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Novel Meshless Method for Solving the Potential Problems with Arbitrary Domain

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Novel meshless method for solving the potential problems

with arbitrary domain

D.L. Young

*

, K.H. Chen, C.W. Lee

Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei 10617, Taiwan Received 16 August 2004; received in revised form 8 March 2005; accepted 8 March 2005

Available online 23 May 2005

Abstract

In this article, a non-singular and boundary-type meshless method in two dimensions is developed to solve the poten-tial problems. The solution is represented by a distribution of the kernel functions of double layer potenpoten-tials. By using the desingularization technique to regularize the singularity and hypersingularity of the kernel functions, the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined. The main difficulty of the coincidence of the source and collocation points then disappears. By employing the two-point function, the off-diagonal coefficients of influence matrices are easily obtained. The numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the results of exact solution, conven-tional MFS and BEM for the Dirichlet, Neumann and mix-type boundary conditions (BCs) of interior and exterior problems with simple and complicated boundaries. Good agreements with exact solutions are observed.

 2005 Elsevier Inc. All rights reserved.

Keywords: Non-singular; Meshless method; Conventional MFS; Singular fundamental solution; Double layer potential; Desingu-larization technique; Singularity; Hypersingularity; Kernel function; Mixed-type BC; Circulants

1. Introduction

During the last decade, scientific researchers have paid attention to the meshless methods in which the mesh or element is free. The meshless methods are the mesh reduction methods with no meshes require-ments and only boundary nodes are necessary. The mesh reduction techniques possess great progresses to compete with the FVM, FEM and FDM as dominant numerical methods. Because neither domain nor surface meshing is required, the meshless methods have become very attractive for engineers in model

0021-9991/$ - see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2005.03.007

*

Corresponding author. Tel./fax: +886 2 23626114. E-mail address:dlyoung@ntu.edu.tw(D.L. Young).

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creation, and important tools for scientific computing. Several meshless methods have been reported in the literature, for example, the smooth particle hydrodynamics (SPH) method[12], the element-free Galerkin (EFG) method[2], the reproducing kernel particle method (RKPM)[16], the method of fundamental solu-tions (MFS)[6,11,18–21], boundary knot method (BKM)[8,9], boundary collocation method (BCM)[3–7]

and boundary node method (BNM)[22–24]. These methods are truly meshless, since no domain or bound-ary meshes are required.

The MFS has been extensively applied to solve some engineering problems[10,11,18–21]. It is a kind of meshless methods, since only boundary nodes are distributed. A comprehensive review of the MFS was gi-ven by Fairweather and Karageorghis[11]. The solution procedure makes use of the fundamental solutions, which satisfies the governing equation in the interested domain. To avoid the singularity problem, the solu-tion is represented as a set of singular kernels or the single layer potentials on non-physical boundary (fic-titious boundary). The kernel function is composed of two-point function which is one kind of the radial basis functions (RBFs). The independent variable of two-point function depends on point position only. A regular singularity-free formulation was obtained as a result, and achieving an attractive truly boundary-type and mathematically simple meshfree method. However because of the controversial artificial boundary (off-set boundary) outside the physical domain, the MFS has not become a popular numerical method. The meaning of off-set boundary is an auxiliary boundary to offset a distance from the real boundary. In general for real engineering problems especially for a complicated geometry, the off-set boundary distance is diffi-cult to determine. The diagonal coefficients of influence matrices are divergent due to the point collocation when the off-set boundary approaches to the real boundary. Despite its gain in singularity free, the influence matrices become ill-posed matrices when the off-set boundary is far away from the real boundary. It results in an ill-posed problem since the condition number for the influence matrix becomes very large. The loca-tion of source and observaloca-tion points is vital to the accuracy of the soluloca-tion by implementing the conven-tional MFS.

An improved approach called the BKM or BCM was introduced very recently, by Chen and his co-workers[3–7], and Kang and his collaborators[14,15] as well as Chen and his co-workers[8,9]. Instead of using the singular fundamental solutions, the non-singular kernels were employed to evaluate the homo-geneous solution. These methods dealt successfully with many kinds of problems and eliminated the well-known drawback of ambiguous off-set boundary. The major differences in these meshless methods come only from the techniques used for the chosen non-singular kernels RBFs. However, the introduction of non-singular kernels may jeopardize the accuracy of the solutions as comparing with using the singular fun-damental solutions. Another improved method is called the Hybrid boundary node method (Hybrid BNM), which combines the moving least squares (MLS) interpolation scheme with the hybrid displacement variational formulation[23,24]. However, some integration is still needed as far as with the BNM or Hybrid BNM.

In these BKM and BCM references, the methods only worked well in regular geometry with the Dirichlet and Neumann BCs. Even though these methods can locate the source points on the physical boundary and use the non-singular kernels, there still accompanies some difficulty at the ill-posed problems. Therefore, the purpose of this paper is to develop a novel meshless method for solving the potential problems based on the potential theory as well as the desingularization of subtracting and adding-back technique[13,17]to regu-larize the singularity and hypersingularity of the kernel functions. The proposed method is to distribute the observation and source points on the coincident locations of the real boundary even using the singular ker-nels (double layer potentials) instead of non-singular kerker-nels and still maintains the spirit of the MFS. The diagonal terms of the influence matrices can be derived by using the proposed technique. Also, the influence coefficients by numerical methods are compared with analytical solutions by using separable kernels[1]and circulants[4]for the circular domain. Finally, a new program of the novel meshless method is constructed to solve the Laplace problems subject to the Dirichlet, Neumann and mix-type problems. This includes con-tinuous or disconcon-tinuous BCs with the smooth and non-smooth simple and complicated boundaries.

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2. Formulation

Consider a boundary value problem with a potential /(x), which satisfies the Laplace equation as follows:

r2/ðxÞ ¼ 0; x2 D ð1Þ

subject to BCs

/ðxÞ ¼ /; x2 B1; ð2Þ

wðxÞ ¼ w; x2 B2; ð3Þ

where $2is the Laplacian operator, D is the domain of the problem. The boundary conditions are described as following: where wðxÞ ¼o/ðxÞon

x and B1is the essential boundary (Dirichlet boundary) in which the potential

is prescribed as /; B2is the natural boundary (Neumann boundary) in which the normal derivative is

pre-scribed as w; and B1and B2construct the whole boundary of the domain D as well as the outside domain D e

as shown inFig. 1. The real physical problems for the Laplace equation contain potential flow problems, torsion bar problems, Stokes equations of the vorticity transport equations, etc. By employing the RBF technique[8,18], the representation of the solution for interior problem can be approximated in terms of the strengths ajof the singularities sjas

Fig. 1. The source point and observation point distributions and definitions of r, h, q, u by using the conventional MFS and the novel meshless method for the interior and exterior problems: (a) interior problem (MFS), (b) exterior problem (MFS), (c) interior problem (proposed method), (d) exterior problem (proposed method).

