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Chap te r | F i ve

Boundary ElementTechnology

CHAPTER OUTLINE

5.1 Introduction................................................................................................................. 1135.2 Linear Elements .........................................................................................................1145.3 The BEM With Linear Boundary Elements........................................................119

5.3.1 Corner points and points of change of boundary conditions ........................ 1215.4 Evaluation of Line Integrals on Linear Elements............................................ 123

5.4.1 Outside integration ...........................................................................................1235.4.2 Inside integration ............................................................................................. 1255.4.3 Indirect evaluation of the diagonal influence coefficients ............................135

5.5 Higher Order Elements ............................................................................................1375.6 Near-Singular Integrals...........................................................................................144

5.6.1 The method of element subdivision................................................................ 1465.7 References .................................................................................................................148Problems..............................................................................................................................149

5.1 INTRODUCTION

In the previous chapter, the boundary value problem described by the

Poisson’s differential equation (3.6a) and the boundary condition (3.6b) was

modeled numerically by the boundary element method (BEM). This formula-

tion originated from the discretization of the boundary integral equation

(3.37) and resulted in the system of linear algebraic equation (4.9), which

approximates the solution of the integral equation. The accuracy of the

approximation and the efficiency of the BEM depends on the boundary dis-

cretization technique (i.e., the type of employed element), and on the method

used for the integration of the kernel functions over the elements.

The constant element was presented in Chapter 4. This element approxi-

mates the actual geometry by a straight line, while the unknown boundary

quantity is assumed to be constant on the element, resulting in a discontinu-

ous distribution on the boundary. A better approximation of the boundary

quantity can be achieved by adopting linear variation over the element. 113

The Boundary Element Method for Engineers and Scientists.

© 2016 Elsevier Ltd. All rights reserved.

Even the linear element is not an ideal one as it cannot approximate accu-

rately a curved boundary. For this reason, higher order elements have been

developed, namely, elements that approximate both the boundary geometry

and the boundary quantities by higher order interpolation polynomials, usu-

ally of second or third degree. Evidently, these elements model more accu-

rately the curved boundaries and the distribution of the boundary quantities

on the element. However, they have the weak point that the functions being

integrated over each element become more complicated and the computer

time is considerably increased.

Generally, the boundary elements may be classified in the following three

categories [1]:

a. Subparametric elements. The polynomial approximating the geometry of

the boundary is of a lower degree than that approximating the variation

of the boundary quantity, for example, the element is a straight line,

whereas the boundary quantity varies parabolically.

b. Isoparametric elements. The geometry and the boundary quantity are

approximated by the same degree polynomials, for example, the linear

and the parabolic elements defined in chapter “Numerical Implementation

of the BEM” are isoparametric.

c. Superparametric elements. The geometry is approximated by a higher

degree polynomial than that approximating the boundary quantity, for

example, the element is modeled by a parabolic arc, whereas the bound-

ary quantity is constant or varies linearly on it.

The use of the subparametric and superparametric elements is quite

limited. Subparametric elements, however, result inevitably as a degenerate

case of parabolic elements for rectilinear boundaries. Superparametric ele-

ments, in which the boundary geometry is modeled by a parabolic arc and

the boundary quantity is constant, have been introduced and widely used by

Katsikadelis and his coworkers [2�6]. This element has the advantages of

the constant element regarding the simplicity of the assembly procedure for

the influence matrices, whereas it approximates the curvilinear boundary with

great accuracy. In general, the isoparametric elements are the most widely

used elements, especially in commercial BEM codes.

The boundary elements are also distinguished in continuous and discon-

tinuous. Continuous elements have nodes at their extreme points, and there-

fore they share nodes with the adjacent elements, while discontinuous

elements have nodes located away from the extreme points. In the sections

that follow, we will discuss the linear and the parabolic elements for continu-

ous and discontinuous modeling.

5.2 LINEAR ELEMENTS

As it was mentioned before, linear elements approximate the geometry of the

boundary by straight lines and the boundary quantity by a linear function on

each element. In order to establish the expression for the variation of the

114 Boundary Element Technology

boundary quantity over an element, its values at two nodal points are

required. For this purpose, it is convenient to introduce a local coordinate

system Ox 0y0 on each element, where 2‘=2# x 0 # ‘=2 with ‘ being the

length of the element. For continuous elements the nodal points are placed at

the extreme points (Fig. 5.1a), whereas for discontinuous elements the nodal

points are placed between the end points (Fig. 5.1b).

Figure 5.2 shows the discretization of an elliptic boundary into 12 cont-

inuous linear elements, whereas Fig. 5.3 shows the discretization of the same

boundary into 12 discontinuous linear elements. From these two figures, it

becomes evident that the discretization of the boundary using discontinuous

elements requires twice as many nodes compared to the continuous elements.

However, a better approximation of the boundary quantity is achieved by

employing the discontinuous elements (see Fig. 5.4). Nevertheless, the

advantage of the discontinuous element is not the improved accuracy as

compared to the continuous element, but rather its capability to overcome

computational problems arising at points where the boundary quantity is

discontinuous, for example, the normal derivative at corner points. Hybrid

or continuous�discontinuous elements have also been invented, that is,

elements having only one of the nodes placed at an extreme point.

