The boundary node method for three-dimensional problems in potential theory

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng. 2000; 47:1523–1547

The boundary node method for three-dimensionalproblems in potential theory

Mandar K. Chati‡ and Subrata Mukherjee∗;†;§

Department of Theoretical and Applied Mechanics; Cornell University; Ithaca; NY 14853; U.S.A.

SUMMARY

The boundary node method (BNM) is developed in this paper for solving potential problems in three dimen-sions. The BNM represents a coupling between boundary integral equations (BIE) and moving least-squares(MLS) interpolants. The main idea here is to retain the dimensionality advantage of the former and themeshless attribute of the later. This results in decoupling of the ‘mesh’ and the interpolation procedure forthe �eld variables.A general BNM computer code for 3-D potential problems has been developed. Several parameters involvedin the BNM need to be chosen carefully for a successful implementation of the method. An in-depth andsystematic study has been carried out in this paper in order to better understand the e�ects of variousparameters on the performance of the method. Numerical results for spheres and cubes, subjected to di�erenttypes of boundary conditions, are extremely encouraging. Copyright ? 2000 John Wiley & Sons, Ltd.

KEY WORDS: meshless; boundary node method; potential theory

1. INTRODUCTION

The task of meshing a 3-D object with complicated geometry can be arduous, time consum-ing and computationally expensive. Although signi�cant progress has been made in 3-D meshingalgorithms, a considerable computational burden is associated with these algorithms.Conventional computational engines such as the �nite di�erence method (FDM), �nite ele-

ment method (FEM) and boundary element method (BEM) can be used, but often with di�cultyhowever, to solve problems that involve, for example, large deformation and crack propagation.The main di�culty in these problems is the task of remeshing a three-dimensional object afterlarge deformation or crack propagation. In recent years, novel computational algorithms have been

∗Correspondence to: Subrata Mukherjee; Department of Theoretical and Applied Mechanics; Cornell University; 212Kimball Hall; Ithaca; NY 14853; U.S.A.

†E-mail: [email protected]‡Graduate student§Professor

Contract=grant sponsor: Ford Motor Company

CCC 0029-5981/2000/091523–25$17.50 Received 6 May 1998Copyright ? 2000 John Wiley & Sons, Ltd. Revised 19 February 1999

1524 M. K. CHATI AND S. MUKHERJEE

proposed which circumvent the problems associated with 3-D meshing. These methods have beencollectively referred to as ‘Meshless’ methods.Nayroles et al. [1] proposed a method which they call the di�use element method (DEM). The

main idea of their work is to replace the usual FEM interpolation by a ‘di�use approximation’.Their strategy is to use a least-squares approximation scheme to interpolate the �eld variables.These interpolants are called moving least-squares (MLS) interpolants. Nayroles et al. [1] haveapplied the di�use element method to 2-D problems in potential theory and linear elasticity.Meshless methods proposed to date include—the element-free galerkin (EFG) method [2], the

reproducing kernel particle method (RKPM) [3], h-p clouds [4; 5], the meshless local Petrov–Galerkin (MLPG) approach [6; 7] and a local boundary integral equation (LBIE) method [8]. Theelement-free Galerkin (EFG) method has been popularized by Belytschko and his co-workers. Themain idea in the EFG method is to use moving least-squares (MLS) interpolants to constructthe trial functions used in the weak form. A wide variety of problems have been solved using theEFG method. In the introductory paper by Belytschko et al. [2], the EFG method was applied to2-D problems of heat conduction and linear elasticity with great success. The method has beenapplied to problems in fracture mechanics with crack growth [9], dynamic fracture [10; 11], platebending [12] and for the analysis of thin shells [13].More recently, the EFG method has been applied to 3-D fracture mechanics [14] and the RKPM

to problems involving large deformation of non-linear structures [15; 16]. A recent special issue ofthe journal Computer Methods in Applied Mechanics and Engineering contains excellent reviewarticles by Belytschko et al. [17] and Liu et al. [18] on meshless methods. Another excellentsource of information on the RKPM is an overview article by Liu et al. [19].Although the EFG method gained immediate recognition and is now a well-established method,

an issue about accurate imposition of natural boundary conditions surfaced early on. The MLSinterpolants lack the delta function property of the usual FEM and BEM shape functions, i.e.

�I (xJ ) 6= �IJ (1)

where �I is the I th shape function evaluated at xJ and �IJ is the Kronecker delta. Belytschkoand his co-workers have employed various approaches for the satisfaction of essential bound-ary conditions. These include the use of Lagrange multipliers [2], use of traction as Lagrangemultipliers [20], and the use of a layer of �nite elements along the boundary where essential con-ditions are prescribed [21]. However, this issue has been addressed successfully by Mukherjee andMukherjee [22]. The strategy proposed in their paper for completely alleviating the above probleminvolves a new de�nition of the discrete norm used for the construction of the MLS interpolants.Chen et al. [15] have independently proposed essentially the same idea to solve the boundary con-dition problem for reproducing kernel particle methods (RKPM). Zhu and Atluri [23] also addressthe problem of satisfaction of essential boundary conditions in the EFG method. The main idea intheir paper [23] is the same as that proposed earlier [22], together with the addition of a penaltyformulation.Recently, Mukherjee and Mukherjee [24] proposed a meshless method which they call the bound-

ary node method (BNM). The BNM involves a coupling between moving least-squares (MLS)interpolants and boundary integral equations (BIE). The BIE has the well-known dimensionality ad-vantage, i.e. for a three-dimensional object meshing is required only on the two-dimensional bound-ing surface. The BNM is particularly attractive for three-dimensional problems since it combinesthe meshless attribute of the MLS interpolants and the dimensionality advantage of the BIE. Also,the input data structure for solving a boundary value problem involves just nodes on the boundary.