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/ðxiÞ ¼X N j¼1 AðiÞðsj; xiÞaj; ð4Þ wðxiÞ ¼X N j¼1 BðiÞðsj; xiÞaj; ð5Þ

where A(i)(sj, xi) is RBF in which the superscript (i) denotes the interior domain, ajare the jth unknown coefficients (strengths of the singularities), sjis jth source point (singularity), xi is ith observation point, N is the numbers of source points and BðiÞðsj; xiÞ ¼oAðiÞðsj;xiÞ

onxi . The coefficientsfa jgN

j¼1 are determined so that

BC is satisfied at the boundary points,fxigN

i¼1. InFig. 1the distributions of source points and observation

points are shown for the interior and exterior problems. The descriptions of the terminology of observation points, source points, field points, collocation points, boundary points, two-point function, off-set bound-ary and strength of singularity can also be found in[4,6].

By collocating N observation points to match with the BCs from Eq.(4)for Dirichlet problems and Eq.

(5)for the Neumann problems, we have the following linear systems of the form a1;1 a1;2    a1;N a2;1 a2;2    a2;N .. . .. . . . . .. . aN ;1 aN ;2    aN ;N 2 6 6 6 6 4 3 7 7 7 7 5 a j f g ¼ A ðiÞf g ¼ /aj n ðiÞo; ð6Þ b1;1 b1;2    b1;N b2;1 b2;2    b2;N .. . .. . . . . .. . bN ;1 bN ;2    bN ;N 2 6 6 6 6 4 3 7 7 7 7 5 a j f g ¼ B ðiÞf g ¼ waj n ðiÞo; ð7Þ where ai;j¼ AðiÞðsj; xiÞ; i; j¼ 1; 2; . . . ; N ; ð8Þ bi;j¼ BðiÞðsj; xiÞ; i; j¼ 1; 2; . . . ; N ð9Þ

For the mixed-type problems, a linear combination of Eqs.(6) and (7) is made to satisfy the mixed-type BCs. After solving the unknown density functionsfajgN

j¼1 with the linear algebraic solver, the solutions

for the interested domain are calculated from the field equations(4) and (5). Similarly for the exterior problems, we have

/ðxiÞ ¼X N j¼1 AðeÞðsj; xiÞaj; ð10Þ wðxiÞ ¼X N j¼1 BðeÞðsj; xiÞaj; ð11Þ

where the superscript of A(e)(sj, xi) denotes the exterior domain. After collocating N observation points with the Dirichlet or Neumann BCs, we obtain

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 a1;1 a1;2    a1;N  a2;1 a2;2    a2;N .. . .. . . . . .. .  aN ;1 aN ;2    aN ;N 2 6 6 6 6 6 4 3 7 7 7 7 7 5 aj f g ¼ A ðeÞf g ¼ /aj n ðeÞo; ð12Þ  b1;1 b1;2    b1;N  b2;1 b2;2    b2;N .. . .. . . . . .. .  bN ;1 bN ;2    bN ;N 2 6 6 6 6 6 4 3 7 7 7 7 7 5 aj f g ¼ B ðeÞf g ¼ waj n ðeÞo: ð13Þ

Similar procedures as the interior problems are undertaken to obtain the field solutions for the exterior problems. According to the dependence of the outward normal vectors in the two kernel functions for inte-rior and exteinte-rior problems, their relationships are

AðiÞðsj; xiÞ ¼ AðeÞðsj; xiÞ; i6¼ j; AðiÞðsj; xiÞ ¼ AðeÞðsj; xiÞ; i¼ j; ( ð14Þ BðiÞðsj; xiÞ ¼ BðeÞðsj; xiÞ; i6¼ j; BðiÞðsj; xiÞ ¼ BðeÞðsj; xiÞ; i¼ j: ( ð15Þ

The chosen RBFs in this study are the double layer potentials in the potential theory and were derived in

Appendix Afor the exterior problems or can be found in[4,5]as

AðeÞðsj; xiÞ ¼nkyk r2 ij ; ð16Þ BðeÞðsj; xiÞ ¼ 2ykylnknl r4 ij nknk  r2 ij ; ð17Þ

where rij ¼ jsj xij, nk is the kth component of the outward normal vector at sj; nk is the kth

compo-nent of the outward normal vector at xi and yk ¼ xik s j

k. The chosen RBF is a kind of the two-point

function.

It is noted that the double layer potentials have both singularity and hypersingularity at the origin, which lead to troublesome singular kernels and controversially auxiliary boundary in the conventional MFS. The off-set distance between the off-set (auxiliary) boundary (B0) and the real boundary (B)

de-fined by, d, as shown in Figs. 1(a) and (b) needs to be chosen deliberately. To overcome the above-mentioned drawback, sj is distributed on the real boundary as shown in Figs. 1(c) and (d) by using the proposed regularization technique. The rationale for choosing double layer potential instead of the single layer potential as used in the proposed method for the form of RBFs is to take advantage of the desingularization of the subtracting and adding-back technique, so that no off-set distance is needed when evaluating the diagonal coefficients of influence matrices as explained in Section 3. The single layer potential will not be chosen as the form of RBFs, because Eqs. (20) and (21) in Section 3 are not satisfied. If the single layer potential is used, the desingularization of subtracting and add-ing-back technique will fail.

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3. Derivation of diagonal coefficients of influence matrices for arbitrary domain

When the collocation point xiapproaches to the source point sj, Eqs.(4) and (5)will become singular.

