The boundary node method for three-dimensional linear elasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng. 46, 1163–1184 (1999)

THE BOUNDARY NODE METHOD FORTHREE-DIMENSIONAL LINEAR ELASTICITY

MANDAR K. CHATI1;†, SUBRATA MUKHERJEE1;∗;‡ AND YU X. MUKHERJEE2;§

1Department of Theoretical and Applied Mechanics; Cornell University; Ithaca; NY 14853; U.S.A2DeHan Engineering Numerics; 95 Brown Road; Box 1016; Ithaca; NY 14853; U.S.A.

SUMMARY

The Boundary Node Method (BNM) is developed in this paper for solving three-dimensional problems inlinear elasticity. The BNM represents a coupling between Boundary Integral Equations (BIE) and MovingLeast-Squares (MLS) interpolants. The main idea is to retain the dimensionality advantage of the former andthe meshless attribute of the later. This results in decoupling of the ‘mesh’ and the interpolation procedure.For problems in linear elasticity, free rigid-body modes in traction prescribed problems are typically eliminatedby suitably restraining the body. However, an alternative approach developed recently for the BoundaryElement Method (BEM) is extended in this work to the BNM. This approach is based on ideas from linearalgebra to complete the rank of the singular sti�ness matrix. Also, the BNM has been extended in the presentwork to solve problems with material discontinuities and a new procedure has been developed for obtainingdisplacements and stresses accurately at internal points close to the boundary of a body. Copyright ? 1999John Wiley & Sons, Ltd.

KEY WORDS: meshless; boundary node method; linear elasticity

1. INTRODUCTION

The task of meshing a 3-D object with complicated geometry can be arduous, time-consumingand 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 Finite Di�erence Method (FDM), Finite Ele-

ment Method (FEM) and Boundary Element Method (BEM) can be used, but often with di�culty,however, to solve problems involving changing domains such as large deformation or crack prop-agation. The main di�culty in these problems is the task of re-meshing a three-dimensional objectafter large deformation or crack propagation. In recent years, novel computational algorithms have

∗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§President

Contract=grant sponsor: Ford Motor Company

CCC 0029-5981/99/321163–22$17.50 Received 6 January 1999Copyright ? 1999 John Wiley & Sons, Ltd. Revised 25 March 1999

1164 M. K. CHATI, S. MUKHERJEE AND Y. X. MUKHERJEE

been proposed that largely circumvent the problems associated with 3-D meshing. These methodshave been collectively 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 DEM 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] and a Local Boundary IntegralEquation (LBIE) method [6]. The Element Free Galerkin (EFG) method has been popularizedby Belytschko and his co-workers. The main idea in the EFG method is to use Moving Least-Squares (MLS) interpolants to construct the trial functions used in the Galerkin weak form. Awide variety of problems have been solved using the EFG method. In the introductory paper byBelytschko et al. [2], the EFG method was applied to 2-D problems of linear elasticity and heatconduction with great success. The method has been applied to problems in fracture mechanicswith crack growth [7], dynamic fracture [8; 9], plate bending [10] and for the analysis of thinshells [11]. More recently, the EFG method has been applied to 3-D fracture mechanics [12].A recent special issue of the journal Computer Methods in Applied Mechanics and Engineeringcontains excellent review articles by Belytschko et al. [13] and Liu et al. [14] on meshless methods.Another excellent source of information on the RKPM is an overview article by Liu et al. [15].Although the EFG method gained immediate recognition and is now a well-established method,

an issue about accurate imposition of essential 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 co-workers have employed various approaches for the satisfaction of essential boundary con-ditions. These include the use of Lagrange multipliers [2], use of traction as Lagrange multipliers[16] and the use of a layer of �nite elements along the boundary where essential conditions areprescribed [17]. However, this issue has been successfully resolved by Mukherjee and Mukherjee[18]. The strategy proposed in their paper for alleviating the above problem involves a new def-inition of the discrete norm used for the construction of the MLS interpolants. Chen et al. [19]have independently proposed essentially the same idea to solve the boundary condition problemfor Reproducing Kernel Particle Methods (RKPM).Recently, Mukherjee and Mukherjee [20] proposed a meshless method which they call the