Copyright ? 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 2000; 47:1523–1547

BOUNDARY NODE METHOD FOR 3-D PROBLEMS 1525

Figure 1. ‘Salted’ Potato: Input data structure for theBNM—cells and collocation nodes.

Figure 2. The evaluation point E lies in the rangeof in uence of nodes 1, 2 and 3 only. The rangeof in uence of node 4 is truncated at the edges

and corners.

The (surface) cells are used for integration only. Their topology can be much simpler than conven-tional boundary elements, in that, for example, some cells can be divided into smaller ones withouta�ecting their neighbours in any way (Figure 1). This data structure considerably reduces the taskof meshing and a solid model of a 3-D body can be directly used as input for stress analysis. Thismethod has been successfully tried for 2-D problems in potential theory [24] and linear elasticity[25]. The present paper is a �rst attempt in applying the BNM to three-dimensional problems.The main contribution of the present paper is the development of appropriate MLS interpolants

for 3-D surfaces and the coupling of these interpolants with the boundary integral equation (BIE)for potential problems in three dimensions. This paper thoroughly investigates the e�ect of thevariation of all important parameters on the performance of the method. Considerable insight hasbeen provided in choosing the various parameters involved.The paper is organized as follows. First, the MLS interpolation scheme is described for prob-

lems in three dimensions. This is followed by a description of the various weight functionsused in this work. The next section brie y describes the well-known BIE for potential problems(e.g. Reference [26]). The following section describes the mapping method for carrying out theweakly singular integration. Finally, exhaustive numerical results are presented for two simplethree-dimensional objects, namely the sphere and the cube. The main motivation for choosingthese objects is that careful comparison is possible between exact and numerical (BNM) solutionsfor various types of boundary conditions. These numerical results indicate that the method hastremendous potential for use in practical applications.

2. MOVING LEAST SQUARES (MLS) INTERPOLATION

An MLS interpolation scheme is developed here for 3-D problems in the BNM. Since the BNMnodes lie only on the boundary @B of a 3-D body B, curvilinear coordinates are necessary to de�nethe MLS interpolants on the surface.



The following surface points are de�ned as:

I a boundary node, I =1; 2; : : :E an evaluation point which can be a Gauss point or a boundary nodes a generic point with curvilinear co-ordinates (s1; s2)

The MLS interpolants for u and q ≡ @u=@n on @B are de�ned as

u(s)=m∑i=1pi(s − sE)ai= pT(s − sE)a (2)

q(s)=m∑i=1pi(s − sE)bi= pT(s − sE)b (3)

Equations (2) and (3) are to be interpreted as the values of u (or q) at a surface point withglobal co-ordinates (s1; s2). The monomials pi (see below) are evaluated in local co-ordinates(s1−sE1 ; s2−sE2 ) where (sE1 ; sE2 ) are the global co-ordinates of an evaluation point E. It is importantto state here that ai and bi are not constants. Their functional dependencies are determined later.The integer m is the number of monomials in the basis used for u and q. In this paper, numericalresults have been obtained using quadratic (m=6) or cubic bases (m=10), i.e.

pT(s1; s2)= [1; s1; s2; s21 ; s22 ; s1s2]; m=6 (4)

pT(s1; s2)= [1; s1; s2; s21 ; s22 ; s1s2; s

31 ; s

32 ; s

21s2; s1s

22]; m=10 (5)

where

si= si − sEi ; i=1; 2 (6)

The coe�cients a and b are obtained by minimizing weighted, discrete L2 norms de�ned as

Ru=n∑I=1w(dI ) [pT(sI − sE)a − uI ]2 (7)

Rq=n∑I=1w(dI ) [pT(sI − sE)b− qI ]

2 (8)

where the summation is carried over the n boundary nodes for which the weight function w(dI ) 6= 0.These n nodes are said to be within the domain of dependence of an evaluation point (Figure 2).Also, (sI1 − sE1 ; sI2 − sE2 ) are the local surface co-ordinates of the boundary nodes with respect tothe evaluation point sE =(sE1 ; s

E2 ) and uI and qI are the approximations to the nodal values uI and

qI . The equations above can be rewritten in compact form as

Ru=(P(sI − sE)a − u)TW(s; sI )(P(sI − sE)a − u) (9)

Rq=(P(sI − sE)b− q)TW(s; sI )(P(sI − sE)b− q) (10)

where

uT = ( u1; u2; : : : ; un) (11)

qT = ( q1; q2; : : : ; qn) (12)



P(sI )=

1 p2(s11; s

12) · · · pm(s11; s

12)

1 p2(s21; s22) · · · pm(s21; s

22)

......

......

1 p2(sn1; sn2) · · · pm(sn1; s

n2)

(13)

and

W(s; sI ) ≡W(dI )=

w(d1) 0 · · · 00 w(d2) · · · 0...

......

...0 0 · · · w(dn)

(14)

where di; i=1; 2; : : : ; n are de�ned in Section 3 on weight functions. The stationarity of Ru andRq, with respect to a and b, respectively, leads to the equations

a(s)=A−1(s)B(s) u (15)

b(s)=A−1(s)B(s) q (16)

where

A(s)=PT(sI − sE)W(s; sI )P(sI − sE) (17)

B(s) = PT(sI − sE)W(s; sI ) (18)

Also, de�ne

CjI = (A−1B)jI (19)

It is noted from above that the coe�cients ai and bi turn out to be functions of s. SubstitutingEquation (15) into Equation (2) and Equation (16) into Equation (3), leads to

u(s) =n∑I=1�I (s)uI (20)

q(s) =n∑I=1�I (s)qI (21)

where the shape functions �I are

�I (s) =m∑j=1pj(s − sE)CjI (s) (22)

At an evaluation point E, one has

u(sE) =n∑I=1�I (sE)uI (23)

with

�I (sE) =m∑j=1pj(0)CjI (sE) (24)



As mentioned previously, u and q are approximations to the actual nodal values u and q. The twosets of values can be related by �nding the number of nodes in the range of in uence of eachcollocation node and then evaluating the shape function at each of these nodes. Thus, by carryingout this procedure for each of the collocation nodes, one gets

[H]{u} = {u} (25)

[H]{q} = {q} (26)

In other words Equations (25) and (26) are the discretized versions of Equations (20) and (21),respectively. They relate the nodal approximations of u and q to their actual nodal values.The reader is cautioned that the method is not as easy to implement as it appears and there

are many pitfalls associated with it. The remarks that follow try to highlight some of the insightgained into the method.

