Applied Mathematical Modelling 37 (2013) 1638–1651

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Applied Mathematical Modelling

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Infinite domain potential problems by a new formulation of singularboundary method

Yan Gu 1, Wen Chen ⇑Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 210098, PR ChinaState Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, PR China

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

Article history:Received 12 September 2011Received in revised form 31 March 2012Accepted 21 April 2012Available online 28 April 2012

Keywords:Infinite domainMeshless boundary collocation methodSingular boundary methodFundamental solutionRegularization technique

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.04.021

⇑ Corresponding author. Address: Department ofNanjing, Jiangsu 210098, PR China. Tel.: +86 025 83

E-mail addresses: guyan1913@163.com (Y. Gu),1 Tel.: +86 15261889263.

This study proposes a new formulation of singular boundary method (SBM) and documentsthe first attempt to apply this new method to infinite domain potential problems. Theessential issue in the SBM-based methods is to evaluate the origin intensity factor. Thispaper derives a new regularization technique to evaluate the origin intensity factor onthe Neumann boundary condition without the need of sample solution and nodes as inthe traditional SBM. We also modify the inverse interpolation technique in the traditionalSBM to get rid of the perplexing sample nodes in the calculation of the origin intensity fac-tor on the Dirichlet boundary condition. It is noted that this new SBM retains all merits ofthe traditional SBM being truly meshless, free of integration, mathematically simple, andeasy-to-program without the requirement of a fictitious boundary as in the method of fun-damental solutions (MFS). We examine the new SBM by the four benchmark infinitedomain problems to verify its applicability, stability, and accuracy.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Infinite domain problems [1–3] are frequently encountered in a broad range of scientific and engineering applications,such as acoustics, aerodynamics, geophysics and meteorology, just to mention a few. Most popular numerical methods suchas the finite element method (FEM) and the finite volume method (FVM) need to truncate infinite domain into an artificialfinite region with subtle artificial boundary conditions [4] or absorbing layers [5,6]. This truncation can be somewhat arbi-trary largely based on various trial–error or empirical approaches. As an alternative approach, the boundary element method(BEM) [7,8] has long been touted to avoid such drawbacks because it applies the fundamental solution, which satisfies thegoverning equation and the boundary condition at infinity, to naturally reduce an infinite domain problem into a finiteboundary problem. As the price paid for such a merit, the standard BEM formulation, however, has to evaluate weakly sin-gular, strongly singular, or hyper-singular integrals over boundary segments, which is usually a cumbersome and non-trivialtask [9]. Moreover, both the standard FEM and BEM schemes share a troublesome problem of mesh generation which is oftencomputationally expensive and mathematically difficult for some complex problems such as complex-shaped or movingboundary problems.

In order to alleviate some of these difficulties, recent decades have witnessed a fast development of meshless/meshfreenumerical techniques [10–13]. Generally, these meshless methods can be divided into the domain-type or boundary-type

. All rights reserved.

Engineering Mechanics, College of Mechanics and Materials, Hohai University, No. 1 Xikang Road,78 6873; fax: +86 025 8373 6860.chenwen@hhu.edu.cn (W. Chen).

Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651 1639

techniques, depending on whether their basis functions satisfy the governing equation of interest. The method of fundamen-tal solutions (MFS) is one of the collocation based boundary-type meshless methods, which was first proposed by Kupradzeand Aleksidze [14] in 1964 but has been revitalized during the past decades and attracts quickly increasing attention [15–17]. It does not require numerical integration and is ‘‘truly meshless’’ in the sense that costly domain or boundary mesh isnot necessary. Hence the method is extremely simple to implement. In order to avoid the singularity of fundamental solu-tions with a strong-form collocation formulation, the MFS, however, places the source points on a fictitious boundary outsideor inside the physical domain, respectively, corresponding to interior or exterior problems. The determination of the distancebetween the physical and the fictitious boundaries is largely based on experiences and presents the most serious drawbackin the MFS applications to the real-world problems [18–20].

In recent years, tremendous efforts have been made aiming to remove this barrier in the traditional MFS, so that thesource points can be placed on the real boundary directly. Generally, these proposed methods can be divided into the fol-lowing two categories. The first kind of algorithms, which originate from the regularization techniques for the evaluationof the singular integrals in the BEM-based method, indirectly calculate the singular kernel functions by establishing new reg-ularized boundary collocation equation, such as the regularized meshless method (RMM) [21,22], the modified method offundamental solution (MMFS) [23], the singular boundary method (SBM) [24–26], and the boundary distributed source(BDS) method [27]. On the other hand, the second kind of algorithms are developed to avoid the use of the singular basisfunctions such as the boundary knot method (BKM) [28,29], the boundary collocation method (BCM) [30], and the colloca-tion Trefftz method (CTM) [31].

Each of the above-mentioned methods has its own merits and drawbacks. The BKM employs the non-singular generalsolutions instead of using the singular fundamental solutions to avoid the singularity of the discretized matrix. However,similar to the MFS, the condition number of its discretization matrix worsens quickly with an increasing number of boundarynodes. This method is also mostly applied to interior Helmholtz and diffusion problems because the non-singular generalsolution is not available in some cases, such as the infinite domain problem. The kernel functions in the RMM are double-layer potentials and lead to troublesome hyper-singular kernels at the origin and jeopardize its overall accuracy. In contrastto the RMM, the MMFS simply uses the single-layer potential as its kernel function and determines the diagonal terms by theintegration of the fundamental solution on line segments formed by using neighboring points and calculate the diagonalcoefficients of the derivatives of the fundamental solution via a constant solution of the problem of interest. This approachis stable but less accurate and computationally more expensive. In the BDS method, the singular fundamental solution isintegrated over small regions covering the source points to evaluate the diagonal elements of the discretized matrix. How-ever, the diagonal coefficients on the Neumann boundary condition have to be indirectly determined, and this method is stillimmature and requires further development.