Eqs.(4) and (5)for the interior problems need to be regularized by using special treatment of the

desingu-larization of subtracting and adding-back technique[13,17]as follows: /ðxiÞ ¼X N j¼1 AðiÞðsj; xiÞajX N j¼1 AðeÞðsj; xiÞai ¼X i1 j¼1 AðiÞðsj; xiÞajþX N j¼iþ1 AðiÞðsj; xiÞajþ X N m¼1 AðiÞðsm; xiÞ  AðiÞðsi; xiÞ " # ai; xi2 B; ð18Þ wðxiÞ ¼X N j¼1 BðiÞðsj; xiÞajX N j¼1 BðeÞðsj; xiÞai ¼X i1 j¼1 BðiÞðsj; xiÞajþX N j¼iþ1 BðiÞðsj; xiÞaj X N m¼1 BðiÞðsm; xiÞ  BðiÞðsi; xiÞ " # ai; xi2 B ð19Þ in which XN j¼1 AðeÞðsj; xiÞ ¼ 0; xi2 B; ð20Þ XN j¼1 BðeÞðsj; xiÞ ¼ 0; xi2 B: ð21Þ

In Appendix A, the detail derivations of Eqs. (20) and (21) are given. The original singular terms of

A(i)(si, xi) and B(i)(si, xi) in Eqs. (4) and (5) have been transformed into regular terms ½PNm¼1AðiÞ ðsm; xiÞ  AðiÞðsi; xiÞ and ½PN

m¼1BðiÞðsm; xiÞ  BðiÞðsi; xiÞ in Eqs. (18) and (19), respectively. In which the

terms of PNm¼1AðiÞðsm; xiÞ and PN

m¼1BðiÞðsm; xiÞ are the adding-back terms and the terms of A (i)

(si, xi) and B(i)(si, xi) are the subtracting terms in the two brackets for the special treatment technique. After using the desingularization of subtracting and adding-back technique[13,17], we are able to remove the singular-ity and hypersingularsingular-ity of the kernel functions. Therefore, the diagonal coefficients for the interior prob-lems can be extracted out as:

Table 1

The properties of the influence matrices for the Laplace equation Kernel function Aðsj; xiÞ ¼ yknk

r2 ij Bðsj; xiÞ ¼2ykylnknl  r4 ij nknk r2 ij

Exterior Interior Exterior Interior

Eigenvalue kl t ðeÞ 0 ¼ 0; t ðeÞ l ¼N2r t ðiÞ 0 ¼Nr; t ðiÞ l ¼2rN d ðeÞ 0 ¼ 0; d ðeÞ l ¼2rNl2 d ðiÞ 0 ¼ 0; d ðiÞ l ¼2rNl2 Analytical solution: 1 N P N1 m¼0

km¼Sum of diagonal termsN ðcircular domain onlyÞ

Diagonal value N1 2r 2prp N Nþ1 2r 2prp N NðN1Þ 4r2  p 2 ð2pr NÞ 2 NðN 1Þ 4r2  p 2 ð2pr N Þ 2 Numerical solution (arbitrary domain)

PN k¼1ai;k ai;i PN k¼1ai;k ai;i ð PN k¼1bi;k bi;iÞ ð PN k¼1bi;k bi;iÞ Where rij¼ jxi sjj, yk¼ xik s j

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1 φ= 0 φ = 2 ( , )x y 0 φ ∇ = y

a=1

1 φ= 2 ( , )x y 0 φ ∇ = x a=1 (a) Source point Collocation point Normal vector -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 (b) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 (c)

Fig. 2. Problem sketch and the nodes distribution (60 nodes) in the case 1.1: (a) problem sketch, (b) nodes distribution, (c) exact solution.

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/i   ¼ PN m¼1 a1;m a1;1 a1;2    a1;N a2;1 P N m¼1 a2;m a2;2    a2;N .. . .. . . . . .. . aN ;1 aN ;2    P N m¼1 aN ;m aN ;N 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 aj f g; ð22Þ (a) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 (b) (c)

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(a) -0.8 -0.4 0 0.4 0.8 0 0.2 0.4 0.6 0.8 1 (x, 0 )

m e shless m eth od (60 nodes) E x act solution B E M (60 elem en ts) M FS ( = .5, 6 0 nodes) Novel -0.8 -0.4 0 0.4 0.8 0 0.2 0.4 0.6 0.8 1 (x, 0 )

m e shless m eth od (60 nodes) m e shless m eth od (60 nodes) E x act solution B E M (60 elem en ts) M FS (d = 0 5 Novel -0.8 -0.4 0 0.4 0.8 0 0.2 0.4 0.6 0.8 1 (x, 0 )

m e shless m eth od (60 nodes) m e shless m eth od (60 nodes) E x act solution B E M (60 elem en ts) M FS ( = .5 Novel -0.8 -0.4 0 0.4 0.8 0 0.2 0.4 0.6 0.8 1 (x, 0 )

m e shless m eth od (60 nodes) m e shless m eth od (60 nodes) E x act solution B E M (60 elem en ts) M FS (d = 0 Novel -0.8 -0.4 0 0.4 0.8 0 0.2 0.4 0.6 0.8 1 (x, 0 )

m e shless m eth od (60 nodes) E x act solu -0.8 -0.4 0 0.4 0.8 0 0.2 0.4 0.6 0.8 1 (x, 0 )

m e shless m eth od (60 nodes) E x act solution B E M (60 elem en ts) M FS ( = .5, 6 0 nodes) Novel -0.8 -0.4 0 0.4 0.8 0 0.2 0.4 0.6 0.8 1 0 )

m e shless m eth od (60 nodes) m e shless m eth od (60 nodes) E x act solution

B E M (60 elem en t

m e shless m eth od (60 nodes) m e shless m eth od (60 nodes) E x act solution B E M (60 elem en ts) M FS (d = 0 5 Novel -0.8 -0.4 0 0.4 0.8 0 0.2 0.4 0.6 0.8 1 0 )

m e shless m eth od (60 nodes) m e shless m eth od (60 nodes) E x act solution

B E M (60 e

m e shless m eth od (60 nodes) m e shless m eth od (60 nodes) E x act solution B E M (60 elem en ts) M FS ( = .5 Novel -0.8 -0.4 0 0.4 0.8 x 0 0.2 0.4 0.6 0.8 1 0 )

m e shless m eth od (60 nodes) m e shless m eth od (60 nodes) E x

m e shless m eth od (60 nodes) m e shless m eth od (60 nodes) E x act solution B E M (60 elem en ts) M FS (d = 0 Novel Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS ( .5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d=0.5 m, 60 nodes).5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS ( .5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS ((d=0.5, 60 nodes).5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS ( .5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d=0.5 m, 60 nodes).5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS ( .5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS ((d=0.5, 60 nodes).5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS ( .5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS ( .5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) Exact solut Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y ) ion BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution

BEM (60 elements) MFS ( .5 m

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS ( .5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y ) ) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution

BEM (60 elements) MFS (d =0.5 m)

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y ) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution

BEM (60 elements) MF

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d=0.5 m, 60 nodes).5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method 0 nodes) meshless method 0 nodes) Exact solution BEM (60 elements) MFS ( .5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method odes) meshless method odes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method odes) meshless method odes) Exact solution BEM (60 elements) MFS ( .5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method odes) Exact solut Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method odes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y ) ion BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method odes) meshless method odes) Exact solution

BEM (60 elements) MFS ( .5 m

meshless method odes) meshless method odes) Exact solution BEM (60 elements) MFS ( .5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y ) ) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method odes) meshless method odes) Exact solution