FIGURE 5.1 Continuous and discontinuous linear elements. (a) Continuous element; (b)Discontinuous element.

5.2 Linear Elements 115

Let us consider the j-th element of the discretized boundary having end

points j and j1 1, and length ‘j (Fig. 5.5). In the local system of axes O0x 0y0

its geometry is described by the equations

x 0 5 x 0 ð2‘j=2# x 0 # ‘j=2Þy0 5 0

FIGURE 5.2 Elliptic boundary modeled by 12 continuous linear elements.

FIGURE 5.3 Elliptic boundary modeled by 12 discontinuous linear elements.


whereas in the global system of axes Oxy by the equations

x5xj11 1 xj

21

xj11 2 xj‘j

x 0 ð5:1aÞ

y5yj11 1 yj

21

yj11 2 yj‘j

x 0 ð5:1bÞ

Recall that the positive direction of x 0 on the element is from point jtowards point j1 1, since the nodes and extreme points of the discretized

FIGURE 5.4 Approximation of the exact function uðxÞ by linear continuous and dis-

continuous elements. (a) Continuous elements; (b) Discontinuous elements.

FIGURE 5.5 Global ðOxyÞ and local ðO0x 0y0Þ systems of axes for the j-th element.

5.2 Linear Elements 117

boundary are numbered in the counter-clockwise sense. The interval

2‘j=2; ‘j=2� �

is normalized by setting

ξ5x 0

‘j=2ð5:2Þ

and then Eq. (5.1) become

x5xj11 1 xj

21

xj11 2 xj2

ξ ð5:3aÞ

y5yj11 1 yj

21

yj11 2 yj2

ξ ð5:3bÞ

where 21# ξ# 1.Equation (5.3) may be rewritten as

xðξÞ5 12ð12 ξÞ xj 1

12ð11 ξÞ xj11 ð5:4aÞ

yðξÞ5 12ð12 ξÞ yj 1

12ð11 ξÞ yj11 ð5:4bÞ

The variation of the boundary quantity u (or un 5 @u=@n) is linear on the

element. Hence, its distribution in the local system of axes is given by the

expression

u5u j11 1 u j

21

u j11 2 u j

‘jx 0

or

u5u j11 1 u j

21

u j11 2 u j

2ξ

or

uðξÞ5 12ð12 ξÞu j 1

12ð11 ξÞu j11 ð5:5Þ

Using local numbering for the nodes, where j and j1 1 are renamed to

1 and 2, respectively, Eqs. (5.4a), (5.4b), and (5.5) may all be expressed

through the following general equation

f ðξÞ5ψ1ðξÞf1 1ψ2ðξÞf2 ð5:6Þ


where f1 and f2 are the values of the function f ðxÞ at the nodes 1 and 2, and

f ðξÞ represents any of the functions xðξÞ, yðξÞ, uðξÞ, or unðξÞ. The functions

ψ1ðξÞ and ψ2ðξÞ are given as

ψ1ðξÞ512ð12 ξÞ ð5:7aÞ

ψ2ðξÞ512ð11 ξÞ ð5:7bÞ

and they express the influence of the nodal values fj and fj11 on the expres-

sion of f ðξÞ for the linear element. They are the functions of the linear

interpolation and they are referred to as linear shape functions. From the

above, one readily concludes that the linear element is isoparametric.

5.3 THE BEM WITH LINEAR BOUNDARY ELEMENTS

In this section, we will present the BEM for the potential equation using

continuous linear elements. The number of the elements is equal to the

number of nodes. The actual boundary is, thus, modeled by an inscribed poly-

gon with the nodes placed at corner points.

In the case of constant elements, the substitute boundary which models

the actual one is always smooth at the nodes and, therefore, the integral

equation (3.29) is employed. In the case of linear elements, however, the

nodes lie at the corners of the polygon and, consequently, the integral

equation (3.28) should be applied instead with α5αi, where αi is the angle

between the elements i2 1ð Þ and i (Fig. 5.6). Thus, after discretizing the

boundary into N linear elements, Eq. (3.28) becomes

FIGURE 5.6 Modeling of the boundary with continuous linear elements.

5.3 The BEM With Linear Boundary Elements 119

εi ui 52XNj51

ðΓj

v un ds1XNj51

ðΓj

u vn ds ð5:8Þ

where εi 5αi=2π and vn 5 @v=@n denotes the derivative of the fundamental

solution in the normal to the boundary direction.

We examine now the integrals over the j-th element. Using the linear

approximation of Eq. (5.6) for the boundary quantities u and un , the line inte-gral appearing in the first sum of Eq. (5.8) may be written as

ðΓj

v un ds5ð121

vhψ1ðξÞ u1

n 1ψ2ðξÞ u2n

i ‘j2

dξ

5 u1n

ð121

v ψ1ðξÞ‘j2dξ1 u2

n

ð121

v ψ2ðξÞ‘j2dξ

5 gij1 u1n 1 gij2 u2

n

ð5:9Þ

where

gij1 5‘j2

ð121

v ψ1ðξÞ dξ ð5:10aÞ

gij2 5‘j2

ð121

v ψ2ðξÞ dξ ð5:10bÞ

and

v512π

lnr ð5:11Þ

r 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½xðξÞ2xi�2 1 ½yðξÞ2yi�2

qð5:12Þ

It should be made clear that the superscript i in the symbols gij1 and gij2indicates the i-th node ðpiÞ where the source is applied, while the superscript

j indicates the element over which the integration is carried out. Finally, the

subscripts 1 and 2 denote in local numbering the points j and j1 1ð Þ,respectively.

