Revisit of Logarithmic Capacity of Line Segments and Double-Degeneracy of BEM/BIEM

Engineering Analysis with Boundary Elements 120 (2020) 238–245

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Engineering Analysis with Boundary Elements

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Correspondence

Revisit of logarithmic capacity of line segments and double-degeneracy of BEM/BIEM

Jeng-Tzong Chen a,b,c,d,e,∗, Shyh-Rong Kuo a, Yi-Ling Huang a, Shing-Kai Kao a a Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan b Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan c Department of Civil Engineering, National Cheng-Kung University, Tainan, 70101, Taiwan d Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan e Bachelor Degree Program in Ocean Engineering and Technology, National Taiwan Ocean University, Keelung, 20224, Taiwan

a r t i c l e i n f o a b s t r a c t

Keywords: Degenerate scale of line segments is derived in this paper. It is found that two formulas in the Rumely’s book Degenerate scale are inconsistent after comparing with our results. We conﬁrm our formulas not only by testing the value of

BEM degenerate scale in the BEM/BIEM but also by ﬁnding the unit logarithmic capacity. Double degeneracy due to

Logarithmic capacity the degenerate boundary (line segment) and degenerate scale in the BEM/BIEM is also examined. By increasing Conformal mapping the number of boundary elements, the value of degenerate scale converges to the corresponding one derived by Double degeneracy using the complex variables instead of Rumely’s formulas. Three cases in the Rumely’s book, include one segment, two equal segments and three segments (L, 2 L, L) with the spacing 3 L, are re-conﬁrmed. However, we found that there are two inconsistent cases, two unequal segments (L, 5 L) with the spacing 3 L and three segments (L, 2 L, L) with the spacing L, respectively, after analytically deriving by using the complex variables and numerically examining by detecting the degenerate scale in the BEM/BIEM.

1. Introduction BIEM/BEM may appear in four aspects: degenerate scale [5] , degenerate boundary [6] , spurious eigenvalue [7] and fictitious frequency [8] . For a long time, potential theory was viewed as only a chapter of Fictitious frequency and spurious eigenvalue appear in the exterior and mathematics. After close connection with the boundary value problem interior Helmholtz problems, respectively. A degenerate boundary oc- (BVP) for the Laplace operator, it led to the creation of single and dou- curs in the problem such as a crack or a cutoff wall etc. A degenerate ble layer potentials. Later, the BEM/BIEM for solving engineering prob- scale relates to a critical geometric size. Degenerate scale can be inter- lems stimulates engineers the interest of potential theory and integral preted in several ways. The first one is that the influence matrix of the equations. In the modern potential theory [1] , the logarithmic capac- single-layer potential is rank deficient in the case of degenerate scale. ity as well as the transfinite boundary (radius) was investigated. Not Mathematically speaking, the degenerate scale occurs when the trivial only compact set of R 2 but also R 3 logarithmic capacity is included in boundary potential result in the nontrivial boundary flux. This special Landkof’s book [2] . Robinson [3] proposed a method for computing ca- scale matches the transfinite boundary [9] . On the other hand, this de- pacities of sets in the real line by introducing the pullback formula in generate scale can map into a unit circle where the leading coefficient of a disguised form. More than twenty examples of logarithmic capacity linear term is one in the Riemann conformal mapping. The definition of for different geometric curves were summarized in Rumely’s book [4] . the logarithmic capacity was given in [10] . Many researchers discussed It includes, circle, polygon, semi-disk, ellipse, star shape, disjoint discs, the issue and linked the logarithmic capacity with the degenerate scale tangent discs, line segments, and other shapes. in the BEM [11–13] . NTOU/MSV group has also revisited a lot of cases By using the software of the BEM for analysis and simulation at [14–16] in the Rumely’s book and the agreement is made. However, the the beginning of the engineering design, it may cause non-unique so- logarithmic capacity of the line segment was not linked yet. lutions. These non-unique solutions are not physically realizable but Line segment in the (BVPs) may have two different boundary con- stems from the zero eigenvalue imbedded in the influence matrix af- ditions. One is the Dirichlet type and the other is the Neumann type. ter discretizing the boundary integral equation (BIE). Therefore, we For the anti-plane elasticity, a rigid line inclusion is specified by the called it “mathematical degeneracy “. Mathematical degeneracy in the Dirichlet B.C. to describe the rigid behavior, while a crack is described by the Neumann B.C. to describe the free traction. No matter which