BEM (60 elements) MFS (d =0.5 m)

meshless method odes) meshless method odes) Exact solution BEM (60 elements) MFS (d =0.5 m) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y ) Novel -0.8 -0.4 0 0.4 0.8 y 0 0.4 0.8 1.2 (0 ,y )

meshless method odes) meshless method odes) Exact solution

BEM (60 elements) MF

meshless method (6 odes) meshless me od odes) Exact solution

BEM (60 elements) MFS ((d=0.5, 60 nodes).5 m)

(b)

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wi   ¼  P N m¼1 b1;m b1;1   b1;2    b1;N b2;1  P N m¼1 b2;m b2;2      b2;N .. . .. . . . . .. . bN ;1 bN ;2     P N m¼1 bN ;m bN ;N   2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 aj f g: ð23Þ

In a similar way, the desingularization of subtracting and adding-back technique was applied to the exterior problems, we then have

/ðxiÞ ¼X N j¼1 AðeÞðsj; xiÞajX N j¼1 AðeÞðsj; xiÞai; ð24Þ wðxiÞ ¼X N j¼1 BðeÞðsj; xiÞajX N j¼1 BðeÞðsj; xiÞai: ð25Þ

After using Eqs. (14) and (15), the diagonal coefficients for the exterior problems can be extracted out as: -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

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0 40 80 120 160 Number of Nodes (N) 1E-006 1E-005 0.0001 0.001 0.01 0 40 80 120 160 Number of Nodes (N) 1E-006 1E-005 0.0001 0.001 0.01 0 40 80 120 160 Number of Nodes (N) 1E-006 1E-005 0.0001 0.001 0.01 (b) (a)

Fig. 6. The error analyses for the case 1.1: (a) relative error with exact solution for entire domain (60 source nodes), (b) norm error (at radius = 0.5) versus the numbers (N) of nodes.

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/i   ¼ PN m¼1 a1;m a1;1 a1;2    a1;N a2;1 P N m¼1 a2;m a2;2    a2;N .. . .. . . . . .. . aN ;1 aN ;2    P N m¼1 aN ;m aN ;N 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 aj f g; ð26Þ 2 ( , )x y 0 φ ∇ = 1 φ= 1 φ= − a=1 x y 2 ( , )x y 0 φ ∇ = 1 φ= 1 φ= − a=1 1 φ= − a=1 x y x y (a) Source point Collocation point Normal vector -1 0 1 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 (b) -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0 .5 -1 .5 0 0.5 1 1.5 (c)

Fig. 7. Problem sketch and the nodes distribution (60 nodes) in the case 1.2: (a) problem sketch, (b) nodes distribution, (c) exact solution.

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wi   ¼  P N m¼1 b1;m b1;1   b1;2    b1;N b2;1  P N m¼1 b2;m b2;2      b2;N .. . .. . . . . .. . bN ;1 bN ;2     P N m¼1 bN ;m bN ;N   2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 aj f g: ð27Þ -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 (a) -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 (b) -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 (c)

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0 0 2 4 6 ϕ -1 -0.5 0 0.5 1 1.5 φ (2 ,)ϕ

Novel meshless method (60 nodes) Exact solution BEM (60 elements) MFS (d = 0.2, 60 nodes) 0 0 0 2 4 6 ϕ -1 -0.5 0 0.5 1 1.5 φ (2 ,)ϕ

Fig. 9. The field solutions, /(2, u), by using the proposed method, BEM and conventional MFS for the case 1.2.

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

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The diagonal terms of the two influence matrices for both interior and exterior problems can also be derived analytically for a circular domain as shown inAppendix B.Table 1shows the properties of the influence matrices for both circular and arbitrary domains.

4. Numerical results

In order to show the accuracy and validity of the proposed method, the potential problems with circular, square and arbitrary domains subject to the Dirichlet, Neumann, and mixed-type problems with continu-ous and discontinucontinu-ous BCs are considered. The results will be compared to the solutions obtained by using the conventional MFS, BEM and exact solutions.

4.1. Example 1: Circular domain cases (cases 1.1 and 1.2)

In cases 1.1 and 1.2, the interior and exterior Dirichlet problems with discontinuous BCs are given. The interior Dirichlet problem is considered in case 1.1. Case 1.2 is the exterior Dirichlet problem. Both figures in the following two cases are the results with 60 source nodes.

Source point Collocation point Normal vector 0 0.5 1 0 0.5 1 (b) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (a) (c)

Fig. 11. Problem sketch and the nodes distribution (120 nodes) in the case 2.1: (a) problem sketch, (b) nodes distribution, (c) exact solution.

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4.1.1. Case 1.1: interior Dirichlet problem

Problem sketch and the nodes distribution employing the proposed method are depicted inFigs. 2(a)

and (b), respectively. The problem is subject to Dirichlet discontinuous BC as follows:

 /ð1; uÞ ¼ 0; 0 < u < p; 1; p <u <2p:  ð28Þ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (a) (b) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (c) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9

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In this case, an analytical solution is found as follows: /ðx; yÞ ¼1 parctan 1 x2 y2 2y   : ð29Þ

The exact field solution is plotted inFig. 2(c). We obtain the results of the field potential solutions by using the conventional MFS for distributing source points (60 nodes) on the fictitious boundary with different

0 0.2 0.4 0.6 0.8 1 (a) (b) x -0.4 -0.2 0 0.2 0.4 (x ,0 .5 )

New meshless method (120 nodes) Exact solution BEM (120 elements) MFS ( 0 0.2 0.4 0.6 0.8 1 x -0.4 -0.2 0 0.2 0.4 (x ,0 .5 )

New meshless method (120 nodes) Exact solution BEM (120 elements) MFS ( 0 0.2 0.4 0.6 0.8 1 -0.4 -0.2 0 0.2 0.4 (x ,0 .5 )

New meshless method (120 nodes) Exact solution BEM (120 elements) MFS (d = 0.5, 120 nodes) x 0 0.2 0.4 0.6 0.8 1 y 0 0.2 0.4 0.6 0.8 1 (0 .5,y )

New meshless method (120 nodes) Exact solution BEM (120 elements) MFS ( 0 0.2 0.4 0.6 0.8 1 y 0 0.2 0.4 0.6 0.8 1 (0 .5,y )

New meshless method (120 nodes) Exact solution BEM (120 elements) MFS ( 0 0.2 0.4 0.6 0.8 1 y 0 0.2 0.4 0.6 0.8 1 (0 .5,y )

New meshless method (120 nodes) Exact solution BEM (120 elements) MFS ( 0 0.2 0.4 0.6 0.8 1 y 0 0.2 0.4 0.6 0.8 1 (0 .5,y )

New meshless method (120 nodes) Exact solution BEM (120 elements) MFS ( 0 0.2 0.4 0.6 0.8 1 y 0 0.2 0.4 0.6 0.8 1 (0 .5,y )

New meshless method (120 nodes) Exact solution BEM (120 elements) MFS(d = 0.5, 120 nodes) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (0 .5,y )

New meshless method (120 nodes) Exact solution

BEM (120 elements) MFS

Fig. 13. The field solutions by using the proposed method, BEM and conventional MFS by adding a rigid body term for the case 2.1: (a) /(x, 0.5), (b) /(0.5, y).