In a similar fashion, the line integral appearing in the second sum of

Eq. (5.8) may be written as

ðΓj

vn u ds5 hij1 u1 1 hij2 u2 ð5:13Þ


where

hij1 5‘j2

ð121

vn ψ1ðξÞ dξ ð5:14aÞ

hij2 5‘j2

ð121

vn ψ2ðξÞ dξ ð5:14bÞ

and

vn 5@v@n

512π

cosφr

ð5:15Þ

Substituting Eqs. (5.9) and (5.13) back into Eq. (5.8), the latter yields

2εi ui 1XNj51

H ij u j 5XNj51

Gij u jn ð5:16Þ

where

H ij 5hi11 1 hiN2 for j5 1

hij1 1 hi;j212 for j5 2; 3; . . . ;N

(ð5:17Þ

Gij 5gi11 1 giN2 for j5 1

gij1 1 gi;j212 for j5 2; 3; . . . ;N

(ð5:18Þ

Equation (5.16) may be written in matrix form as

½H �fug5 ½G�fung ð5:19Þ

in which it has been set

½H �52 ½ε�1 ½H � ð5:20Þ

½ε� is a diagonal matrix with elements of the coefficients εi.

5.3.1 Corner points and points of change of boundaryconditions

In formulating Eq. (5.19), it was assumed that the quantities u and un 5 @u=@nhad a unique value. However, this is not always the case. For example, un is

5.3 The BEM With Linear Boundary Elements 121

not continuous at the corner points of the actual boundary, since its value is

generally different before and after the corner. Likewise, in mixed boundary

conditions different values are prescribed at nodal points, where the boundary

conditions change type. At corner points, we may distinguish the following

cases for the boundary conditions [7]:

a. Known: un before and after the corner. Unknown: u at the corner.

b. Known: u at the corner and un before the corner. Unknown: un after the

corner.

c. Known: u at the corner and un after the corner. Unknown: un before the

corner.

d. Known: u at the corner. Unknown: un before and after the corner.

In all the above cases, u is assumed to be continuous at the corner points,

having therefore a unique value at these points. The terms before or after

refer to the value of the quantity just before or right after the corner point

according to the positive sense on the boundary (see Section 4.7).

The unknown boundary quantities are determined under the assumption

that un may be discontinuous at all nodal points, which means that we are

dealing with 2N values of un . With this in mind, we can write Eq. (5.19) as

½H �fug5 ½G��fu�ng ð5:21Þ

where fu�ng is a vector containing the 2N values of the normal derivative

(two at each node) and ½G�� is a N 3 2N matrix whose elements are

defined as

G�i;2j21 5 gij1

G�i;2j 5 gij2 ðj5 1; 2; . . . ;N Þ

ð5:22Þ

For the first three cases of the corner boundary conditions—cases (a), (b),

and (c)—only one boundary quantity is unknown. Therefore, rearranging the

unknowns on the basis of the boundary conditions, Eq. (5.21) produces a

system of N linear equations, which can be solved for the N unknown

boundary quantities. The unknowns are rearranged by examining all the

nodes. If the value of u is unknown at a node, then the respective column

remains at the left-hand side of Eq. (5.21), otherwise this column is multi-

plied by the known value of u, its sign is switched, and it is shifted to the

right-hand side of the equation. Similarly, if the value of un is unknown, then

the respective column of ½G�� is shifted with opposite sign to the left-hand

side of the equation. The two consecutive columns corresponding to un are

added together if the node is not a corner point of the actual boundary. After

completing this process, the right-hand side of the equation contains only

known quantities and, thus, the matrix multiplication results in a single vec-

tor. Corner points may also be treated using one-sided discontinuous elements

before and after the corner (Fig. 5.7). Thus, two separate nodes appear in the

equations at which two different values of un are computed.


It should be noted that an abrupt change in the boundary’s slope

(especially reentrant corners) or a change in the type of boundary conditions

causes a local singularity in the behavior of the solution, which may even

“pollute” the numerical results over the whole domain. A remedy to this

problem is the refinement of the elements near the point of singularity.

Nevertheless, this technique is not always successful in giving a reliable

solution, especially for its derivative, and recourse to special techniques is

unavoidable [8].

5.4 EVALUATION OF LINE INTEGRALS ON LINEARELEMENTS

The matrices ½G� and ½H � appearing in Eq. (5.19) require the computation of

the line integrals (5.10) and (5.14) whose integrands are products of the fun-

damental solution v or its normal derivative vn 5 @v=@n and the linear shape

functions ψ1ðξÞ and ψ2ðξÞ. The integrations are carried out over the interval

½21;11�. Two cases are considered for the linear elements as it was done for

the constant elements. These cases are dictated by the behavior of the func-

tions (5.11) and (5.15). Specifically, when the j-th element, over which the

integration is performed, does not contain the source point i, that is, i 6¼ j,then it is always r 6¼ 0 and the integral is regular. On the other hand, when

the source point lies on the j-th element, that is, i5 j, then the distance rtakes also the value r 5 0 and the behavior of the integral is singular. The

integration for the first case i 6¼ jð Þ will be referred to as outside integration,

whereas for the second case i5 jð Þ will be referred to as inside integration.