Remark 1 (Invertibility of A). The matrix A is an m × m matrix, composed of the matricesP and W (Equation (17)). It needs to be invertible for the construction of the shape functions.It is also desirable that A be well conditioned. From a well-known fact in linear algebra aboutranks of products of matrices, it is necessary that the rank of matrix P be m. However, if n¡m,i.e. the number of nodes (n) in the domain of dependence of an evaluation point is less thanthe order of the polynomial basis (m), then matrix A would be rank de�cient and would becomenon-invertible. So, it is essential to choose the parameter which controls the range of in uence ofa node, namely d, such that n¿m. However, even if the condition n¿m is satis�ed, but the nnodes in the domain of dependence of the evaluation point E lie on a straight line on the surface,then the matrix A becomes singular. Also, it has been observed that choosing n ∼ m may lead toan unacceptably large condition number of the matrix A.Remark 2 (Matrix H). As noted above, the matrix H relates the actual nodal values to their

nodal approximations. It is observed through numerical experiments that the matrix H has meigenvalues equal to unity. The associated m eigenvectors are described by the monomials usedin the bases for constructing the approximation. Thus, when looking for solutions that cannot bespanned by the monomials used in the bases, the matrix H plays a signi�cant role in the successof the method.Remark 3 (Boundary conditions). The H matrix plays a crucial role in the satisfaction of

essential boundary conditions in the EFG method [21] and in the satisfaction of all boundaryconditions in the BNM [23; 24].Remark 4 (De�ne a panel). Curvilinear co-ordinates (s1; s2) are used to measure distances over

curved surfaces. However, real life objects consist of piecewise smooth surfaces, referred to asPanels in this work, and de�ning curvilinear co-ordinates across edges and corners is a formidabletask. In this work, collocation nodes are placed inside panels, and, in order to circumvent theproblem of ‘reaching over edges’, it was decided that the range of in uence of each node wouldbe truncated at an edge or corner (Figure 2). It will be seen through numerical experiments thatrestricting the range of in uence of a node to the panel to which it belongs still yields acceptableresults.Remark 5 (The nature of s1; s2). The co-ordinates (s1; s2) are the curvilinear co-ordinates mea-

sured along the bounding surface @B. These co-ordinates are local and not global. In other words,these are constructed with the origin at the evaluation point E, i.e. these curvilinear co-ordinateswill always be (0,0) at the evaluation point E. This simpli�es the computation of the shape



functions to some extent. Since, (s1 = 0; s2 = 0), one has p1 = 1 and pi = 0 for i = 2; : : : ; m.This further implies that the shape function is just the �rst row of the matrix C.

3. WEIGHT FUNCTIONS

In meshless methods, there is an associated range of in uence for each node. The parameter ddecides the ‘compact support’ associated with each node. It could be chosen to be identical for allthe nodes (homogeneous) or it could be varied from node to node (non-homogeneous). For thekind of numerical examples tried in this paper, homogeneous d has served the purpose and so thepossibility of non-homogeneous d has not been explored.The parameter d needs to be chosen carefully for successful implementation of the method. As

mentioned in the previous section, for the matrix P to be of full rank it is necessary that n¿m,i.e. the parameter d needs to be chosen such that for a given evaluation point E, at least m nodesare in its domain of dependence. On the other hand, the requirement of a well-conditioned matrixH places another restriction on the parameter d. As will be seen below, the suggested weightfunctions are monotonically decreasing functions of distance. Thus, if d is chosen such that a verylarge number of nodes (n ∼ 10m) are in the range of in uence of a node, then the matrix H tendsto be ill-conditioned. Thus, the order of the polynomial bases (m) places a lower bound on d andthe requirement of a ‘reasonable’ condition number of H places an upper bound on the parameterd. In this work, it has been observed that choosing d such that n ∼ 2m− 3m leads to acceptableresults.A variety of weight functions have been suggested in the past for meshless methods. In order

to gain a better understanding of the method, a detailed numerical study of these various weightfunctions has been carried out in this paper. Four weight functions have been used in this paper.These are:

(i) Gaussian (referred to as—WFA):

w(d) ={e−(d=c)

2for d61

0 for d¿1(27)

(ii) Exponential (referred to as—WFB):

w(d) =

e−(dI =c)

2 − e(d=c)2

1− e(d=c)2 for d61

0 for d¿1(28)

(iii) Cubic Spline (referred to as—WFC):

w(d) =

2=3− 4d2 + 4d3 for d61=24=3− 4d+ 4d2 − (4=3)d3 for 1=2¡d61

0 for d¿1(29)

(iv) Quartic Spline (referred to as—WFD):

w(d) ={1− 6d2 + 8d3 − 3d4 for d61

0 for d¿1(30)



where d = dI =d = g(s; sI )=d and c is a constant. dI is the minimum distance between the evaluationpoint E and the node I. This minimum distance is the geodesic measured on the bounding surface@B. In this paper, the examples considered involve very simple geometries like spheres and cubesfor which the calculation of geodesics is trivial. For a sphere the geodesics are great circles. For acube, the six faces are considered as six independent panels. In other words, the range of in uenceof a node is arbitrarily truncated at the edges of the cube. Thus, the calculation of the geodesicis reduced to calculating the minimum distance between two points lying on a plane which is astraight line. Again, it is restated that the arbitrary truncation of the range of in uence of a nodeat the edges still yields overall acceptable results.