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off-set distances as depicted inFigs. 3(a)–(c). It is obvious that the relative errors of the conventional MFS comparing with the exact solution inFig. 2(c) for d = 0.01 and d = 1.0 are larger than d = 0.5, where d is the distance between the off-set (auxiliary) boundary (B0) and the real boundary (B). This illustrates the

draw-back that the location of source is dubious by using the conventional MFS. The comparisons of results by using the proposed novel method, the conventional MFS (d = 0.5), the BEM, and the analytical solution are displayed inFig. 4(a)for /(x,0) andFig. 4(b) for /(0, y), respectively. The field solution of the present method is plotted inFig. 5. The results match the exact solutions very well by using the present meshless method. To see the sensitivity analysis of the boundary layer effect, the relative error with exact solution in the interested domain field with 800 inner points is given inFig. 6(a).Fig. 6(b) shows the norm error of the numerical results plotted versus number of nodes and displays the changes of norm error at radius = 0.5 with the increase of source nodes N. The norm error is defined as R02pj/exactðq ¼ 0:5; uÞ

/ðq ¼ 0:5; uÞj2du inFig. 6(b).

4.1.2. Case 1.2: exterior Dirichlet problem

In this case, we investigate a circular domain with the Dirichlet discontinuous BC given as follows: 

/ð1; uÞ ¼ 1; 0 < u < p; 1; p <u <2p: 

ð30Þ Problem sketch and the nodes distribution using the proposed method are depicted inFigs. 7(a) and (b), respectively. In this case, an analytical solution is available as following:

/ðx; yÞ ¼2 parctan 2y x2þ y2 1   : ð31Þ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

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The exact field solution is plotted in Fig. 7(c). We obtain the results of the field potential solutions by using the conventional MFS (60 nodes) for different off-set distances to boundary as shown in Figs.

8(a)–(c). The relative errors of the conventional MFS are larger for d = 0.001 and d = 0.5 than

d = 0.2. This clearly illustrates the drawback of the well-known ill-posed influence matrices by using the conventional MFS. The field results, /(2,u) by using the present novel method, the conventional MFS (d = 0.2), the BEM and the analytical solutions are plotted in Fig. 9. In Fig. 10 the field solution of the proposed technique is plotted. The present numerical results are very close to the exact solutions by using the proposed novel method.

4.2. Example 2: square domain cases (cases 2.1 and 2.2)

In cases 2.1 and 2.2, the interior Dirichlet and mixed-type problems are given for the square domain, respectively. The two problems considered here are all with discontinuous BCs.

(a) 0 h= h= m 0 h= h= m

x

y

0 h= h= h=π h=π

x

y

Source point Collocation point Normal vector 0 0.5 1 0 0.5 1 (b) 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 (c) 1

Fig. 15. Problem sketch and the nodes distribution (120 nodes) in the case 2.2: (a) problem sketch, (b) nodes distribution, (c) exact solution.

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4.2.1. Case 2.1: interior Dirichlet problem (discontinuous BC)

A square domain (1· 1) subject to the Dirichlet BC is considered as

/ðx; 0Þ ¼ x; /ðx; 1Þ ¼ /ð0; yÞ ¼ /ð1; yÞ ¼ 0: ð32Þ

Problem sketch and the nodes distribution using the proposed method are depicted inFigs. 11(a) and (b), respectively. An analytical solution is available as follows:

(a) 0 0.5 1 1.5 2 2.5 3 (b) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 (c)

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0 1 2 x 3 0 0.4 0.8 1.2 1.6 2

meshless method (120 nodes) Exact solution BEM (120 elements) MFS (d = .5 , 120 nodes) Novel 0 1 2 x 3 0 0.4 0.8 1.2 1.6 2

meshless method (120 nodes) meshless method (120 nodes) Exact solution BEM (120 elements) MFS (d .5 , 120 nodes) Novel 0 1 2 x 3 0 0.4 0.8 1.2 1.6 2

meshless method (120 nodes) meshless method (120 nodes) Exact solution BEM (120 elements) MFS (d .5 , 120 nodes) Novel 0 1 2 x 3 0 0.4 0.8 1.2 1.6 2

meshless method (120 nodes) meshless method (120 nodes) Exact solution BEM (120 elements) MFS (d 0.5 , 120 nodes) Novel (a) (b) Novel 0 1 2 y 3 0 0.2 0.4 0.6 0.8 1 φ (0.5 π ,y) φ (x , 0.5 π )

meshless method (120 nodes) Exact solution BEM (120 elements) MFS , 120 nodes) Novel 0 1 2 y 3 0 0.2 0.4 0.6 0.8 1

meshless method (120 nodes) meshless method (120 nodes) Exact solution BEM (120 elements) MFS (d = , 120 nodes) Novel 0 1 2 y 3 0 0.2 0.4 0.6 0.8 1

meshless method (120 nodes) meshless method (120 nodes) Exact solution BEM (120 elements) MFS 5, 120 nodes) Novel 0 1 2 y 3 0 0.2 0.4 0.6 0.8 1

meshless method (120 nodes) meshless method (120 nodes) Exact solution BEM (120 elements) MFS (d = , 120 nodes) Novel 0 1 2 y 3 0 0.2 0.4 0.6 0.8 1

meshless method (120 nodes) meshless method (120 nodes) Exact solution BEM (120 elements) MFS , 120 nodes) Novel 0 1 2 y 3 0 0.2 0.4 0.6 0.8 1

meshless method (120 nodes) meshless method (120 nodes) Exact solution BEM (120 elements) MFS (d = , 120 nodes) Novel 0 1 2 y 3 0 0.2 0.4 0.6 0.8 1

meshless method (120 nodes) meshless method (120 nodes) Exact solution BEM (120 elements) MFS , 120 nodes) Novel 0 1 2 y 3 0 0.2 0.4 0.6 0.8 1

meshless method (120 nodes) meshless method (120 nodes) Exact solution

BEM (120 elements) MFS (d = 0. , 120 nodes)

Fig. 17. The field solutions by using the proposed method, BEM and conventional MFS by adding a rigid body term for the case 2.2: (a) /(x, 0.5p), (b) /(0.5p, y).

