On the linkage between influence matrices in the BIEM and BEM to explain the mechanism of degenerate scale for a circular domain

In this paper, we proposed two ways to understand the rank deficiency in the continuous system (boundary integral equation method, BIEM) anddiscretesystem(boundaryelementmethod,BEM)foracircularcase.Theinfinite-dimensionaldegreeoffreedomforthecontinuoussystemcanbereducedtofinite-dimensionalspaceusingthegeneralizedFouriercoordinates.Thepropertyofthesecond-ordertensorfortheinfluencematrixunderdifferentobserversisalsoexamined.Ontheotherhand,thediscretesystemintheBEMcanbeanalyticallystudied,thankstothespectralpropertyofcirculantmatrix.Weadoptthecirculantmatrixofodddimension,(2 N + 1) by (2 N + 1), instead of the previous even one of 2 N by 2 N to connect the continuous system by using the Fourier bases. Finally, the linkage of influence matrix in the continuous system (BIE) and discrete system (BEM) is constructed. The equivalence of the influence matrix derived by using the generalized coordinates and the circulant matrix are proved by using the eigen systems (eigenvalue and eigenvector). The mechanism of degenerate scale for a circular domain can be analytically explained in the discrete system.


INTRODUCTION
The boundary element method (BEM) or the boundary integral equation method (BIEM) is efficient and accurate to solve the two-dimensional (2D) Laplace equation. However, it may give nonunique solutions by using the software of the BEM for analysis and simulation at the beginning of the engineering design. These nonunique solutions are not physically realizable but stem from the zero eigenvalue embedded in the influence matrix after discretizing the BIE. Therefore, we called it "mathematical degeneracy". Mathematical degeneracy in the BIEM/BEM may appear in four aspects. Here, we focus on degenerate scale [1][2][3][4]. Degenerate scale is related to the Gamma contour [5] or the logarithmic capacity [6,7]. Petrovsky [8] pointed out a very simple circular example to demonstrate the failure of singlelayer representation for the Laplace solution since it disobeys the maximum principle. Jaswon and Symm [5] also pointed out this phenomenon. Chen et al. [9][10][11][12][13][14] conducted a series of research works on the degenerate scale problems for the anti-plane and in-plane elasticity.
Mathematically speaking, only the 2D Laplace equation subject to the Dirichlet boundary condition will result in the degenerate scale by using the BEM. In engineering practice, solving the steady-state heat conduction problem with the specified temperature along the boundary may face the degenerate scale problem. For geomechanics, the settlement problem of the Dirichlet boundary condition may also encounter this degeneracy.
By employing the single-layer representation to solve the 2D Dirichlet problem, the unknown boundary density (constant base) cannot be determined due to the degenerate scale. The solution representation may lose the constant term in the singlelayer integral operator. Adding a rigid body mode in the fundamental solution, ln r + τ [15], is the most direct way to avoid the degenerate scale. Nevertheless, this way just avoids the original degenerate scale but results in another degenerate scale. Based on the objectivity, Hu [16] proposed a necessary and sufficient BIE. Later, Chen et al. [17] proposed a new self-regularization technique by using the bordered matrix and the singular value decomposition. Besides, some regularization techniques to ensure the unique solution, namely hypersingular formulation, rank promotion by adding the boundary flux equilibrium, CHEEF method (direct BEM) and Fichera's method (indirect BEM) were proposed in the literature [15,[17][18][19][20][21][22]. The comparison of these techniques is shown in Table 1. This is an important and popular issue in the BIEM/BEM.

