Linkage of logarithmic capacity in potential theory and degenerate scale in the BEM for two tangent discs

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Abstract

Degenerate scales of different shapes have been studied by many researchers. BEM/BIEM may lead to a degenerate scale for a two-dimensional Laplace problem. In this paper, we study the problem of the two tangent discs with different radii. We use the conformal mapping of complex variables and the unit logarithmic capacity to examine the degenerate scale. Although the degenerate scale of infinite plane containing two circular holes has been derived by using the degenerate kernel in terms of bipolar coordinates, this shape cannot be solved by using the degenerate kernel. Finally, the numerical results of BEM were also compared with those of analytical formula.

Introduction

Boundary element method (BEM) or boundary integral equation method (BIEM) is widely used to solve the two-dimensional Laplace equation. However, by using the software of the BEM for analysis and simulation at the beginning of the engineering design, it may cause incorrect results for a certain size. This size is called a degenerate scale. Mathematically speaking, there are two ways to understand the degenerate scale. One is the non-uniqueness solution in the BIEM/BEM, where trivial boundary potential may result in nontrivial flux. The other is the unit logarithmic capacity corresponding to the conformal radius in the complex analysis [1]. Hille [2] gave the definition of the logarithmic capacity and it was addressed by mathematicians. Many researchers continued to investigate this issue, the linkage between the degenerate scale of BEM and the logarithmic capacity of potential theory was even discussed in [3], [4], [5]. Rumely [6] derived the logarithmic capacity of many shapes by using the conformal mapping. In 2013, Kuo et al. [7] studied the regular N-gons problem and linked the unit logarithmic capacity as shown in [8]. These two studies both employed the Riemann conformal mapping. In 2015, Chen et al. [9] studied the degenerate scale of a semi-circular disc by using the BEM as well as the complex variable. These shapes [7], [9] are discussed in the present paper.

For the two-circles problem of the degenerate scale, there are four geometries to study, two tangent or non-tangent circles with equal or unequal radius. In the part of deriving the degenerate scale of the infinite plane problem containing two equal holes, various researchers used different methods. In 2007, Chen and Shen [10] studied the eccentric case by using degenerate kernels and Fourier series in the null-field integral equation. Chen et al. [11] used the degenerate kernel in terms of the bipolar coordinates. Kuo et al. [12] used the conformal mapping technique, where a dictionary [13] of conformal mapping can be consulted.

In this paper, we devoted to study the degenerate scale and logarithmic capacity. Although we have analytically derived the degenerate scale for the infinite plane containing two holes using the degenerate kernel in terms of bipolar coordinates, this idea cannot be directly applied to solve the case of two tangent discs. The reason is that bipolar coordinates cannot exactly describe the geometry of tangent discs. Therefore, an alternative approach of the conformal mapping of complex variables is resorted. Finally, we revisit the same problem but different radii and compare the present results with those of the BEM. Analytical results and numerical data are also examined.

Section snippets

Analytical derivation of degenerate scale for two tangent discs of various radii by using the conformal mapping

In order to analytically derive the degenerate scale of two tangent discs of various radii, we map the tangent discs to a unit circle through four successive mappings (f1(z), f2(w1), f3(w2) and f4(w3)). First, we define the right circle in the z plane as z=a(1+eiθ),zCR,while the left circle is described by z=αa(1+eiθ),zCL.

Therefore, the first mapping from the z plane to the w1 plane is w1=f1(z)=azα14α,According to the mapping, two circular discs in the z plane (Fig. 1(a)) are mapped to two

Conclusions

This study focused on the degenerate scale problem of the two tangent discs of various radii. Since the bipolar coordinates cannot describe the geometry of two tangent discs, we employed the conformal mapping combining with the unit logarithmic capacity to analytically derive the degenerate scale. This is a demonstrative example to verify that the method of using complex variables is more flexible than that using the degenerate kernel for determining degenerate scales. We found that the two

Acknowledgements

The authors wish to thank the financial support from the Ministry of Science and Technology, Taiwan under Grant No. MOST106-2221-E-019-009-MY3.

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