Boundary Node Method (BNM). The BNM involves a coupling between MLS interpolants andBIE. The BIE has the well-known dimensionality advantage, i.e. for a three-dimensional objectmeshing is required only on the two-dimensional bounding surface of a body. The BNM is par-ticularly attractive for three-dimensional problems since it combines the meshless attribute of theMLS interpolants with the dimensionality advantage of the BIE. Also, the input data structure forsolving a boundary value problem involves nodes just on the bounding surface of a body, whilethe (surface) cells are used for integration only. The topology of the cells can be much simplerthan conventional boundary elements, in that, for example, some cells can be divided into smallerones without a�ecting their neighbours in any way (Figure 1). This data structure considerablyreduces the task of meshing and a solid model of a 3-D body can be directly used as input forstress analysis. This method has been successfully tried for 2-D problems in potential theory [20]

Copyright ? 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 46, 1163–1184 (1999)

THE BOUNDARY NODE METHOD FOR THREE-DIMENSIONAL LINEAR ELASTICITY 1165

Figure 1. ‘Salted potato’ input data structure for the BNM—cells and collocation nodes.

and linear elasticity [21]. Very recently, the method has been extended to solve three-dimensionalproblems in potential theory by Chati and Mukherjee [22]. The present paper extends the frontiersof BNM into solving problems in three-dimensional linear elasticity.It is a well-known fact in �nite element and boundary element methods that problems with

tractions prescribed on the entire boundary lead to singular matrices at the discretized level. Theusual way to circumvent this problem is to apply su�cient restraint on the body by prescribingdisplacements at suitable points on the boundary. An alternative approach is to suitably modifythe singular matrix using ideas from linear algebra. This idea has been originally suggested byVerchery [23] for singular symmetric sti�ness matrices and has been extended by Lutz et al. [24]for singular non-symmetric system matrices arising in the boundary element method. In this paper,this idea is extended to solve purely traction prescribed problems using the BNM.In this paper, the BNM has also been extended to treat problems with material discontinuities.

The EFG method has been applied to two-dimensional problems with material discontinuities byCordes and Moran [25]. The method developed in their paper produced considerable oscillationsin stress and strain as compared to the exact solution. However, the application of BNM to thisclass of problems yields promising results.Another important new development here is an algorithm for obtaining accurate displacements

and stresses, at internal points close to the boundary of a body, by the BNM.The main contributions of this paper can be summarized as follows:

1. Extension of the BNM to solve three-dimensional problems in linear elasticity.2. Developing the BNM to solve purely traction prescribed problems by using ideas in linearalgebra.

3. Treatment of problems in linear elasticity with material discontinuities.4. Accurate determination of displacements and stresses at internal points close to the boundaryof a body.

This paper is organized as follows. First, a surface-based MLS interpolation scheme is describedfor problems in three dimensions. The next section couples the MLS with the well-known BIE forproblems in linear elasticity. Numerical results are presented for a variety of examples includingthe classical problems of Lam�e and Kirsch in three dimensions. The main motivation for choosingthese problems is that a careful comparison between exact and numerical (BNM) solutions is



possible. Finally, the scheme for numerically eliminating rigid-body modes in completely tractionprescribed problems (Neumann-type problems), by using ideas in linear algebra, is described inthe appendix.

2. MOVING LEAST-SQUARES (MLS) APPROXIMATION

An MLS interpolation scheme developed by Chati and Mukherjee [22] for three-dimensional BNMproblems is applied to linear elasticity. Since the BNM nodes lie only on the boundary @B of a3-D body B, curvilinear co-ordinates are necessary to de�ne the MLS interpolants on the surface.The variables on the boundary that need to be interpolated include displacement, ui, and trac-

tion, ti; i = 1; 2; 3. However, for the sake of brevity of index notation, the interpolation scheme isdescribed below using the variables u and t to represent any particular component of the displace-ment and traction, respectively.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 node.s: A generic point with curvilinear co-ordinates (s1; s2).

The MLS interpolants for u and t on @B are de�ned as

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

t(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 t) 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 t. In this paper, numericalresults have been obtained using quadratic interpolants, i.e.