4. BOUNDARY INTEGRAL EQUATIONS AND DISCRETIZATIONS

The Laplace’s equation in three dimensions is written as (see Reference [26])

∇2u =@2u@x2

+@2u@y2

+@2u@z2

= 0 (31)

along with prescribed boundary conditions. The problem can be recast into an integral equationon the boundary. The well-known primary BIE for potential problems in 3-D is

u(p) =∫@B[G(p;Q)q(Q)− F(p;Q)u(Q)] dSQ (32)

where u is the potential and the well-known kernels are

G(p;Q) =1

4�r(p;Q)(33)

F(p;Q) =@G(p;Q)@nQ

(34)

The BIE (Equation (32)) is written for the source point p inside the domain B and the �eld pointQ on the boundary @B. Also, r is the Euclidean distance between the source and �eld points andnQ is the unit normal to @B at a �eld point Q. To solve a boundary value problem the sourcepoint p needs to be taken to the boundary in the limiting sense. Taking the limit, as p → P inEquation (32) and using the rigid body mode, we have

0 =∫@B[G(P;Q)q(Q)− F(P;Q)(u(Q)− u(P))] dSQ (35)

The MLS interpolations derived in Section 2 will be used to approximate u and q on theboundary @B. The bounding surface @B is discretized into cells. A variety of shape functions havebeen used to interpolate the geometry. In particular, the linear (T3) triangle, bilinear (Q4) elementand quadratic (T6) triangle have been used. These ‘geometric’ shape functions can be found inany standard text on the �nite element method [27; 28].Substituting the expressions for u and q (Equations (20) and (21)) into Equation (35) and

dividing @B into Nc cells, one gets

0 =Nc∑k=1

∫@Bk

[G(P;Q)

nQ∑I=1�I (Q)qI − F(P;Q)

{ nQ∑I=1�I (Q)uI −

nP∑I=1�I (P)uI

}]dSQ (36)



where �I (P) and �I (Q) are the contributions from the I th node to the collocation point P and�eld point Q, respectively. Also, nQ nodes are in the domain of dependence of the �eld pointQ and nP nodes are in the domain of dependence of the source point P. Now, to evaluate theintegrals over each cell, regular Gaussian integration can be used. However, since the kernelsbecome singular as Q → P, special integration techniques need to be employed. When P and Qbelong to the same cell, the cell is treated as a singular cell and the special techniques developedin the next section are used to carry out the integration.The discretized form of Equation (36) becomes

[A(u)]{u}+ [A(q)]{q} = {0} (37)

where {u} and {q} contain the approximations to the nodal values of u and q at the NB boundarynodes. A well-posed boundary value problem can be solved using Equation (37), in combinationwith Equations (25) and (26). There are a total of 4NB quantities on the boundary, i.e. 2NB valuesof u and its nodal approximation and so also for q. For a well-posed problem, values of eitheru or q are known at each node on the boundary, so NB nodal values are given. Therefore, 3NBequations are needed to solve for the 3NB remaining unknowns. Equation (37) consists of NBequations and Equations (25) and (26) consist of NB equations each. Thus, the boundary valueproblem can be solved by appropriate switching of the columns.To obtain the gradient of the potential u at an internal point, the derivative of the primary BIE

(Equation (32)) needs to be taken with respect to the source point. This can be written as

@u(p)@xm(p)

=∫@B

[@G(p;Q)@xm(p)

q(Q)− @F(p;Q)@xm(p)

u(Q)]dSQ (38)

It is a well-known fact in the boundary element method (BEM) literature that the calculation ofthe potential and its gradient at internal points close to the boundary can yield very poor results.This is due to the fact that the kernels become nearly singular=hypersingular as p→ P. And theproblem is even more severe for the computation of gradients as the kernels in the hypersingularBIE (HBIE) have a stronger singularity than the kernels in the primary BIE. This problem hasbeen overcome in the present work by employing a novel regularization approach. Details aregiven in Section 6.1.

5. WEAKLY SINGULAR INTEGRATION SCHEME

In order to solve the boundary value problem, Equation (35) needs to be integrated. Since theintegrals are over the 2-D bounding surface @B, regular Gaussian integration can be used. However,the kernels become singular as the source point and collocation point come closer, i.e. as Q → P.The order of singularity associated with the integrand is O(1=r), which is a weak singularityfor two-dimensional surface integrals. There have been various methods proposed in the past tohandle weakly singular integrals arising in BEM. However, the method suggested by Nagarajanand Mukherjee [29] is very attractive and is used to carry out the weakly singular integration inBNM. The details follow.