∗ Corresponding author. E-mail address: [email protected] (J.-T. Chen). https://doi.org/10.1016/j.enganabound.2020.08.003 Received 6 March 2020; Received in revised form 3 August 2020; Accepted 6 August 2020 0955-7997/© 2020 Elsevier Ltd. All rights reserved. J.-T. Chen, S.-R. Kuo and Y.-L. Huang et al. Engineering Analysis with Boundary Elements 120 (2020) 238–245

3 3 2 B.C. is specified, both one belong to the degenerate boundary since a 𝑤 𝑒 = − 𝛼 𝐿 + 3 𝜆 𝛼𝐿, (7) double-valued function on the line segment is defined. Since 1988, the dual BEM/BIEM for solving the BVP with degenerate boundaries was 𝑤 𝛼 3𝐿 3 𝜆2 𝛼 𝐿, proposed by Hong and Chen [17] . Until 1999, a review article can be 𝑒 = ( 5 − ) − 3 (5 − ) (8) found by Chen and Hong [18]. In the literature, the degenerate scale for as shown in Fig. 1 (b), respectively. The two unknowns, 𝛼 and 𝜆 can be the line segment was investigated by Chen et al. [19–23] for the rigid determined by solving the Eqs. (6)–(8) , we have line inclusion instead of the line crack. However, the degenerate scale 2 for the line segment still exists. It is coined the “double degeneracy ” 𝛼 = , (9) 3 since the degenerate boundary and the degenerate scale may occur at the same time in the implementation of BEM. and 7 In this paper, we focus on the derivation of the degenerate scale 𝜆 = 𝐿, (10) of line segments by using the conformal mapping. Five cases in the 3 Rumely’s book [4] will be considered and the corresponding rank defi- respectively. Substituting Eqs. (9) and (10) into Eq. (6) or (7) or (8) , the ciency in the BEM will be examined to re-confirm our analytical result rightmost endpoint we in the w plane as shown in Fig. 1(b) yields using the conformal mapping of complex variables. The result of the 286 3 𝑤 𝑒 = 𝐿 . (11) Rumely’s three examples, one segment, two equal segments and three 27 segments (L, 2 L, L) with the spacing 3 L, agree well with our analytical Substituting Eq. (10) into Eq. (1) , we have and numerical solutions. However, the result of the other two examples, 49 two unequal segments (L, 5 L) with the spacing 3 L and three segments 𝑤 = 𝑧 3 − 3 𝜆2𝑧 = 𝑧 3 − 𝐿 2𝑧, (12) 3 (L, 2 L, L) with the spacing L are not consistent with those of present where L is imbedded in the mapping function, while Rumely does not methods. The formula of these two examples derived by the present have. To obtain the length of the line segment in the w plane, we need to conformal mapping and the numerical evidences (BEM results) are co- determine the next w . Since z and z are both mapped to the leftmost incident. Then, both present ways, analytical and numerical approach b m 1 endpoint w of the line segment in the w plane as shown in Fig. 1 (b), indicate that two formulas in the Rumely’s book [4] are not consistent b and the zero slope at z in Fig. 1 (a), we have with our results. These analytical formulas derived by using the confor- m ( ) mal mapping and the numerical evidence will be provided to support 𝑑 𝑓 𝑅 = 3 𝑧 2 − 𝜆2 = 0 . (13) our finding in the present paper. The error in the Rumely’s book is also 𝑑𝑧 examined here. For the positive root, we select z = 𝜆. Hence, w can be obtained by m b 𝑧 𝜆 7 𝐿 substituting 𝑚 = = into Eq. (12) as shown below: 2. Derivation of the degenerate scale of line segments by using 3 3 2 3 686 3 the conformal mapping 𝑤 𝑏 = 𝑧 − 3 𝜆 𝑧 𝑚 = −2 𝜆 = − 𝐿 . (14) 𝑚 27

In this section, we employ the conformal mapping to study the de- Since z1 also maps to the wb as shown in Fig. 1(b), we have generate scale. Since the degenerate scale relates to the logarithmic ca- 𝑧 3 𝜆2𝑧 𝜆3. − 3 1 = −2 (15) pacity, we will map the boundary to a unit circle. Based on this idea, a 1