In this study, we focus on the singular boundary method (SBM), recently proposed by Chen and his collaborators [24–26].The key idea of this method is to introduce the concept of the origin intensity factor to isolate the singularity of the funda-mental solutions, and then an inverse interpolation technique is developed to determine the origin intensity factors fromboth the fundamental solution and its derivative. Consequently, the source points coincide with the collocation points di-rectly on the real boundary. This clearly differs from the traditional MFS which requires a fictitious boundary to place thesource points. This approach is mathematically simple, easy-to-program and truly meshless but the inverse interpolationtechnique requires the placement of a cluster of sample nodes inside or outside the physical domain for either interior orexterior problems. Our numerical experiments indicate that the overall accuracy of this SBM formulation may be, to a certaindegree, sensitive to the location of such sample nodes.

To remedy the above-mentioned drawback, this paper presents an improved SBM (ISBM) formulation and documents thefirst attempt to apply this new method to infinite domain potential problems. We derive a new regularization technique toevaluate the origin intensity factor on the Neumann boundary condition without the need of sample solution and nodes as inthe traditional SBM. We also modify the inverse interpolation technique in the traditional SBM to get rid of the perplexingsample nodes in the calculation of the origin intensity factor on the Dirichlet boundary condition. It is noted that the pro-posed ISBM remedies the major drawbacks of the traditional SBM, while maintaining its merits being truly meshless, freeof integration, mathematically simple, and easy-to-program.

A brief outline of the rest of this paper is as follows. The ISBM formulation and its numerical implementation for infinitedomain potential problems are presented in Section 2. In Section 3, the accuracy and stability of the proposed method aretested using four benchmark 2D potential examples, in which the proposed method is compared with the MFS, BEM, RMM,and traditional SBM. Finally, the conclusions and remarks are provided in Section 5.

2. ISBM formulation for infinite domain potential problems

We introduce the proposed ISBM formulation with Laplace equation governing potential problems in a 2D infinite domainXe

r2uðxÞ ¼ 0; x 2 Xe; ð1Þ

subject to the following boundary conditions

1640 Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651

uðxÞ ¼ �uðxÞ; x 2 CD; ð2Þ

qðxÞ ¼ @u@nðxÞ ¼ �qðxÞ; x 2 CN ; ð3Þ

limkxk2!1

uðxÞ ¼ const; ð4Þ

where u is the potential field, C = CD + CN represents the boundary of Xe which we shall assume to be piecewise smooth, ndenotes the outward normal, and the barred quantities indicate the given values on the boundary. In Eq. (4), kxk2 representsthe Euclidean distance, and const is a finite constant. It is noted that the solution u(x) satisfies not only Dirichlet (2) and Neu-mann (3) boundary conditions but also the boundary condition (4) at infinity.

The MFS approximates the solution u(x) and qðxÞ ¼ @uðxÞ@nx

by a linear combination of fundamental solutions with respect todifferent source points sj, in the form:

uðxiÞ ¼XN

j¼1

aju�ðxi; sjÞ; ð5Þ

qðxiÞ ¼ @uðxiÞ@nxi

¼XN

j¼1

aj @u�ðxi; sjÞ@nxi

; ð6Þ

where xi is the ith collocation point, sj is the jth source point, aj denotes the jth unknown coefficient of the distributed sourceat sj, N represents the numbers of source points, and

u�ðxi; sjÞ ¼ � 12p

ln kxi; sjk2; ð7Þ

is the fundamental solution of the two-dimensional Laplace equation.In the traditional MFS, a fictitious boundary slightly outside the problem domain is required in order to place the

source points fsjgNj¼1 and avoid the singularity of the fundamental solutions. These source points are either pre-assigned

or taken to be part of the unknowns of the problem along with the coefficients fajgNj¼1. In either case, the unknowns

are determined so that the approximations (5) and (6) satisfy, in some sense, the boundary conditions (2) and (3) as wellas possible. Usually, this is done by collocating the boundary conditions at a chosen set of boundary points fxigM

i¼1. In theearly applications of the MFS [20], the locations of the source points were not specified, which leads to a non-linear sys-tem of equations that can be solved using existing non-linear least-squares minimization software for such systems. Thisapproach, however, has attracted limited attention primarily because of its high computational cost and the criticism thatone often transforms a linear boundary value problem to a non-linear discrete problem [32]. In the now more establishedapproach in which the source points are pre-assigned, collocation leads to a linear system of M equations in N unknownswhich can be solved by a least-squares solver. It should be noted that in case the locations of the source points are pre-assigned, they are usually located on a fictitious boundary, preferably taken to be a circle or a curve similar to the realboundary, see e.g. Fig. 1(a). However, despite many years of focused research, the determination of the distance betweenthe physical boundary and the fictitious boundary is based on experience and therefore troublesome, especially for prob-lems in complicated geometries and higher dimensions. This drawback severely limits the applicability of the MFS to real-world applications.

Similar to the MFS, the SBM also uses the fundamental solution as the kernel function of its approximation. In contrast tothe MFS, the collocation and source points of the SBM are coincident and are placed on the physical boundary without theneed of using a fictitious boundary, as shown in Fig. 1(b). The SBM interpolation formulation for general potential problemscan be expressed as [25]

x

y

o

∞Γ

Fictitious boundary

Source point

x

y

o

∞Γ

Physical boundary

Source point

(b) (a)

Fig. 1. The source point distributions in the MFS and the SBM for exterior problems: (a) the MFS, (b) the SBM.

Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651 1641

uðxiÞ ¼XN

j¼1;i–j

aju�ðxi; sjÞ þ aiuii; xi 2 CD; sj 2 C; ð8Þ

qðxiÞ ¼XN

j¼1;i–j

aj @u�ðxi; sjÞ@nxi

þ aiqii; xi 2 CN ; sj 2 C; ð9Þ

where uii and qii are defined as the origin intensity factor, i.e., the diagonal elements of the SBM interpolation matrix. Thefundamental assumption of the SBM is the existence of the origin intensity factor upon the singularity of the coincidentsource-collocation nodes for mathematically well-posed problems. Our experimental findings [24–26] are that the originintensity factors do exist, and they have a finite value depending on the distribution of discrete boundary nodes and theirrespective boundary conditions.