Method Formulation Extra constraint Remark
The self-regularization technique The boundary flux equilibrium The hypersingular formulation In this paper, revisiting of degenerate scale by using the degenerate kernel in BIE [23] and the circulant matrix in the BEM [24] is obtained. Using the degenerate kernel in the BIE, five advantages are obtained: (1) singularity free; (2) boundary layer effect free; (3) exponential convergence; (4) well-posed model; and (5) the null-field point on the real boundary. Golberg [25] pointed out the potential power of degenerate kernel in the BIE as well as BEM, while Galybin [26] solved the crack problems by using the degenerate kernel. Chen and Chiu [27] studied the spectral properties in a discrete system for a circular cavity when a uniform constant element scheme is adopted. The circulant matrices and their properties have been used to solve plane elasticity problems by Wu and Yang [28]. The BIE was turned into an algebraic equation by expanding the excitation and the solution into the Fourier series for the annular domain in [29]. Mathematically speaking, the integral equation is nothing more than the linear algebra [30], once the degenerate kernel is available. Here, we employ the degenerate kernel to examine the mechanism of the rank deficiency in the influence matrix for a circular case. For the 2D Laplace problem subjected to the Dirichlet boundary condition, the rank-deficiency problem may occur by using the BEM or BIEM. No matter the shape of the domain is circular or not, the degenerate scale still exists. The key point is the fundamental solution. The mathematical degeneracy mechanism of integral equation and numerical evidence have been extensively investigated. More numerical examples can be found in [1,3]. The mechanism of degenerate scale can be analytically explained not only in the continuous system but also in the discrete system. Although the mechanism of the degenerate scale has been investigated for decades by using BEM, degenerate kernel and circulant matrix separately, the linkage between these three ways seems not to be deeply investigated. The main focus of this paper is to construct the linkage of three influence matrices derived by BEM, degenerate kernel and circulant matrix.

DERIVATION OF INFLUENCE MATRIX BY USING THE GENER ALIZED FOURIER COORDINATES
First, we give an example to demonstrate the transformation of the influence matrix [U] between the Fourier continuous system of BIEM and the discrete system of BEM. In the indirect BIEM, the field solution can be represented as shown below: where B is the boundary with the enclosing domain D and α(s) is the unknown boundary density. We consider a circle to be an example as shown in Fig. 1, where a is the radius of the circle. The kernel function can be expanded into the degenerate form by using the polar coordinates as shown below: where the field point x = (ρ, φ) and the source point s = (R, θ ). It must be noted that the superscripts of the "i" and "e" kernels are used for the interior and exterior problems, respectively. Convergence of the degenerate kernel in Eq. (2) is discussed in Table 2. For the case of R = ρ, the two series can be proved to be convergent by using the ratio test. For the case of R = ρ and θ = φ, the series converges to lnr since r is not zero (r = 0). When x coincides with s, the series diverges to lnr since r = 0. However, the single-layer potential due to lnr converges after boundary

Positions of s and x Kernel function Ratio test
integration. The Dirichlet boundary condition is given below: The unknown boundary density and the given boundary condition can be expressed as the Fourier series as shown below: a n cos(nθ ) andū (x) = p 0 + ∞ n=1 p n cos(nφ) respectively. By employing the degenerate kernel and the orthogonal property, the integral equation of using Eq. (1) to satisfy Eq. (3) involves nothing more than a linear algebraic system as shown below: ⎡ where N is the number of truncation term of the Fourier series. It is found that the influence matrix may be rank deficient for a = 1. The value a = 1 is the so-called degenerate scale for a circular domain. Equation (6) can be written as a matrix form Column vector {v 0 } = 1 · · · 1 T {v 2 j−1 } = cos( jθ 0 ) cos(jθ 1 ) · · · cos( jθ 2N ) T {v 2 j } = sin( jθ 0 ) sin(jθ 1 ) · · · sin( jθ 2N ) T j = 1, 2, . . . , N where {a} is the vector of Fourier coefficient and {p F } is the forcing vector in terms of Fourier coefficient. For the discrete system in the indirect BEM, the corresponding linear algebraic system of Eq. (1) is where {a} and {p B } are obtained by using the constant boundary element for the boundary density and the vector of boundary data. The influence matrices [U F ] and [U B ] are obtained in the continuous system and the discrete system, respectively. Transformation law of the unknown boundary density in two observer coordinates yields where θ 0 , θ 1 ,…, and θ 2N are the uniformly distributed angles along the circular boundary and 1 cos θ 0 sin θ 0 · · · cos Nθ 0 sin Nθ 0 1 cos θ 1 sin θ 1 · · · cos Nθ 1 sin Nθ . (10) Therefore, the relation between {p B } and {p F } is also the same form: By substituting Eqs (9) and (11) into Eq. (8), and comparing with Eq. (7), we have where in which each element D i of the diagonal matrix [D] is It is noted that Eq. (12) is the transformation law for the influence matrix between the Fourier continuous system and the discrete system of indirect BEM. By using the property of determinant, we have where The lengths of the column vectors in [Q] are all 1 after normalization. Therefore, Eq. (12) can be normalized to The comparison between [Q F ] and [Q] is shown in Table 3. Table 4 shows the transformation law of the matrix and vector between the continuous system and the discrete system. It is found that the influence matrix has the property of second-order tensor.