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

where

si = si − sEi ; i=1; 2 (5)

It has been observed, through numerical experiments (see [22]) that using a cubic basis hasrelatively little e�ect towards improving the accuracy of the results. Hence, only a quadratic basisis used in this paper.The coe�cients ai and bi are obtained by minimizing weighted discrete L2 norms de�ned as

Ru=n∑I=1w(dI )[pT(sI − sE)a − uI ]2 (6)

Rt =n∑I=1w(dI )[pT(sI − sE)b− tI ]2 (7)



Figure 2. The nodes 1, 2 and 3 lie within the domain of dependence of E. The ranges of in uence of nodes 1–4 areshown as gray circles—the last one is truncated at the edges of the body

where the summation is carried over the n boundary nodes for which the weight functionw(dI ) 6=0. The quantity dI = g(s; sI ) is the length of the geodesic on @B between s and sI—seealso equation (26). These n nodes are said to be within the domain of dependence of a point s(Figure 2). Also, (sI1 − sE1 ; sI2 − sE2 ) are the local surface co-ordinates of the boundary nodes withrespect to the evaluation point sE =(sE1 ; s

E2 ) and uI and tI are the approximations to the nodal

values uI and tI . These equations above can rewritten in compact form as

Ru = (P(sI − sE)a − u)TW(s; sI )(P(sI − sE)a − u) (8)

Rt = (P(sI − sE)b− t)TW(s; sI )(P(sI − sE)b− t) (9)

where

uT = (u1; u2; : : : ; un) (10)

tT= (t1; t2; : : : ; tn) (11)

P(sI ) =

1 p2(s11; s12) · · · pm(s11; s

12)

1 p2(s21; s22) · · · pm(s21; s

22)

......

......

1 p2(sn1; sn2) · · · pm(sn1; s

n2)

(12)

and

W(s; sI ) ≡W(dI ) =

w(d1) 0 · · · 0

0 w(d2) · · · 0

......

......

0 0 · · · w(dn)

(13)



The stationarity of Ru and Rt , with respect to a and b, respectively, leads to the equations

a(s) = A−1(s)B(s)u (14)

b(s) = A−1(s)B(s)t (15)

where

A(s) =PT(sI − sE)W(s; sI )P(sI − sE) (16)

B(s) =PT(sI − sE)W(s; sI ) (17)

Also, de�ne

CjI = (A−1B)jI (18)

It is noted from above that the coe�cients ai and bi turn out to be functions of s. Substitutingequation (14) into equation (2) and equation (15) into equation (3), leads to

u(s) =n∑I=1�I (s) uI (19)

t(s) =n∑I=1�I (s)tI (20)

where the shape functions �I are

�I (s) =m∑j=1pj(s − sE)CjI (s) (21)

At an evaluation point E, one has

u(sE) =n∑I=1�I (sE) uI (22)

with

�I (sE) =m∑j=1pj(0; 0)CjI (sE) (23)

As mentioned previously, u and t are approximations to the actual nodal values u and t. 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, carrying outthis procedure for each of the collocation nodes, one gets

[H]{uk} = {uk}; k = 1; 2; 3 (24)

[H]{tk} = {tk}; k = 1; 2; 3 (25)

Equations (24) and (25) relate the nodal approximations of the displacement and traction to theactual nodal values of the same.There have been a variety of weight functions proposed in the past for meshless methods. It

has been observed (see [22]) that the Gaussian weight function yields the best results for potential



problems in three dimensions using the BNM. Hence, the numerical results in this work have beenobtained using only the Gaussian weight function described below:

w(d) ={e−(d=c)

2for d61

0 for d¿1(26)

where d = dI =d = g(s; sI )=d and c is a constant. Here dI = g(s; sI ) is the minimum distance,measured on the surface @B, between a point s and the collocation node I and d determines therange of in uence associated with each node I . This minimum distance is the geodesic measuredon the bounding surface @B. In this paper, the focus is on very simple geometries like spheres andcubes for which the calculation of geodesics is trivial. For a sphere the geodesics are great circles.For a cube, results have been obtained treating each face of the cube as independent ‘Panels’.In other words, the range of in uence of a node is arbitrarily truncated at edges and corners ofthe cube (Figure 2). Thus, the calculation of a geodesic is reduced to calculating the minimumdistance between two points lying on a plane which is a straight line. It has been numericallyobserved (see [21]) that an arbitrary truncation of the range of in uence at the edges still yieldsoverall acceptable results.The reader is cautioned that there are many potential pitfalls that must be avoided in order to

carry out successful implementation of the method. The comments below try to highlight someof the issues. Also, the reader is referred to the paper by Chati and Mukherjee [22] for a moreelaborate explanation.