5.4.1 Outside integration

The integrals (5.10) and (5.14) may be evaluated analytically using symbolic

languages (e.g., MAPLE). This process, however, yields very lengthy expres-

sions, which in some cases may cover several pages, making them computa-

tionally impractical. A very practical and accurate approach is the numerical

FIGURE 5.7 Discontinuous elements adjacent to a corner of the actual boundary(un 5 @u=@n).

5.4 Evaluation of Line Integrals on Linear Elements 123

integration. Any integration rule may be utilized for this purpose, for exam-

ple, trapezoidal rule, Simpson’s rule, or Newton-Cotes integration formulae.

But, the most suitable method for the numerical evaluation of the BEM

integrals is the Gaussian quadrature (see Appendix B). This method approxi-

mates the integral with great accuracy using the least number of values of the

integrand. The numerical integration should not be performed “blindly.” The

accuracy depends not only on the number of integration points, but also on

how the integrand varies within the integration interval. A smooth variation

of the integrand gives more accurate results. Therefore, the integration

process requires a thoughtful consideration and special care is required when

the integrand exhibits intense changes. In integrals (5.10) and (5.14), the

shape functions vary smoothly and, consequently, the behavior of the inte-

grand is dominated by the functions lnr and 1=r .In order to have a better insight into the variation of the integrand, we

consider the domain of Fig. 5.8 and examine the function gðξÞ5ψ1ðξÞ lnr on

the element with extreme points ð4; 4Þ and ð3; 4Þ. Three locations for the

source are considered:

i. location A, relatively far from the element,

ii. location B, at a relatively moderate distance from the element, and

iii. location C , relatively close to the element.

For the particular element, the transform equation (5.4a) and (5.4b)

become

xðξÞ5�12ð12 ξÞ

�41

�12ð11 ξÞ

�35 3:52 0:5 ξ

yðξÞ5�12ð12 ξÞ

�41

�12ð11 ξÞ

�45 4

FIGURE 5.8 Points A, B, and C where the source is applied.


and according to Eq. (5.12), the relative distance from the source point is

rðξÞ5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�xðξÞ2xi

�21�yðξÞ2yi

�2q

where xi and yi are the global coordinates of the source point.The behavior of the integrand associated with points A, B, and C is illus-

trated graphically through the plots of gðξÞ depicted in Fig. 5.9. It should be

noticed that the variation of the function gðξÞ is quite different when the source

is located close to the element over which the integration is performed.

Therefore, an efficient programming of the BEM should take into account such

a behavior by using Gaussian quadratures with variable number of integration

points (i.e., increasing number of points as the source is getting closer to the

element). The number of integration points should be chosen in a way to

ensure sufficient accuracy. However, an unnecessarily large number of integra-

tion points should be avoided, in order to keep the computation cost low.

5.4.2 Inside integration

In this case the source lies on the element over which the integration is per-

formed. As the integration point runs along the whole element, it will coincide

inevitably with the source point. There, the distance r vanishes and the inte-

grands of Eqs. (5.10) and (5.14) exhibit a singular behavior, because the factors

lnr and cosφ=r become infinite for r 5 0 (see Eqs. (5.11) and (5.15)). These

integrals are known as singular integrals. Their value exists and is determined

through special integration techniques, either analytical or numerical. Even an

indirect method has been invented to circumvent the evaluation of the singular

FIGURE 5.9 Behavior of the integrand gðξÞ5ψ1ðξÞlnr for different locations of thesource.


integrals by computing directly the coefficients Hii and Gii (see Section 5.4.3).

In the sequel, we will first study the integrals with logarithmic singularity, and

then those with Cauchy type singularity (1=r).

INTEGRALS WITH LOGARITHMIC SINGULARITYAnalytical integrationWe consider the general case of the discontinuous linear element shown in

Fig. 5.10. The continuous element results as a special case, when the nodes are

shifted to the extreme points. The two nodes of the linear element are designated

locally by the numbers 1 and 2, while their global coordinates are denoted by

ðx1; y1Þ and ðx2; y2Þ. Using this notation, it can be easily proved that the coordinatetransformation from the local to the global system is expressed by the equations

xðξÞ5 κ1x2 1κ2x1κ

1x2 2 x1

κξ ð5:23aÞ

yðξÞ5 κ1y2 1κ2y1κ

1y2 2 y1

κξ ð5:23bÞ

where 21# ξ# 1 and

κ5κ1 1κ2; ξ5x 0

‘=2ð5:24Þ

Furthermore, Eqs. (5.23) may be rewritten as

xðξÞ5ψ1ðξÞ x1 1ψ2ðξÞ x2yðξÞ5ψ1ðξÞ y1 1ψ2ðξÞ y2

FIGURE 5.10 Discontinuous linear element in local coordinate systems.


in which the shape functions ψ1ðξÞ and ψ2ðξÞ are given by the expressions

ψ1ðξÞ51κðκ2 2 ξÞ ð5:25aÞ

ψ2ðξÞ51κðκ1 1 ξÞ ð5:25bÞ

with 0#κ1; κ2 # 1.It is apparent that for κ1 5κ2 5 1, the shape functions, Eqs. (5.25), reduce to

Eqs. (5.7), which represent the shape functions of the continuous linear element.