Consider evaluating the primary BIE (Equation (35)) over a cell as shown in Figure 3(a). Thiscan be represented as

I =∫@BO(1=r) dSQ (39)

The cell shown contains the source point P, so that the �eld point Q could coincide with thecollocation point P. In this example, quadratic (T6) triangles are used to describe the geometryof the bounding surface. But, the method being developed can be easily extended for variousother kinds of geometric interpolations. Firstly, the cell could be mapped into the parent space(Figure 3(b)) using the well-known shape functions for T6 triangles. This would involve a Jacobianand the integral I takes the form

I =∫ t=1

t=0

∫ s=1−t

s=0O(1=r)J1 ds dt (40)

Now, in the parent space the triangle is divided into six pieces. Each individual triangle is mappedinto the �1 − �2 space (Figure 3(c)) using the mapping for linear (T3) triangles. The integral Ican now be written as

I =6∑i=1

∫ �2=1

�2=0

∫ �1=1−�2

�1=0O(1=r)J1J

(i)2 d�1 d�2 (41)

where J (i)2 is the Jacobian for each of the triangle. Now, consider the mapping

�1 = � cos2 �; �2 = � sin2 � (42)

which maps the at triangle from the �1 − �2 co-ordinate system into a rectangle in �− � space(Figure 3(d)). The integral I takes the form

I =6∑i=1

∫ �=�=2

�=0

∫ �=1

�=0O(1=r)J1J

(i)2 � sin � d� d� (43)

As � is a measure of the distance between the source point and the �eld point, the integral I isnow regularized. In other words, the � in the numerator cancels the O(1=r) singularity. Now, toevaluate the integral I , as in Equation (43), regular Gaussian integration can be used. The �nalmapping involves the use of quadratic (Q4) shape functions to map the rectangle from the �− �space into the standard square in �1 − �2 space (Figure 3(e)). The �nal form of the integral I is

I =6∑i=1

∫ �2=1

�2=−1

∫ �1=1

�1=−1O(1=r)J1J

(i)2 J3� sin � d�1 d�2 (44)

where J3 is the Jacobian of the �nal transformation. Finally, regular Gaussian integration can beused to evaluate the above integral I .

6. NUMERICAL RESULTS

A detailed numerical study has been carried out in order to better understand the BNM for three-dimensional problems. The following issues have been investigated:

(i) Choice of Weight functions,



Figure 3. Mapping for weakly singular integrals. Figure 4. Mapping for linear (T3) and quadratic (T6)triangles to the parent space.

(ii) Location of nodes in a cell,(iii) Interpolation of geometry,(iv) Truncation of the range of in uence of a node at an edge,(v) Number of nodes per cell.

The following two error criteria have been used to evaluate the performance of the methodproposed:

Re=un − ueue

100 per cent (45)

�=100|u|max

√1N

N∑i=1(un − ue)2 per cent (46)

where Re is the percentage relative error in u and � is the ‘global’ percentage L2 error over Nnodes. Also, un and ue refer to the numerical and exact solutions, respectively, and |u|max is themaximum value of u over N nodes.The numerical method has been tested throughly on two simple three-dimensional geometrical

objects, namely the sphere and the cube. The exact solutions presented below have been used totest the performance of the proposed method.

(i) Linear solution

u= x + y + z (47)



Table I. Variation in L2 error in @u=@n for Dirichlet problemson a sphere for various locations of the collocation point in the

parent space.

s= t u= linear u=quadratic-1 u=cubic

0.05 63.469 94.647 346.470.1 25.833 17.562 53.1870.15 2.443 3.615 9.2410.2 1.864 2.695 7.7340.25 2.009 4.403 6.7980.3 1.129 1.947 3.0741=3 1.067 1.696 2.0180.35 3.765 4.121 4.4690.4 1.198 3.006 5.1990.45 1.952 4.191 9.852

(ii) Quadratic solution-1

u= xy + yz + zx (48)

(iii) Quadratic solution-2

u= − 2x2 + y2 + z2 (49)

(iv) Cubic solution

u= x3 + y3 + z3 − 3yx2 − 3xz2 − 3zy2 (50)

The numerical results presented in this paper are very attractive and encouraging.

6.1. Dirichlet problems on a sphere

A variety of problems, using Equations (47)–(50), have been solved on a sphere. The usual curvi-linear co-ordinates � and � are used. In each case, Dirichlet boundary conditions corresponding tothe exact solution have been imposed on the surface of the sphere. Numerical results have beenobtained using linear (T3) triangles and quadratic (T6) triangles for interpolating the geometry.To carry out the integration, each of these triangles are mapped into a unit triangle in the parentspace (Figure 4). The results have been obtained for four di�erent meshes: (a) Crude Mesh—72cells interpolated using linear (T3) triangles, (b) Fine mesh—288 cells interpolated using linear(T3) triangles, (c) crude mesh—72 cells interpolated using quadratic (T6) triangles and (d) �nemesh—288 cells interpolated using quadratic (T6) triangles.It has been observed that an important ingredient in the success of this method is the choice of

the locations of the collocation nodes on each cell. Table I presents the global L2 error in @u=@nfor the linear, quadratic-1 and cubic solutions (imposed as Dirichlet boundary conditions on thesphere surface) for various locations of the collocation node in the parent space (s− t). The resultshave been obtained using a crude mesh (72 T6 triangles) with one node per cell and a Gaussianweight function (c=0:4). It can be clearly seen that, as expected, placement of the collocationnode at the centroid of the triangle in the parent space, i.e. s= t= 1

3 , yields excellent results. All



Figure 5. E�ect of changing d on the L2 error in @u=@n for a Dirichlet problem on a sphereusing a linear solution.

Table II. L2 error in @u=@n on a sphere, for the Dirichlet problem: u= x + y + z.

Weight functions Crude mesh (T3) Fine mesh (T3) Crude mesh (T6) Fine mesh (T6)

WFA 4.237 2.083 1.067 0.497WFB 4.237 2.086 1.069 0.539WFC 5.029 2.104 1.031 0.584WFD 4.057 2.253 1.246 0.757

Table III. L2 error in @u=@n on a sphere, for the Dirichlet problem: u= xy + yz + zx.