Comment 1 (Invertibility of matrix A). The matrix A, composed of the matrices P and W(equation (16)), needs to be invertible for the construction of the shape functions. It is necessaryto choose the parameter d, such that n¿m, where n is the number of nodes in the domain ofdependence of an evaluation point and m is the order of the polynomial basis.

Comment 2 (Matrix H). As noted above, the matrix H relates the actual nodal values totheir 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.

Comment 3 (The nature of s1; s2). The use of local curvilinear co-ordinates (s1 − sE1 ; s2 − sE2 )to de�ne the monomials pi is vital for the success of this numerical scheme. Also, this choicesimpli�es the computation of the shape functions �I (sE) to some extent.

3. BOUNDARY INTEGRAL EQUATION AND DISCRETIZATION

The standard boundary integral equation for 3-D linear elasticity [26], in regularized form, and inthe absence of body forces, can be written as

0 =∫@B[Uik(P;Q)tk(Q)− Tik(P;Q)(uk(Q)− uk(P))] dSQ (27)



where uk and tk are the components of the displacement and traction respectively, and the well-known Kelvin kernels are

Uik =1

16�(1− �)Gr [(3− 4�)�ik + r; ir; k ] (28)

Tik =−1

8�(1− �)r2[{(1− 2�)�ik + 3r;ir;k} @r@n − (1− 2�)(r; ink − r; kni)

](29)

where r is the distance between the source point P to a �eld point Q and ni are the components ofthe unit normal at the �eld point Q. A comma denotes a derivative with respect to a �eld point, i.e.

r; i =@r@yi

=yi(Q)− yi(P)

r(30)

The MLS interpolants derived in Section 2 are used to approximate ui and ti on the boundary@B. In order to carry out the integrations, the bounding surface is discretized into cells. A varietyof shape functions have been used to interpolate the geometry. In particular, the bilinear (Q4)element and quadratic (T6) triangle have been used. These ‘geometric’ shape functions can befound in any standard text on the FEM (see [27; 28]).Substituting the expressions for ui and ti (equations (19) and (20)) into equation (27) and

dividing @B into Nc cells, one gets

0 =Nc∑m=1

∫@Bm

[Uik(P;Q)

nQ∑I=1�I (Q)tkI − Tik(P;Q)

{ nQ∑I=1�I (Q) ukI −

nP∑I=1�I (P) ukI

}]dSQ (31)

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 point Qand nP nodes are in the domain of dependence of the source point P. In order to evaluate thenon-singular integrals a 7 point and a 3×3 Gauss quadrature is used over triangles and rectangles,respectively. However, as Q→P the kernels Uik and Tik become weakly and strongly singular,respectively, and special integration techniques need to be used to evaluate these weakly singularintegrals (see [32]).The discretized form of equation (31) becomes

[A( u )]{ u}+ [A(t )]{t} = {0} (32)

For NB nodes on the bounding surface, there are a total of 12NB quantities on the boundary i.e. 3NBvalues each of ui and its nodal approximation ui and so also for ti. For a well-posed problem, valuesof either ui or ti are known at each node on the boundary, so 3NB nodal values are given. Therefore,9NB equations are needed to solve for the 9NB remaining unknowns. Equation (32) consists of 3NBequations and equations (24) and (25) consist of 3NB equations each. Thus, well-posed boundaryvalue problem can be solved using equation (32), in combination with equations (24) and (25).To obtain the stress distribution within a body, the derivative of the primary BIE needs to be

taken with respect to a source point inside the body. It is well known that the Boundary IntegralEquation (BIE) and the Hypersingular Boundary Integral Equation (HBIE) can yield poor resultswhen used to compute the displacement and stress at internal points close to the boundary of abody. This is due to the fact that the kernels become nearly singular=hypersingular as p→P. Andthe problem is even more severe for the computation of stresses as the kernels in the HypersingularBIE (HBIE) have a stronger singularity than the kernels in the primary BIE. This problem has been



overcome in the present work by employing a novel regularization approach. Attention is drawnto equations (3) and (22) in the paper by Cruse and Richardson [29]. Now, in these equations,let p∈B be an internal point close to @B and P ∈ @B be a regular boundary point close to p.A novel application of these equations has been carried out in the present paper in order to �ndui(p) and �ij(p). It should be stated that these equations were used for di�erent purposes in thepaper by Cruse and Richardson [29].