If the source lies on the J -th ðJ 5 1; 2Þ local node of the element, its

coordinates are going to be

xJ 5κ1 x2 1κ2 x1

κ1

x2 2 x1κ

ξJ

yJ 5κ1 y2 1κ2 y1

κ1

y2 2 y1κ

ξJ

where

ξ1 52κ1; ξ2 5κ2

and the relative distance of Eq. (5.12) becomes

rðξÞ5 ‘j2

ξ2 ξJ�� ð5:26Þ

where ‘j is the length of the j-th element.

Having described the discontinuous linear element, we shall study now

the integrals of Eqs. (5.10) for the case where the source node i lies on the

integration element j. The integral (5.10a) may then be written for a discon-

tinuous element as

4π‘j

gij1 5

ð121

ψ1ðξÞlnr dξ

5

ðξJ21

ψ1ðξÞlnr dξ1ð1ξJψ1ðξÞlnr dξ

5 I1 1 I2

ð5:27Þ

where, the i-th node coincides with the J -th local node J 5 1; 2ð Þ of element

j j5 1; 2; . . .;Nð Þ. Note, that for discontinuous elements the number of nodes

cannot be equal to the number of elements N . For example, if all the ele-

ments are discontinuous, the total number of nodes will be 2N , which is the

maximum possible number of nodes for a linear element discretization.


The transformation

ξ52ð11 ξJ Þ z1 ξJ ð5:28Þ

maps the integration interval ½21; ξJ � of I1 onto the interval ½0; 11�.Substituting into the expression of I1 which is given in Eq. (5.27), we have

I1 5ðξJ21

ψ1ðξÞlnr dξ

5

ð10

1κ

ðκ2 2 ξJ Þ1 ð11 ξJ Þ z� �

ln�‘j2ð11 ξJ Þ z

�ð11 ξJ Þ dz

Introducing the quantities

θ5‘j2ð11 ξJ Þ z; θ1 5

‘j2ð11 ξJ Þ

the above integral becomes

I1 5ðθ10

1κ

2‘j

ðκ2 2 ξJ Þ12‘jθ

� �lnθ dθ ð5:29Þ

which is of the form

I1 5ðθ10ða1 bθÞ lnθ dθ ð5:30Þ

where the two constants are

a51κ

2‘jðκ2 2 ξJ Þ and b5

1κ

2‘j

� 2

The integral of Eq. (5.30) can be readily integrated by parts to yield

I1 5 aθ ðlnθ21Þ 1 bθ212lnθ2

14

� � �θ10

ð5:31Þ

or using the definitions of the constants a and b, it yields

I1 51κðκ2 2 ξJ Þð11 ξJ Þ

ln�‘j2ð11 ξJ Þ

�2 1

�

11κð11ξJ Þ2

12ln�‘j2ð11 ξJ Þ

�2

14

� ð5:32Þ


Referring now to the second integral of Eq. (5.27), the transformation

ξ5 ð12 ξJ Þ z1 ξJ ð5:33Þ

maps the integration interval ½ξJ ;11� of I2 onto the interval ½0;11�. Thus,the new expression of the integral is

I2 5ð1ξJψ1ðξÞlnr dξ

5

ð10

1κ

ðκ2 2 ξJ Þ2 ð12 ξJ Þ z� �

ln�‘j2ð12 ξJ Þ z

�ð12 ξJ Þ dz

Furthermore, by setting

θ5‘j2ð12 ξJ Þ z; θ1 5

‘j2ð12 ξJ Þ

the above integral may also be written as

I2 5ðθ10

1κ

2‘j

ðκ2 2 ξJ Þ22‘jθ

� �lnθ dθ ð5:34Þ

The integral of Eq. (5.34) has exactly the same form as that of Eq. (5.30),

but with different values of the two constants, which are now defined as

a51κ

2‘jðκ2 2 ξJ Þ and b52

1κ

2‘j

� 2

Substituting this set of constants into Eq. (5.31), we obtain the final

expression for the second integral of Eq. (5.27)

I2 51κðκ2 2 ξJ Þð12 ξJ Þ ln

�‘j2ð12 ξJ Þ

�2 1

8<:

9=;

21κð12ξJ Þ2


�2

14

8<:

9=;

ð5:35Þ

The results of Eqs. (5.32) and (5.35) are combined into Eq. (5.27) to pro-

duce the analytical expression for the influence coefficient gij1 when the node

i lies on the element j (i.e., singular case).


The other influence coefficient defined by Eq. (5.10b), is written for a

discontinuous element as

4π‘j

gij2 5

ð121

ψ2ðξÞlnr dξ

5

ðξJ21

ψ2ðξÞlnr dξ1ð1ξJψ2ðξÞ lnr dξ5 I3 1 I4

ð5:36Þ

where the indices i, j, and J are defined as for the case of Eq. (5.27).