𝑔 𝜉 𝜉 1 Therefore, the roots of the cubic polynomial equation of one variable z line segment of 4 is a degenerate scale since the mapping is ( ) = + 𝜉 . 1

It is found that the leading coeﬃcient of the linear term in the Riemann yield conformal mapping is one. This matches that the logarithmic capacity of 𝑧 𝜆 , 𝑧 𝜆 . 1 = (double roots) 1 = −2 (singleroot) (16) a circle and a line segment is the radius a and 1 𝐿 , respectively, where 4 𝜆 𝜆 𝜆 L is the length of line segment. We select the negative z1 = −2 because zm = . By substituting z1 = −2

To map the two line segments to one segment, we employ the fol- into Eq. (12), we can also obtain the same value of wb for the double lowing mapping function check. Both ways are consistent to each other. Then, we have the length of the line segment 3 2 𝑤 = 𝑓 𝑃 ( 𝑧 ) = 𝑧 − 3 𝜆 𝑧, (1) 3 𝐿 𝑑 = 𝑤 𝑒 − 𝑤 𝑏 = 36 𝐿 , (17) where the linear boundary along the axis in the z plane maps to the axis in the w plane. The mapping scheme is shown in Fig. 1 . For easily in the w plane. For a line segment, we have a degenerate scale of Ld = 4. understanding the mapping function, the axis of the linear boundary in Thus, we have the w plane versus the one in the z plane is shown in Fig. 1(a). The local 36 𝐿 3 = 4 . (18) ∗ maximum w and the local minimum wb are in the region of z2 z3 and ∗ Therefore, the degenerate scale occurs at the z3 z4 , respectively. The corresponding points in the z plane are z and 2 z . We set that z map to w . Therefore, there exists a point w , w − m 1 b e b 𝐿 3 . ≤ ≤ ∗ = 3 (19) we w , to satisfy fp (z2 ) = fp (z3 ) = fp (z4 ) = we as shown in Fig. 1(b). Following the pattern of two unequal segments (L,5 L) with the spacing for the two unequal segments (L, 5 L) with the spacing 3 L. In this case, 3 L as showing in Fig. 1 (c), we set the series conformal mapping from the exterior circle in the 𝜉 plane to

𝑧 𝛼𝐿, the exterior line segments domain in the z plane as shown in Fig. 1(d) 3 = − (2) can be expressed in terms of the series expansion as the following form then 1 𝑏 𝑏 𝑧 = − ( 4 + 𝛼) 𝐿, (3) 𝑧 = 𝑔 ( 𝜉) = 𝑎 𝜉 𝑚 + 𝑏 + 1 + 2 + ..., (20) 1 1 1 0 1 2 𝜉 𝑚 𝜉 𝑚

𝑧 𝛼 𝐿, 2 = −(3 + ) (4) where m = 3. Equation (20) is not a standard power series but it looks like a Puiseux series [24–25] . They were ﬁrst introduced by Newton in

𝑧 𝛼 𝐿, 1676 and rediscovered by Puiseux in 1850. Multiple imaginary roots of 4 = (5 − ) (5) the Puiseux series is a fundamental issue. In the last decades, researchers 𝛼 where will be determined later. Since z2 , z3 and z4 map to the end tip devoted to compute the critical characteristic roots located on the imag- we of the line segment through Eq. (1), we have inary axis and their behaviors with respect to the delay parameters [26] . 3 3 2 𝑤 𝑒 = − ( 3 + 𝛼) 𝐿 + 3 𝜆 ( 3 + 𝛼)𝐿, (6) Chudnovsky and Chudnovsky determined coeﬃcients of the power and

239 J.-T. Chen, S.-R. Kuo and Y.-L. Huang et al. Engineering Analysis with Boundary Elements 120 (2020) 238–245

Fig. 1. Mapping scheme for the two unequal segments (L, 5 L) with the spacing 3 L.