4. NUMERICAL RESULTS AND DISCUSSION

Several numerical results have been obtained using the BNM for some well-known examples inlinear elasticity. In order to evaluate the performance of the numerical method, an L2 error normis de�ned on the surface of the body as follows:

� =100|u|max

√1N

N∑i=1(un − ue)2 % (33)

where � is the percentage L2 error over N nodes. Also, un and ue refer to the numerical and exactsolutions respectively and |u|max is the maximum value of u over N nodes. Here u could be acomponent of the displacement or traction depending on the boundary conditions of the problem.Numerical results have been obtained using the Gaussian weight function with c = 0·4. The

parameter d which decides the compact support associated with each node is assumed to behomogeneous, i.e. it is constant for all the nodes. The invertibility of matrix A places a lowerbound on d, i.e. d should be such that n¿m and the condition number of H places an upperbound of d. In this work, d has been chosen such that n ∼ 2m− 3m (see also [22]).The example in Section 4.1 shows the numerical results for the patch test on a cube. Section 4.2

demonstrates the capability of the BNM to solve problems with mixed boundary conditions. The3-D Lam�e and Kirsch problems described in Sections 4.3 and 4.4 have been solved by prescribingtractions over the entire boundary. The singular matrix for each problem is regularized by themethod outlined for the BEM in [24] and also summarized for the BNM in the appendix. Finally,the solution to the bimaterial problem is presented in Section 4.5. The numerical results presentedin this paper are very encouraging.

4.1. Patch test on a cube

The patch test involves imposing a linear function of co-ordinates as an exact solution for thedisplacements. The satisfaction of the patch test then requires that the displacements within thebody be described by the same linear function and that the stresses be constant. The followingexact solution is used for the patch test

ux =2x + y + z

2; uy =

x + 2y + z2

; uz =x + y + 2z

2(34)

The cube is bounded by the planes x= ±1; y= ±1 and z= ±1. The boundary value problem issolved with the corresponding tractions prescribed on all faces of the cube. The scheme presentedin the appendix is used to eliminate the rigid-body modes numerically. Numerical results havebeen obtained using 96 bilinear Q4 cells for interpolating the geometry. One node is used per cell.The material constants were chosen to be E=1·0; � = 0·25.



Figure 3. ux displacement along the x-axis of the cube for the patch test

Figure 4. �yy and �xy along the x-axis of the cube for the patch test

Figures 3 and 4 show the x-component of the displacement (ux), the constant normal stress(�yy) and shearing stress (�xy) along the x-axis of the cube. These are internal points. Figures 3and 4 clearly show that BNM passes the patch test successfully.It has been observed (see [22]) that an important ingredient for the success of BNM is the

choice of locations of collocation nodes within each cell. As mentioned before, one node per cellis used in this example. Since each cell is mapped from the physical space to the parent space(Figure 5), the location of this node within a cell is chosen in the parent space. Figure 6 showsthe variation in the displacement L2 error norm for the patch test for various locations of the



Figure 5. Mapping to the parent space for a bilinear (Q4) cell

Figure 6. Variation in L2 error in displacement (ux) with change in location of the (only) node in each cell

collocation node in the parent space. It can be seen that placing the node at the centroid of eachcell gives the best numerical results.

4.2. One-dimensional strain problem

Figure 7 describes the one-dimensional strain problem solved using the three-dimensional BNM.The main idea behind solving this elementary problem is to demonstrate the ability to solveproblems with mixed boundary conditions using the BNM.An interesting feature that has been explored in the meshless literature is the irregular placement

of nodes within a cell (see [2]). In order to gain an understanding of irregularity in the nodalstructure, the numerical results obtained using a regular as well as an irregular nodal arrangementare compared for the one-dimensional strain problem. Figures 8 and 9 show the regular and



Figure 7. One-dimensional strain problem

Figure 8. Cells and regular nodal structure used for theone-dimensional strain problem on a cube

Figure 9. Cells and irregular nodal structure used for theone-dimensional strain problem on a cube

irregular nodal structure used to generate the numerical results. Figure 10 compares the exactsolution for the internal displacement (ux), along the x-axis, with the BNM solutions obtainedby using a regular nodal structure and an irregular nodal structure. It can be seen that there ishardly any noticeable di�erence between the internal displacements obtained using the two nodalcon�gurations considered. The material and geometric parameters used to obtain the numericalresults were E = 1·0; � = 0·25; a = 2·0; �0 = 1·0.

