Exponentialbasisfunctionsinsolutionofstaticandtimeharmonic ... · Published online 1 September 2009 in Wiley InterScience (). DOI: 10.1002/nme.2718 ... In engineering mechanics, where

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2010; 81:971–1018Published online 1 September 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2718

Exponential basis functions in solution of static and time harmonicelastic problems in a meshless style

B. Boroomand1,∗,†, S. Soghrati2 and B. Movahedian1

1Department of Civil Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran2Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, U.S.A.

SUMMARY

In this paper, exponential basis functions (EBFs) are used in a boundary collocation style to solveengineering problems whose governing partial differential equations (PDEs) are of constant coefficienttype. Complex-valued exponents are considered for the EBFs. Two-dimensional elasto-static and timeharmonic elasto-dynamic problems are chosen in this paper. The solution procedure begins with firstfinding a set of appropriate EBFs and then considering the solution as a summation of such EBFswith unknown coefficients. The unknown coefficients are determined by the satisfaction of the boundaryconditions through a collocation method with the aid of a consistent and complex discrete transformationtechnique. The basis and various forms of the transformation have been addressed and discussed. We shallpropose several strategies for selection of EBFs with the aid of the basis explained for the transformation.While using the transformation, the number of EBFs should not necessarily be equal to (or less than) thenumber of boundary information data. A library of EBFs has also been presented for further use. The effectof body forces is included in the solution via construction of particular solution by the use of the discretetransformation and another series of EBFs. A number of sample problems are solved to demonstrate thecapabilities of the method. It has been shown that the time harmonic problems with high wave numbercan be solved without much effort. The method, categorized in meshless methods, can be applied to manyother problems in engineering mechanics and general physics since EBFs can easily be found for almostall problems with constant coefficient PDEs. Copyright q 2009 John Wiley & Sons, Ltd.

Received 21 January 2009; Revised 24 June 2009; Accepted 25 June 2009

KEY WORDS: fundamental solution; exponential basis; elasto-static; elasto-dynamic; discretetransformation; meshless method; exponential functions

1. INTRODUCTION

Along with the advances in technology and science the number of demands for highly accuratesimulations is increasing. In engineering mechanics, where stress analysis plays a key role in the

∗Correspondence to: B. Boroomand, Department of Civil Engineering, Isfahan University of Technology, Isfahan84156-83111, Iran.

†E-mail: [email protected]

Copyright q 2009 John Wiley & Sons, Ltd.

972 B. BOROOMAND, S. SOGHRATI AND B. MOVAHEDIAN

simulation of the problem, several methods have so far been developed as the simulation tools.Among them are the finite element method (FEM) [1] and the boundary integral method [2] (orboundary element method (BEM) [3]). BEM is nowadays receiving attention by many scientistssince it is capable of giving highly accurate results. However, availability of Green’s functions isone of the restrictions in the method.

Both FEM and BEM need a meshing procedure, one for the whole domain and another for theboundary. Meshless methods are also emerging in two forms, one utilizing points in the wholedomain with the aim of being used in place of FEM [4–8], and another requiring points on theboundaries with the aim of being used in place of BEM. The method of fundamental solutions(MFS) is classified in the latter category [9, 10]. Both BEM and MFS are sometimes categorizedin the so-called ‘Trefftz’ type of methods [11]. In such a category of methods a series of basisfunctions, satisfying the governing differential equation, are found and the solution is obtainedby approximately satisfying the boundary conditions. In the case of BEM, the boundaries ofthe solution domain are discretized into elements and the boundary conditions are approximatelysatisfied in a weighted residual approach through some integral expressions. In MFS, however,the boundaries are discretized into points and the boundary conditions are satisfied in a colloca-tion approach, while the sources of the fundamental solutions are placed outside of the domainsolution.

It is well understood that BEM and MFS both need Green’s/fundamental functions. Thesefunctions are not available for a wide range of problems, considering just, for instance, those withconstant coefficient partial differential equations (PDEs) as the governing equations. On the otherhand, it is also mathematically understood that exponential basis functions (EBFs) can play the roleof bases for the solution of PDEs with constant coefficients [12, 13]. This can be regarded as anadvantage of EBFs when compared with Green’s or fundamental functions. Therefore, constructionof a solution procedure with the use of EBFs may be considered as a gateway for the solution ofmany PDEs with constant coefficients.

It may be noted that trigonometric functions can individually be written as a summation of specialforms of EBFs. Therefore, in this sense solution methods using the Fourier series (or integrals)may be considered in the category of methods using EBFs. Such an interpretation of using EBFscovers many analytical solution methods available in the mathematical sources [12, 13] or thosein the structural/mechanical engineering fields [14], which are basically given for problems withspecific domain shapes (e.g. on rectangle, sector, etc.) and boundary conditions. Disregarding sucha general interpretation for the use of EBFs, here we are interested in employing them in a widerrange of problems whose solutions can only be obtained numerically. In this form of approach,the EBFs are not necessarily orthogonal functions.

In the literature there are some studies focusing on the improvement of some numerical methodsby employing EBFs. For instance in the realm of wave propagation modelling via FEM in acousticproblems, exponential wave bases are used to enrich the ordinary shape functions so that thedispersion effect is reduced (see [15, 16] for instance). As another case one can find exponentialwave bases in the improvement of the finite difference method while modelling wave problems[17]. In such studies exponential wave bases, known as plane waves, have been locally used withfixed choices of bases. The exponents of the wave bases are either purely real or purely imaginary.As will be explained later, in this study the EBFs are used in a more general way. Moreover,the exponents of the EBFs are considered to include complex-valued parameters, which are to bechosen in the Gaussian plane. This provides infinitely many choices for EBFs and thus the EBFselection becomes a challenging task.

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2010; 81:971–1018DOI: 10.1002/nme

SOLUTION OF STATIC AND TIME HARMONIC ELASTIC PROBLEMS 973

In this paper we shall propose a method based on the use of EBFs as the bases for 2D elasticityproblems. To this end we shall first find the appropriate EBFs and then impose the boundaryconditions through a collocation approach. With these features, the method may be categorized inthe meshless Trefftz type of methods. As will be seen later, the formulation is capable of beingconverted to a boundary-mesh-based method though it is not intended so in this paper. Here theproblems are treated as the generic ones in general physics whose PDEs are of constant coefficienttype. By doing so, we show the way that the solution of other problems can be dealt with usingthe proposed method.

As will be seen later, we shall employ a complex discrete transformation technique for impositionof the boundary conditions. The transformation, proposed in [18–20], allows us to use a largernumber of basis functions compared with the number of boundary information. Here we shallexplain the basis of the transformation and its various possible forms. The transformation will alsobe used in the construction of the particular solution when body forces are present in the problem.The particular part is included in the solution in a way similar to the dual reciprocity method(DRM) introduced in MFS and BEM methods [21, 22]. However, unlike the DRM where someradial basis functions (RBF) are employed, here we construct the particular solution by series ofEBFs. In either of the cases, that is satisfaction of boundary conditions or construction of theparticular part of the solution, the question regarding the way one should select the EBFs willbe answered by considering the basis of the proposed transformation. We propose two forms ofstrategies for EBF selection, one based on the projection of the boundary information on boundaryvalues of the basis and another heuristically based on numerical experiments.

The layout of the paper is as follows. In Section 2 the models used for elasticity problems arepresented and the corresponding EBFs are found. The transformation employed for the impositionof the boundary conditions is explained in Section 3. The way we select the parameters neededin the selection of the EBFs is described in Section 4. We give a step-by-step summary of theprocedure in Section 5. For problems with body forces, we give the extension of the formulationin Section 6. The numerical results of the method are given in Section 7. Finally in Section 8 wesummarize the conclusions.

2. THE MODELS OF ELASTIC PROBLEMS

In this section the models used for elasticity problems are given. In all cases attacked in this paper,the material is assumed to be isotropic, however, as will be illustrated later this is not a restrictionfor the method.

2.1. Time harmonic elasto-dynamic problems

We consider a linear elastic material occupying a 2D bounded domain � with boundaries as��=�u∪�t while �u∩�t =∅. Here �u and �t represent boundaries with prescribed displacementsand tractions, respectively. In the absence of martial damping, assuming time harmonic motionswith frequency of �, the governing equation of the motion in the system may be written in avector notation as

STDSu+��2u+b=0 in � (1)



which is to be solved with the following boundary conditions:

u=uB on �u (2)

and

nDSu= t on �t (3)

In the above relations, u is the vector of time harmonic displacements, S is the well-known operatorfor defining strains as e=Su, D is the matrix of material constants in 2D elasticity problems, � isthe material density, b is the vector of time harmonic body forces, uB and t are prescribed timeharmonic displacements and tractions and n is a matrix containing the components of the unitvector normal to the boundary for defining the tractions. Here we consider 2D isotropic problemswhose matrix of material constants D can be presented as

D=⎡⎢⎣D1 D2 0

D2 D1 0

0 0 D3

⎤⎥⎦ (4)

As will be seen later in the paper, isotropy is not a restriction for the method we present and in factany form of material property, as long as it is constant over the whole domain, can be consideredhere (note that with definition (4) some orthotropic materials are still possible to be modelled).

Denoting ��2 by K 2, Equation (1) is written as

(STDS+K 2I)u+b=0 (5)

in which I is an identity matrix.

2.1.1. Exponential basis functions. Neglecting the body forces in (5) at this stage, the equationbecomes as

(STDS+K 2I)uH=0 (6)

The subscript ‘H’ is used to denote the homogeneous part of the solution. The EBFs may beobtained by assuming

uH(�,�) =h(�,�)e�x+�y (7)

where x and y are coordinates of a generic point in �, � and � are two parameters known as wavenumbers in elasto-dynamic problems. Here � and � can take on complex values, that is (�,�)∈C2.Also h(�,�) =[h1 h2]T is an appropriate vector. By inserting (7) in (6), the following system ofequations is resulted:(

D1�2+D3�

2+K 2 (D2+D3)��

(D2+D3)�� D3�2+D1�

2+K 2

){h1

h2

}=0 (8)

In order to obtain a non-trivial solution for the problem, the determinant of the generic coefficientmatrix in (8) must be set to zero, which gives

(D1�2+D3�

2+K 2)(D3�2+D1�

2+K 2)−(D2+D3)2�2�2=0 (9)



From the above characteristic equation one may find � in terms of � or vice versa. The characteristicvector h may be found by one of the two equations in (8). For instance when � is to be found interms of �, four distinct roots are calculated as �l = f l(�), l=1, . . . ,4. The solution of (6) is thenwritten as

uH=∫A�

{4∑

l=1dl( f l ,�)

hl( f l ,�)

e f l (�)x+�y

}dA� (10)

where A� is an appropriate region in the Gaussian plane since � can take on complex values. In(10) the functions dl

( f l ,�), l=1, . . . ,4, play the role of constants (are not functions of x and y)

and are to be selected so that the boundary conditions (2) and (3) are satisfied. It is clear thatsuch an integral form of solution, if not impossible for certain cases, is not explicitly applicablein all problems of this kind. Therefore, one may use a numerical integration scheme and convertintegrals in (10) to a summation sign as

uH=∑i

wi

{4∑

l=1dli h

lie

f l (�i )x+�i y

}(11)

In (11) wi represents the weight for i th quadrature point, while �i is the location of the quadraturepoint on the Gaussian plane. By combining the coefficients dli and wi , as cli , one may arrive at thefollowing expression:

uH=∑i

{4∑

l=1clih

lie

�li x+�i y

}(12)

In (11) and (12) the characteristic vectors hli , l=1, . . . ,4, are found in terms of �i . A similarexpression can be written when � is expressed in terms of �.

Let E and � denote Young’s modulus and the Poisson ratio of the material. Here we considerisotropic plane stress and plane strain problems.