Puiseux series expansion in the neighborhood of regular singularities by chosen, the logarithmic capacity |a1 | is the same. Although the argu- using explicit linear recurrences [27–28] . Here, we focus on the loga- ment of z is diﬀerent, the modulus is the same. For example, the pullback rithmic capacity. By introducing 𝜉 = v m into Eq. (20) , we obtain formula z 2 used by Rumely [4] can also be solved. For the two√ equal 𝑏 𝑏 line segment as shown in Fig. 2 , choosing either 𝑧 = ℎ ( 𝜉) = ℎ ( 𝜉) or 1 2 √ 1 𝑧 = 𝑔 ( 𝜉) = 𝑓( 𝑣 ) = 𝑎 𝑣 + 𝑏 + + + … .. (21) 𝑧 ℎ 𝜉 ℎ 𝜉 1 1 0 𝑣 𝑣 2 = 1 ( ) = − ( ) of m = 2, the logarithmic capacity |a1 | is the same. Following the above procedure, the logarithmic capacity | a | can be de- Equation (21) yields the form of the Riemann conformal mapping of 1 termined only if the series expansion z = h ( 𝜉) from the exterior circle v → f ( v ) as mentioned in [13] . Thus, we take the leading coeﬃcient of v 1 𝜉 |𝑎 | in the plane to the exterior line segments domain in the z plane can term, a1 , is the logarithmic capacity of the domain. In this case, 1 = 1 be expressed. 𝐿 9 3 . The degenerate scale occurs when |a1 | = 1, yielding the result as shown in Eq. (19) . 3. The comparison between the Rumely’s book and the present

For the unit logarithmic capacity |a1 | = 1, we obtain the degenerate method scale. This can support that our result is correct and is consistent by two ways, the analytical derivation of the unit logarithmic capacity, In Rumely’s book [4] , he employed a function as given below: and the later BEM check. The linkage of the logarithmic capacity and 𝑓 ( 𝑧 ) = 𝑧 3 − 3 𝑧. (22) the degenerate scale was discussed in [9,10 , 29,30] . 𝑅 𝜉 Although z = g1 ( ) is not a univalent function, we can still determine It is obvious that the mapping function oscillates 3 times between −2 the logarithmic capacity |a1 |. The reason is that no matter which z is and 2 in the interval [−2, 2]. In particular, the non-one to one roots are

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Fig. 2. A series conformal mapping to show that the pullback formula is a special case in our system.

we assume L = 1 for the two unequal segments (L, 5 L) with the spacing −1 𝑓 𝑅 (−2) = (−2 , 1) , (23) 3 L and set the origin at the leftmost endpoint as shown in Fig. 3 , the logarithmic capacity of the three segments (L, 2 L, L) with the spacing L ( ) ( ) 2 −1 286 −13 2 11 can be obtained under the pullback formula p(z) = z as shown below: 𝑓 𝑅 − = , , , (24) 343 7 7 7 1 𝛾 ( 𝐸) = 3 3 𝐿. (32) ( ) ( ) ∞

𝑓 −1 286 −11 , −2 , 13 , 𝑅 = (25) Since the result is obtained by using the pullback formula, the second 343 7 7 7 error in the book is a direct consequence of the previous error. That is, the logarithmic capacity for the three segments (L, 2 L, L) with the 𝑓 −1(2) = (−1 , 2) . (26) 𝑅 spacing L should be the result as shown in Eq. (32) , instead of the ﬁnal 1 Eqs. (23) and (25) are chosen for calculating the logarithmic capacity 𝛾 𝐸 𝐿 result ∞( ) = 3 6 as shown in Rumely’s book [4]. By using the same of two unequal segments (L, 5 L) with the spacing 3 L. We have technique, the logarithmic capacity for the three segments (L, 2 L, L) ([ ]) [ ] [ ] 286 −11 −2 13 with the spacing 3 L can be calculated, too. First, we consider 𝑓 −1 −2 , = −2 , ∪ , . (27) 𝑅 ([ ]) [ ] [ ] 343 7 7 7 −286 −13 2 11 𝑓 −1 −2 , = −2 , ∪ , . (33) Robinson’s hypotheses [3] implies that 𝑅 343 7 7 7