The integrals I3 and I4 are evaluated following the procedure applied for I1and I2. The resulting expressions for these two integrals are



�2 1

�

21κð11ξJ Þ2


�2

14

� ð5:37Þ

and



�2 1

�

11κð12ξJ Þ2


�2

14

� ð5:38Þ

which may be combined into Eq. (5.36) to give the analytical expression for

the singular case of the influence coefficient gij2 .The influence coefficients for continuous elements may be deduced from

the foregoing equations by setting ξ1 521 and ξ2 5 1. In this case,

Eqs. (5.27) and (5.36) give:

i. for ξJ 521, it is κ1 5 κ2 5 1, κ5 2 and

gij1 5‘j4π

ln‘j 2 1:5�

gij2 5‘j4π

ln‘j 2 0:5� ð5:39aÞ

ii. for ξJ 511, it is κ1 5κ2 5 1, κ5 2 and

gij1 5‘j4π

ln‘j 2 0:5�

gij2 5‘j4π

ln‘j 2 1:5� ð5:39bÞ


Numerical integrationAccording to the previous discussion, integrals with logarithmic singularity,

like those of Eqs. (5.27) and (5.36), can be set in the form

I 5ð10f ðxÞ lnx dx ð5:40Þ

Special Gauss integration schemes have been developed for its numerical

evaluation. Stroud and Secrest [9] approximated this type of integral as

follows

ð10f ðξÞ ln 1

ξ

� dξ �

Xnk51

f ðξkÞ wk ð5:41Þ

and produced tables with the integration points ξk and the corresponding

weights wk (see Appendix B and Refs. [9,10]). It should be emphasized here

that the integration interval must always be reduced to ½0;11�. Therefore,when the source lies inside the element, the integration interval must be split

into two intervals, ½21; ξJ � and ½ξJ ;11�. Consequently, the influence coeffi-

cients of Eq. (5.10) become

4π‘j

gijα 5

ð121

ψαðξÞlnr dξ

5

ðξJ21

ψαðξÞlnr dξ1ð1ξJψαðξÞ lnr dξ ðα5 1; 2Þ

ð5:42Þ

where ξJ ðJ 5 1; 2Þ denotes the node of the j-th element where the source

node i is located and r is the relative distance between the source and the

integration point given in Eq. (5.26).

The transformations (5.28) and (5.33) map the two integration intervals

½21; ξJ � and ½ξJ ; 11� of Eq. (5.42) onto the interval ½0; 11�, respectively.Eq. (5.42) is then written as

4π‘j

gijα 5

ð10ψð1Þα ðzÞð11 ξJ Þ ln

�‘j2ð11 ξJ Þz

�dz

1

ð10ψð2Þα ðzÞð12 ξJ Þ ln

�‘j2ð12 ξJ Þz

�dz


or by expanding the logarithms

4π‘j

gijα 5 ð11 ξJ Þ ln�‘j2ð11 ξJ Þ

� ð10ψð1Þα ðzÞ dz

1ð11 ξJ Þð10ψð1Þα ðzÞ lnz dz

1ð12 ξJ Þ ln�‘j2ð12 ξJ Þ

� ð10ψð2Þα ðzÞ dz

1ð12 ξJ Þð10ψð2Þα ðzÞ lnz dz

5 I1 1 I2 1 I3 1 I4

ð5:43Þ

where ψð1Þα ðzÞ and ψð2Þ

α ðzÞ ðα5 1; 2Þ are transformed shape functions obtained

from ψαðξÞ by expressing the variable ξ in terms of z according to

Eqs. (5.28) and (5.33), respectively. The integrals I1 and I3 are regular and

can be evaluated either analytically or by applying the conventional Gauss

integration, while the I2 and I4 are singular and can be evaluated numerically

by virtue of Eq. (5.41).

Integration by extracting the singularityThe integrals of Eq. (5.10) may also be written as

4π‘j

gijα 5

ð121

ψαðξÞlnr dξ

5

ð121½ψαðξÞ2ψαðξJ Þ�lnr dξ1ψαðξJ Þ

ð121

lnr dξ

ð5:44Þ

The integrand of the first integral vanishes for ξ5 ξJ . Indeed, this inte-

grand can be written as

lnrðξÞ1

ψαðξÞ2ψαðξJ Þ

� �

It can be easily noticed that this expression takes for ξ5 ξJ the indetermi-

nate form NN. Thus, applying consecutively L’Hopital’s rule and, taking into

account that the derivatives ψ0αðξÞ5 dψαðξÞ=dξ and r 0ðξÞ5 drðξÞ=dξ are

finite and do not vanish, the integrand in question yields

limξ-ξJ

½ψαðξÞ2ψαðξJ Þ� lnrðξÞ� �

5 0


This result shows that the first integral of Eq. (5.44) is regular and can be

evaluated using conventional Gaussian quadrature. It is worth mentioning

that the accuracy of this numerical integration is increased if it is carried out

on two separate subintervals, ½21; ξJ � and ½ξJ ;11�. Apparently, this is

dictated by the shape of the integrand as it is depicted in Fig. 5.11. The sec-

ond integral of Eq. (5.44) exhibits a logarithmic singularity, but it can be

readily evaluated analytically. In closing we could say that the method of

extracting the singularity simplifies the evaluation of the integral (5.44),

though it does not avoid the evaluation of singular integrals.