Case (a). Plane stress problems: In this case D1=E/(1−�2), D2=�D1, D3=E/(2(1+�)).When �i is to be written in terms of �i :

�i = ±1

√E�2i +2K 2(1+�)

E, h1=±1

√E�2i , h2=

√E�2i +2K 2(1+�)

�i = ±1

√E�2i −K 2(−1+�2)

E, h1=±1

√E�2i −K 2(−1+�2), h2=

√E�2i

(13)

When �i is to be written in terms of �i :

�i = ±1

√E�2i +2K 2(1+�)

E, h1=±1

√E�2i +2K 2(1+�), h2=

√E�2i

�i = ±1

√E�2i −K 2(−1+�2)

E, h1=∓1

√E�2i , h2=

√E�2i −K 2(−1+�2)

(14)



Case (b). Plane strain problems: In this case

D1= E(1−�)

(1+�)(1−2�), D2= �

(1−�)D1, D3= E

2(1+�)


�i = ±1

√E�2i +2K 2(1+�)

E, h1=±1

√E�2i , h2=

√E�2i +2K 2(1+�)

�i = ±1

√(E�2i +K 2)(−1+�)+2�2K 2

E(−1+�), h1=±1

√(E�2i +K 2)(−1+�)+2�2K 2

h2 =√E(−1+�)�2i

(15)


�i = ±1

√E�2i +2K 2(1+�)

E, h1=∓1

√E�2i +2K 2(1+�), h2=

√E�2i

�i = ±1

√(E�2i +K 2)(−1+�)+2�2K 2

E(−1+�), h1=∓1

√E(−1+�)�2i

h2 =√

(E�2i +K 2)(−1+�)+2�2K 2

(16)

In the above relations 1=√−1. The summary of the results for special cases with multiple rootsis given in Appendix A.

2.2. Elasto-static problems

For static problems, we set �=0 in (1). Thus

STDSu+b=0 (17)

The boundary conditions are similar to (2) and (3) noting that in this case they refer to the static-prescribed displacements and tractions. The EBFs are found in the same fashion as described inSection 2.1.1 but with some small differences.

2.2.1. Exponential basis functions. Here again, at this stage, we neglect the body forces in (17)and rewrite the equation as

STDSuH=0 (18)

By defining uH as (7) and inserting it in (18), noting that from now on we consider point-wisevalues of �i and �i , the following system of equations is resulted:(

D1�2i +D3�

2i (D2+D3)�i�i

(D2+D3)�i�i D3�2i +D1�

2i

){h1

h2

}=0 (19)



The non-trivial solution is obtained by finding �i and �i so that the determinant of the genericcoefficient matrix in (19) vanishes. The characteristic equation in this case is similar to (9) butwith K =0.

When �i is calculated in terms of �i , for both isotropic plane stress and plane strain cases, weobtain

�i =±1�i (folded roots), h1=±1, h2=1 (20)

or when �i is calculated in terms of �i

�i =±1�i (folded roots), h1=∓1, h2=1 (21)

The missed EBFs are found by considering another set of solutions for the problem, that is

uiH={[

a1i

a2i

]x+

[b1i

b2i

]y+

[d1i

d2i

]}e�i x+�i y (22)

Introducing (22) in (18) results in the following system of equations:

A

[a1i

a2i

]x e�i x+�i y+A

[b1i

b2i

]y e�i x+�i y+

{A

[d1i

d2i

]+B

[a1i

a2i

]+C

[b1i

b2i

]}e�i x+�i y =0 (23)

with A being the same coefficient matrix as in (19) and

B=(

2D1�i (D2+D3)�i

(D2+D3)�i 2D3�i

), C=

(2D3�i (D2+D3)�i

(D2+D3)�i 2D1�i

)(24)

Since the determinant of A vanishes, one may choose

�i = ±1�i →[a1i

a2i

]=a

[±1

1

],

[b1i

b2i

]=b

[±1

1

]

�i = ±1�i →[a1i

a2i

]=a

[∓1

1

],

[b1i

b2i

]=b

[∓1

1

] (25)

Now setting as zero the last term in (23) leads to

�i = ±1�i →�i

( −D1+D3 ±1(D2+D3)

±1(D2+D3) −D3+D1

)[d1i

d2i

]

= −{−2D1+D2+D3

±1(D2+3D3)

}a−

{ ±1(3D3+D2)

−(D2+D3)+2D1

}b

�i = ±1�i →�i

(D1−D3 ±1(D2+D3)

±1(D2+D3) D3−D1

)[d1i

d2i

]

= −{±1(D2+D3−2D1)

D2+3D3

}a−

{3D3+D2

±1(2D1−(D2+D3))

}b

(26)



Table I. Evaluated d1i and d2i for elasto-static problems (to be used in (22)).

Plane strain problems Plane stress problems

a =0, b=0, d2i =a �i =±1�i

⎡⎣d1i

d2i

⎤⎦=

⎡⎣− (3−4�)

�i± 1

1

⎤⎦a

⎡⎣d1i

d2i

⎤⎦=

⎡⎣− (3−�)

�i (1+�) ± 1

1

⎤⎦a

�i =±1�i

⎡⎣d1i

d2i

⎤⎦=

⎡⎣∓ 1(−3+4�)

�i∓ 1

1

⎤⎦a

⎡⎣d1i

d2i

⎤⎦=

⎡⎣∓ 1(�−3)

�i (1+�) ∓ 1

1

⎤⎦a

a=0, b =0, d2i =b �i =±1�i

⎡⎣d1i

d2i

⎤⎦=

⎡⎣∓ 1(−3+4�)

�i± 1

1

⎤⎦b

⎡⎣d1i

d2i

⎤⎦=

⎡⎣∓ 1(−3+�)

�i (1+�) ± 1

1

⎤⎦b

�i =±1�i

⎡⎣d1i

d2i

⎤⎦=

[− (3−4�)�i

∓ 1

1

]b

⎡⎣d1i

d2i

⎤⎦=

⎡⎣− 1(3−�)

�i (1+�) ∓ 1

1

⎤⎦b

In each case shown above, the two relations are proportional and thus the system of equationsplays the role of just one equation (note that the determinant of the coefficient matrix is zero).Now, choosing d2i for instance, d1i can be found in terms of arbitrary parameters a and b. To thisend, one can consider two cases as (a =0 and b=0) and (a=0 and b =0) for which d1i and d2iare evaluated as shown in Table I.

By combining and renaming the arbitrary coefficients one can show that the solution, for when�i =±1�i and the plane strain case for instance, may be written as

uH =∑⎧⎪⎨⎪⎩c1i

[1

1

]e�i (1x+y)+c2i

[−1

1

]e�i (−1x+y)+c3i

⎛⎜⎝[

1

1

]x+

⎡⎢⎣− (3−4�)

�i+ 1

1

⎤⎥⎦⎞⎟⎠ e�i (1x+y)

+c4i

⎛⎜⎝[

1

1

]y+

⎡⎢⎣− (−3+4�)

�i (1+�)+1

1

⎤⎥⎦⎞⎟⎠ e�i (1x+y)+c5i

⎛⎜⎝[−1

1

]x+

⎡⎢⎣− (3−4�)

�i− 1

1

⎤⎥⎦⎞⎟⎠ e�i (−1x+y)

+c6i

⎛⎜⎝[−1

1

]y+

⎡⎢⎣− (−3+4�)

�i (1+�)+ 1

1

⎤⎥⎦⎞⎟⎠ e�i (−1x+y)

⎫⎪⎬⎪⎭ (27)

The solution for other cases can be written analogously. It can be seen that six basis functionscontribute to the solution in each case. Our experience shows that combining two pairs of them



still leads to excellent results. In this paper we take a=b, which gives the following forms

uH =∑⎧⎪⎨⎪⎩c1i

[1

1

]e�i (1x+y)+c2i

[−1

1

]e�i (−1x+y)

+c3i

⎛⎜⎝[

1

1

]x+

[1

1

]y+

⎡⎢⎣

1�i (1+�)+(1− 1)(−3+�)

�i (1+�)

1

⎤⎥⎦⎞⎟⎠ e�i (1x+y)

+c4i

⎛⎜⎝[−1

1

]x+

[−1

1

]y+

⎡⎢⎣

−1�i (1+�)+(1+ 1)(−3+�)

�i (1+�)

1

⎤⎥⎦⎞⎟⎠ e�i (−1x+y)

⎫⎪⎬⎪⎭ (28)

for plane stress problems (�i =±1�i ) and

uH =∑⎧⎪⎨⎪⎩c1i

[1

1

]e�i (1x+y)+c2i

[−1

1

]e�i (−1x+y)

+c3i

⎛⎜⎝[

1

1

]x+

[1

1

]y+

⎡⎢⎣

1(�i +(1− 1)(3−4�))

�i

1

⎤⎥⎦⎞⎟⎠ e�i (1x+y)

+c4i

⎛⎜⎝[−1

1

]x+

[−1

1

]y+

⎡⎢⎣

−1(�i +(1− 1)(3−4�))

�i

1

⎤⎥⎦⎞⎟⎠ e�i (−1x+y)

⎫⎪⎬⎪⎭ (29)

for plane strain problems.The reader may note that when �i =0 (or �i =0) four roots will be found for � (or �). The

summary of the calculations is given in Appendix B for such a case.

3. IMPOSITION OF BOUNDARY CONDITIONS

In this section we describe the way that the boundary conditions can be imposed through acollocation approach. To this end, first we rewrite uH in its general form considering all the EBFsevaluated either in the form of (12) or in the form of (28) as

uH=N∑j=1

c jh j e� j x+� j y (30)



where N is the total number of EBFs. Note also that in the above expression the EBFs obtainedfrom calculating � j in terms of � j and vice versa are included. Using a collocation approach atM boundary points leads to

U=N∑j=1

C jV j (31)

In the above relation, U is a vector containing all point-wise boundary conditions,V j is a normalizedvector that contains contribution of each exponential basis to the boundaries arranged in the samemanner as U and C j is proportional to coefficient c j in (30) considering the normalization factorused in V j . For instance supposing that m boundary points, out of M , are allocated to Dirichlettype of conditions and n=M−m points are allocated to Neumann conditions, then U can bearranged in the following form:

U={(uTB)1 (uTB)2 . . . (uTB)m |tT1 tT2 . . . tTn }T (32)

In a similar manner V j can be arranged as

V j = 1

s j{(uTj )1 (uTj )2 . . . (uTj )m |(tTj )1 (tTj )2 . . . (tTj )n}T (33)

where s j is a scaling factor for normalization and

(u j )k =[h j e� j x+� j y]x=xk ,y=yk ∀(xk, yk)∈�u, k=1, . . . ,m (34)

and

(t j )k =[nkDS(h je� j x+� j y)]x=xk ,y=yk ∀(xk, yk)∈�t , k=1, . . . ,n (35)

The scaling factor s j may be defined in different ways. One is the use of the vector length

s j =|V j | (36)

and another is the maximum element of the vector

s j =maxk1

(|V k1j |), k1=1, . . . ,M (37)

with V k1j being the k1th element of V j (in the above relations |.| denotes the Hermitian length).

This latter form has been used in the numerical examples in this paper. Note that with definingscaling factor s j , coefficients c j in (30) are now related to coefficients C j in (31) as

c j = 1

s jC j (38)

Now the question regarding the evaluation of coefficients C j must be answered. These coefficientsmay be evaluated as explained in Section 3.2.



3.1. Other weighted residual methods in place of collocation

The reader may note that the use of the boundary collocation method described above is nota restriction in our approach. In fact other weighted residual methods are also applicable. Forinstance in a Petrov–Galerkin approach one may write∫

�u

wk(uH−uB)d�=0, k=1, . . . ,m (39)

and ∫�t

wk[nDSuH−t]d�=0, k=1, . . . ,n (40)

where wk and wk represent two sets of appropriate weight functions defined on boundaries �uand �t , respectively. With such a weighed residual approach, Equation (31) is rewritten with newvector definitions as

U={(�uTB)1 (

�uTB)2 . . . (

�uTB)m |�

t T1�

t T2 . . .�

t Tn }T (41a)

where

(�u B)k =

∫�u

wkuB d�,�

t k =∫

�t

wktd� (41b)

and

V j = 1

s j{(�uT

j )1 (�uT

j )2 . . . (�uT

j )m |(�

t Tj )1 (�

t Tj )2 . . . (�

t Tj )n}T (42)

The elements of V j are defined in a manner analogous to expressions (34) and (35) by defining

(�u j )k and (

�

t j )k as

(�u j )k =

∫�u

wkh j e� j x+� j y d�, k=1, . . . ,m (43)

and

(�

t j )k =∫

�t

wk[nDSh j e� j x+� j y]d�, k=1, . . . ,n (44)

The scaling factor s j is defined according to (36) or (37). It may be noted that by choosing theweight functions as the Dirac delta function, the above formulation will convert to the collocationpresented in (32)–(35).