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/ðx; yÞ ¼X

1

n¼1

Cnsinhðnpð1  yÞÞ sinðnpxÞ; ð33Þ

where Cn¼

2ð1Þnþ1

ðnpÞ sinhðnpÞ: ð34Þ

The exact field solution is plotted in Fig. 11(c). After distributing 120 nodes, we obtain the results by using the conventional MFS for different off-set distances to boundary (d) as depicted in Fig. 12, where d is the off-set distance between the off-set (auxiliary) boundary (B0) and the real boundary (B). It is

obvious that the relative errors of the conventional MFS comparing with the exact solution in Fig. 11(c) for d = 0.1 and d = 1 are larger than d = 0.5. This illustrates the important fact that the location of source is vital to the accuracy of the solution by using the conventional MFS. In such a situation, the conventional MFS does not yield reliable and consistent solutions. The field solutions of / (x, 0.5) and / (0.5, y) by employing the proposed novel method, the conventional MFS (d = 0.5), the BEM and the analytical results are plotted in Figs. 13(a) and (b), respectively. The present method predicts the accu-rate solutions after comparing with the analytical solutions as shown in Fig. 13. Good match is ob-served from the comparison of the two solutions. Thus the selection of the off-set distances in the conventional MFS is avoided by adopting the present study. The field solution by using the proposed method is plotted in Fig. 14.

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3

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4.2.2. Case 2.2: interior mixed-type problem (discontinuous BC)

A square domain (p· p) subject to the mixed-type BC is considered as

/ðp; yÞ ¼ 1; wðx; 0Þ ¼ /ðx; pÞ ¼ wð0; yÞ ¼ 0: ð35Þ

Problem sketch and the nodes distribution using the proposed method are depicted inFigs. 15(a) and (b), respectively. An analytical solution is available as follows:

/ðx; yÞ ¼X 1 n¼1 Dncosh ð2n  1Þx 2   cos ð2n  1Þy 2   ; ð36Þ where Dn¼ 4ð1Þnþ1 ð2n  1Þp coshð2n1Þp2 : ð37Þ The field solution of the exact solution is plotted in Fig. 15(c). By collocating 120 nodes, we derive the results by using the conventional MFS for different values of d as obtained in Fig. 16. It is obvious that the results of the conventional MFS for d = 0.01 and d = 1.0 are larger than d =0.5 after comparing with the exact solution in Fig. 15(c). The results of /(x, 0.5p) and /(0.5p, y) by using the proposed novel meshless method, the conventional MFS (d = 0.5), the BEM and the analytical solutions are plotted

Fig. 19. The error analyses for the case 2.2: (a) relative error with exact solution for entire domain (120 source nodes), (b) norm error at lineðx;p

2Þ, (c) norm error at line ð p 2; yÞ.

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in Figs. 17(a) and (b), respectively. The field solution by using the proposed method is plotted in Fig. 18. To investigate the error analysis, we plot Figs. 19(a)–(c). In Fig. 19(a), the relative error with exact solution in the interested domain with 400 inner points is plotted. MeanwhileFigs. 19(b) and (c) display

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the norm error along line y = p/2 and x = p/2 versus the number N of boundary nodes. The norm errors are defined as R0pj/exactðx; y ¼ p=2Þ  /ðx; y ¼ p=2Þj

2

dx in Fig. 19(b) and R0pj/exactðx ¼ p=2; yÞ

/ðx ¼ p=2; yÞj2dy in Fig. 19(b), respectively. The boundary layer effect is observed in Fig. 6(a) for case 1.1 and Fig. 19(a) for case 2.2. When the observation points are calculated, in the proximity of the inte-rior points to the boundary owing to the singularities of the double layer kernels, the precision would deteriorate quickly. In order to reduce the boundary layer effect to some extent, the remedy may be to refine the local source points. This has been verified from Fig. 6(b) for case 1.1 andFigs. 19(b) and (c) for case 2.2. As the node points are refined the boundary layer effects become less obvious.

Source point Collocation point Normal vector -1.5 -1 -0.5 0 0.5 1 1.5 -1 0.5 0 0.5 1 1.5 (a) Source point Collocation point Normal vector -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 (b) (c) Source point Collocation point Normal vector -4 -2 0 2 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

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4.3. Example 3: arbitrary domain cases (cases 3.1–3.3)

In cases 3.1–3.3, the interior Dirichlet problems with peanut, armor-unit and gear wheel shapes for more complex boundaries are undertaken. Figs. 20 and 21, respectively, depict the geometry sketch and the node distributions of these three problems. The BCs and analytical solutions for the chosen

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problems are also shown in Fig. 20. The three field potential analytical solutions are plotted in Figs.

22(a)–(c), respectively, while Figs. 23(a)–(c) plot the three numerical results by using the proposed

no-vel method. Fig. 23 shows good numerical results are obtained after comparing with the exact solutions.

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5. Conclusions

In this study, we implement a novel meshless method to solve the Laplace problems for arbitrary domains subject to the Dirichlet, Neumann and mixed-type BCs. Only the boundary nodes on the real boundary are required. The major difficulty of the coincidence of the source and collocation points in the conventional MFS is then circumvented. Furthermore, the controversy of the artificial (off-set) boundary outside the physical domain by using the conventional MFS no longer exists. Although it results in the singularity and hypersingularity due to using the double layer potential, the finite values of the diagonal terms for the influence matrices have been extracted out by the proposed desingulariza-tion technique to regularize the singularity and hypersingularity of the kernel funcdesingulariza-tions. The ill-posed influence matrices generated by using the conventional MFS are eliminated when using the off-set boundary far from the real boundary. The numerical results were obtained by using the developed pro-gram for three category examples with different BCs and shapes of domain. Solutions were compared very well with the analytical solutions or other numerical methods such as BEM and conventional MFS.

Acknowledgments

Financial support from the National Science Council of Taiwan is gratefully acknowledged. It is granted to the National Taiwan University under Grant No. NSC-92-2281-E-002-020. We thank the reviewersÕ very constructive comments about this paper.

Appendix A. The detail derivations of Eqs.(20) and (21)

The null-fields of the boundary integral equations (BIEs) based on the direct method are 0¼ Z B oUðeÞðs; xiÞ ons /ðsÞ dBðsÞ  Z B UðeÞðs; xiÞo/ðsÞ ons dBðsÞ; xi2 De; ðA:1Þ 0¼ Z B o2UðeÞðs; xiÞ onsonxi /ðsÞ dBðsÞ  Z B oUðeÞðs; xiÞ onxi o/ðsÞ ons dBðsÞ; xi2 De; ðA:2Þ

where the superscript (e) denotes the exterior domain, U is the single layer potential, and is equal to lnðrijÞ.