The traditional SBM employs an inverse interpolation technique to determine the above-mentioned origin intensity fac-tors for both the Dirichlet and Neumann boundary equations. However, in order to carry out this technique, this method hasto place a cluster of sample nodes inside or outside the physical domain for either interior or exterior problems [24,25]. Ournumerical experiments illustrate that the solution accuracy of this SBM formulation is, to a certain degree, sensitive to thelocation of such sample nodes. In the following Sections 2.1 and 2.2, a new SBM formulation will be given to extract out thefinite values of the origin intensity factors, respectively, for Neumann and Dirichlet boundary condition equations withoutusing the troublesome sample nodes.

2.1. Origin intensity factor on Neumann boundary

When the collocation point xi approaches the source point sj, the distance between these two boundary nodes tends tozero which would cause Eqs. (8) and (9) present various orders of singularity. By adopting the subtracting and adding-backtechnique [21,33], we construct a new non-singular boundary collocation equation for Neumann boundary conditions (9) asfollows:

qðxiÞ ¼XN

j¼1

aj @u�ðxi; sjÞ@nxi

¼XN

j¼1

ðaj � aiÞ @u�ðxi; sjÞ@nxi

þ aiXN

j¼1

@u�ðxi; sjÞ@nxi

þ @u�Iðxi; sjÞ@nsj

� �� ai

XN

j¼1

@u�Iðxi; sjÞ@nsj

; ð10Þ

in which

XN

j¼1

@u�Iðxi; sjÞ@nsj

¼ �1L; ð11Þ

and u⁄I(xi,sj) denotes the fundamental solution of the interior finite domain problems, L represents the distance of the jthsource point and the (j + 1)th source point. The detailed derivation of Eq. (11) is given in Appendix A.

According to the dependency of the outward normal vectors on the two kernel functions of interior and exterior prob-lems, we can obtain the following relationships [21]:

@u�ðxi ;sjÞ@n

sj¼ � @u�Iðxi ;sjÞ

@nsj

; i – j;

@u�ðxi ;sjÞ@n

sj¼ @u�Iðxi ;sjÞ

@nsj

; i ¼ j:

8<: ð12Þ

Noting that

@u�ðxi; sjÞ@nxi

þ @u�ðxi; sjÞ@nsj

¼ @u�ðxi; sjÞ@x1

ðn1ðxiÞ � n1ðsjÞÞ þ @u�ðxi; sjÞ@x2

ðn2ðxiÞ � n2ðsjÞÞ; ð13Þ

where the indicial notation for the coordinates of points xi, i.e. (x1,x2), is employed. If the boundary is a straight line, theabove equation is explicitly equal to zero since ni(xi) is equal to ni(sj) at all points. For problems of practical interest witharbitrary smooth geometries, we assume that the source point sj gets increasingly close to the collocation point xi along aline segment, and thus, we have

limsj!xi

@u�ðxi; sjÞ@nxi

þ @u�ðxi; sjÞ@nsj

¼ 0: ð14Þ

With the help of Eqs. (11), (12) and (14), the regularized Neumann boundary Eq. (10) can be rewritten as:

qðxiÞ ¼XN

j¼1;i–j

ðaj � aiÞ @u�ðxi; sjÞ@nxi

þ aiXN

j¼1;i–j

@u�ðxi; sjÞ@nxi

� @u�ðxi; sjÞ@nsj

� �þ ai

L

¼XN

j¼1;i–j

aj @u�ðxi; sjÞ@nxi

� aiXN

j¼1;i–j

@u�ðxi; sjÞ@nsj

þ ai

L¼XN

j¼1;i–j

aj @u�ðxi; sjÞ@nxi

þ ai 1L�XN

j¼1;i–j

@u�ðxi; sjÞ@nsj

!: ð15Þ

1642 Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651

It can be seen from the above equation that the original singular terms @u�ðxi ;sjÞ@nxi

when i = j have been transformed into the fol-lowing regular terms

qii ¼1L�XN

j¼1;i–j

@u�ðxi; sjÞ@nsj

; ð16Þ

which is the aforementioned origin intensity factor qii on the Neumann boundary condition.The matrix form of discretization Eq. (15) can be written as:

fqðxiÞg ¼

1L �XN

j¼2

@u�ðx1 ;sjÞ@n

sj

@u�ðx1 ;s2Þ@nx1

� � � @u�ðx1 ;sN Þ@nx1

@u�ðx2 ;s1Þ@nx2

1L �

XN

j¼1;j–2

@u�ðx2 ;sjÞ@n

sj� � � @u�ðx2 ;sN Þ

@nx2

..

. ... . .

. ...

@u�ðxN ;s1Þ@nxN

@u�ðxN ;s2Þ@nxN

� � � 1L �XN�1

j¼1

@u�ðxN ;sjÞ@n

sj

2666666666666664

3777777777777775

fajg:: ð17Þ

By using the procedure described above, the origin intensity factor on the Neumann boundary condition have been extractedout without using sample solution and any sample nodes.

It should be noted that the RMM also uses a subtracting and adding-back technique to remove the singularities of thediagonal terms of the influence matrix. However, the kernel functions in the RMM are the double-layer potentials in the po-tential theory, because Young and his collaborators [21] consider that the subtracting and adding-back technique will failwith the single-layer potentials. This view is not true as verified in this study.

In addition, it is also noted that the double-layer potentials have both singularity and hyper-singularity at the origin,which lead to troublesome singular kernels and jeopardize the overall accuracy of the RMM. The proposed ISBM formulationavoids the above-mentioned drawback of the RMM and makes a breakthrough in that the subtracting and adding-back tech-nique can be used to single-layer potentials in the strong-form collocation discretization methods. And to the best of ourknowledge, no similar approach has, as yet, been found in the literature.

2.2. Origin intensity factor on Dirichlet boundary

The regularized expressions for the Dirichlet boundary Eq. (8) can be performed by using the strategy proposed by Sarler[23], where the diagonal terms are considered as an average value of the integration of the fundamental solution on line seg-ments formed by using neighboring points. However, the integral calculation makes this strategy more complex and less effi-cient. In this paper, the origin intensity factor on the Dirichlet boundary condition is carried out in a new indirect way,namely an improved inverse interpolation technique (IIT), which is different from the original IIT [25,26] in that it doesnot require the sample nodes.