Similar to other numerical methods based on weighted residual approaches, the choice of weightfunctions in (39) and (40) may have some effects on the stability and convergence of the solutionmethod. For instance, it has been experienced by many authors that in meshless methods basedon a collocation approach some instability effects may be seen in the numerical solution (see[23] and references therein). Such instability effects are dependent on the interpolation used andthe way that boundary conditions are satisfied. The readers may note that collocation methodsare growing fast in two forms; those using global shape functions such as radial basis functionsin [6–8] and those using local shape functions like the ones in [23] for instance (for the former set



mathematical basis can be found for stability and convergence of the method [24–26]; however,for the latter set just a few studies are traceable in the literature, for example [27], in this regard).It has also been experienced that other integral-based meshless methods may show better stablebehaviour (see [28, 29]). In the method presented in this paper there is no problem pertaining tointerpolation; however, there may be some discussions on the treatment of the boundary conditionregarding the stability of the solution. We shall refer to this effect after addressing the issue ofevaluation of coefficients C j in the next subsection.

3.2. The use of a discrete transformation

In this paper, we use a transformation technique for finding unknown coefficients C j . Note thatwe aim at choosing N and M so that N =M (and not necessarily N�M). The procedure startsfrom assuming that in (31) C j can be evaluated by finding the projection of V j on U as

C j =VTjRU (45)

where R plays the role of a projection matrix that is assumed to be suitable for all j =1, . . . ,N .By inserting (45) in (31) we find

U=N∑j=1

V jVTjRU=GRU, G=

N∑j=1

V jVTj (46)

In the above relations, G is a symmetric M×M matrix. Since the rank of G might be less thanM , R is evaluated as

R=G+ (47)

where G+ is pseudo inverse of G [30]. Having found C j as (45), we find the homogeneous partof the solution, uH, form (30) as

uH=�{(

N∑j=1

1

s je� j x+� j yh jVT

j

)RU

}(48)

In the above equation, �(.) denotes the real part of the quantity.The transformation given above has successfully been used in some other applications

(see [18, 20]) in its primitive form and without elaboration on its limitations. Here we explain thebasis in the following remark.

Remark 1The basis of the transformation may be explained as follows. We assume that C j is proportionalto the direct projection of V j on U as

C j =�VTj U (49)

In the above expression, � is a scalar coefficient assumed to be independent of j . Substitution of(49) in (31) results in the following equation:

U=�N∑j=1

V jVTj U=�GU (50)



or

(G−�I)U=0, �= 1

�(51)

This means that (50) holds just for when U is expressed in terms of eigenvectors of (51) and infact several values exist for � and �. Since G is a symmetric matrix, its eigenvectors are orthogonaland thus it can be decomposed as

G=Q�QT (52)

where K is a diagonal matrix of the eigenvalues of G, and Q contains the eigenvectors. Nowprovided that all eigenvalues are non-zero, one can write the generalized form of (49) as

C j =VTj

M∑i=1

ai�i/i =VTj

M∑i=1

ai1

�i/i (53)

where /i is the i th eigenvector playing the role of U in (49) assuming that U can be written interms of orthogonal bases Q. In that case ai is a coefficient denoting the projection of /i on U,that is

ai =/Ti U (54)

Insertion of (54) in (53) leads to

C j =VTj

M∑i=1

1

�i/i/

Ti U=VT

jQK−1QTU (55)

If some eigenvalues of (51) vanish, for instance �k =0, relation (53) will not be valid except forwhen the associated coefficients identically vanish, that is ak =/Tk U=0. In that case

C j =VTj

M∑i=1

1

�i/i/

Ti U, M�M (56)

or

C j =VTj QK

−1QTU (57)

in which Q is column reduced of Q and K is the associated diagonal matrix. The reader may note

that QK−1

QT is equivalent to the pseudo inverse of G and thus (45) holds.The above discussion shows that when G is a rank deficient matrix, expression (45) can only be

written for some special forms of U (i.e. those having no projection on null space vectors of G).This provides a criterion for suitability of vectors V j in construction of R and thus applicability ofthe transformation in reproducing U with the use of particular bases. Hence an adaptive solutionmethod may be constructed by re-evaluation of R by adding some EBFs with new �i , �i . Thishowever is beyond the scope of this paper.

Remark 2The reader may note that when D is a real-valued matrix in (6) or (18), the components of uH willalso be real. This means that the real and imaginary parts of the EBFs in (30) can individually be



considered as the basis for the solution, that is

uH=N∑j=1

{C1j�(h j e

� j x+� j y)+C2j (h je

� j x+� j y)} (58)

with C1j and C2

j being two real coefficients. Again in above, �(.) and (.) represent real andimaginary parts of the quantity. In that case (31) can be rewritten as

U=N∑j=1

{C1j�(V j )+C2

j (V j )} (59)

Then, analogous to (45), one can write

C1j =�(VT

j )RU, C2j = (VT

j )RU (60)

where

R=G+, G=N∑j=1

{�(V j )�(VTj )+ (V j ) (VT

j )} (61)

which is a real-valued matrix. Therefore the rest of the procedure is performed with real-valuedquantities.

Remark 3Considering the simple relations (46) and (45), other forms of (45) seem to be possible. Forinstance one may consider

C j =VTjP jRU (62)

where P j is a projector matrix defined just for V j (i.e. dependent on � j , � j ). By inserting (62)in (31) one may conclude that

R=G+, G=N∑j=1

V jVTjP j (63)

However, this latter form may not be interpreted from direct projection of V j on U.

Remark 4In conjunction with Remark 3, one may use a projector matrix as

P j =g(� j ,� j )I (64)

with I being an identity matrix and g being a scalar function. In that case the R matrix will besymmetric:

R=[

N∑j=1

V jVTj g(� j ,� j )

]+(65)

and thus one can follow the interpretation given in Remark 1 by assuming that

C j =�g(� j ,� j )VTj U (66)

and analogously proceed from (50) to (57).



Remark 5Another special case for using projector matrix as in Remark 3 is when we assume

C j =(V∗j )TRU, (V∗

j )T=VT

jP j (67)

in which V∗j is a complex conjugate of V j and P j is an appropriate diagonal matrix (and thus (67)

may be interpreted as a special case of (62)). In that case G in (46) will be a Hermitian matrix

R=G+, G=N∑j=1

V j (V∗j )T (68)

and again one can interpret the transformation by considering

C j =�(V∗j )TU (69)

which finally gives

C j =(V∗j )TQK

−1(Q∗)TU (70)

In (70) Q and Q∗ are column reduced of Q and Q∗, respectively (note that Q is a matrix containingthe basis vectors of G while Q∗ is its complex conjugate, and in this case (Q∗)TQ=0). It can be

seen that QK−1

(Q∗)T in (70) is again the pseudo inverse of G.

Remark 6In problems with mixed Dirichlet–Neumann boundary conditions the D matrix in (35) plays therole of a multiplier compared with the values in (34). This is mainly because in practice Young’smodulus, E , is a large value and may introduce some numerical errors in the computationsmentioned above. The effect may be removed by defining

V j =EV j , U=EU, E=[Im×m 0

0 E×In×n

], ET=E (71)

with I being identity matrix (note that the scaling factor defined in (36) or (37) is now calculatedfor V j in this case). Now with U and V j , in a manner analogous to expressions (31) and (45), wemay write

U=N∑j=1

C j V j , C j = VTj RU (72)

which gives

U=(

N∑j=1

V j VTj

)RU=GRU, G=

N∑j=1

V j VTj , R=G+ (73)

Then, in view of (72) and (71), C j is calculated as

C j =VTjE

−TG+E−1U (74)

which can be used in (38) for evaluation of c j (note again that here s j is used for scaling).



3.3. Stability of the procedure

As discussed in Section 3.1, some numerical procedures are prone to instability effects. Most ofthe instability effects are seen as spurious modes in the evaluation of the unknown coefficients.One of the major reasons for confronting the spurious modes is the round-off errors during thesolution of systems of equations with ill-conditioned coefficient matrices. From a numerical standpoint of view, the capability of achieving accurate solution for such systems is dependent on theprecision of the computation. The problem becomes prominent when the mathematical model isnot well-posed, that is when the exact solution is not unique and is very sensitive to any smallperturbation of the given data. Such situations happen in the solution of Cauchy problems (see[31, 32] for instance). The use of some stabilization procedures has been recommended by manyauthors for Cauchy problems when the MFS is employed as the numerical solution tool (see[33, 34] for MFS and [35] for BEM).

For well-posed problems, like those presented in this paper, the numerical solution method isexpected to show stability (i.e. approaching to a unique solution while altering auxiliary parameters).In the context of using boundary collocation methods like MFS, the readers may find such studiesin References [36, 37]. It may be noted that in a well-posed problem, where a unique solutionis expected, the use of boundary methods may lead to an accurate solution provided that theboundary conditions are satisfied. This is in contrast with the cases in which the governing equationsand the boundary conditions are approximately satisfied during the solution process, for examplemeshless methods based on collocation [6–8, 23, 28, 29]. The reader may consult [24–27, 38] andthe references therein for discussion on stability of other collocation methods.

The stability analysis of the presented collocation method is an open issue that should beaddressed in a separate paper. However, here we shall preliminarily address the issue by consideringthe content of Remark 1. This can be performed by looking into the procedure of the pseudoinverse used in (57). Apart from the cases in which �k =0 in (55), there may be some cases where�k ≈0 causing magnification of spurious modes when ak =T

k U =0. In such a case the contributionof the mode is ignored in the computation when �k is less than a user-specified value. In otherwords one may compute C j in (56) with the following condition:

C j =VTj

M∑i=1

1

�i/i/

Ti U ∀ |�i |�×max

j(|� j |) (75)

where is a prescribed small value. However, the reader may note that such a condition maynot work satisfactorily when is chosen as a very large value (i.e. much larger than the machineepsilon order). A thorough discussion on the effects of such a regularization technique is beyondthe scope of this paper. Nevertheless we shall present some studies on the stability of the solutionby altering in the section of numerical examples.

Here it is worthwhile to mention that the stability of a method inherently affects the convergenceof its results. Having studied the stability of our method, we shall address the convergence ofthe results by some numerical experiments. The reader may note that the convergence studiesof collocation methods, including MFS, are performed either by finding error bounds throughmathematical proofs or by presenting the results of numerical experiments. The latest studies onmethods using RBF show that error bounds may be found just under specific assumptions [26].The situation is worse for collocation methods using local interpolation similar to that in [23].This is perhaps the reason that in all studies of this kind the results of numerical experiments areusually given as some evidences for convergence and stability by varying the solution parameters.



In MFS, convergence and stability studies are essentially performed in the latter form. However,after more than 40 years since the development, latest studies in this regard are emerging bypresenting mathematical basis followed by numerical experiments for some special classes ofproblems (the readers may consult [39] and the references therein). In this paper, without presentingany mathematical proof, we shall proceed to give some numerical evidences for the convergenceof our method.

4. SELECTION OF � AND � FOR THE HOMOGENEOUS PART

It is obvious that parameters � and � play important roles in the variation of EBFs throughout thesolution domain and thus have a prominent effect on the projection matrix R. These parameterscan be complex values, that is

�=a+ 1b, �=c+ 1d (a,b,c,d)∈R4 (76)

However, it is known that a, b, c and d are not independent and they should satisfy the characteristicequation, for example (9), which leads to explicit relations as (20) or (21) for K =0. Suppose that� is calculated in terms of �, we select the pair (a,b) and then evaluate (c,d). However, we notethat b controls the periodic part of the exponential function and a determines its amplitude (i.e.in the x direction). On this basis one can decide on max(|b|) and max(|a|) by considering thenode spacing at the boundaries, in this case along x axis. The first limit determines the oscillationof the function between the nodes and the second one shows the decay of the function from itsmaximum in the domain (note that the basis functions are normalized as (1/s j )e

� j x+� j y , see (48)).To give an insight into the variation of the normalized EBFs on a rectangular solution domain andfor different pairs of (a,b), schematic presentations of the functions are given in Figure 2 (u or v

for a sample plane stress problem).The same discussion is valid for (c,d) from which a decision is made for max(|c|) and max(|d|).

However, it is clear that max(|a|), max(|b|), max(|c|) and max(|d|) are not independent. Forinstance in elasto-static problems, when �=±1� as in (20) and (21), max(|a|) is controlled bymax(|d|) and max(|c|) is controlled by max(|b|). This means that the decision on the imaginaryparts of � and � plays a prominent role in defining the feasible domains.

