Analytical derivation and numerical experiments of degenerate scales for regular N-gon domains in two-dimensional Laplace problems

Abstract

Degenerate scale of a regular N-gon domain is studied by using the boundary element method (BEM) and complex variables. Degenerate scale stems from either the non-uniqueness of BIE using the logarithmic kernel or the conformal radius of unit logarithmic capacity in the complex variables. Analytical formula and numerical results for the degenerate scale are obtained by using the conformal radius and boundary element program, respectively. Analytical formula of the degenerate scale contains the Gamma function for the Gamma contour which can be derived from the Schwarz–Christoffel mapping. Based on the dual BEM, the rank-deficiency (mathematical) mode due to the degenerate scale (mathematics) is imbedded in the left unitary vector for the influence matrices of weakly singular (U kernel) and strongly singular (T kernel) integral operators. On the other hand, we obtain the common right unitary vector corresponding to a rigid body mode (physics) in the influence matrices of strongly singular (T kernel) and hypersingular (M kernel) operators after using the singular value decomposition. To deal with the problem of non-unique solution, 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. Null field for the exterior domain and interior nonzero field are analytically derived and numerically verified for the normal scale while the interior null field and nonzero exterior field are obtained for the homogeneous Dirichlet problem in the case of the degenerate scale. It is found that the contour of nonzero exterior field for the degenerate scale using the BEM matches well with that of Schwarz–Christoffel transformation. Both analytical and numerical results agree well in the demonstrative examples of right triangle, square, regular 5-gon and regular 6-gon. It is straightforward to extend to general regular N-gon case.