Weight functions Crude mesh (T3) Fine mesh (T3) Crude mesh (T6) Fine mesh (T6)

WFA 4.022 1.788 1.696 0.886WFB 4.022 1.789 1.697 1.011WFC 5.314 1.785 1.722 1.199WFD 5.391 1.893 2.103 1.942

further results for the sphere have been obtained with one node per cell with the collocation nodeplaced at the centroid of the triangle.As mentioned previously, the parameter d needs to be chosen such that a ‘reasonable’ number

of nodes lie in the domain of dependence of an evaluation point E. In Figure 5, the L2 error in@u=@n has been recorded for increasing d. It is observed that choosing the smallest possible dyields the lowest L2 error. For the optimum choice of d, there were about 12–14 nodes in therange of in uence of each node, which is about 2m− 3m, for a quadratic polynomial basis m=6.Tables II, III and IV present a convergence study that has been carried out for linear, quadratic-1



Table IV. L2 error in @u=@n on a sphere, for the Dirichlet problem: u= x3 + y3 + z3 − 3yx2 − 3zy2 − 3xz2.Weight Crude mesh (T3) Fine mesh (T3) Crude mesh (T6) Fine mesh (T6)functions m=6 m=10 m=6 m=10 m=6 m=10 m=6 m=10

WFA 5.641 5.944 2.284 2.319 2.018 2.962 0.909 0.485WFB 5.641 5.944 2.285 2.318 2.019 2.963 1.002 0.485WFC 6.007 5.525 2.298 2.301 2.254 3.003 1.120 0.574WFD 7.896 5.674 2.345 2.318 2.456 3.570 1.733 0.796

Table V. Relative error in @u=@n on a sphere, for the Dirichlet problem:u= xy + yz + zx.

Co-ordinates @u=@ncomputed @u=@nexact Relative error

(2.0, 0.0, 0.0) 0.0 0.0 —(1.5, 0.866, 1.0) 3.650 3.665 −0:413(0.5, 0.866, 1.732) 2.852 2.799 1.897(1.0, 1.0, 1.414) 3.847 3.828 0.475(1.732, 1.0, 0.0) 1.689 1.732 −2:484(1.414, 1.414, 0.0) 1.998 2.000 −0:118(0.707, 0.707, 1.732) 3.053 2.949 3.513(1.225, 1.225, 1.0) 3.912 3.949 −0:954

and cubic solutions, respectively, for various weight functions proposed in the literature. It can beseen that a crude mesh with 72 quadratic (T6) cells already yields acceptable results. Also, theGaussian weight function (c=0:4) seems to have an edge over the other weight functions used.The results for the linear and quadratic-1 solutions have been obtained using a polynomial basism=6, while, for the cubic solution, polynomial bases m=6 and 10 have been used. It can beseen from Table IV that using a higher-order basis has only a marginal e�ect on the solution,except for the case with the �ne mesh with T6 triangles.Upon solving the Dirichlet boundary value problem using the boundary node method (BNM),

the value of @u=@n is known at the NB boundary nodes. However, to compute q at any loca-tion other than the collocation nodes, the BNM-shape functions, Equation (21), need to be used.Table V presents the results for @u=@n at a few points on the boundary that are not the collocationnodes. The results are obtained upon imposing a quadratic-1 solution on a crude mesh (72 T6cells) with the Gaussian weight function (c=0:4). It can be seen that the solution obtained is wellwithin acceptable limits.Figures 6 and 7 show variation in the potential and its directional derivative for points inside

the sphere. The Dirichlet boundary value problem is solved upon imposing the cubic solution ona crude mesh (72 T6 cells) with the Gaussian weight function and a quadratic basis (m=6). Thegradient is dotted with the diagonal (x=y= z) in order to get the directional derivative along thisline. Values of u and ∇u, at internal points that are close to the surface of the body, are obtainedby application of Equations (17:25) and (17:34) in Kane [30]. It should be noted that the aboveequations in [30] were used to regularize the BIE, and HBIE, respectively. Use of these equationsto obtain the primary variable and its derivative, at internal points close to the boundary, is a newapplication in this paper.



Figure 6. Variation in potential u along the line x= y= z for a sphere using a cubic solution.

Figure 7. Variation of directional derivative of potential u along the line x= y= z for a sphereusing a cubic solution.

The method has also been tried on more challenging problems which cannot be described bypolynomial approximations. The following exact solution has been tried on the sphere:

u=2r2

R2cos2 �− 2r2

3R2− 13

(51)



Figure 8. Comparison of L2 errors for the BNM and the BEM for a Dirichlet problem on a sphere. N is thenumber of nodes (one node per cell is used in the BNM).

where R is the radius of the sphere and � is the angle measured from the z-axis. The Dirichletboundary condition on the surface then becomes

u|(r=R) = cos 2� (52)

A crude mesh (72 T6 cells) with the Gaussian weight function (d=1:616; c=0:4) have beenused to obtain the results. The global L2 error for @u=@n is 0:203 per cent for a quadratic basis(m=6). The results obtained for this problem clearly demonstrate the robustness of the method.The results obtained by the boundary node method (BNM) have also been compared to those

from the conventional boundary element method (BEM) for a Dirichlet problem on a sphere withthe exact solution as in Equation (51). Table VI compares the wall-clock times between a serialBEM code with a serial and a parallel version of the BNM code. The parallel version is an earlyone in which only the assembly of the BNM matrices, not the solution phase, has been parallelized.The parallel BNM code is run on 4, 16, and 32 processors using the message passing interface(MPI) standard on the IBM SP2 (R6000 architecture, 120 MHz P2SC Processor). Certainly, theserial boundary node method (BNM) is considerably slower than the conventional (serial) BEMand this is because the shape functions in the BNM need to be generated at each point unlike inthe BEM (similar observations have been made by other researchers regarding the performance ofthe EFG compared to the �nite element method (FEM)). However, with computers getting fasterand faster this issue should not be a major hindrance. Also, the BNM is very easy to parallelize,and, as shown in Table VI, parallel versions drastically reduce wall clock times. Figure 8 shows acomparison in the L2 error for BEM and BNM and it can be clearly seen the two methods yieldcomparable results and have almost identical rates of convergence.