DERIVATION OF INFLUENCE MATRIX BY USING THE CIRCUL ANT M ATRIX
Now we are going to explain the mechanism of the degenerate scale in the discrete system for a circular domain. Based on the circular symmetry, the influence matrix of the discrete system is found to be circulant with the following form [23,24]: where the influence coefficient in the matrix is defined as shown below: θ n ; a, 0) a θ, n = 0, 1, 2, . . . , 2N, in which θ = 2π/(2N+1) and θ n = n θ. By introducing the following bases for the circulant matrix, where C 2N+ 1 is the cyclic permutation matrix . (22) The jth eigenvalue of [U C ] is given by where α j is the jth eigenvalue of C 2N+ 1 and given by The corresponding eigenvector {ψ j } can be obtained by the definition where By substituting Eqs (20) and (24) into Eq. (23), we have When N approaches infinity, the Riemann sum in Eq. (27) can be transformed to the following integral: Equation (28) indicates when a = 1, λ 0 = 0. That is, det[U C ] = 0. It is proved that the influence matrix [U C ] of the indirect BEM is singular when a is the degenerate scale (a = 1). It is also found that N can be infinite in Eq. (28) to show that it is equivalent to the analytical solution in the infinite-dimensional space.

EQUIVALENCE OF INFLUENCE MATRIX DERIVED BY T WO DIFFERENT METHODS
Last but not least, the mechanism of the degenerate scale for a circle is stated above by two different methods. Here, we intend to prove that the influence matrices of two methods are the same.
Similarly, the eigen equation of circulant matrix [U C ] is where [ ] is constructed by 2N + 1 linearly independent eigenvectors of [U C ], and is given by In addition, it is interesting to find that Therefore, Eq. (30) can be expressed as Equations (29)   we have It is interesting that the relation of eigenvectors between [Q] and [˜ ] can be linked by using characteristics of double roots of eigenvalues. The properties of the circulant matrices can be found in [31,32]. Hence, we have The equivalence of the influence matrices is proved through the normalization of corresponding eigenvectors.

CONCLUSIONS
The mechanism of degenerate scale of solving a 2D Laplace problem by using the BIEM/BEM was analytically derived for a simple circular domain from two ways. One is the generalized coordinates by using the degenerate kernel and Fourier bases in the continuous system. The other is adopting the constant element scheme in conjunction with the circulant matrix in the discrete system. It is interesting that the influence matrices constructed by using the two proposed ways have a linkage. Their equivalence and the transformation law of the secondorder tensor are found through the normalization of corresponding eigenvectors. Therefore, the rank deficiency of the influence matrix is clearly examined for the case of degenerate scale for a unit circle from both the continuous and discrete systems. The linkage of the three influence matrices is summarized in Fig. 2.

ACKNOWLEDGMENT
Financial support from the Ministry of Science and Technology, Taiwan, under grant no. MOST-106-2221-E-019-009-MY3 for the National Taiwan Ocean University is gratefully acknowledged.