First, let us assume a pure Neumann problem with all the boundary values set as �qðxÞ ¼ @�uðxÞ@nx

, where �uðxÞ, to be named assample solution in this paper, is an arbitrary known particular solution, such as

�uðxÞ ¼ x1 þ x2

x21 þ x2

2

; ð18Þ

for Laplace infinite problems.Then, substituting this sample solution into the regularized Neumann boundary Eq. (15), we have

�qðxiÞ ¼XN

j¼1;i–j

bj @u�ðxi; sjÞ@nxi

þ bi 1L�XN

j¼1;i–j

@u�ðxi; sjÞ@nsj

!; i ¼ 1;2; . . . ;N; : ð19Þ

where fbjgNj¼1 are unknown coefficients and can be calculated directly by solving the above equation.

Finally, substituting the calculated coefficients fbjgNj¼1 into the Dirichlet boundary Eq. (8), we get the following algebraic

equations:

�uðxiÞ ¼XN

j¼1;i–j

bju�ðxi; sjÞ þ biuii þ c; i ¼ 1;2; . . . ;N; ð20Þ

where c is a constant which can be solved by using an arbitrary field point inside the domain. It is noted that only the originintensity factors uii are unknown in the above equation. Thus, the origin intensity factor uii can be calculated as

uii ¼1bi

�uðxiÞ �XN

j¼1;i–j

bju�ðxi; sjÞ � c

" #; i ¼ 1;2; . . . ;N: ð21Þ

Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651 1643

It is noted that the traditional SBM needs to place a cluster of sample nodes inside the domain to indirectly evaluate the un-known coefficient bj. In contrast, the proposed new method directly uses the regularized Neumann boundary Eq. (15) to cal-culate these unknown coefficients bj without a need of sampling nodes on the Dirichlet boundary.

It is also stressed that the origin intensity factor only depends on the distribution of the source points, and therefore, theorigin intensity factor is independent of the sample solution (18) used in the inverse interpolation technique.

It should be noted that, for Dirichlet boundary conditions, the proposed ISBM has to solve two linear N � N algebraic sys-tems (19) and (8) to compute b and a coefficients, respectively. It differs from the standard SBM in that it does not requireperplexing sample nodes in the calculation of origin intensity factors. Thus, the ISBM may be computationally slightly expen-sive that the MFS. But nevertheless the origin intensity factors for Dirichlet boundary equations can be directly determined inconjunction with the strategy proposed by Sarler [23]. Consequently, the computational cost of the proposed ISBM would befurther improved.

Using the procedure described above, the origin intensity factors for both the Neumann and Dirichlet boundary equationshave been extracted out. For the mixed-type boundary problems, a linear combination of Neumann and Dirichlet boundaryequations can be made to satisfy the mixed-type boundary conditions.

Finally, it is worth noting that like the MFS and the indirect BEM, we always add a constant term to the original SBM for-mulation to guarantee the uniqueness of the interpolation matrix. Our new formulation with a constant term is given by:

uðxiÞ ¼XN

j¼1;i–j

aju�ðxi; sjÞ þ aiuii þ C; ð22Þ

with the constraint satisfies

XN

j¼1

aj ¼ 0: ð23Þ

3. Numerical results and discussions

In this section, four benchmark 2D potential examples are presented to verify the method developed above. The effect ofthe proposed regularization technique, the stability, and the convergence with respect to the number of boundary nodes arecarefully investigated. The numerical results obtained using the traditional SBM, MFS, BEM, and RMM are also given for thepurpose of comparison. Unless otherwise specified, the fictitious boundaries in the MFS are taken to be a curve similar to thereal boundary. To measure the accuracy of the numerical solution, the relative errors defined as below is employed

Relative Error ¼ 1W

XWk¼1

Iknumerical � Ik

exact

Ikexact

!224

35

1=2

; ð24Þ

where W represents the numbers of calculated points, Iknumerical and Ik

exact denote the numerical and analytical solutions at thekth calculated point, respectively.

3.1. A circular domain with Dirichlet boundary condition

First, we investigate a circular domain with the Dirichlet discontinuous boundary conditions given as follows:

�uð1; hÞ ¼1; 0 < h < p�1; p < h < 2p

�: ð25Þ

The problem sketch and the configuration of the nodes distribution of the SBM are depicted in Fig. 2. In this case, an analyt-ical solution is available as follows:

x

y

o

1r =

∞Γ

Fig. 2. A circle domain with Dirichlet boundary condition.

Fig.

1644 Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651

uðx; yÞ ¼ 2p

arctan2y

x2 þ y2 � 1

� �; ð26Þ

@uðx; yÞ@x

¼ � 2p

4xy

4y2 þ ðx2 þ y2 � 1Þ2: ð27Þ

To investigate the convergence rate of relative error versus the mesh size, the number of boundary nodes N used here variesfrom 100 to 1000. It should be noted that the numerical results in this case are strongly dependent on the placement of theboundary nodes surrounding discontinuous boundary condition point. If nodes are placed near the discontinuities(h = 0,p,2p) the errors become much larger. Thus, the boundary nodes adopted here are as far as possible from discontinu-ities, i.e., hi = Dh/2 + (i � 1)Dh in which Dh = 2p/N and hi denotes the polar coordinate of the boundary nodes. Relative error isperformed using W = 200 calculation points uniformly distributed on a circle with radius r = 2 and center at the origin. Thelocation of the calculation points is specified as hc

i ¼ Dhc=2þ ði� 1ÞDhc in which Dhc = 2p/W and hci denotes the polar coor-

dinate of the calculation points.The accuracy of numerical temperatures u(x) is illustrated in Fig. 3 via relative error versus the number of boundary nodes

in the log–log scale. The numerical results obtained by using the traditional SBM, RMM, BEM, and MFS are also provided forthe purpose of a fair comparison. The BEM solutions are obtained using the indirect boundary integral equations with dis-continuous quadratic elements. It is worth noting that in most engineering applications, the quadratic element is ideal be-cause it can approximate the geometry of curvilinear boundaries with sufficient accuracy. For the MFS simulation, thedistance d of the fictitious boundary away from the physical boundary is quantified by the unit s which is the interval spacingof nodes on the physical boundary. It is stressed that the number of boundary nodes is the same in the SBM, RMM, BEM, andMFS.