Table VI. Comparison of wall-clock times for the BNM and BEM for a Dirichletproblem on a sphere.

Parallel BNMSerial BEM Serial BNM 4 Procs 16 Procs 32 Procs

Meshes (s) (s) (s) (s) (s)

72 T6 cells 3.3 29.5 10.0 2.4 1.3128 T6 cells 9.3 106.3 35.5 7.6 4.4288 T6 cells 47.5 690.7 249.0 53.0 27.3

Figure 9. Variation of L2 error in @u=@n, for the Dirichlet problem: u= x+y+z, for a cube, with the change inthe location of the collocation node in the parent space.

6.2. Dirichlet and mixed problems on a cube

A variety of Dirichlet and mixed problems have been solved on a cube. The cube faces arex= ±1; y= ±1 and z= ±1, respectively. The cell geometry has been interpolated using quadraticQ4 elements. The numerical results have been obtained using three meshes: (a) crude mesh(24 cells), (b) �ne mesh (96 cells) and (c) �nest mesh (384 cells). As mentioned previously,the collocation nodes are placed inside the cube faces, and, to circumvent the problem of ‘reach-ing over edges’, each face of the cube is considered as a panel and the range of in uence of eachnode is restricted to the panel to which it belongs. Truncation of the range of in uence of thenodes at the edge does not e�ect the overall results adversely, but the results are inaccurate nearthe edges.As mentioned in the previous section, choosing the location of collocation nodes is crucial for

obtaining any meaningful results. Figures 9 and 10 show the variation in the global L2 error on acube with variation in the location of the collocation nodes in the parent space (s− t). It can beseen from Figure 9 that for the linear solution the location of the collocation nodes at s= t=0:54



Figure 10. Variation of L2 error in @u=@n, for the Dirichlet problem: u= x3 + y3 + z3 − 3yx2 − 3zy2 − 3xz2,for a cube, with the change in the location of the collocation node in the parent space.

Table VII. L2 error in @u=@n on each face of a cube, for the Dirichlet problem:u= x + y + z.

Weight Crude mesh Fine mesh Finest meshfunctions nc=2 nc=4 nc=1 nc=2 nc=1

WFA 1.188 1.604 0.965 1.454 0.704WFB 1.189 1.604 0.966 1.458 0.704WFC 1.185 1.585 0.954 1.504 0.710WFD 1.188 1.604 0.951 1.476 0.725

Table VIII. L2 error in @u=@n on each face of a cube, for the Dirichletproblem: u= x3 + y3 + z3− 3yx2 − 3zy2 − 3xz2.Crude mesh Fine mesh Finest mesh

Weight nc=4 nc=1 nc=1functions m=6 m=10 m=6 m=10 m=6 m=10

WFA 2.561 1.994 1.597 2.377 1.410 1.495WFB 2.561 1.994 1.597 2.422 1.425 1.495WFC 2.300 1.998 1.824 2.440 1.470 1.485WFD 2.274 1.999 1.879 2.368 1.514 1.480

in the parent space yields the lowest L2 error in @u=@n, while, for the the cubic solution (Figure 10),the location s= t=0:0 yields the lowest global L2 error in @u=@n. A possible explanation for theseresults could be that the e�ect of arbitrary truncation of the range of in uence of a node at an edgeor a corner is being compensated for by the collocation points being away from the centroids ofthe rectangles. Tables VII and VIII show the variation in the global L2 error in @u=@n for various



Table IX. L2 error in @u=@n on the face x=±1 of a cube, for mixed boundaryconditions: u=− 2x2 + y2 + z2.

Weight Crude mesh Fine mesh Finest meshfunctions nc=2 nc=4 nc=1 nc=2 nc=1

WFA 0.836 0.809 0.794 0.747 0.449WFB 0.838 0.809 0.794 0.747 0.449WFC 0.837 0.810 0.799 0.753 0.452WFD 0.840 0.809 0.800 0.756 0.457

Table X. L2 error in u on the faces y= ± 1 and z= ± 1 of a cube, for mixed boundaryconditions: u= − 2x2 + y2 + z2. These errors are the same on each of the four faces.Weight Crude mesh Fine mesh Finest meshfunctions nc=2 nc=4 nc=1 nc=2 nc=1

WFA 1.969 2.001 1.220 1.256 0.925WFB 1.969 2.001 1.220 1.254 0.929WFC 1.970 2.003 1.243 1.200 0.958WFD 1.971 2.003 1.255 1.182 1.012

meshes with di�erent choice of weight functions. It is again observed that the Gaussian weightfunction (c=0:4) yields the best results among the various weight functions that have been triedhere. The e�ect of placing more than one node per cell (denoted as nc) has also been explored.It appears that having more than 1 node per cell has an adverse e�ect on the solution. A similartrend was also observed in a paper by Mukherjee and Mukherjee [21] where it was noted that thebest h re�nement strategy for the EFG appeared to be one in which one increases the number ofintegration cells while keeping the ratio of nodes per cell small and roughly the same. In Table VIIIthe e�ect of using a higher polynomial basis seems to yield only marginally better results.A boundary value problem with mixed boundary conditions has also been solved. The quadratic-2

solution has been used here. Dirichlet conditions are imposed on faces x= ± 1 of the cube andNeumann conditions on faces y= ± 1 and z= ± 1. Tables IX and X present the L2 errors in@u=@n on the faces x= ± 1 and L2 errors in u on the faces y= ± 1 and z= ± 1, respectively. Itcan be clearly seen that excellent results have been obtained even for a crude mesh consisting ofonly 24 cells.Figures 11 and 12 present the calculation of the potential and its x-derivative, at internal points

along the x-axis inside the cube, for the imposed cubic solution. Again, very accurate results areobtained.In order to better understand the e�ect of truncation of the range of in uence of a node, the

potential u was calculated along an edge (x=y=1). For an evaluation point on an edge, thepotential can be calculated by considering it to be a part of either of the two adjoining faces.Figure 13 gives a comparison in the values of u, calculated on each of the faces, with the exactsolution. These results obtained for a quadratic polynomial basis, m=6, are inaccurate. However,if a cubic polynomial basis, m=10 is chosen, then accurate results are obtained (Figure 14).To further understand the e�ect of truncation of a range of in uence of a node, the value of @u=@n