4.3. 3-D Lam�e problem

The 3-D Lam�e problem consists of a hollow sphere under internal pressure. Figure 11 showsa schematic of the problem under consideration. The numerical solution for this problem wasobtained using material parameters: E = 1·0; � = 0·25 and geometric parameters a = 1·0; b = 4·0with internal pressure pi = 1. The numerical solution was obtained by prescribing tractions allover the boundary and then modifying the resulting singular matrix using the ideas presented in the



Figure 10. Comparison of internal displacement (ux) along the x-axis for a regular and irregular nodal data structure forthe one-dimensional strain problem

Figure 11. 3-D Lam�e problem: hollow sphere under internal pressure

appendix. The entire surface of the hollow sphere was modelled without considering the inherentsymmetry within the problem. The exact solution for the radial displacement, radial and tangentialstresses is given as [30]

ur =pia3r

E(b3 − a3)[(1− 2�) + (1 + �) b

3

2r3

]

�rr =pia3(b3 − r3)r3(a3 − b3) (35)

�tt =pia3(2r3 + b3)2r3(b3 − a3)



Figure 12. Internal radial displacement along the x-axis for the 3-D Lam�e problem

Table I. Convergence study of the L2 error in radial displacement (ur) at the inner and outer surface for the3-D Lam�e problem

L2 error Mesh 1 (144 cells) Mesh 2 (256 cells) Mesh 3 (576 cells)Per cent per cent per cent

Outer surface 1·414 0·428 0·261Inner surface 0·574 0·263 0·155

The cell=nodal structure used to obtain the numerical results consists of 72 quadratic T6 cells oneach surface of the hollow sphere with 1 node per cell. Figure 12 shows the radial displacementalong the x-axis compared with the analytical solution. The radial and tangential stresses arecompared to the exact solution in Figure 13. It can be clearly seen that the numerical results arein excellent agreement with the analytical solution. Also, a convergence study is carried out todemonstrate the robustness of the proposed numerical method. Table I shows the L2 error in theradial displacement (ur) at the inner and outer surface of the sphere. One node per cell is usedin all calculations.

4.4. 3-D Kirsch’s problem

The 3-D Kirsch’s problem consists of examining the stress distribution in the vicinity of asmall spherical cavity in a cube subjected to far-�eld uniform tension (Figure 14). The mate-rial parameters were chosen to be E=1·0; �=0·25 and geometric parameters were chosen asa=1·0; b=10·0. Again, the loading is applied without restraining any rigid-body modes andthe scheme presented in the appendix is used to obtain meaningful numerical results. The exactsolution for the normal stress (�zz), for points in the plane z=0, is given as [30]

�zz = �0

[1 +

4− 5�2(7− 5�)

(ar

)3+

92(7− 5�)

(ar

)5](36)



Figure 13. Radial and tangential stress distributions inside the sphere for the 3-D Lam�e problem

Figure 14. 3-D Kirsch problem: a cube with a spherical cavity loaded in far-�eld uniaxial tension

Figure 15 shows a comparison between the BNM solution and the exact solution for the normalstress (�zz) along the x-axis. Again, it can be clearly seen that the BNM solution is in excellentagreement with the analytical solution. The cell structure consists of 96 Q4 cells modelling thecube and 72 T6 cells modelling the spherical cavity, again with one node per cell. It should benoted that the new algorithm mentioned in Section 3 is essential for obtaining accurate values ofstresses near the surface of the cavity.

