Theoretical derivationusing the Riemann conformal mapping with the unitlogarithmic capacity and the degenerate scale inBEM/BIEM both indicate the null field in the interiordomain analytically and numerically. References[1] J.T. Chen, J.H. Lin, S.R. Kuo and Y.P. Chiu: Analyticalstudy and numerical experiments for degenerate scaleproblems in boundary element method using degeneratekernels and circulants, Eng. Anal. Bound. Elem., 25(9),pp. 819-828, 2001.[2] J.T. Chen, S.R. Kuo and J.H. Lin: Analytical study andnumerical experiments for degenerate scale problems inthe boundary element method for two-dimensionalelasticity, Int. J. Numer. Meth. Eng., 54(12),pp.1669-1681, 2002.[3] J.T. Chen, C.F. Lee, I.L. Chen and J.H. Lin: Analternative method for degenerate scale problems inboundary element methods for the two-dimensionalLaplace equation, Eng. Anal. Bound. Elem., 26(7), pp.559-569, 2002.[4] J.T. Chen, W.C. Chen, S.R. Lin and I.L. Chen, Rigidbody mode and spurious mode in the dual boundaryelemnent formulation for the Laplace equation, Comput.Struct., 81(13), pp. 1395-1404, 2003.[5] J.T. Chen, S.R. Lin and K.H. Chen: Degenerate scaleproblem when solving Laplace equation by BEM and itstreatment, Int. J. Numer. Meth. Eng., 62(2), pp.233-261,2005.[6] J.T. Chen, C.S. Wu, K.H. Chen and Y.T. Lee:Degenerate scale for analysis of circular plate using theboundary integral equations and boundary elementmethod, Comput. Mech., 38, pp 33-49, 2006.[7] J.T. Chen, Y.T. Lee, S.R. Kuo and Y.W. Chen:Analytical derivation and numerical experiments ofdegenerate scale for an ellipse in BEM, Eng. Anal.Bound. Elem., 36(9), pp. 1397-1405, 2012.