Figure 11. Calculation of potential u, along the x-axis inside a cube, for the Dirichlet problem: u= x3 + y3

+z3 − 3yx2 − 3zy2 − 3xz2, with m=6.

Figure 12. Calculation of @u=@x, along the x-axis inside a cube, for the Dirichlet problem: u= x3 + y3 + z3

−3yx2 − 3zy2 − 3xz2, with m=6.

was computed at points close to the edge on the x=1 face of the cube. Figures 15 and 16 showthe results for cubic solution and linear solution respectively. It can be seen from Figure 15 thatthe results are not accurate for points close to the edge even after using a cubic polynomial basis,m=10. However, very accurate results are obtained for the linear solution (Figure 16).



Figure 13. Variation of u along the edge (x=1; y=1) of a cube, for the Dirichlet problem: u= x3 + y3

+z3 − 3yx2 − 3zy2 − 3xz2, with m=6.

Figure 14. Variation of u along the edge (x=1; y=1) of a cube, for the Dirichlet problem: u= x3 + y3

+z3 − 3yx2 − 3zy2 − 3xz2, with m=10.

In addition to the polynomial representations of the exact solution, a problem has been triedwith the following exact solution:

u= sinh�x2sin

�y

2√2sin

�z

2√2

(53)



Figure 15. Variation of @u=@n on the x=1 face of a cube, for the Dirichlet problem: u= x3 + y3 + z3− 3yx2−3zy2 − 3xz2, with m=10.

Figure 16. Variation of @u=@n on the face x=1 of a cube, for the Dirichlet problem:u= x + y + z, with m=6.

again, as before, imposed as Dirichlet boundary conditions on the surface of a cube. The numericalresults summarized in Table XI are in excellent agreement with the exact solution, especially form=10.Figure 17 shows the cell structure with the collocation nodes within each cell (one node per

cell) for a cube. The sparsity pattern of matrix H is shown in Figure 18. The e�ect of restricting



Table XI. L2 error in @u=@n on the cube.

Order of the polynomial basis m=6 m=10

L2 error in @u=@n on x= ± 1 0.389 1.458L2 error in @u=@n on y= ± 1 4.939 1.425L2 error in @u=@n on z= ± 1 4.939 1.425

Figure 17. Cell structure for a cube together with thecollocation nodes.

Figure 18. Sparsity pattern of the matrix Hfor a cube.

the range of in uence of each node to the face to which it belongs, results in a block diagonalform for the matrix H.

7. CONCLUSIONS

The boundary node method (BNM) has been extended to solve three-dimensional problems inpotential theory. The numerical method proposed is tested on simple 3-D geometrical objects likespheres and cubes but with complicated imposed boundary conditions on their bounding surfacesin many cases. The primary reasons for choosing the geometry and boundary conditions in theexamples given in this paper are that (1) many of the examples be non-trivial and (2) exactsolutions, for comparison purposes, exist for all cases.In this work, collocation nodes are placed inside smooth panels, and not on edges and corners.

Collocation on edges and corners is planned for the future.Since curvilinear co-ordinates are used in the MLS interpolation scheme, de�ning these over

the edges=corners was considered a formidable task. So, the range of in uence of a node wastruncated at an edge=corner but this arbitrary truncation still yields overall acceptable results.However, inaccurate results are obtained close to the edges.



It is necessary that n¿m for the matrix A, used in the construction of the MLS interpolants, tobe invertible. The matrix H plays a crucial role in the successful implementation of the method,for solutions that cannot be spanned by the monomials used in the bases.The parameter d which decides the range of in uence of each node needs to be chosen carefully.

The order of the polynomial bases (m) places a lower bound on d and the requirement of a‘reasonable’ condition number of H places an upper bound on d. In this work it was observedthat choosing d, such that n∼ 2m− 3m yields the best results, where n is the number of nodes inthe range of in uence of a node.Results were obtained using a variety of weight functions with Dirichlet and mixed boundary

conditions and the Gaussian weight function seems to yield the best results. It was also observedthat having only one node per cell yields better results than having more nodes per cell. Thelocation of the collocation node (within a cell) is also an important component in the successfulimplementation of the method. Usually, placing a collocation node at the centroid of the rectangleor triangle in the parent space yields the best results for most of the cases, especially for asphere which does not have any edges or corners. The situation with edges and corners is morecomplicated and needs further investigation.Overall, tremendous insight has been gained into the boundary node method (BNM) for problems

in three dimensions. The exhaustive numerical results presented in this paper are most encouraging.The serial and parallel computer code developed in this work can be applied to solve problemswith complicated 3-D geometries. This is planned for the future.The boundary node method has also been recently extended for solving three-dimensional prob-

lems in linear elasticity [31]. The results presented in this paper are also very encouraging.

ACKNOWLEDGEMENTS

This research has been supported by a University Research Programs (URP) grant from Ford Motor Companyto Cornell University. The computing for this research was carried out at the Cornell Theory Center.

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The boundary node method for three-dimensional problems in potential theory