Having defined a feasible domain on Gaussian plane, we define a grid of points on it. Thesolution accuracy may differ by changing the grid; however, as we shall show in the numericalresults, the accuracy obtained is usually very high in all the cases studied. Nevertheless, this isan open issue for further studies through which some adaptive strategies might be devised. Butsupposing that a decision is to be made on the fineness of the grid, in this paper we suggest twosimple strategies in this regard; one with mathematical basis and another heuristically based onsome numerical experiments.

Strategy INoting that the coefficients of EBFs, that is C j , are indirectly dependent on the projection V j onU as in (49), we propose the following strategies to use such an effect.

Strategy I-aAs we saw earlier, the size of the feasible domain on Gaussian plane is dependent on the pointspacing at the boundary of the main domain �. Suppose that the average distance between adjacent



points on the boundary is denoted by dave.

dn =[(xn−xn−1)2+(yn− yn−1)

2]1/2, dave=∑

n dnM−1

�∑

n dnM

(77)

The minimum period of variation of the exponential functions may be considered to be dependenton dave. Therefore we may write

− �

2dave�b� �

2davewhen � is written in terms of � (78)

and

− �

2dave�d� �

2davewhen � is written in terms of � (79)

assuming that a single oscillation of the function is allowed within a length of 4dn . In (78) and (79)each interval represents a segment along the imaginary axis in the Gaussian plane.

Let the feasible domains for � and � be defined by the segments defined in (78) and (79), thatis �=±1b and �=±1d . A grid of points is considered within each segment. Having defined thegrids, we try to find the points with the most contribution to the solution. To this end for eachpoint of the grids the exponential functions are defined. The maximum absolute value of the directprojections of such functions on the nodal boundary values is calculated as

p�k1 =max(|(V�,�1

k1)TU|, |(V�,�2

k1)TU|, |(V�,�3

k1)TU|, |(V�,�4

k1)TU|)

k1=1, . . . , (No. of grid points for �) (80)

and

p�k2

=max(|(V�1,�k2

)TU|, |(V�2,�k2

)TU|, |(V�3,�k2

)TU|, |(V�4,�k2

)TU|)

k2=1, . . . , (No. of grid points for �) (81)

with |.| again denoting the Hermitian length of the quantity. Note that V�,� jk1

, V�,� jk2

, j=1, . . . ,4,are normalized vectors (with scaling factor as (36) or (37)) when girds of � and � are used,respectively (note also that for each grid point, e.g. for �, four basis functions are found as given

in (10) or (28)). The maximums of the two series of numbers p�k1, p�

k2are found as

p�max=max

k1(p�

k1), p�max=max

k2(p�

k2) (82)

We choose all grid points whose projection values are greater than a specified value as � times themaximum projections defined above, that is we choose pk so that

p�k1��× p�

max, p�k2

��× p�max, 0<�<1 (83)

We shall give some graphs for the variations of p�k1

or p�k2

in one of the numerical examples.



Strategy I-bThe approach proposed in Strategy I-a is based on a pre-processing algorithm to find those EBFsthat have more projection on U. In order to avoid such pre-processing procedures, one may usethe weighting approach proposed in Remark 3. To this end, instead of (45) we write

C j =|VTj U|rVT

jRU, r ∈R, r�0 (84)

which leads to

R=[

N∑j=1

V jVTj |VT

j U|r]+

(85)

In (84) and (85), r is an exponent that controls the sharpness of the weight variation. For instance,using very large value for r corresponds to considering just few EBFs that have maximum or nearlymaximum projections. On the other hand, using very small value for r corresponds to consideringalmost all EBFs with |VT

j U| =0 (in this paper we have used r =0.25).If � (or �) is confined to be an imaginary value, then the intervals chosen for b in (78) (or

for d in (79)) may be used here again. In that case the point spacing along the imaginary axis inGaussian plane may be calculated by requiring that the total number of EBFs be proportional tothe total number of boundary information, that is N = (2M). This gives

�b= 4�

Mdavd

(and �d= 4�

Mdavd

)(86)

when four EBFs are used for each grid point. Our experience shows that an interval of 1� �2 isappropriate for the proportionality factor.

Strategy IIAlthough not presented in this paper, we have used the method to solve different engineeringproblems, such as those with Helmholtz equations and bi-harmonic equations, etc. Based onnumerical experiments, on a variety of such problems defined on different solution domains, it hasbeen found that the following strategy for defining a suitable grid of points works very well:

�=(a+ 1b)=± m

L

(k

N+ 1

), m=1, . . . , M, k=1, . . . , N (87)

In (87) we choose (M, N )∈N2, ∈R and L is a characteristic length. The following bounds arefound to be appropriate for many cases:

5.6� �7.2 (say =2�), L=1.6max(Lx , Ly), Mmin=4, Nmin=2, Nmax=8 (88)

where again Lx and Ly are the dimensions of the circumscribing rectangle (see Figure 1(b)). Asimilar formula may be used for �. It can be seen that the formula given above is completelyheuristic. However, one may find it useful, as a rule of thumb, for a preliminary grid selectionwithout putting much effort on this part. We shall present the results in the section of numericalresults.



Figure 1. Schematic presentation of the solution domain and the boundary points: (a) thedomain � with its boundary discretized to a number of points for Neumann and Dirichlet

types and (b) the circumscribing rectangle.

Figure 2. Variation of an EBF with �=1.2+121, �= 1� for a plane stress problem shownon a rectangular domain: (a) variation of the real part of u displacement in the first part of(28); (b) variation of v in the first part of (28); (c) variation of u in the third part of (28);

and (d) variation of v in the third part of (28).

5. THE STEP BY STEP PROCEDURE OF THE METHOD

To give an insight into the implementation of the method, we present the step by step procedure.Here we choose Strategy I-b for this purpose. After defining the EBFs as a series of libraryfunctions,

1. Choose a series of points on the boundary of the domain. At kinked boundaries choose twoclose points to define distinct normal vectors (if necessary).

2. Evaluate dave as defined in (77).3. Evaluate E as defined in (71).4. Evaluate the collocated values of the boundary conditions in a vector as defined in (32),

that is U.5. Evaluate �b (or �d) by (86).6. Choose two grids of points in Gaussian plane. For Strategy I-b we have just chosen the

imaginary axes. In this case two sets of points with spacing of �b (for �) and �d (for �) areselected (see (86)). The bounds of the values are preliminarily determined by considering



the oscillation of the boundary values, for example relations (78) and (79) (the grid may berefined and extended in an adaptive manner if needed).

7. For each point of the first grid in step 6, that is for �, evaluate the set of EBFs as given in (28)for instance. (The reader may note that if it is necessary to write the code in real-valuednumbers, the computation may be performed purely by real-valued functions by writingeach complex-valued EBF in the form of two real functions, see Remark 2).

8. For each EBF of step 7, evaluate the contribution of the EBF on the boundaries, that is V j .9. Evaluate V j =E−1V j .

10. Evaluate V j U (or V j U).11. Evaluate V j VT

j and add it to other similar vector products to evaluate G=∑ V j VTj |VT

j U|r(or G=∑ V j VT

j |VTj U|r ). In this paper we have used r =0.25.

12. Repeat from 8 to consider all EBFs associated with each �.13. Repeat from 7 to include the contributions of all points of the grid for �.14. Repeat steps 7–13 for the points considered for grid of �.15. Evaluate G+. In commercial codes the minimum singular value is chosen automatically

(e.g. 1E−13 for double precision computations). One may also use subroutines providedin [40] for singular value decomposition of square matrices. In that case the minimumsingular value should be specified by the user (see the numerical examples for appropriatevalues known as truncation factor ).

16. Evaluate R=E−TG+E−1.17. Evaluate the homogeneous part of the solution as (48), which in case of using Strategy I-b

takes the form of

uH=�{(

N∑j=1

1

s j|VT

j U|r e� j x+� j yh jVTj

)RU

}(89)

or

uH=�{(

N∑j=1

1

s j|VT

j U|r e� j x+� j yh jVTj

)RU

}(90)

For evaluation of stresses or other quantities the above expressions are used directly.

A similar procedure is applicable for other engineering problems such as plates, etc. The maindifference lies in the definitions of the EBFs and the boundary conditions. Therefore, a routinewritten for one category may be used for others just by adding the corresponding library functions.The reader may have noted that the method introduced above can easily be extended to problemsinvolving time as a new independent variable and thus with small efforts the written routine maybe used for dynamic problems.

5.1. Error indicator

Although not intended in this paper, here we briefly address the way that one can construct anadaptive formulation for the presented method. Similar to other boundary mesh based or meshlessmethods, here deviation of the numerical solution from the exact one may be understood byevaluating the discrepancy of the approximated boundary values from the exact ones. It may benoted that since the EBFs satisfy the governing equations, that is to say there is no residual in the



interior parts of the domain, the residuals of the boundary conditions can play the role of suitableerror indicator. Considering the content of Sections 3.3 and 4, an error indicator may be devised by

1. Re-evaluation of the boundary collocated values. This means that, for instance by (89) in

hand, one may find approximated vector of U named here as ˜U so that

˜U={(uTH)1 (uTH)2 . . . (uTH)m |tT1 tT2 . . . tTn }T (91)

where

(uH)k = [uH]x=xk ,y=yk (92)

(t)k = [nkDSuH]x=xk ,y=yk (93)

By collocated errors as

e= U− ˜U (94)

one may define an error norm as an indicator for errors in the collocated boundary values.Such an indicator is suitable for judging the performance of the transformation techniqueused in Section 3.2. The smallness of the error norm indicates the suitability of the EBFsused. By finding those /k , see (75), whose associated singular values are small one may seekfor new EFBs that have projection on /k .

2. Re-evaluation of boundary values at points different from those chosen for collocation. It isclear that (in our method) if the uniqueness of the exact solution is assured, which is thecase for our problems in this paper, reproduction of the boundary values can be interpretedas achieving a highly accurate solution. In the other way round, non-zero residuals indicateapproximation in the solution. The reduction of the error may be performed by adding newpoints at the boundary as well as using new sets of EBFs. The new EBFs may be chosen byfinding their projections on the boundary residuals (similar to relations (80) and (81) with Ureplaced with the collocated residuals at all points).

The above two points may be used for the construction of an adaptive strategy consistent withthe formulation. However, this is beyond the scope of this paper.

6. PROBLEMS WITH BODY FORCE

When body forces are present in a problem the solution to Equation (5), as a general formconvertible to (17), may be performed by finding a particular part as up so that

(STDS+K 2I)up+b=0 (95)

At this stage, up does not need to satisfy the boundary conditions (2) and (3). This means that theform of up may not be unique. This in fact is similar to assumptions used in the DRM initiallyproposed in [21]. However, as will be seen next, instead of using radial basis functions we shallemploy another set of EBFs for the construction of up. To satisfy the boundary conditions, thehomogeneous part is found so that

uH+up=uB on �u (96)



and

nDS(uH+up)= t on �t (97)

with uH defined as (30). For imposition of the boundary condition, one may rewrite (31) as

U=N∑j=1

C jV j +Up (98)

Up is a vector with an arrangement similar to U in (32), that is

Up={(uTp )1 (uTp )2 . . . (uTp )m |(tTp )1 (tTp )2 . . . (tTp )n}T (99)

where

(uTp )k =[up]x=xk ,y=yk ∀(xk, yk)∈�u, k=1, . . . ,m (100)

and

(tTp )k =[nkDSup]x=xk ,y=yk ∀(xk, yk)∈�t , k=1, . . . ,n (101)

The unknown coefficients in (98) are found in a manner similar to expression (45) for instance

C j =VTjR(U−Up) (102)

The final solution thus becomes as

u=�{up+

(N∑j=1

1

s je� j x+� j yh jVT

j

)R(U−Up)

}(103)

Remark 7In a manner analogous to (67), one may replace (102) with the following relation:

C j =(V∗j )TR(U−Up) (104)

In that case V j in (103) is also replaced with V∗j . We shall present some results for such a

formulation in the section of numerical results.

Remark 8In conjunction with the content of Remark 6, when normalization with respect to E is of concern,the following relation must be used in place of (102):

C j =VTjE

−TG+E−1(U−Up) (105)

with G+ defined in (73). The final solution is written in a manner analogous to expression (103). Asimilar form may also be used for when V j in (103) is replaced with V∗

j as discussed in Remark 7.

