Revisit of a degenerate scale: A semi-circular disc

Boundary 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.