Revisit of a degenerate scale: A semi-circular disc

Abstract

a b s t r a c tBoundary 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.

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

“…There are several terminologies for the degenerate scale, e.g., the critical value (Yan and Sloan, 1988), the transfinite diameter (Yan and Sloan, 1988; Christiansen, 2001), the Gamma contour and the unit logarithmic capacity (Hille, 1962; Hayes and Kellner, 1972; Constanda, 1993; Dijkstra and Mattheij, 2007; Corfdir and Bonnet, 2013; Kuo et al , 2013a, 2013b) in the modern potential theory. In addition to the degenerate scale of a circle (Petrovsky, 1954; Vodička and Mantič, 2004; Vodička and Mantič, 2008; Lee et al , 2013), Chen et al had studied the degenerate scales of an elliptic case (Chen et al , 2012), a regular N -gon (Kuo et al , 2013a, 2013b) and a semi-circular disc (Chen et al , 2015) by using the unit logarithmic capacity through the complex variables and the BEM. In the BEM, the degenerate scale stems from the rank deficiency problems due to the fundamental solution ln(r) for the 2D Laplace equation subject to the Dirichlet boundary condition (Chen et al , 2001, 2002, 2003b, 2005a; Han et al , 2015).…”

Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

“…This causes troubles in seeking the true solution as well as the numerical solutions, which is called the degenerate scale problems (or degenerate scales). Recently, the study of degenerate scales becomes active, see [7,[9][10][11][13][14][15][16]18]. The singularity of the BIE and its numerical algorithms may be different.…”

“…Chen and Y.Z. Chen in [8][9][10][11][12][13][14][15][16] seems to be more focused on the degenerate scale problems. Our efforts are mainly paid for establishing effective algorithms by following [23,27].…”

“…In Ang and Kang [1], complex boundary elements are studied. For circular and elliptic domains, the NFM is developed by Chen's group, see [8][9][10][11][12][13]. Recently, the explicit collocation equations and semi-analytic solutions are derived in Li et al [25], and the conservative schemes are developed in Lee et al [19].…”