In order to find up, three cases are considered here.Case 1. Body forces expressed in polynomial forms. Since the system of differential equation is

of constant coefficient, the particular solution may be found by assuming polynomial forms. Forinstance in elasto-static problems K =0, if b is expressed as

b={a1x

2+a2y2+a3x+a4y+a5

b1x2+b2y

2+b3x+b4y+b5

}(106)



then the associated particular solution may be found as

up=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

a1x4

12D1+ a2y4

12D3+ a3x3

6D1+ a4y3

6D3+ a5x2

D1

b1x4

12D3+ b2y4

12D1+ b3x3

6D3+ b4y3

6D1+ b5x2

D3

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(107)

This is performed by inserting the polynomial forms, assumed for up, in differential Equation (17)and calculating the associated coefficients.

Case 2. The use of the Fourier series in expressing the body forces. In this case the componentsof the body forces are written in terms of the complex Fourier series

b ={b1(x, y)

b2(x, y)

}=

∞∑m=−∞

∞∑n=−∞

dmne(21m�x/L x+21n�y/L y)

dmn = 1

L x L y

∫L x

∫L y

be−((21m�x/L x )+21n�y/L y) dx dy

(108)

where L x and L y are two appropriate lengths along x and y. For instance one may takeL x=1.2Lx and L y =1.2Ly where Lx and Ly are the edge sizes of the circumscribing rectangle(see Figure 1(b)). The reason for choosing the length scales greater than the circumscribing rect-angle is to obtain smoother values for up at boundaries. The reader may notice that the functionsdefining b may be extended beyond the boundaries of the domain without loss of generality. Bydenoting

�m = 21m�

L x, �n = 21n�

L y(109)

the particular part of the solution is expressed as

up=∑m,n

hpmn e(�m x+�n y) (110)

where vectors hpmn are to be found by insertion of (110) and (108) in (95). This leads to

∑m,n

⎧⎨⎩⎛⎝D1�

2m+D3�

2n+K 2 (D2+D3)�m �n

(D2+D3)�m �n D3�2m+D1�

2n+K 2

⎞⎠[(hpmn)1

(hpmn)2

]+[

(dmn)1

(dmn)2

]⎫⎬⎭ e(�m x+�n y) =0 (111)

from which (hpmn)1 and (hpmn)2 can be calculated, provided that the determinant of the coefficientmatrix in (111) does not vanish:

(D1�2mn+D3�

2mn+K 2)(D3�

2mn+D1�

2mn+K 2)−(D2+D3)

2�2mn�2mn =0 (112)

For those pairs of (�mn, �mn) for which the determinant vanishes, a new form of solution shouldbe found:

hpmn = fpmn x+gpmn y+hpmn (113)

where fpmn and gpmn are two more vectors to be found by satisfaction of (95).



Case 3. Approximation of the body forces by a series of EBFs. Our experience shows that theFourier series explained above usually needs excessive number of terms to represent the bodyforces. Moreover, the integration required for the evaluation of dmn in (108) may not be takenanalytically since the information of the body forces might be in the form of discrete data at aseries of points. Therefore, as an alternative, one may employ the transformation introduced inSection 3. To this end the body force components are expressed in terms of EBFs as

b={b1(x, y)

b2(x, y)

}=∑

r,scrs e�r x+�s y (114)

where �r and �s are two independent complex numbers, that is �r =ar + 1br , �s =cs+ 1ds ,(ar ,br ,cs,ds)∈R4 and crs=[c1rs c2rs]T is a vector containing the coefficients that are to be evaluatedthrough the discrete transformation. In (114) �r and �s are to be chosen so that the determinantof the coefficient matrix in (8) does not vanish:

srs=(D1�2r +D3�

2s +K 2)(D3�

2r +D1�

2s +K 2)−(D2+D3)

2�2r�2s =0 (115)

By choosing a number of points, that is a grid of points in the domain defined by L x and L y , thecomponents of ckl are evaluated as

c1rs=(Vb)TrsRb1, c2rs=(Vb)

TrsRb2 (116)

where

R=[∑r,s

(Vb)rs(Vb)Trs

]+(117)

If the number of points for the body force data is considered as p, then in this form we are allowedto choose nr +ns = p where nr and ns are the numbers of parameters �r and �s , respectively. Thenumbers nr and ns may be altered until b1(x, y) and b2(x, y) are reproduced satisfactorily, that iswhen an L2 error norm becomes less than a specified value. In (116) and (117)

b1 = {b1(x1, y1) b1(x2, y2) . . . b1(xp, yp)}Tb2 = {b2(x1, y1) b2(x2, y2) . . . b2(xp, yp)}T

(118)

and also

(Vb)rs={e�r x1+�s y1 e�r x2+�s y2 . . . e�r xp+�s yp}T (119)

Having found the coefficients c1rs and c2rs, we evaluate the particular solution as

up=∑r,s

hprs e�r x+�s y (120)

where hprs is calculated by inserting (120) and (114) in (95) that yields to

hprs= 1

srs

{(D2+D3)�r�sc

2rs−(D3�

2r +D1�

2r +K 2)c1rs

(D2+D3)�r�sc1rs−(D1�

2r +D3�

2s +K 2)c2rs

}(121)

with srs defined as (115).



Remark 9While working with case 3, in order to comply condition (115), some particular combinations of�r and �s should be avoided. Moreover, it may happen that the expression (115) does not vanishwhile its value is very small, that is srs≈0. This latter form causes some difficulties in evaluationof hprs in (121). To remove such an unwanted effect, one may follow the idea given in Remark 4and instead of (116) write

c1rs=(srs)2(Vb)

TrsRb1, c2rs=(srs)

2(Vb)TrsRb2 (122)

which leads to

R=[∑r,s

(Vb)rs(Vb)Trs(srs)

2]+

(123)

and

hprs=srs

{(D2+D3)�r�s c

2rs−(D3�

2r +D1�

2s +K 2)c1rs

(D2+D3)�r�s c1rs−(D1�

2r +D3�

2s +K 2)c2rs

}, c1rs=(Vb)

TrsRb1, c2rs=(Vb)

TrsRb2 (124)

From (123) and (124) it may be seen that the contribution of the troublesome modes is automaticallyeliminated when srs→0.

For selection of the pairs (�r ,�s), one may note that in this case there is no specific relationbetween �rand �s . This means that two separate grids in Gaussian plane may be used for the twoparameters. Here in this paper we follow two strategies:

Case 3-a In this form we choose ar =0, cs =0 and thus

(�r ,�s) = 1(r�b,s�d), (�b,�d)∈(R+)2

(r,s) ∈ {−Nr , . . . ,−1,0,1, . . . ,Nr }∪{−Ns, . . . ,−1,0,1, . . . ,Ns}(125)

The integers (Nr ,Ns)∈N2 as well as the spacing between the points �b and �d are determined byinspection (until b1(x, y) and b2(x, y) are reproduced satisfactorily). This form is somewhat similarto the form of the Fourier series, but it may be noted that the basis functions are not orthogonal.

Case 3-b In this form again we suggest a heuristic formulation to define a grid of points inGaussian plane for (�r ,�s) as follows:

(�r ,�s) ∈ {�=± p(i+ j 1),�=± p(i+ j 1), i=1,2, . . . ,np, j =1,2, . . . ,np}∪{�=± p(i+ j 1),�=∓ p(i+ j 1), i=1,2, . . . ,np, j =1,2, . . . ,np}∪{�=± p(i+ j 1),�=± p( j+i 1), i=1,2, . . . ,np, j =1,2, . . . ,np}∪{�=± p(i+ j 1),�=∓ p( j+i 1), i=1,2, . . . ,np, j =1,2, . . . ,np}∪{�=± p(i− j 1),�=± p(i− j 1), i=1,2, . . . ,np, j =1,2, . . . ,np}∪{�=± p(i− j 1),�=∓ p(i− j 1), i=1,2, . . . ,np, j =1,2, . . . ,np}∪{�=± p(i− j 1),�=± p( j−i 1), i=1,2, . . . ,np, j =1,2, . . . ,np}∪{�=± p(i− j 1),�=∓ p( j−i 1), i=1,2, . . . ,np, j =1,2, . . . ,np} (126)



where p is chosen as

p= �pnp×max{Lx , Ly} (127)

Appropriate values for np and �p are suggested here as

np = 5

�p = max{9−(1+K )2,3} (128)

None of the grid points of (126) matches those of the one defined in (87). This heuristic formulationmay be used for a wide range of problems in elasticity and acoustics (as our experience shows).

7. NUMERICAL RESULTS

In this part we present the results of the method applied to some benchmark problems. The deviationfrom the exact solutions is measured by defining error norms for the displacement components as

�u =(∑DP

k=1 (uk−uexactk )2∑DPk=1 (uexactk )2

)1/2

×100, �v =(∑DP

k=1 (vk−vexactk )2∑DPk=1 (vexactk )2

)1/2

×100 (129)

and

emaxu = maxk(|uk−uexactk |)(∑DP

k=1 |uexactk |/DP)×100, emax

v = maxk(|vk−vexactk |)(∑DPk=1 |vexactk |/DP

)×100 (130)

where DP is the number of points selected inside the domain for calculation of the errors. In allexamples solved, for the pseudo inverse procedure, we use the truncation factor =1E−13 as adefault value except otherwise stated (see (75)).

Example 1As the first benchmark problem, we consider the Timoshenko cantilever beam [14] with an endload as shown in Figure 3. The exact solution is as

u = P

6EI

(y− h

2

)[(6Lx−3x2)+(2+�)(y2−hy)]

v = − P

6EI

[(3Lx2−x3)+3�(L−x)

(y− h

2

)2

+ 4+5�

4h2x

] (131)

where u and v are displacement components along x and y, � is Poisson ratio, E is elasticitymodulus, L and h are the height and length of the beam and I =h3/12. In this example we useE=10, �=0.2, h=3, L=10 and P=1. The boundary of the beam has been discretized with 64points (10 segments along the height and 20 segments along the length). The boundary conditionsat the boundary points are found from the exact solution (Dirichlet type at the fixed end andNeumann type at the other three). While selecting EBFs we do not include the polynomial basesgiven in Appendix B (i.e. relations (B2) and (B3)). This means that we intend to find the solutionjust with EBFs (the reader may note that in the case that the polynomial bases are included, theexact solution is reproduced). For selection of the EBFs we follow both strategies described in



Figure 3. The Timoshenko beam and the boundary points used.

Figure 4. Variations of p�k1

and p�k2

defined in (80) and (81) for Example No. 1:

(a) variation of p�k1

and (b) variation of p�k2.

Section 4. For Strategy I-a, we have chosen a series of points along the imaginary axis in Gaussian

plane. Figure 4 depicts the variations of p�k1

and p�k2

defined in (80) and (81). We have chosen�=0.2 for selection of the EBFs. This leads to the following intervals for a and b :

�= 1b, b∈{[−2,−ε]∪[ε,2]}, �b=0.25, ε=0.01 (132)

and

�= 1d, d∈{[−3,−ε]∪[ε,3]}, �d=0.25, ε=0.01 (133)

As is seen, we have excluded the origin of the imaginary axis by choosing ε=0.01. The totalnumber of EBFs selected in this strategy is 160, while the number of data available at the boundariesis 128 (note that at the corner points we have used two close points at the two intersectingboundaries). The error norms defined in (129) and (130) are reported in Table II. A grid of 6×20points (DP=120 in (129) and (130)) is used for the calculation of the errors. The numbers in theparentheses are obtained from the application of formula (67), Remark 5, in the calculation ofEBF’s coefficients.



Table II. The error norms calculated for Example No. 1 (numbers in theparentheses are from application of (67)).

Strategy I-a Strategy I-b Strategy II

� % emax % � % emax % � % emax %

For u 0.00014 0.0003 0.0077 0.023 0.00015 0.00065(0.000044)∗ (0.00012)∗ (0.0052)∗ (0.012)∗ (0.00039)∗ (0.0010)∗

For v 0.00011 0.00031 0.0437 0.103 0.00011 0.00029(0.000044)∗ (0.00011)∗ (0.0077)∗ (0.022)∗ (0.00040)∗ (0.00040)∗

The results of Strategy I-b are also reported in the table. In this strategy the intervals and theincrements needed for b and d are determined automatically with the aid of dave and formulas(77), (79) and (86). Here we use =1.5 in (86). It can be seen that the two strategies have ledto small error norms; however, those obtained from Strategy I-b are larger. Interesting to note isthat in both cases the results of formula (67) are superior to those obtained by relation (45). Wehave included the results of Strategy II in the same table (M=4, N =3, =2� giving 192 EBFs).The order of the error norms in this latter strategy seems to be the same as that of Strategy I-a.Moreover, the results from (67) and (45) are of nearly similar orders.