4.5. 3-D bimaterial problem

The BNM has been extended to solve problems involving material discontinuities. Figure 16shows a schematic of two perfectly bonded spheres of two di�erent materials. Numerical results



Figure 15. Normal stress distribution along the x-axis ahead of the hole for the 3-D Kirsch problem

Figure 16. Bimaterial sphere subjected to uniform external radial displacement (u0)

for this model have been obtained by prescribing displacements on the outer boundary. One couldalso prescribe tractions over the entire outer surface and then appropriately modify the schemepresented in the appendix for solving traction prescribed problems. This is planned for the future.Upon prescribing a radial displacement (u0) on the outer surface the exact solution for the radial

displacement and stresses in each material is given below. For material 1,

u(1)r = A1r1− 2�1E1

(37)

�(1)rr = �(1)tt =A1 (38)



Figure 17. Radial displacement within the bimaterial sphere along the x-axis

where the constant A1 is de�ned below. For material 2,

u(2)r =rE2

[A2(1− 2�2)− B2

2r3(1 + �2)

](39)

�(2)rr = A2 +B2r3; �(2)tt =A2 −

B22r3

(40)

where the constants are

A1 = A2 +B2R31

(41)

�=E1(1− 2�2)E2(1− 2�1) ; �=

E1(1 + �2)2E2(1− 2�1) (42)

A2 =u0E2R2R31

1 + �C(1 + �)

; B2 = − u0E2R2C

(43)

C =(1− 2�2)(1 + �)R31(1− �)

+1 + �22R32

(44)

The material and geometric parameters chosen for the two materials were: for material 1, E1 =1·0; �1 = 0·28; R1 = 1·0 and for material 2, E2 = 2·0; �2 = 0·33; R2 = 4·0. A constant radialdisplacement is prescribed on the outer boundary of material 2 (u0 = 1·0). Figure 17 shows acomparison between the numerical solution and the exact solution for the radial displacementwithin the two materials. Figures 18 and 19 show the radial stress and tangential stress along theline x-axis. It can be seen that the jump in the tangential stress at the bimaterial interface is verywell captured by the BNM.



Figure 18. Radial stress distribution within the bimaterial sphere along the x-axis

Figure 19. Tangential stress distribution within the bimaterial sphere along the x-axis

5. CONCLUSIONS

In this paper, the BNM has been extended to solve three-dimensional problems in linear elasticity.The method has been tested on a variety of classical examples in linear elasticity.It is seen that the matrices A and H play a crucial role in the successful implementation of

the method. The BNM passes the patch test for linear elasticity successfully. It has been observedthat the location of collocation nodes is an important ingredient for the success of the method.For most cases, using one node per cell and placing this node at the centroid of a cell yields



the best numerical results. Also, choosing the location of the collocation node from a randomuniform distribution yields almost identical numerical results as those obtained from a regularnodal distribution, at least for the cube problem.The idea of eliminating rigid-body modes at the discretized level has been successfully imple-

mented. This totally circumvents the problem of having to restrain the body appropriately so as toeliminate the rigid-body modes. The BNM has also been tested on problems with material discon-tinuities. The stress discontinuities across a bimaterial interface have been successfully capturedby the BNM.A new algorithm, for obtaining accurate displacements and stresses at internal points close to the

boundary, has been developed, and successfully implemented in several of the numerical examples.A ‘non-conforming’ collocation approach is used in this work, i.e. all the collocation points

are placed inside the boundary cells. Also, the domains of dependence of evaluation points aretruncated at edges and corners (Figure 2). While the method has been successfully implemented fora cube example with mixed boundary conditions (Section 4.2), it is suspected that the above twofacts are the cause of numerical di�culties that have been encountered when trying to model theLam�e problem on a one-eighth sphere using symmetry boundary conditions. Collocation on edgesand corners will provide better communication between piecewise smooth ‘panels’ that typicallyde�ne the boundaries of bodies in practical problems, and is expected to enable modelling ofproblems such as that of the one-eighth sphere, by the BNM. This is planned for the future.Overall, the frontiers of the BNM have been extended to solve problems of great engineering

interest. Due to its considerable exibility in mesh generation and relative ease in implementation,this method has tremendous potential for e�ciently solving a wide range of industrial problems.This is planned for the future.