𝐸 = 𝑓 −1( [ 𝑎, 𝑏 ] ) , (28) By substituting Eq. (33) into Eq. (29) , we have the logarithmic capacity, and the logarithmic capacity yields (( ) ) 1 ( ) 1 −286 1 3 100 3 𝛾 ( 𝐸) = − (−2) = , (34) [ ] 1 ∞ 𝑏 𝑎 𝑛 343 4 343 𝛾 𝐸 − , ∞( ) = (29) 4 for the two unequal segments ( 1 , 9 ) with the spacing 15 . By scaling 7 7 7 where E is a union of closed segments in ℝ , a monic polynomial f ( z ) is through a factor 7, we assume L = 1 and set the origin at the leftmost real for all z ∈ E , with a ≤ f ( z ) ≤ b , and 𝑓( 𝑧 ) ∈ ℝ ( 𝑧 ) of the degree, n , endpoint as shown in Fig. 4 . Based on the pullback formula p ( z ) = z 2, we which means "oscillates n times between a and b on E ". have

By substituting Eq. (28) into Eq. (29), we have the logarithmic capacity, ( ) 1 ( ) 1 2 100 3 1 𝛾 ( 𝐸) = 7 𝐿 = 10 3 𝐿. (35) (( ) ) 1 ( ) 1 ∞ 286 1 3 243 3 343 𝛾 ( 𝐸) = − (−2) = , (30) ∞ 343 4 343 The logarithmic capacity for the three segments (L, 2 L, L) with the spac- for the two unequal segments ( 3 , 15 ) with the spacing 9 . By scaling, 7 7 7 ing 3 L is obtained. Although Rumely’s method can find the logarithmic the logarithmic capacity for the two unequal segments (L, 5 L) with the capacity by scaling, the extension may have difficulty. However, our spacing 3 L can be obtained as present function is more general than that mentioned in Rumely’s book 𝜆 ( ) 1 [4] since we introduce one slack variable, . The comparison between 243 3 7 2 𝛾 ( 𝐸) = 𝐿 = 3 3 𝐿. (31) the Rumely’s function and the present one is shown in Table 1 . We con- ∞ 343 3 sidered the scaling in the formulation at the beginning. We demonstrate We believe that the error in Rumely’s book [4] is a minor one in the final a systematic procedure for the case of two unequal segments (L, 5 L) step of the computation. The logarithmic capacity for the two unequal with the spacing 3 L. By using the present function and setting the lo- segments (L, 5 L) with the spacing 3 L yields the result as shown in Eq. cation of the three lengths (two segments and one spacing) as shown 1 𝛾 𝐸 𝐿 (31), instead of the final result ∞( ) = 3 3 in Rumely’s book [4]. If in Fig. 1(c), the logarithmic capacity can be determined by following

241 J.-T. Chen, S.-R. Kuo and Y.-L. Huang et al. Engineering Analysis with Boundary Elements 120 (2020) 238–245

Fig. 3. The schematic diagram of pullback formula from two unequal segments (L, 5 L) with the spacing 3 L to three segments (L, 2 L, L) with the spacing L.

Fig. 4. The schematic diagram of pullback formula from two unequal segments (L, 9 L) with the spacing 15 L to three segments (L, 2 L, L) with the spacing 3 L.

Table 1 Comparison of the present result and Rumely’s formulas.

∗ 𝜆 = 7 𝐿. 3

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Table 2 Comparison of the present result and Rumely’s formulas in the degenerate scale of ﬁve cases.