It is worthwhile to give some insights to the effects of the truncation parameter in (75). This isperformed by altering from a rather large value to a small one. Figure 5 shows the variations ofthe error norms versus −1 in logarithmic scales. The problem has been solved with 144 boundarypoints using two formulations for the evaluation of the coefficients (i.e. formulas (45) and (67)).It can be seen that although for various strategies the achieved accuracy differs, for almost allcases the smallest choice of , that is close to machine epsilon order, is the best one. However, itmay be thought that such a small value may increase the sensitivity of the results with respect tounwanted noises in the boundary data. This will be discussed in the next example.

Example 2A simply supported beam under uniform normal traction q at the top surface and varying tractionsat the two ends is considered here. The variations of normal stresses at the two ends are of cubicorder in terms of y coordinate (but produce no bending moment). The variations of shearingstresses are of quadratic order whose resultants are equal to the reaction forces at the two supports.The exact solution may be found in [14] as

u = q

2EI

[(1

4L2x− x3

3

)y+x

(2

3y3− 1

10h2y

)+�x

(1

3y3− h2

4y+ h3

12

)]

v = − q

2EI

{1

12y4− 1

8h2y2+ 1

12h3y+�

[1

2

(1

4L2−x2

)y2+ 1

6y4− 1

20h2y2

]}

− q

2EI

[1

8L2x2− 1

12x4− 1

20h2x2+ 1

4

(1+ 1

2�)

)h2x2

]

+ 5

384

qL4

EI

[1+ 12

5

h2

L2

(4

5+ �

2

)](134)



0.00001

0.0001

0.001

0.01

0.1

1

10

100

1.E+063

1/

1/

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 1.E+12 1.E+13

1.E+063 1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 1.E+12 1.E+13

(a)

(b)

Strategy I-a formula (45)Strategy I-a formula (67)Strategy I-b formula (45)Strategy I-b formula (67)

Strategy II formula (45)Strategy II formula (67)

Strategy I-a formula (45)Strategy I-a formula (67)Strategy I-b formula (45)Strategy I-b formula (67)

Strategy II formula (45)Strategy II formula (67)

Figure 5. The effect of truncation factor on the accuracy of the solution in Example No. 1: (a) variationof error norm of u displacement and (b) variation of error norm of v displacement.

In the above formula, the origin of the coordinates x and y is considered at the centre of thebeam (−L/2�x�L/2,−h/2�y�h/2). The definitions of L , h, I , E and � are the same as inExample No. 1 (also the chosen values are the same). We use q=1 in this example. Here againthe boundary of the beam has been discretized with 64 points in a similar fashion explained forthe previous example. The boundary conditions at the boundary points are found from the exactsolution (Neumann type at the four surfaces while u=v=0 at x=−L/2 and v=0 at x= L/2).The error norms defined in (129) and (130) are reported in Table III. A grid of 6×20 points(DP=120 in (129) and (130)) is used for the calculation of the errors. The results of Strategies I-aand I-b are given in the table. Here again the errors are very small; however, those from StrategyI-b are larger. The results of Strategy II are also shown in the same table (M=4, N =3, =2�giving 192 EBFs), which are of the same order as those from Strategy I-a. As shown in the table,



Table III. The error norms calculated for Example No. 2 (numbers in theparentheses are from application of (67)).


� % emax % � % emax % � % emax %

For u 0.00036 0.0009 0.00093 0.0042 0.00022 0.00051(0.00033)∗ (0.0009)∗ (0.0052)∗ (0.012)∗ (0.00078)∗ (0.0023)∗

For v 0.000019 0.000042 0.00066 0.0010 0.000046 0.00007(0.000025)∗ (0.00009)∗ (0.0077)∗ (0.022)∗ (0.00023)∗ (0.00023)∗

Figure 6. Variations of displacement components and the corresponding errors computed in ExampleNo. 2 by Strategy I-b: (a) variation of u; (b) variation of eu ; (c) variation of v; and (d) variation of ev .

except for Strategy I-a, the results of formula (45) are superior to those obtained from formula(67). Variations of the displacement components and the associated error distributions, for StrategyI-b, are presented in Figure 6.

In order to investigate the sensitivity of the solution with respect to perturbation of the boundarydata, here we add some randomly generated noises to the data. To this end, inspired by sensitivityanalysis in [33, 37], we employ a routine for generating a set of random real-valued data for theelements of an array as r, which is of the same length as U in (31), such that ri ∈[−1,1] fori=1, . . . ,M . The array is then multiplied by a fraction of the maximum element of U as

Ur =(�U )r, U =maxi

(|Ui |) (135)



Table IV. Effects of noise added to the data in Example No. 2 using truncation parameter =1E−13(64 points are used at the boundaries).


Noise level � % �u % �v % �u % �v % �u % �v %

0 0.0004 1.9E−5 0.00093 0.00066 0.00022 4.4E−50.005 0.0043 0.0016 0.0098 0.0035 0.0039 0.00160.05 0.033 0.011 0.13 0.062 0.037 0.0130.1 0.098 0.049 0.14 0.09 0.09 0.0480.5 0.29 0.12 1.37 0.22 0.27 0.111 0.45 0.19 3.35 0.8 0.34 0.14

Table V. Effects of noise added to the data in Example No. 2 using truncation parameter =1E−11(64 points are used at the boundaries).


Noise level � % �u % �v % �u % �v % �u % �v %

0 0.00039 0.00097 0.021 0.030 0.0056 0.00170.005 0.0054 0.0019 0.20 0.030 0.076 0.00270.05 0.032 0.011 0.033 0.085 0.037 0.0140.1 0.10 0.051 0.33 0.087 0.095 0.0490.5 0.29 0.11 0.45 0.24 0.28 0.111 0.45 0.19 1.6 0.38 0.42 0.17

where Ur is the array of the boundary noise data to be added to the true boundary values and � isthe noise level (in percent). Table IV shows the results of adding different levels of the noises tothe boundary data. To investigate the effect of choosing different sets of noise data, we report themaximum error norms for three sets of generated noises in each case. This means that the resultspresented in each row of the table are the maximum error norms of three different solutions withdifferent noise data but similar levels. The table shows that the errors are nearly proportional tothe level of the added noises for all strategies. The analysis is repeated for another value of thetruncation parameter in order to examine its effects on the sensitivity of the solution. The resultsfor =1E−11 are reported in Table V. A comparison of the results of the two tables indicates thatfor 1E−13��1E−11, the order of error norms does not change much for higher noise levels.However, compared with Table IV, less sensitivity is observed in Table V for very small valuesof �, that is 0��0.005%.

Based on the above discussion, one may be interested to know about the effects of on theconvergence of the solution. Table VI demonstrates the error norms obtained from Strategy I-afor three selected values of the truncation parameter. As is seen in the table, the best results areobtained for =1E−13. Convergence plots for the variations of the relative errors with respectto the number of boundary nodes are given in Figure 7. The results indicate that for rather courseboundary grids, exponential convergence may be expected; however, when the grids become finersuch a high rate is lost. This means that just with a few boundary nodes one may obtain a ratherhigh accuracy, which seems to be sufficient from an engineering standpoint of view (note that all



Table VI. Convergence of the solution in Example No. 2 using Strategy I-a (numbers in theparentheses are from application of (67)).

=1E−13 =1E−11 =1E−9

No. of DOFs �u % �v % �u % �v % �u % �v %

56 7.1E−1 6.2E−1 7.1E−1 6.3E−1 7.1E−1 6.3E−1(1.8E−1)∗ (9.5E−2)∗ (1.8E−1)* (9.5E−2)∗ (1.8E−1)∗ (9.5E−2)∗

128 3.6E−4 1.9E−5 3.9E−3 9.7E−4 1.5E−2 5.6E−3(3.3E−4)∗ (2.5E−5)∗ (2.2E−4)∗ (5.1E−5)∗ (2.6E−4)∗ (1.8E−4)∗

248 1.4E−4 1.6E−5 3.9E−4 6.9E−5 1.2E−2 2.9E−3(1.8E−4)∗ (1.6E−5)∗ (1.2E−4)∗ (1.8E−5)∗ (2.7E−4)∗ (1.2E−4)∗

368 1.9E−4 1.33E−5 3.5E−4 8.0E−5 1.3E−2 2.9E−3(2.7E−5)∗ (4.4E−6)∗ (9.9E−5)∗ (1.3E−5)∗ (2.7E−4)∗ (1.1E−4)∗

488 9.4E−5 9.0E−6 2.8E−4 5.3E−5 7.0E−3 1.7E−3(4.3E−5)∗ (3.9E−6)∗ (8.8E−5)∗ (1.1E−5)∗ (2.6E−4)∗ (1.1E−4)∗

928 7.8E−5 7.8E−6 2.0E−4 4.3E−5 5.4E−3 1.2E−3(2.2E−5)∗ (5.5E−6)∗ (5.5E−6)∗ (5.9E−5)∗ (2.4E−4)∗ (9.9E−5)∗

1

9

1

1

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

01

Log(M)

Lo

g(

u)

%

Lo

g(

v) %

Strategy I-a formula (45)


12.5

1

10.5

-6

-5

-4

-3

-2

-1

0

Log(M)



(a) (b)

1.5 2 2.5 3 1 1.5 2 2.5 3

Figure 7. Convergence plots in logarithmic scales for the relative errors versus the number of nodes atthe boundaries in Example No. 2 using Strategy I-a, =1E−13: (a) for u and (b) for �.

error norms are given in per cent). However, when the boundary grid becomes finer, a similarefficiency is not easily attainable. The reason for such a high rate of convergence for course gridsmay lie in the fact that we are using exponential bases (and this is in contrast with other methodssuch as FEM). However, at the moment we do not have any mathematical explanation for suchan effect, but the readers are referred to similar evidences in the MFS method reported in [39].As another observation we may note that the convergence of the solution is not monotonic. This



Figure 8. The perforated plate under tension: (a) the geometry and the solution domain; (b) the boundarypoints used; and (c) the domain points used for calculation of errors.

Table VII. Number of points used at the boundaries in Example No. 3.

No. of points Boundary 1 Boundary 2 Boundary 3 Boundary 4 Boundary 5

Set No. 1 21 28 28 21 12Set No. 2 31 41 41 31 15

is an inherent effect in many collocation methods as experienced in [23]. The weighted residualmethod addressed in Section 3.1 may be a clue for elimination of such an effect.

Example 3The well-known perforated plate under tension is revisited here. The geometry of the problem isshown in Figure 8 (L=4 in this paper). Plane strain condition is considered in this example. Theexact solution in polar coordinate can be found in many texts.

ur = tx4G

{r

[(�−1)

2+cos(2)

]+ r20

r

[1+(1+�)cos(2)

]− r40r3

cos(2)

}

u = tx4G

[(1−�)

r20r

−r− r40r3

]sin(2)

(136)

where

G= E

2(1+�), �= 3−�

1+�(137)

In a rectangular coordinate one may write uexact=cos()ur −sin()u, vexact=sin()ur +cos()u. In this example we use E=1, �=0.3 r0=0.5 and tx =1. The exact solution is usedfor defining the tractions at the bounded domain used for the solution. A sample of boundarypoints and the grid of points used for error calculation are shown in Figure 8. According to thenumbering of the boundaries in Figure 8(b), two sets of boundary point arrangements, as given inTable VII, are considered.



Table VIII. The results of Strategy I for the perforated plate, Example No. 3 (numbers in the parenthesesare from application of (67)).