It can be observed in Fig. 3 that the proposed ISBM performs as well as the RMM and outperforms the traditional SBM,BEM, and MFS in terms of accuracy, e.g. when the value of N reaches 900, the ISBM is more accurate than the traditional SBMby about two orders of magnitude. It is also interesting to see in this figure that in general, the convergence rate of the testednumerical methods does not lie in a straight line and somewhat varies as the number of boundary nodes increases. It impliesthat the convergence rate is a little sensitive to the boundary nodes number N. This may be because the Dirichlet boundaryconditions in this case are discontinuous when h = 0, p, and 2p. The mean value of convergence rate for ISBM is approxi-mately 2.56, as shown by the dash-dot line in Fig. 3, namely, O(N�2.56). Generally, the convergence rate is approximately1.66 for traditional SBM, 2.67 for RMM, 2.79 for BEM, 0.97 for MFS (d = 0.5s), and 0.95 for MFS (d = s).

It is found that the solution accuracy of the MFS with the fictitious boundary d = 0.5s dramatically deteriorates comparedwith that of d = s. This clearly illustrates the essential role of the fictitious boundary in the determining of the MFS solutionaccuracy. Although an appropriate or optimal placement of the fictitious boundary in the MFS can result in a very accuratesolution, it remains an open issue to find this appropriate fictitious boundary for the real-word problems.

The results of the fluxes @u/@x at the field points are listed in Table 1. As demonstrated in this table, the proposed ISBMhas the fast convergence rate than all the other tested numerical methods.

100 200 300 400 600 800 1,00010

-6

10-5

10-4

10-3

10-2

Improved SBM

Traditional SBM

RMM

BEM

MFS (d=0.5s)

MFS (d=s)

Number of boundary nodes

Rel

ativ

e er

ror

slope=2.56

3. Convergence rate of computed temperature for a circle domain generated using the improved and traditional SBMs, RMM, BEM, and MFS.

Table 1Relative errors of fluxes @u/@x at interior points along the circle r = 2 with 0 6 h 6 2p.

N ISBM SMB RMM BEM MFS (d = 0.5s) MFS (d = s)

100 2.375E�3 5.138E�3 2.402E�3 3.368E�3 3.661E�3 4.512E�4300 3.266E�4 6.063E�4 3.431E�4 4.388E�4 1.304E�3 5.639E�5500 2.198E�5 3.254E�4 2.874E�5 4.332E�5 7.937E�4 4.258E�5700 1.033E�5 1.423E�4 1.211E�5 2.397E�5 5.703E�4 3.384E�5900 6.726E�6 5.298E�5 7.328E�6 9.170E�6 4.450E�4 2.790E�5

Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651 1645

3.2. A square domain with mixed boundary conditions

As a second problem in this section, a square domain (2 � 2) subject to the mixed-type boundary conditions is considered.The problem sketch and the boundary conditions are depicted in Fig. 4. The mixed boundary conditions are imposed on thefour edges with the following analytical solution:

uðx; yÞ ¼ 2xþ yx2 þ y2 þ 1; ð28Þ

@uðx; yÞ@x

¼ 2x2 þ y2 �

2xð2xþ yÞðx2 þ y2Þ2

: ð29Þ

For the numerical implementation, the number of boundary nodes N used here varies from 100 to 700. To show the numer-ical results, W = 200 evenly distributed calculation points are selected along the rectangle edge (5 � 5). It is noted that theBEM solutions are obtained using the indirect boundary integral equations with linear discontinuous boundary elementsince it can represent a straight line boundary exactly.

The solutions of computed temperatures are illustrated in Fig. 5 in the form of relative error versus the number of bound-ary nodes in the log–log scale. As shown in this figure, the proposed ISBM agrees pretty well with the BEM and yields moreaccurate results than the traditional SBM and the RMM. In fact, when using 700 boundary nodes, the ISBM is more accuratethan the traditional SBM and the RMM by about one order of magnitude. The mean value of convergence rate of relative errorversus the value of N is around 2 for ISBM, 1.41 for traditional SBM, 1.04 for RMM, 1.96 for BEM, and 4.31 for MFS. In com-parison, the proposed ISBM has faster convergence rate than the traditional SBM, RMM, and BEM. Although the solutionaccuracy and convergence rate of the BEM and the proposed ISBM are very close, it is worth noting that the proposed methodis inherently free of mesh and integration, and thus, is far more computationally efficient, easier to program, and mathemat-ically simple than the BEM. In comparison with the MFS and the traditional SBM, the present ISBM converges monotonicallywith increasing number of boundary nodes without any oscillation, indicating that it works stably with mixed boundary con-ditions. As in the previous case, it is noted that the MFS can obtain very accurate solution if the artificial boundary is carefullychosen. But in many real-world applications, the placement of an appropriate artificial boundary is a difficult task.

The results of the fluxes @u/@x at the field points are listed in Table 2. It can be observed in this table that the proposedISBM produces accurate solutions in comparison with the other numerical methods. Again, although the MFS with d = 0.05yields more accurate solution in this case than the ISBM, the fictitious boundary severely damages its applicability to thereal-world problems.

To study the stability, Fig. 6 shows the condition numbers of interpolation matrix of tested numerical methods. From thenumerical results, we can observe that the RMM has the largest condition numbers which imply the worst stability. Whenthe number of boundary nodes increases, the condition number of the MFS increases sharply leading the resulting matrixequation arising from this method to be severely ill-conditioned. In addition, the condition numbers of the MFS withd = 0.05 grow far more rapidly than those of the MFS with d = 0.01, indicating that the stability rapidly deteriorates as the

∞Γ

x

y

o

2

21

1

yu

y

+= ++

2

21

1

yu

y

−= ++

2

2 2

4 1

( 1)

x xq

x

− + +=+

2

2 2

4 1

( 1)

x xq

x

+ −=+

Fig. 4. A square domain with mixed boundary conditions.