Table 5.1 presents the values of the integrals gijα . They have been com-

puted: (1) analytically from Eqs. (5.27) and (5.36); (2) using the special

Gauss integration of Eq. (5.43) with 8 integration points; and (3) by extract-

ing the singularity according to Eq. (5.44). In the last case, the regular inte-

gral was computed using eight Gauss points on each of the subintervals

½21; ξJ � and ½ξJ ;11�, while the singular one was evaluated analytically. For

all cases the element data are x1 5 3:0, y1 5 2:0, x2 5 1:0, y2 5 3:0, and

κ1 5κ2 5 0:5.

FIGURE 5.11 Variation of the function f ðξÞ5 ½ψ1ðξÞ2ψ1ðξJ Þ� lnrðξÞ for ξJ 52 0:5.

TABLE 5.1 Values of the Integrals gijα , when the Source i Lies on the Element

InfluenceCoefficient

AnalyticalIntegration

Gauss Integration,Eq. (5.43)

Extracting theSingularity, Eq. (5.44)

g1j1 20.2970889377 20.2970889376 20.2970889377

g1j2 0.0274675251 0.0274675250 0.0274675251

g2j1 0.0274675251 0.0274675250 0.0274675251

g2j2 20.2970889377 20.2970889376 20.2970889377


Integrals with Cauchy type singularityIt has already been mentioned that the computation of the diagonal elements

H ii and Gii of matrices ½H � and ½G� requires an inside integration. In the

preceding subsection we presented techniques for the evaluation of the coeffi-

cients Gii which are line integrals with logarithmic singularity. The coeffi-

cients H ii, however, are determined by evaluating integrals of the form (see

Eqs. (5.14) and (5.15))

ð121

ψαðξÞcosφr

dξ; r 5‘j2

jξ2 ξJ j ð5:45Þ

where ψαðξÞ ðα5 1; 2Þ are given in Eq. (5.25) and r in Eq. (5.26).

Linear elements approximate the geometry by a straight line. Hence it is

cosφ5 cos 6π2

� �5 0

along the whole element, since φ5 angle ðr;nÞ (see Appendix A). Therefore,

the integrand of Eq. (5.45) becomes

ψαðξÞcosφr

5

0; ξ 6¼ ξJ

00; ξ5 ξJ

8>><>>:

Applying L’Hopital’s rule, the above expression yields

limξ-ξJ

ψαðξÞ cosφrðξÞ 5 lim

ξ-ξJ

ψ0αðξÞ cosφr 0ðξÞ 5 0 ð5:46Þ

because the derivative ψ0αðξÞ5 ð21Þα=κ is constant (see Eq. 5.25) and finite

in the interval ½21; 11�, and the same is true for the derivative r 0ðξÞ, whichaccording to Eq. (5.26) is given as

r 0ðξÞ5 ‘j2sgnðξÞ ð5:47Þ

where sgnðξÞ (signum of ξ) is the function defined as

sgnðξÞ511; ξ. 0

21; ξ, 0

8<: ð5:48Þ


Consequently, the value of integral (5.45) is

ð121

ψαðξÞcosφr

dξ5 0 ð5:49Þ

It should be noted that this result is not valid for higher order elements,

since for these elements cosφðξÞ 6¼ 0.The integrand in (5.45) behaves as 1=r and becomes infinite when r 5 0.

This singularity is known as Cauchy type singularity. Analytical and numeri-

cal, as well as hybrid techniques have been developed for the evaluation of

integrals with this type of singularity. Their use and programming require

special care. In addition to the diagonal elements H ii, the coefficients εi must

also be computed (see Eqs. 5.16 or 5.20), which, of course, increases the

computational task. However, it is possible to evaluate directly the elements

Hii 5 H ii 2 εi by an indirect method that avoids the evaluation of any singu-

lar integral.

5.4.3 Indirect evaluation of the diagonal influencecoefficients

The matrices ½G� and ½H � in Eq. (4.7) for constant elements, or in Eq. (5.19)

for linear elements, are affected only by the boundary geometry, its discreti-

zation and the employed type of element. Hence, these matrices do not

depend on the boundary conditions, that is, they remain unchanged for given

boundary geometry, boundary discretization, and type of boundary elements.

The indirect integration is based on the fact that the function

u5αx1 by1 c is a solution to the Laplace equation, namely,

r2u5 0 in Ω

with boundary conditions

u5α x1 b y1 c on Γ ð5:50aÞ

and

un 5ruUn5α nx 1 b ny on Γ ð5:50bÞ

a. Evaluation of the elements Hii

Let us assume that a5 b5 0 and c5 1. In this case, it will be u5 1 and

un 5 0 on the boundary. Obviously, these values must satisfy Eq. (5.19),

which gives, after substitution,

½H �f1g5 0


where f1g is the vector whose elements are all equal to 1. The above equation

may also be written as

XNj51

Hij 5 0

or

Hii 52XNj51i 6¼ j

Hij ði5 1; 2; . . .;N Þ ð5:51Þ

Equation (5.51) states that the diagonal element in the i-th row of matrix

½H � is equal to the negative sum of the remaining elements in this row. It

should be noted that Eq. (5.51) is valid only for a closed domain Ω. For infinitedomains, it is not valid, because a constant value for u violates the regularity

condition at the infinity. Nevertheless, it is possible even in this case to evalu-

ate Hii using an indirect approach (see Ref. [11]).