∗Denotes inconsistent Rumely’s results in comparison with those of present methods. our procedures. Therefore, it indicates that we can find the logarithmic capacity more flexibly for the three lengths (two segments and one spacing). We can solve not only the examples mentioned in Rumely’s book [4] , but also the examples that Rumely’s formula may not be feasible. Since the degenerate scale and the logarithmic capacity of two unequal segments (L, 5 L) with the spacing 3 L were solved, we used the same technique (pullback formula [3] ) to solve the corresponding three segments (L, 2 L, L) with the spacing L and three segments (L, 2 L, L) with the spacing 3 L as in Rumely’s book [4] . We only show the final result in Table 2 . Similarly, the logarithmic capacity for any value of the five Fig. 5. Boundary element mesh (8 elements). lengths (three segments and two spacing) can be determined once the mapping function is available. When the degenerate scale and the degenerate boundary simultaneously occur, the rank of the influence matrix is seriously deficient. It is worth noting that the rank deficiency may be triggered by both the degener- 4. Numerical results and discussions for double degeneracy ate scale and the degenerate boundary. The drop of the rank order in the influence matrix caused by the degenerate boundary depends on N , Many researchers have investigated these four rank-deficiency prob- where N is the number of boundary elements and the boundary element lems as mentioned in Section 1 due to the singular matrix or ill-posed mesh is shown in Fig. 5 . Once N is chosen, the drop of the rank order model. When the scatter or radiator is shrunk to a thin-break down due to the degenerate boundary is correspondingly confirmed to be N geometry, the fictitious frequency becomes infinite. It seems that the /2. A detailed discussion can be found in [31] . However, the drop of the fictitious frequency disappears. However, it turns out to be the non- rank order for the influence matrix of single-layer potential due to the uniqueness due to the degenerate boundary. For the statics, there exists degenerate scale is only one when the logarithmic capacity is equal to a degenerate (critical) scale resulting in the non-uniqueness solution. one. By using the singular value decomposition as in the previous paper When the boundary is shrunk to a thin-break, a degenerate scale may [32] , the influence matrix [ U ]of single-layer kernel can be expressed by still exist and the critical scale is finite. Since dynamic and static problems cannot occur at the same time, degenerate scale and fictitious fre- 𝑈 𝑇 . [ ]𝑁×𝑁 = [Φ]𝑁×𝑁 [Σ]𝑁×𝑁 [Ψ]𝑁 𝑁 (36) quency, or degenerate scale and spurious eigenvalue, can not appear × simultaneously. Besides, spurious eigenvalue of interior problems and where T denotes the transpose and fictitious frequency of exterior problems cannot occur at the same time. ⎡ 𝜎𝑁 ⋯ ⋯ ⋯ ⋯ 0 ⎤ The possible combination of these four kinds of degenerate problem to ⎢ ⋮ ⋱ ⋮ 0 ⋮ ⎥ ⎢ ⎥ appear together is only the degenerate scale and the degenerate bound- ⋮ ⋮ 𝜎𝑁 0 ⋮ ⋮ [Σ] = ⎢ ∕2+1 ⎥ , ⎢ ⋮ ⋮ 𝜎 ⋮ ⋮ ⎥ ary. Although the BEM/BIEM have been developed more than 40 years, 0 𝑁∕2 ⎢ ⎥ ⋮ ⋮ ⋮ ⋱ ⋮ double-degeneracy problem is still an issue that few researchers noticed. ⎢ 0 ⎥ ⎣ 0 ⋯ ⋯ ⋯ ⋯ 𝜎 ⎦ Since we deal with the rigid line in the present paper, the degenerate 1 𝑁×𝑁 𝜎 ≥ 𝜎 ≥ 𝜎 ≥ 𝜎 ≥ scale and the degenerate boundary occur simultaneously. We call this 𝑁 𝑁−1 𝑁−2 … 1 0 is a diagonal matrix corresponding kind of problem as “double-degeneracy problem ”. to singular values of [ U ],[ Φ] and [ Ψ] are left and right singular

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Fig. 6. The drop of the zero singular value caused by the degenerate scale in the BEM. matrices, respectively. Degenerate boundary results in a series of zero increasing the number of the boundary elements, the drop location of 𝜎 𝜎 𝜎 singular values of 1 , 2 , ..., N /2 and the degenerate scale results in the degenerate scale gradually converges to the exact solution and the 𝜎 only one zero singular value of N /2 + 1 . Since the degenerate scale peak is more obvious. Numerical evidences of BEM are shown in Fig. and the degenerate boundary may appear at the same time, the rank 7 (a), the single-line segment case and (b), the two unequal segments 𝑁 order of the influence matrix is totally reduced to + 1 for the (L, 5 L) with spacing 3 L case. Rumely’s formula, our analytical solution 2 double-degeneracy problem. Figure 6 shows the drop of the singular and the BEM results of the five cases were shown in Table 2 . After value in the BEM of four cases, (a) two equal segments, (b) two unequal comparing these solutions, we find that the formulas in Rumely’s book segments (L, 5 L) with spacing 3 L, (c) three segments (L, 2 L, L) with [4] are incorrect in cases of the two unequal segments (L, 5 L) with the spacing L, and (d) three segments (L, 2 L, L) with the spacing 3 L. By spacing 3 L and the three segments (L, 2 L, L) with the spacing L.

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procedure for the case of two unequal segments (L, 5 L) with the spacing 3 L and the extension to general case is promising.

Declaration of Competing Interest

None.

Acknowledgments

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 and MOST109-2221-E-019-001-MY3 . References

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