� % emax %

Satisfac. of bound. cond. Boundary grid No. For u For v For u For v

Strategy I-a Set No. 1 0.04 (0.04)∗ 0.11 (0.11)∗ 0.22 (0.29)∗ 0.68 (0.76)∗Set No. 2 0.04 (0.04)∗ 0.13 (0.12)∗ 0.21 (0.23)∗ 0.65 (0.61)∗

Strategy I-b Set No. 1 0.042 (0.010)∗ 0.13 (0.034)∗ 0.25 (0.029)∗ 0.79 (0.10)∗Set No. 2 0.040 (0.011)∗ 0.13 (0.033)∗ 0.11 (0.029)∗ 0.43 (0.099)∗

Here again for selection of the EBFs, we follow both strategies described in Section 4. ForStrategy I-a, we have chosen a series of points along the imaginary axis in Gaussian plane (�=0.2)within the intervals of

�= 1b, b∈{[−15,−ε]∪[ε,15]}, �b=1, ε=0.5 (138)

and

�= 1d, d∈{[−10,−ε]∪[ε,10]}, �d=1, ε=0.5 (139)

The results are summarized in Table VIII for the two sets of boundary nodes. Unlike the twoprevious examples, in this example the error norms are of higher order but are still very small(note that the errors are given in a per cent scale). While using formula (45) the error norms forboth strategies are of similar orders. However, for Strategy I-b the results obtained by formula(67) are of less error compared with those from formula (45). As shown in Table IX, the errorscalculated by the application of Strategy II are less than those from the other two strategies. Also,the results of formulas (45) and (67) are of similar orders.

Example 4In this example we consider elasto-dynamic problems with body forces. The exact solutions areconsidered to be in the form of polynomials. The aim is to examine and compare the resultsobtained from the use of the Fourier series, Case 2, and those explained in Cases 3-a-b in Section 6(we disregard Case 1 since its results just convey the accuracy of the homogeneous part, which hasalready been discussed in three examples). We shall also consider two domains for the solutions.The first domain is of square shape and the second one is a sector as shown in Figure 9. Schematicboundary conditions in each case are shown around the domains. The parameters used, the exactsolutions and associated body forces are given in Table X.

Before solving the problems, it is worthwhile to examine the performances of the Fourier seriesand the EBFs in reproduction of the body forces (i.e. formulas (108) and (114)). Figures 10 and 11depict the variations of the errors (computed via (129) while replacing u and v with bx and by)versus the number of functions used in each case. For Case 3 the sampling points for the calculationof the reproduced functions are different from those used for the evaluation of the coefficientfactors. It can be seen that the error norms from the proposed method are of three-logarithmic-orderless than those from the Fourier series.

Having expressed the body forces in terms of the basis functions, we find the solutions to theproblems defined in Table X first through intermediate particular solutions as (110) or (120) and



Table IX. The results of Strategy II for the perforated plate M=4, N =3 (numbers in the parenthesesare from application of (67)).

M=4, N =3 M=7, N =5

Boundary grid No. � % for u � % for v � % for u � % for v

Set No. 1 0.025 (0.027)∗ 0.081 (0.089)∗ 0.004 (0.0034)∗ 0.013 (0.012)∗Set No. 2 0.033 (0.036)∗ 0.10 (0.12)∗ 0.0028 (0.0033)∗ 0.0086 (0.010)∗

Figure 9. The domain shapes used for Example No. 4: (a), (b) a square with 80 boundary pointsand (c), (d) a sector with 60 boundary points.

Table X. Plane strain elasto-dynamic problems with E=20, �=0.3, K =20 (L=10 forthe square and L=4 for the sector).

Body force

No. of exact solution u v bx by

1 −2.5x2 −5y2 5D1+2.5K 2x2 10D1+5K 2y2

2 x2y xy2 2y(D1+D2+D3+0.5K 2x2) 2x(D1+D2+D3+0.5K 2y2)

then via formula (103). The error norms obtained in each case are reported in Tables XI–XIII. InTable XI we have considered two sets of solutions for each problem; one with 1681 terms for theFourier series (produced by considering −20�n�20 and −20�m�20 in (108)) and another with6561 terms (produced by considering −40�n�40 and −40�m�40). The homogeneous part ofthe solution in all cases are found through Strategy II (M=4, N =3).

Example 5An infinitely long vibrating beam with height h=1 and material properties as E=29000, �=0.3is considered here. The density of the material and the frequency of the excitation are consideredso that K =√��2=11.81455. The top and bottom surfaces of the beam are traction free and nobody force is present in the problem. The exact solution of the problem is as

u = e−0.982925y{0.497751(e1.47853y−e0.487325y)+0.493701(e0.4956y−e1.47025y)}sin(0.5x)v = e−0.982925y{0.493372(e1.47853y−e0.487325y)+0.506542(e0.4956y+e1.47025y)}cos(0.5x)

(140)



Results for body Force1 on 214 points

0.0001

0.001

0.01

0.1

1

10

0

The number of functions

No

rm o

f E

rro

r %

200 400 600 800 1000 1200 1400 1600 1800

bx-Fourier

by-Fourier

bx-Case3-a

by-Case3-a

bx-case3-b

by-case3-b

Figure 10. Error norms computed from the Fourier series, formula (108), and our method, formula (114),in reproduction of the body force No. 1 in Table X.

Results for body Force2 on 214 points

0.001

0.01

0.1

1

10

100

0

The number of functions

No

rm o

f E

rro

r %

200 400 600 800 1000 1200 1400 1600 1800

bx-Fourier

by-Fourier

bx-Case3-a

by-Case3-a

bx-case3-b

by-case3-b

Figure 11. Error norms computed from the Fourier series, formula (108), and our method, formula (114),in reproduction of the body force No. 2 in Table X.

For the numerical solution the problem has been considered with finite length of L=20. Dirichletboundary conditions are considered at the two cut-off ends obtained by the exact solution. Variationsof the two components of the displacement are shown in Figures 12(a) and (c).

The problem is solved with two sets of boundary grid points. The number of points used alongeach boundary is given in Table XIV. For Strategy I-a, we have chosen a series of points alongthe imaginary axis in Gaussian plane (�=0.2) within the intervals of

�= 1b, b∈{[−(0.6+ε),−ε]∪[ε,(0.6+ε)]}, ε=0.01 (141)



Table XI. Results obtained through the use of the Fourier transformation (case 2) for particular solutionand Strategy II for satisfaction of boundary conditions (numbers in the parentheses are from application

of (104) for satisfaction of boundary conditions).

Domain No. Exact solution No. The number of terms in (108) � % for u � % for v

A 1 1681 0.298 (0.296)∗ 0.239 (0.237)∗6561 0.0212 (0.0223)∗ 0.0168 (0.0164)∗

2 1681 34.31 (36.14)∗ 35.95 (38.60)∗6561 28.37 (29.58)∗ 26.57 (29.82)∗

Table XII. Results obtained through the use of EBFs, case 3-a for particular solution (Nr =Ns =6,�b=�d=0.1 in (125)) and Strategy II for satisfaction of boundary conditions (numbers in the

parentheses are from application of (67)).

The number ofdomain points

Domain Exact The number ofNo. solution No. functions used For body forces For errors � % for u � % for v

A 1 169 256 240 0.0078 (0.0027)∗ 0.0066 (0.0015)∗2 169 256 240 0.0062 (0.0019)∗ 0.0064 (0.0046)∗

B 1 169 256 214 0.045 (0.051)∗ 0.0064 (0.011)∗2 169 256 214 0.011 (0.025)∗ 0.0055 (0.0083)∗

Table XIII. Results obtained through the use of EBFs, case 3-b for particular solution(np=5 in (126) and (127)) and Strategy II for satisfaction of boundary conditions

(numbers in the parentheses are from application of (67)).

The number ofdomain points

Domain Exact The number ofNo. solution No. functions used For body forces For errors � % for u � % for v

A 1 360 256 240 0.0094 (0.0018)∗ 0.0036 (0.0020)∗2 360 256 240 0.0079 (0.0021)∗ 0.016 (0.0031)∗

B 1 360 256 214 0.029 (0.066)∗ 0.0048 (0.0096)∗2 360 256 214 0.018 (0.015)∗ 0.0062 (0.0068)∗

and

�= 1d, d∈{[−(2+ε),−ε]∪[ε,(2+ε)]}, ε=0.01 (142)

Tables XIV and XV show the errors of computations for the two sets. Similar to the results of theprevious examples, the errors are very small. Table XV shows that while using (45) the resultsform Strategy I-a are of less errors compared with those from Strategy I-b though the number ofEBFs used is less in Strategy I-a. However, the situation is other way round while using (67).Table XVI also shows that the results from Strategy II are nearly in the same order of accuracy asthose in Table XV. In this latter case the errors obtained from (45) are less than those from (67).



Figure 12. Variations of displacement components and the corresponding errors computed inExample No. 5 using Strategy II, Set No. 2, M=4, N =3: (a) variation of u; (b) variation of eu ;

(c) variation of v; and (d) variation of ev .

Table XIV. Number of points used at the boundaries in Example No. 5.

Boundary 1 Boundary 2 Boundary 3 Boundary 4

Set No. 1 9 41 9 41Set No. 2 19 81 19 81

Overall conclusion from these tables is that the three strategies give nearly similar results whileusing either (45) or (67).

Here again to give some insights into the effects of the truncation parameter in (75), a series ofanalyses are performed by altering from a rather large value to a small one. Figure 13 shows thevariations of the error norms versus −1 in logarithmic scales. The problem has been solved with200 boundary points using two formulations for the evaluation of the coefficients (i.e. formulas(45) and (67)). Similar to the results shown in Figure 5 for the first example, it can here be seenthat although the achieved accuracy differs for various strategies, for almost all cases the smallestchoice of is the best one. This may mean that the method is also a good candidate for higherprecision computing schemes, which may be implemented in the next generation of software andcomputers.

The sensitivity of the solution with respect to the noise in the boundary data is also investigatedfor this example by adding some randomly generated noises to the data. The arrays of noise areconstructed in a fashion similar to the second example (i.e. according to (135)). Table XVII shows



Table XV. The results of Strategy I for the vibrating beam, Example No. 5 (numbers in the parenthesesare from application of (67)).

� % emax %

Satisfac. of B. C. Bound. grid No. No. of basis For u For v For u For v

Strategy I-a Set No. 1 96 0.0044 0.0024 0.014 0.0045�b=0.1�d=0.5 (0.032)∗ (0.033)∗ (0.12)∗ (0.063)∗

Set No. 2 176 0.0054 0.0049 0.022 0.011�b=0.05�d=0.25 (0.011)∗ (0.011)∗ (0.041)∗ (0.018)∗

Strategy I-b Set No. 1 304 0.0076 0.0084 0.032 0.02(0.0046)∗ (0.0030)∗ (0.032)∗ (0.005)∗

Set No. 2 608 0.018 0.019 0.067 0.037(0.0055)∗ (0.0057)∗ (0.018)∗ (0.0096)∗

Table XVI. The results of Strategy II for the vibrating beam (numbers in the parenthesesare from application of (67)).

M=4, N =3 M=7, N =5

Number of basis =192 Number of basis =560

Boundary grid No. � % for u � % for v � % for u � % for v

Set No. 1 0.0087 (0.071)∗ 0.0094 (0.076)∗ 0.024 (0.0039)∗ 0.026 (0.0039)∗Set No. 2 0.0078 (0.085)∗ 0.0085 (0.091)∗ 0.026 (0.005)∗ 0.028 (0.0053)∗

the results of adding different levels of the noises to the boundary data. Here to illustrate the effectof different sets of the noise data, we report the error norms for the three sets of the generatednoises in each case. The table shows that here again errors are nearly proportional to the levelof the added noises for all strategies, but the rates are larger than those reported in the secondexample. It may be noted that the lengths of the traction-free boundaries in this example are morethan those of the second example while at the same time the non-zero boundary values, whichcontrol the magnitude of the noise by relation (135), are more than those in the second example.Also interesting to note is that, for a certain level of noise in each strategy, the orders of errornorms are nearly similar. This means that the solution is not much sensitive to the arrangement ofthe noise data.

The results of the convergence study for this example, using Strategies I-b and II, are given inTable XVIII. As is seen in the table for Strategy II, here again the best results are obtained bysetting =1E−13. Similar results are obtained for Strategy I-b and thus we have just includedthose of =1E−13. Convergence plots of the errors are given in Figure 14. Similar to the results inthe second example, for both strategies exponential convergence is seen for rather course boundarygrids although as the grids become finer such an effect is reduced. All conclusions made in thesecond example are valid here noting that the convergence rates for the course grids are even morethan those in Figure 7.