APPENDIX: ELIMINATION OF RIGID-BODY MODES

The discretized form of the boundary integral equation can be written as

[A(u)]{u}+ [A(t)]{t}= {0} (45)

where u and t are approximations to the nodal values of displacement and traction, respectively.The nodal approximations can be related to the actual nodal values,

[H]{u}= {u}; [H]{t}= {t} (46)

Equations (45) and (46) can be combined to form

Au=Bt (47)

where

A= [A(u)][H]−1; B= [A(t)][H]−1 (48)

Equation (48) is the �nal form that can be used to solve boundary value problems in linearelasticity. However, it is well known that unique displacements cannot be obtained for completelytraction prescribed problems. This is due to the fact that the matrix A is singular (note that His a well-conditioned non-singular matrix). The scheme presented below suitably updates the rank



of the matrix A so as to obtain unique displacements for completely traction prescribed problems.This procedure is described in detail in [24] and is summarized below.Firstly, note that the null space of matrix A consists of those displacements that produce zero

traction. Thus, the three rigid-body translations along and three rotations about the x; y and z axesconstitute the null space of A, i.e.

AU= 0 (49)

where U can be written as

U= [u1 u2 u3 u4 u5 u6] (50)

In the above, u1; u2; u3 represent the rigid-body translations and u4; u5; u6 are the rotations aboutthe x; y and z axes, respectively. Note that the dimension of the null space of A is 6. Thus, thegeneral solution to equation (47) can be represented as

u= z +Uy (51)

Now, by construction, we restrict z such that

UTz= 0 (52)

In order to better understand the matrix A, a singular-value decomposition (SVD) of matrix A isconsidered. This can be represented as

A=W�YT (53)

where

W= [w1 w2 · · · wn−6 v1 v2 v3 v4 v5 v6]Y= [y1 y2 · · · yn−6 u1 u2 u3 u4 u5 u6] (54)

�= diag[�1 �2 · · · �n−6 0 0 0 0 0 0]where wi ; i=1; 2; : : : ; n− 6 constitutes the range space of matrix A and ui ; i=1; 2; : : : ; 6 consti-tutes the null space of matrix A. Similarly, note that the columns vi ; i=1; 2; : : : ; 6 consist of thenull space of AT. In order to accomplish a rank 6 update of matrix A, the following modi�cationis suggested:

C=A+VUT (55)

where the columns of U span the null space of matrix A and the matrix V needs to be chosen suchthat its columns span the null space of AT. However, it su�ces to choose the columns of V to lieoutside the range space of matrix A. For the physical problem at hand, this amounts to choosinga traction �eld for which no displacement solution exists. This can be easily accomplished bychoosing tractions that fail to satisfy equilibrium. Let Q be such a matrix, so that

Q= [q1 q2 q3 q4 q5 q6]

V=BQ =∈Range space of A (56)



Now, if q1 is chosen such that tractions are prescribed only along the x1 direction then it wouldcertainly violate equilibrium. But, the displacement vector u1 consists of precisely those entriesthat are desirable in q1. Hence, one can take

q1 = u1 (57)

Similarly, extending the above argument to other components of qi, we get,

qi= ui ; i=1; 2; : : : ; 6 (58)

Thus, the desired matrix V is given as

V=BU (59)

At this point, it still remains to show that by updating the rank of matrix A, using equation (55),one is able to solve traction prescribed problems. The few lines of algebra below establishes thedesired result.

Au = A(z −Uy) From equation (51)

= Az From equation (49)

= (C− VUT)z From equation (55)

= Cz From equation (52)

Thus, equation (47) can be replaced by

Cz=Bt (60)

Note that matrix C is non-singular and so one can solve problems in linear elasticity with tractionsprescribed all over the boundary.The additional computational cost for updating the rank of matrix A can be easily computed.

First, matrix V needs to be formed and then matrix C is formed using equation (55). In orderto form matrix V, using equation (59), 12n2 ops are required and then to form matrix C, usingequation (55), 13n2 ops are required. Here, n is the size of the square matrix A. Hence, a totaladditional cost of 25n2 ops is required to perform a rank 6 update of matrix A. However, sincethe computational cost to solve the linear system is O(n3), this additional cost of O(n2) is marginal.

ACKNOWLEDGEMENTS

This research has been supported by a University Research Programs (URP) grant from FordMotor Company to Cornell University. The computing for this research was carried out usingthe resources of the Cornell Theory Center, which receives funding from Cornell University,New York State, the National Center for Research Resources at the National Institutes of Health,the National Science Foundation, the Defence Department Modernization program, the UnitedStates Department of Agriculture, and corporate partners.

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The boundary node method for three-dimensional linear elasticity