LetoUðeÞonðs;xiÞ

s ¼ A

ðeÞðs; xiÞ; ando2UðeÞðs;xiÞ

onsonxi ¼ B

ðeÞðs; xiÞ. By employing the simple test method (o/(s)/on

s= 0 when

/(s) = 1), we can write Eqs.(A.1) and (A.2)as follows: Z B AðeÞðs; xiÞ dBðsÞ ¼ 0; xi2 De; ðA:3Þ Z B BðeÞðs; xiÞ dBðsÞ ¼ 0; xi2 De: ðA:4Þ

When the field point xiapproaches the boundary, we can discretize Eqs.(A.3) and (A.4)and as follows: XN

j¼1

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XN j¼1

BðeÞðsj; xiÞ‘j¼ 0; xi2 B; ðA:6Þ

where ‘jis the half of distance of the (j 1)th source point and the (j + 1)th source point. When the distri-bution of nodes is uniform, we are able to reduce Eqs.(A.5) and (A.6)to the following

XN j¼1 AðeÞðsj; xiÞ ¼ 0; xi2 B; ðA:7Þ XN j¼1 BðeÞðsj; xiÞ ¼ 0; xi2 B; ðA:8Þ where AðeÞðsj; xiÞ ¼oUðeÞðsj; xiÞ ons ¼nkyk r2 ij ; ðA:9Þ BðeÞðsj; xiÞ ¼o 2 UðeÞðsj; xiÞ onsonxi ¼ 2ykylnknl r4 ij nknk r2 ij ; ðA:10Þ

where rij¼ jsj xij, nkis the kth component of the outward normal vector at sj; nk is the kth component of

the outward normal vector at xiand yk¼ xi k s

j

k. Eqs.(A.7) and (A.8)are Eqs.(20) and (21)in the text of

Section 3; and Eqs. (A.9) and (A.10)are Eqs.(16) and (17)in the text of Section 2.

Appendix B. Analytical derivation of diagonal coefficients of influence matrices for circular domain by using separable kernels and circulants

By adopting the addition theorem[1], we can expand the two kernels in Eqs.(16) and (17)for exterior problems and also the corresponding two kernels for the interior problems into separable kernels which separate the field point, xi, and source point, sj, as follows:

Aðsj; xiÞ ¼o lnðrijÞ or ¼ r q cosðh  uÞ r2þ q2 2rq cosðh  uÞ¼ AðiÞðsj; xiÞ ¼1 rþ P1 m¼1 qm rmþ1cosðmðh  uÞÞ; r >q; AðeÞðsj; xiÞ ¼ P1 m¼1 rm1 qm cosðmðh  uÞÞ; q > r; 8 > > < > > : ðB:1Þ Bðsj; xiÞ ¼o 2lnðr ijÞ oqor ¼ 2rq þ ðr2þ q2Þ cosðh  uÞ ðr2þ q2 2rq cosðh  uÞÞ2 ¼ BðiÞðsj; xiÞ ¼ P1 m¼1 mqm1 rmþ1 cosðmðh  uÞÞ; r >q; BðeÞðsj; xiÞ ¼ P1 m¼1 mrm1 qmþ1 cosðmðh  uÞÞ; q > r; 8 > > < > > : ðB:2Þ where sj= (r, h) and xi= (q, u) in the polar coordinates. The definitions of r, h, q, u for the interior and exterior problems are plotted inFigs. 1(c) and (d). Since the rotation symmetry is preserved for a circular boundary, the two influence matrices in Eqs. (6) and (7)are the circulants with the elements

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where the kernel K can be A or B in Eqs.(6) and (7)for the interior problems and Eqs.(12) and (13)for the exterior problems, hj, uiare the angles of source and collocation points, respectively. By superimposing N

lumped strength along the boundary, we have the following influence matrices:

½K ¼ k0 k1    kN1 kN1 k0    kN2 .. . .. . . . . .. . k1 k2    k0 2 6 6 6 6 4 3 7 7 7 7 5; ðB:4Þ

where the elements of the first row can be obtained by

kj¼ kðr; hj;q;0Þ; ðB:5Þ

in which u = 0 is assigned without loss of generality. The matrix [K] in Eq.(B.4)is found to be a circulant since the rotational symmetry for the influence coefficients is considered. By introducing the following bases for the circulants, I, (CN)1, (CN)2, . . . , and (CN)N 1, we can expand [K] into

½K ¼ k0Iþ k1ðCNÞ 1 þ k2ðCNÞ 2 þ    þ kN1ðCNÞ N1 ; ðB:6Þ

where I is an unit matrix and

CN ¼ 0 1 0    0 0 0 0 1    0 0 .. . .. . .. . . . . .. . .. . 1 0 0    0 0 2 6 6 6 6 4 3 7 7 7 7 5 NN : ðB:7Þ

Based on the circulant theory[4], the eigenvalues for the influence matrix, [K], are found as follows: kl¼ k0þ k1slþ k2ðslÞ

2

þ    þ kN1ðslÞN1; l¼ 0; 1; 2; . . . ; N  1; ðB:8Þ

where kland slare the eigenvalues for [K] and [CN], respectively. It is easily found that the eigenvalues slfor

the circulant [CN] are the roots for sN= 1 as shown below:

sl¼ ei

2pl

N; l¼ 0; 1; 2; . . . ; N  1: ðB:9Þ

Substituting Eq.(B.9)into Eq.(B.8), we have kl¼ XN1 m¼0 kmsml ¼ X N1 m¼0 kmei 2pml N ; l¼ 0; 1; 2; . . . ; N  1: ðB:10Þ

According to the definition for kmin Eq.(B.5), we obtain

km¼ kNm; m¼ 0; 1; 2; . . . ; N  1: ðB:11Þ

Substitution of Eq.(B.11)into Eq.(B.10)it yields kl¼ XN1 m¼0 kmcos 2pml N   ; l¼ 0; 1; 2; . . . ; N  1: ðB:12Þ

By setting u = 0 without loss of generality, the Riemann sum of infinite terms reduces to the following integral kl¼ 1 Dh N!1lim XN1 m¼0 KðmDh; 0Þ cosðmlDhÞDh  N 2p Z 2p 0 cosðlhÞKðh; 0Þ dh; ðB:13Þ

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where Dh¼2p N:

B.1. Interior problem

By employing the separable kernel A(i)(sj, xi) for interior problem (r > q) in Eq.(B.1)and the orthogonal conditions, Eq.(B.13)reduces to

mðiÞl ¼ N r; l¼ 0; N 2r; l¼ 0; 1; 2; . . . ; N  1: ( ðB:14Þ Similarly, we have dðiÞl ¼ 0;Njlj l¼ 0 2r2; l¼ 0; 1; 2; . . . ; N  1; ( ðB:15Þ where mðiÞl and dðiÞl are the eigenvalues of [A(i)] and [B(i)] matrices, respectively. By employing the invariant prop-erty for the influence matrices, the first invariant is the sum of all the eigenvalues. The diagonal coefficients for the two matrices for the interior problem are obtained by adding all the eigenvalues and can be shown below:

Najj ¼ XN1 m¼0 mðiÞm ðj no sumÞ; ðB:16Þ Nbjj ¼ XN1 m¼0 dðiÞm: ðB:17Þ

Hence, the diagonal elements are easily determined from the first invariant as follows: ajj¼ Nþ 1 2r  p 2pr N ; N 1; ðB:18Þ bjj¼ NðN  1Þ 4r2  p2 ð2pr N Þ 2; N  1: ðB:19Þ B.2. Exterior problem

Similarly, we have the diagonal terms of the influence matrices for the exterior problem as follows:  ajj¼ N 1 2r  p 2pr N ; N 1; ðB:20Þ  bjj¼ NðN  1Þ 4r2  p2 ð2pr N Þ 2; N  1: ðB:21Þ

The properties of the influence matrices for interior and exterior problems are shown inTable 1.

References

[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulation, Graphs and Mathematical Tables, Dover, New York, 1972.

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[2] T. Belytschko, L. Gu, Y. Lu, Fracture and crack growth by element-free Galerkin methods, Model. Simul. Mater. Sci. Eng. 2 (1994) 519–534.

[3] J.T. Chen, S.R. Kuo, K.H. Chen, Y.C. Cheng, Comments on vibration analysis of arbitrary shaped membranes using non-dimensional dynamic influence function, J. Sound Vibr. 235 (1) (2000) 156–171.

[4] 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 Vibr. 257 (4) (2002) 667–711.

[5] J.T. Chen, M.H. Chang, K.H. Chen, Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function, Comput. Mech. 29 (2002) 392–408.

[6] J.T. Chen, I.L. Chen, C.S. Wu, On the equivalence of MFS and Trefftz method for Laplace problems, in: Proceedings of the Global Chinese Workshop on Boundary Element and Meshless Method, Hebei, China, 2003.

[7] J.T. Chen, I.L. Chen, K.H. Chen, Y.T. Yeh, Y.T. Lee, A meshless method for free vibration of arbitrarily shaped plates with clamped boundaries using radial basis function, Eng. Anal. Bound Elem. 28 (2004) 535–545.

[8] W. Chen, M. Tanaka, A meshfree, integration-free and boundary-only RBF technique, Comput. Math. Appl. 43 (2002) 379–391. [9] W. Chen, Y.C. Hon, Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz and convection–diffusion problems, Comput. Methods Appl. Mech. Eng. 192 (2003) 1859–1875. [10] A.H.D. Cheng, D.L. Young, C.C. Tsai, The solution of PoissonÕs equation by iterative DRBEM using compactly supported,

positive definite radial basis function, Eng. Anal. Bound. Elem. 24 (7) (2000) 549–557.

[11] G. Fairweather, A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math. 9 (1998) 69–95.

[12] R.A. Gingold, J.J. Maraghan, Smoothed particle hydrodynamics: theory and applications to non-spherical stars, Man. Not. Astron. Soc. 181 (1977) 375–389.

[13] W.S. Hwang, L.P. Hung, C.H. Ko, Non-singular boundary integral formulations for plane interior potential problems, Int. J. Numer. Meth. Eng. 53 (2002) 1751–1762.

[14] S.W. Kang, J.M. Lee, Y.J. Kang, Vibration analysis of arbitrary shaped membranes using non-dimensional dynamic influence function, J. Sound Vibr. 221 (1) (1999) 117–132.

[15] S.W. Kang, J.M. Lee, Application of free vibration analysis of membranes using the non-dimensional dynamic influence function, J. Sound Vibr. 234 (3) (2000) 455–470.

[16] W.K. Liu, S. Jun, S. Li, J. Adee, T. Belytschko, Reproducing kernel particle methods for structural dynamics, Int. J. Numer. Meth. Eng. 38 (1995) 1655–1679.

[17] M.A. Tournour, N. Atalla, Efficient evaluation of the acoustic radiation using multipole expansion, Int. J. Numer. Meth. Eng. 46 (1999) 825–837.

[18] C.C. Tsai, Meshless numerical methods and their engineering applications, Ph.D. Dissertation, National Taiwan University, Taipei, Taiwan, 2002.

[19] C.C. Tsai, D.L. Young, A.H.D. Cheng, Meshless BEM for three-dimensional Stokes flows, Computer Modeling in Engineering and Science (CMES) 3 (2002) 117–128.

[20] Y.S. Smyrlis, A. Karageorghis, Some aspects of the method of fundamental solutions for certain harmonic problems, J. Scientific Comput. 16 (3) (2001) 341–371.

[21] D.L. Young, C.C. Tsai, T.I. Eldho, A.H.D. Cheng, Solution of Stokes flow using an iterative DRBEM based on compactly-supported, positive definite radial basis function, Comput. Math. Appl. 43 (2002) 607–619.

[22] Y.X. Mukherjee, S. Mukherjee, The boundary node method for potential problems, Int. J. Numer. Meth. Eng. 40 (1997) 797–815. [23] J.M. Zhang, M. Tanaka, T. Matsumoto, Meshless analysis of potential problems in three dimensions with the hybrid boundary

node method, Int. J. Numer. Meth. Eng. 59 (2004) 1147–1160.

數據

Fig. 1. The source point and observation point distributions and definitions of r, h, q, u by using the conventional MFS and the novel meshless method for the interior and exterior problems: (a) interior problem (MFS), (b) exterior problem (MFS), (c) interi
Fig. 2. Problem sketch and the nodes distribution (60 nodes) in the case 1.1: (a) problem sketch, (b) nodes distribution, (c) exact solution.
Fig. 3. The field solutions by using the conventional MFS (60 nodes) for the case 1.1: (a) d = 0.01, (b) d = 0.5, (c) d = 1.0.
Fig. 4. The field solutions by using the proposed method, BEM and conventional MFS for the case 1.1; (a) /(x, 0), (b) /(0, y).
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