Note that by setting M=4, N =3 for Strategy II, the number of EBFs is maintained for allsolutions (192 EBFs), while by setting =1.5 for Strategy I-b the number of EBFs is growing



0.001

0.01

0.1

1

10

100

1.E+10 1.E+11 1.E+12 1.E+13

1/

1/1.E+10 1.E+11 1.E+12 1.E+13

0.001

0.01

0.1

1

10

100Strategy I-a formula (45)


Strategy I-b formula (45)

Strategy I-b formula (67)

Strategy II formula (45)

Strategy II formula (67)

Strategy I-a formula (45)Strategy I-a formula (67)Strategy I-b formula (45)Strategy I-b formula (67)Strategy II formula (45)Strategy II formula (67)

(a)

(b)

Figure 13. The effect of truncation factor on the accuracy of the solution in Example No. 5: (a) variationof error norm of u displacement and (b) variation of error norm of v displacement.

by increasing the number of boundary nodes. This means that in Strategy II, for instance, bya fixed number of EBFs one may achieve a certain level of accuracy (which is high enough inthe engineering analysis); however, by further refining the boundary grid, for example when thenumber of boundary information exceeds that of EBFs, no more improvement may be expected.The reader may note that when the number of boundary information exceeds the number of EBFs,the matrix G in (46) or (63) becomes singular; however, this has no effect on the solution as isseen in the results. This makes a desirable flexibility for the method, since the user is not confinedto obey a limit for the number of basis functions.

To demonstrate the potential of the method we reconsider this last example with higher frequency.Here again the top and bottom surfaces of the beam are traction free and no body force is present inthe problem. The problem is reanalysed with K =√��2=11809.0996, which is about a thousandtimes larger than the first case. Other characteristics of the problem are as before. Such a frequency



Table XVII. Effects of noise added to the data in Example No. 5 using truncation parameter =1E−13(100 points are used at the boundaries).


Noise level �% Random set label �u % �v % �u % �v % �u % �v %

0 A0 0.0044 0.0024 0.0076 0.0084 0.0087 0.00940.005 A0.005 0.11 0.089 0.17 0.14 0.11 0.081

B0.005 0.076 0.073 0.067 0.080 0.060 0.079C0.005 0.17 0.17 0.18 0.25 0.16 0.17

0.05 A0.05 0.99 0.87 1.49 1.48 1.21 0.89B0.05 1.13 1.12 1.83 1.93 1.09 1.12C0.05 1.03 0.98 1.35 1.22 1.09 1.03

0.1 A0.1 2.30 2.24 4.07 4.07 2.39 2.29B0.1 1.38 1.93 1.62 1.07 1.38 1.86C0.1 4.57 0.68 5.25 1.27 4.80 0.89

0.5 A0.5 12.75 6.81 14.24 9.10 11.08 6.76B0.5 20.68 17.07 22.90 18.06 23.21 17.49C0.5 14.77 14.07 11.47 15.82 12.54 14.20

Table XVIII. Convergence of the solution in Example No. 5.

Strategy II, M=4, N =3 Strategy I-b, =1.5

=1E−13 =1E−11 =1E−9 =1E−13

No. of DOFs �u % �v % �u % �v % �u % �v % �u % �v %

48 6.01E1 5.94E1 6.01E1 5.94E1 6.01E1 5.94E1 1.44E2 1.43E280 1.34E−2 1.40E−2 6.04E−2 5.52E−2 8.12E−2 7.40E−2 1.05E−1 1.09E−1128 7.81E−3 8.46E−3 1.55E−2 1.69E−2 3.99E−2 4.18E−2 6.21E−3 2.55E−3224 8.42E−3 9.15E−3 1.53E−2 1.66E−2 3.93E−2 3.89E−2 3.33E−2 3.69E−2400 7.84E−3 8.48E−3 1.56E−2 1.69E−2 5.40E−2 5.60E−2 1.81E−2 1.89E−2528 7.84E−3 8.51E−3 1.66E−2 1.79E−2 6.20E−2 6.53E−2 1.12E−2 1.26E−2808 7.80E−3 8.48E−3 1.77E−2 1.93E−2 7.33E−2 7.80E−2 5.86E−3 6.13E−3

value produces a combination of elastic waves, surface and body waves, with relatively smalllengths (for instance the shearing wavelength is 0.056). The exact solution is given as

u = 1500{A[cos(ax−by)+cos(ax+by)]+B[cos(ax−cy)+cos(ax−cy)]}v = 1500{C[cos(ax−by)−cos(ax+by)]+D[cos(ax+cy)−cos(ax−cy)]} (143)

with

A = 0.0988634714190, B=0.053598448083427

C = 0.0044252326676896, D=0.707092934002198

a = 5.000000917333702, b=111.70428425041774, c=65.96208369966548



14

1

11.5

-3

-2

-1

0

1

2

3

1

Log(M)

Lo

g(

) %

u (Strategy I-b)v (Strategy I-b)u (Strategy II)v (Strategy II)

1.5 2 2.5 3

Figure 14. Convergence plot in logarithmic scales for relative errors versus the number of nodes at theboundaries in Example No. 5 using strategies II and I-b, =1E−13.

A length of L=6 is considered for numerical solution. Here again Dirichlet boundary conditionsare considered at the two cut-off ends obtained by the exact solution. Variations of the twocomponents of the displacement are shown in Figures 15(a) and (b). Strategy I-b is employed( =1), while 151 and 31 points are, respectively, used along the length and height of the layer.The error norms calculated are

�u =0.0148%, �v =0.0038%

The reader may note that similar problem cannot easily be solved with mesh-based method suchas the finite element for instance (following a rule of thumb as 10 node per wavelength [41], thenumber of nodes in FEM for this problem roughly amounts to 190 000 nodes that gives 380 000degrees of freedom; however, achieving similar accuracy will be difficult due to dispersion effect).The error distributions of the displacements are shown in Figures 15(c) and (d).

8. CONCLUSIONS

In this paper we have presented a method in which a series of EBFs have been used to solve someelasticity problems. The EBFs are found by defining characteristic equations from the governingdifferential equations in elasto-static and time harmonic elasto-dynamic problems. The boundaryconditions are imposed through a collocation approach and thus the method can be categorized inmeshless types. In the method we presented, the number of EBFs does not need to be equal to thatof the boundary information. A transformation technique, proposed in our previous works [18, 20],has been employed for the evaluation of the unknown coefficients. The basis of the transformationhas been explained in detail. It has been shown that the result of the transformation techniquedepends upon the EBFs used for the solution. A consistent parameter tuning method has beenproposed to find the appropriate EBFs. A heuristic formula has also been given for those who wantto obtain preliminary results. It has been shown that, with a similar approach, problems with bodyforces can also be solved with the use of EBFs. Several examples have been solved to examine theperformance of the method, while discussions have been included on the stability and convergenceof the numerical solution. All examples show excellent performance of the method.



(a) (b)

(c) (d)

Figure 15. Variations of displacement components and the corresponding errors computed inExample No. 5 with K =√��2=11809.0996 using Strategy I-b: (a) variation of u; (b) variation

of v; (c) variation of eu ; and (d) variation of ev .

The success of the method in the solution of some engineering problems with constant coefficientPDEs, in this paper, indicates its capabilities for the solution of multi-dimensional and multi-physics problems with similar type of PDEs. In some parallel studies, we are currently working onthe application of the method to elastic problems with fully incompressibility behaviour, thin andthick plates under static/dynamic lateral loads, problems with Helmholtz and Poisson equations,potential flow in tanks involving very non-linear sloshing modes. In all such problems we haveso far obtained similar accuracy to that presented in this paper for elasticity problems. In eachcategory of problems one just needs to evaluate the EBFs, which are easily found; the rest ofthe formulation remains unchanged. Such a feature of the method provides flexibility in using acomputer program for many applications.

APPENDIX A

For some special cases, the roots found in (13)–(16) are folded. For instance in (13) if �i ischosen as

�i =±1

√2K 2(1+�)

E(A1)



then two-folded values are calculated as �i =0 and therefore one of the EBFs is missed. Themissed EBF for one of the values for � is written, for instance, as

uiH(�i=0,�i=1

√2K2(1+�)/E)

=hi (x, y)e1√

(2K 2(1+�)/E)y, hi (x, y)={a1i

b1i

}x+

{a2i

b2i

}y+

{a3i

b3i

}(A2)

Coefficients a1i to b3i are found by substitution of (A2) in (6). We have summarized the resultsas follows.

Missed EBFs for plane stress problems. For �i in terms of �i :

�i = 0 (folded), �i =±1

√2K 2(1+�)

E, hi (x, y)=

{1

0

}x+

{0

1�−1

}

�i = 0 (folded), �i =±√2K 2(�2−1)

E, hi (x, y)=

{0

1

}x+

{1�−1

0

} (A3)

For �i in terms of �i :

�i = 0 (folded), �i =±1

√2K 2(1+�)

E, hi (x, y)=

{0

1

}y+

{1�−1

i

0

}

�i = 0 (folded), �i =±√2K 2(�2−1)

E, hi (x, y)=

{1

0

}y+

{0

1�−1i

} (A4)

Missed EBFs for plane strain problems. For �i in terms of �i :

�i = 0 (folded), �i =±1

√2K 2(1+�)

E, hi (x, y)=

{1

0

}x+

{0

1�−1

}

�i = 0 (folded), �i =±√

K 2(1−�−2�2)

E(�−1), hi (x, y)=

{0

1

}x+

{�−1

0

} (A5)

For �i in terms of �i :

�i = 0 (folded), �i =±1

√2K 2(1+�)

E, hi (x, y)=

{0

1

}y+

{1�−1

i

0

}

�i = 0 (folded), �i =±√

K 2(1−�−2�2)

E(�−1), hi (x, y)=

{1

0

}y+

{0

�−1i

} (A6)



APPENDIX B

Summary of the results for the evaluation of the characteristic vector introduced in (22) for aspecial case, that is when �i =�i , is given here. In this case there will be four multiple roots as�i =�i =0. The basis function is written as

u0H(�=0,�=0) ={a10+a20x+a30 y+a40x

2+a50xy+a60 y2+a70x

3+a80x2y+a90xy

2+a100 y3

b10+b20x+b30y+b40x2+b50xy+b60y

2+b70x3+b80x

2y+b90xy2+b100 y3

}(B1)

Coefficients a10 to b100 are found by inserting (B1) in (18) leading to the following results:Plane strain

h01 =

⎧⎪⎪⎨⎪⎪⎩

1

3(−1+2�)y3

2(1−�)

3(−1+2�)x3+xy2

⎫⎪⎪⎬⎪⎪⎭ , h02=

⎧⎪⎪⎨⎪⎪⎩

(1−2�)

6(−1+�)x3+xy2

1

6(−1+�)y3

⎫⎪⎪⎬⎪⎪⎭ , h03=

⎧⎪⎪⎨⎪⎪⎩

1

6(−1+�)x3

(1−2�)

6(−1+�)y3+x2y

⎫⎪⎪⎬⎪⎪⎭

h04 =

⎧⎪⎪⎨⎪⎪⎩

2(1−�)

3(−1+2�)y3+x2y

1

3(−1+2�)x3

⎫⎪⎪⎬⎪⎪⎭ , h05=

{4(−1+�)xy

y2

}, h06=

{y2

2(−1+2�)xy

}

h07 ={2(−1+2�)xy

x2

}

h08 ={

x2

4(−1+�)xy

}, h09=

{0

y

}, h010=

{y

0

}, h011=

{0

x

}, h012=

{x

0

}

h013 ={0

1

}, h014=

{1

0

}

(B2)

Plane stress

h01 =

⎧⎪⎪⎨⎪⎪⎩

(1+�)

3(−1+�)y3

2

3(−1+�)x3+xy2

⎫⎪⎪⎬⎪⎪⎭ , h02=

⎧⎪⎪⎨⎪⎪⎩

(−1+�)

6x3+xy2

−(1+�)

6y3

⎫⎪⎪⎬⎪⎪⎭ , h03=

⎧⎪⎪⎨⎪⎪⎩

−(1+�)

6x3

(−1+�)

6y3+x2y

⎫⎪⎪⎬⎪⎪⎭

h04 =

⎧⎪⎪⎨⎪⎪⎩

2

3(−1+�)y3+x2y

(1+�)

3(−1+�)x3

⎫⎪⎪⎬⎪⎪⎭ , h05=

⎧⎨⎩

−(1+�)

4x2

xy

⎫⎬⎭ , h06=

⎧⎨⎩y2− (−1+�)

2x2

0

⎫⎬⎭ (B3)

h07 =⎧⎨⎩

0

x2− (−1+�)

2y2

⎫⎬⎭



h08 ={

xy

−(1+�)y2

}, h09=

{0

y

}, h010=

{y

0

}, h011=

{0

x

}, h012=

{x

0

}

h013 ={0

1

}, h014=

{1

0

}

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