100 200 300 400 500 600 70010

-7

10-6

10-5

10-4

10-3

10-2

10-1

Improved SBM

Traditional SBM

RMM

BEM

MFS (d=0.02)

Number of boundary nodes

Rel

ativ

e er

ror

slope=2

Fig. 5. Convergence rate of the computed temperature for square domain generated using the improved and traditional SBMs, RMM, BEM, and MFS.

Table 2Relative errors of fluxes @u/@x at interior points along the rectangle (5 � 5).

N ISBM SMB RMM BEM MFS (d = 0.01) MFS (d = 0.05)

100 5.581E�4 2.506E�3 2.636E�3 1.175E�3 2.546E�1 2.027E�3200 2.200E�4 5.825E�4 1.275E�3 1.306E�4 1.406E�1 1.079E�6300 8.975E�5 4.433E�4 8.385E�4 1.307E�4 6.192E�2 2.675E�7400 5.029E�5 3.333E�4 6.238E�4 9.819E�5 2.127E�2 2.379E�8500 2.951E�5 3.270E�4 4.964E�4 7.369E�5 6.197E�3 2.627E�7600 1.512E�5 2.306E�4 4.121E�4 5.412E�5 1.591E�3 1.067E�8700 9.374E�6 3.702E�4 3.522E�4 4.311E�5 3.574E�4 2.201E�9

20 60 100 140 180 220 260 30010

0

105

1010

1015

1020

Improved SBM

Traditional SBM

RMM

MFS (d=0.01)

MFS (d=0.05)

Number of boundary nodes

Con

ditio

n nu

mbe

rs

Fig. 6. The condition numbers of interpolation matrix.

1646 Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651

fictitious boundary moves away from the physical boundary. For the solution of tested problems with exact data, this doesnot pose great challenges. However, in real-world applications, small errors in the measured data may lead to large errors in

-5 -4 -3 -2 -1 0 1 2 3 4 5

-6

-4

-2

0

2

4

6

Fig. 7. An epitrochoid-shape domain with Dirichlet boundary condition.

100 200 300 400 600 800 1,00010

-10

10-8

10-6

10-4

10-2

Improved SBM

Traditional SBM

BEM

MFS (d=0.1)

MFS (d=0.02)

Number of boundary nodes

Rel

ativ

e er

ror

slope=1.91

Fig. 8. Convergence rate of the temperature for epitrochoid-shape domain generated using the improved and traditional SBMs, BEM, and MFS.

Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651 1647

the solution due to the high instability. The condition number of the traditional SBM also increases sharply with increasingnumber of boundary nodes. In a sharp contrast, the condition number of the proposed ISBM interpolation matrix grows veryslowly with the number of boundary nodes and remains moderate and is significantly smaller than other tested methods,indicating that the proposed method can reduce ill-conditioning of the MFS and traditional SBM considerably.

3.3. Dirichlet problem in epitrochoid-shape domain

Here, we consider an epitrochoid-like domain [34] defined by

qðhÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaþ bÞ2 þ 1� 2ðaþ bÞ cosðah=bÞ

q; ð30Þ

xðhÞ ¼ q cosðhÞ; yðhÞ ¼ q sinðhÞ; ð31Þ

with a = 4 and b = 1, which is plotted in Fig. 7. The exact solution of this case is

100 200 300 400 600 800 1,00010

-10

10-8

10-6

10-4

10-2

100

Improved SBM

Traditional SBM

BEM

MFS (d=0.1)

MFS (d=0.02)

Number of boundary nodes

Rel

ativ

e er

ror

slope=1.97

Fig. 9. Convergence rate of the computed flux for epitrochoid-shape domain generated using the improved and traditional SBMs, BEM, and MFS.

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-3

-2

-1

0

1

2

3

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

(a) (b)

Fig. 10. Arbitrary domain cases: (a) armor-unit shape domain, (b) gear wheel shape domain.

1648 Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651

u ¼ ex

x2þy2 cosy

x2 þ y2

� �; ð32Þ

@u@x¼ y2 � x2

ðx2 þ y2Þ2e

xx2þy2 cos

yx2 þ y2

� �þ 2xy

ðx2 þ y2Þ2e

xx2þy2 sin

yx2 þ y2

� �: ð33Þ

The number of boundary nodes used here varies from 100 to 1000 and the total number of calculated field points along thecircle with radius r = 7 and center at the origin is set to W = 200. To investigate the convergence rate of the tested methods,the solution accuracy of temperatures u(x) and heat fluxes @u/@x is illustrated in Figs. 8 and 9 via relative error versus thenumber of boundary nodes in the log–log scale. As in the first example, the BEM solutions are obtained using the indirectboundary integral equations with discontinuous quadratic elements.

It can be seen from Figs. 8 and 9 that the errors of the ISBM and BEM are the two parallel straight lines with an errorestimated of O(N�1.91) for temperatures and O(N�1.97) for fluxes, as shown by dash-dot lines in Figs. 8 and 9. Moreover,we note that the traditional SBM is not stable and does not have a monotonic convergence curve. In sharp contrast to thetraditional SBM, the ISBM is of stably monotonic convergence with an increasing number of boundary nodes. In the MFS,

Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651 1649

the solution accuracy with the fictitious boundary d = 0.02 deteriorates, while the MFS with d = 0.1 fictitious boundary getsthe most fast convergence rate in this case. Again, this clearly illustrates the essential role of the fictitious boundary in deter-mining the MFS solution accuracy.

3.4. Arbitrary domain cases

Finally, we consider two more complex-shaped exterior Dirichlet problems with armor-unit and gear wheel shapes. Theirregular shapes, as shown in Fig. 10, are defined by

Fig. 11

Fig. 12.

qðhÞ ¼ 1n2 ½n

2 þ 2nþ 2� 2ðnþ 1Þ cosðnhÞ�; ð34Þ

xðhÞ ¼ qðhÞ cos h; yðhÞ ¼ qðhÞ sin h; ð35Þ

100 300 500 700 90010

-6

10-5

10-4

10-3

10-2

Improved SBM

Traditional SBM

MFS (d=0.01)

100 300 500 700 90010

-5

10-4

10-3

10-2

Improved SBM

Traditional SBM

MFS (d=0.01)

Rel

ativ

e er

ror

Rel

ativ

e er

ror

Number of boundary nodes

(a) Temperature results

Number of boundary nodes

(b) Flux results

slope=2.12 slope=2.17

. Convergence rate of the computed temperature and flux for armor-unit domain generated using the improved and traditional SBMs, and MFS.

100 300 500 700 90010

-5

10-4

10-3

10-2

10-1

Improved SBM

Traditional SBM

MFS (d=0.01)

100 300 500 700 90010

-4

10-3

10-2

10-1

Improved SBM

Traditional SBM

MFS (d=0.01)

Rel

ativ

e er

ror

Rel

ativ

e er

ror

Number of boundary nodes

(a) Temperature results

Number of boundary nodes

(b) Flux results

slope=2.05 slope=1.92

Relative error curves of the computed temperature and flux for gear wheel domain generated using the improved and traditional SBMs, and MFS.

1650 Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651

with n = 3 for armor-unit shape (Fig. 10a) and n = 13 for gear wheel shape (Fig. 10b). The exact solution is the same as in theprevious example 3, i.e. functions (32) and its derivative (33), and the corresponding Dirichlet boundary condition canaccordingly be derived by the exact solution. Here the tested points W = 200 are uniformly distributed along the circle withradius r = 5 and center at the origin.

Figs. 11 and 12 depict the numerical errors of the temperature and flux in these two problems, respectively. We can ob-serve that the traditional SBM has an oscillatory convergence curve of relative errors with the increase of the boundary nodenumber. The present ISBM remedies this drawback and performs stably with a fast convergent rate, indicating that the meth-od works well with complicated domain problems. In this example, the ISBM has better accuracy than the MFS. As shown inthe previous examples, the MFS can be used to obtain very accurate solution if the artificial boundary is carefully chosen.Unfortunately, determining the optimal artificial boundary is a difficult task, and therefore, the artificial boundary severelydamages the applicability of the MFS to real-world applications.

4. Conclusions

In this paper, we present a new formulation of singular boundary method and demonstrate the efficiency of the improvedSBM for solving infinite domain potential problems. The origin intensity factors on the Neumann boundary condition havebeen extracted out and accurately evaluated by using a novel regularization technique without the need of sample solutionand nodes as in the traditional SBM. The origin intensity factors on the Dirichlet boundary condition have also been accuratelyevaluated via a modified inverse interpolation technique without the requirement of perplexing sample nodes. The proposedISBM formulation thus circumvents the troublesome sampling points, one of major problems in the traditional SBM formu-lation, while retaining all its merits being truly meshless, integration-free, mathematically simple, and easy-to-program.

Four benchmark numerical examples demonstrate that the new SBM formulation is capable of solving infinite domainpotential problems with reasonably good accuracy and convergence behaviors. In a summary of overall performances ofour four numerical experiments, the proposed ISBM scheme is superior over the BEM, RMM, and MFS in terms of accuracyand stability. Below are some key features of the proposed method:

(1) a boundary discretization technique with no requirement of domain or surface discretization(2) truly meshless and no numerical integration(3) no ‘‘fictitious boundary issue’’ encountered in the MFS-based methods(4) high stability with small condition number(5) quick convergence rate for smooth data and domain(6) mathematically simple and easy-to-program

Overall, it can be concluded that the proposed scheme is computational efficient, robust, accurate, stable, and convergentwith respect to increasing the number of boundary nodes. In comparison with existing methods for solving numerically infi-nite domain potential problems, the proposed scheme could be considered as a competitive alternative.

On the downside, the remaining issues with the ISBM are as follows:

(1) The origin intensity factors on the Dirichlet boundary condition still need to be indirectly determined by the inverseinterpolation technique via sample solution of problem of interest.

(2) Eq. (15) is derived by the subtracting and adding back technique and is based on Eq. (11) which is derived in the fol-lowing Appendix. However, Eq. (11) is only valid if the distance between boundary source nodes in the boundary isconstant. For two-dimensional irregular domain problems, we can use parameter-based CAD technique to get the uni-form boundary nodes. However, this strategy will not work for the three-dimensional irregular domain cases. Novelapproaches are now under intense study along this line and the further results will be reported in a subsequent paper.

Acknowledgements

The work described in this paper was supported by the National Basic Research Program of China (973 Project No.2010CB832702), the National Science Funds for Distinguished Young Scholars of China (11125208), the R& D Special Fundfor Public Welfare Industry Hydrodynamics, (Project No. 201101014), and Jiangsu Province Graduate Students Research andInnovation Plan (No. CXZZ11_0424).

Appendix A. The detail derivation of Eq. (11)

This section presents the detail derivations of Eq. (11) based on the following direct boundary integral equation:

uðxiÞ ¼Z

Cu�Iðxi; sÞ @uðsÞ

@ns� uðsÞ @u�Iðxi; sÞ

@ns

� �dCðsÞ; xi 2 XI ðA:1Þ

Y. Gu, W. Chen / Applied Mathematical Modelling 37 (2013) 1638–1651 1651

where the superscript (I) denotes the interior domain, u⁄I(xi,s) is the fundamental solution of interior potential problems, s isthe source point located on the physical boundary, xi is the field point. Substituting the simple solution u(s) = 1, @u(s)/@nx = 0into the Eq. (A.1), we can obtain the following equation:

Z

C

@u�Iðxi; sÞ@ns

dCðsÞ ¼ �1; xi 2 XI ðA:2Þ

When the field point xi approaches the boundary, we can discretize the Eq. (A.2) using the quadrature rule as follows:

ZC

@u�Iðxi; sÞ@ns

dCðsÞ ¼XN

j¼1

ZCj

@u�Iðxi; sÞ@ns

dCjðsÞ �XN

j¼1

@u�Iðxi; sjÞ@nj

s

Lj ¼ �1; xi 2 @XI ðA:3Þ

where Lj is the distance of the jth source point and the (j + 1)th source point. When the distribution of boundary nodes isuniform, we are able to reduce the (A.3) to the following form:

XN

j¼1

@u�Iðxi; sjÞ@nsj

¼ �1L; xi 2 @XI ðA:4Þ

which is the Eq. (11) in the text of Section 2.

References

[1] S.V. Tsynkov, Numerical solution of problems on unbounded domains. A review, Appl. Numer. Math. 27 (1998) 465–532.[2] G.H. Miller, An iterative boundary potential method for the infinite domain Poisson problem with interior Dirichlet boundaries, J. Comput. Phys. 227

(2008) 7917–7928.[3] M. Cross, T.N. Croft, G. Djambazov, K. Pericleous, Computational modeling of bubbles, droplets and particles in metals reduction and refining, Appl.

Math. Model. 30 (2006) 1445–1458.[4] D. Givoli, Recent advances in the DtN FE Method, Arch. Comput. Methods Eng. 6 (1999) 71–116.[5] F. Medina, R.L. Taylor, Finite element techniques for problems of unbounded domains, Int. J. Numer. Methods Eng. 19 (1983) 1209–1226.[6] L. Dedek, J. Dedkova, J. Valsa, Optimization of perfectly matched layer for Laplace’s equation, IEEE Transactions on Magnetics 38 (2002) 501–504.[7] A.H.D. Cheng, D.T. Cheng, Heritage and early history of the boundary element method, Eng. Anal. Bound. Elem. 29 (2005) 268–302.[8] J.C.F. Telles, C.A. Brebbia, Boundary elements: new developments in elastoplastic analysis, Appl. Math. Model. 5 (1981) 376–382.[9] V. Sládek, J. Sládek, Boundary integral equation method in micropolar elasticity, Appl. Math. Model. 7 (1983) 433–440.

[10] C.S. Chen, G.R. Liu, Preface to: mesh reduction techniques–Part I, Eng. Anal. Bound. Elem. 28 (2004) 423–424.[11] S. Atluri, H. Liu, Z. Han, Meshless local Petrov–Galerkin (MLPG) mixed collocation method for elasticity problems, Comput. Model. Eng. Sci. 4 (2006)

141–152.[12] M.A. Golberg, The method of fundamental solutions for Poisson’s equation, Eng. Anal. Bound. Elem. 16 (1995) 205–213.[13] J.-T. Chen, H.-C. Shieh, J.-J. Tsai, J.-W. Lee, A study on the method of fundamental solutions using an image concept, Appl. Math. Model. 34 (2010) 4253–

4266.[14] V.D. Kupradze, M.A. Aleksidze, The method of functional equations for the approximate solution of certain boundary value problems, USSR Comput.

Math. Math. Phys. 4 (1964) 82–126.[15] A. Karageorghis, Modified methods of fundamental solutions for harmonic and biharmonic problems with boundary singularities, Numer. Methods

Partial. Differential. Equat. 8 (1992) 1–19.[16] G. Fairweather, A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math. 9 (1998) 69–95.[17] C.S. Chen, M.A. Golberg, Y.C. Hon, The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations, Int. J. Numer. Methods

Eng. 43 (1998) 1421–1435.[18] C.S. Chen, H.A. Cho, M.A. Golberg, Some comments on the ill-conditioning of the method of fundamental solutions, Eng. Anal. Bound. Elem. 30 (2006)

405–410.[19] L. Marin, Treatment of singularities in the method of fundamental solutions for two-dimensional Helmholtz-type equations, Appl. Math. Model. 34

(2010) 1615–1633.[20] R.L. Johnston, G. Fairweather, The method of fundamental solutions for problems in potential flow, Appl. Math. Model. 8 (1984) 265–270.[21] D.L. Young, K.H. Chen, C.W. Lee, Novel meshless method for solving the potential problems with arbitrary domain, J. Comput. Phys. 209 (2005) 290–

321.[22] D.L. Young, K.H. Chen, C.W. Lee, Singular meshless method using double layer potentials for exterior acoustics, J. Acoust. Soc. Am. 119 (2005) 96–107.[23] B. Sarler, Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double

layer fundamental solutions, Eng. Anal. Bound. Elem. 33 (2009) 1374–1382.[24] W. Chen, F.Z. Wang, A method of fundamental solutions without fictitious boundary, Eng. Anal. Bound. Elem. 34 (2010) 530–532.[25] W. Chen, Z. Fu, A novel numerical method for infinite domain potential problems, Chin. Sci. Bull. 55 (2010) 1598–1603.[26] Y. Gu, W. Chen, C.Z. Zhang, Singular boundary method for solving plane strain elastostatic problems, Int. J. Solids. Struct. 48 (2011) 2549–2556.[27] Y.J. Liu, A new boundary meshfree method with distributed sources, Eng. Anal. Bound. Elem. 34 (2010) 914–919.[28] W. Chen, M. Tanaka, A meshless, integration-free, and boundary-only RBF technique, Comput. Math. Appl. 43 (2002) 379–391.[29] Y.C. Hon, W. Chen, Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry, Int. J. Numer.

Methods Eng. 56 (2003) 1931–1948.[30] J.T. Chen, M.H. Chang, K.H. Chen, S.R. Lin, The boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional

cavities using radial basis function, J. Sound. Vib. 257 (2002) 667–711.[31] C.S. Liu, A modified Trefftz method for two-dimensional Laplace equation considering the domain’s characteristic length, Comput. Model. Eng. Sci. 21

(2007) 53–65.[32] A. Karageorghis, D. Lesnic, L. Marin, A survey of applications of the MFS to inverse problems, Inverse. Probl. Sci. Eng. 19 (2011) 309–336.[33] W.S. Hwang, L.P. Hung, C.H. Ko, Non-singular boundary integral formulations for plane interior potential problems, Int. J. Numer. Methods Eng. 53

(2002) 1751–1762.[34] C.S. Liu, A highly accurate solver for the mixed-boundary potential problem and singular problem in arbitrary plane domain, Comput. Model. Eng. Sci.

20 (2007) 111–122.