b. Evaluation of the elements Gii

Once the matrix ½H � has been computed, the diagonal elements Gii of

matrix ½G� can also be computed using an indirect method, which avoids the

evaluation of any integrals, either regular or singular. To this end, Eq. (5.19)

is applied for the function

u5 a x1 b y

giving

½G�fung5 ½H �fug

or

XNj51

Gij ujn 5

XNj51

Hij uj ði5 1; 2; . . .;N Þ

The above equation is solved for Gii, yielding

Gii 51uin

2XNj51j 6¼ i

Gij ujn 1

XNj51

Hij uj

0BB@

1CCA ð5:52Þ

The boundary values uj and ujn appearing in Eq. (5.52) are computed

from the following expressions

uj 5 a xj 1 b yj ð5:53aÞ


ujn 5 a nj

x 1 b njy ð5:53bÞ

where ðxj ; yj Þ are the coordinates of the j-th node, and ðnjx ;n

jyÞ are the compo-

nents of the unit vector normal to the j-th element. The constants a and b are

chosen arbitrarily, but under the condition that uin 6¼ 0. This means according

to Eq. (5.53b), that the vector with components the coefficients a and b must

not be normal to the vector nj ðj5 1; 2; . . .;N Þ, or, in other words, must not

be parallel to the j-th boundary element. This may be achieved by setting

a5 1, b5λ. 0 and choosing λ so that

λ 6¼ yj11 2 yjxj11 2 xj

�� ð5:54Þ

5.5 HIGHER ORDER ELEMENTS

Constant and linear elements cannot approximate with sufficient accuracy the

geometry of curvilinear boundaries. For this reason it is recommended to use

curvilinear elements, for which the interpolating polynomials are of a degree

higher than one. In general, their form in the normalized interval ½21;11� isgoing to be

f ðξÞ5 ao 1 a1 ξ1 a2 ξ2 1 a3 ξ3 1?1 an ξn ð2 1# ξ# 1Þ ð5:55Þ

For n5 2, Eq. (5.55) yields the interpolation function for the quadratic or

parabolic element, for n5 3 that of the cubic element and so forth. In what

follows we will limit our presentation to the parabolic element. For higher

order elements or for a general theory on isoparametric elements the reader is

advised to look in Ref. [12].

The boundary quantities u and un in the line integrals of Eq. (5.8) are

functions of the arc length measured from some origin. When a parabolic

variation is assumed the quantities u or un will be expressed by a polynomial

of the form

f ðsÞ5αo 1α1s1α2s2 ð5:56Þ

The coordinates of point ðx; yÞAΓ, which varies during the integration,

are also functions of s, that is, x5 xðsÞ, y5 yðsÞ. Hence

r 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðsÞ2xi½ �2 1 yðsÞ2yi½ �2

q5 rðsÞ

and

v5 vðsÞ; vn 5@v@n

5 vnðsÞ

5.5 Higher Order Elements 137

Consequently, the integral to be evaluated is of the form

I 5ðΓj

wðsÞ ds ð5:57Þ

Of course, the integration can be performed by first establishing the trans-

formation s5 sðξÞ and then substituting it in the integral (5.57). Although

this method is conceptually simple, its implementation requires the establish-

ment of complicated expressions of s and for this reason it is not the most

suitable. Instead, another method is presented below, which considerably sim-

plifies the integration procedure.

The integration is performed over the normalized interval ½21; 11� withrespect to the intrinsic coordinate ξ, and thus, the integral (5.57) becomes

I 5ð121

w�ðξÞjJ ðξÞjdξ ð5:58Þ

where J ðξÞ�� is the Jacobian of the transformation which maps the parabolic

arc Γj of the xy-plane onto the straight line segment with 21# ξ# 1 and

η5 0 of the ξη-plane (see Fig. 5.12).The boundary quantity f (u or un) is approximated directly in the interval

½21; 11� by a second order polynomial in ξ, namely,

f ðξÞ5α0 1α1ξ1α2ξ2 ð5:59Þ

The coefficients α0, α1, and α2 are determined from the requirement

that the function f ðξÞ takes the nodal values f1, f2, f3 at points ξ521; 0; 1,respectively (see Fig. 5.12). Hence,

f ð21Þ5 f1

f ð0Þ5 f2

f ð1Þ5 f3

ð5:60Þ

FIGURE 5.12 Parabolic element in global and local coordinate systems.


Applying conditions (5.60) to Eq. (5.59), we get the following system of

equations for the unknown coefficients

α0 2α1 1α2 5 f1α0 5 f2

α0 1α1 1α2 5 f3

whose solution is

α0 5 f2

α1 5f3 2 f1

2

α2 5f1 2 2f2 1 f3

2

ð5:61Þ

Introducing Eq. (5.61) into Eq. (5.59), we obtain the expression of the

boundary quantity in terms of the three nodal values of element

f ðξÞ5 f2 1f3 2 f1

2ξ1

f1 2 2f2 1 f32

ξ2 ð5:62Þ

Equation (5.62) may further be written in the form

f ðξÞ5ψ1ðξÞ f1 1ψ2ðξÞ f2 1ψ3ðξÞ f3 ð5:63Þ

or

f ðξÞ5X3α51

ψαðξÞ fα ð5:64Þ

where

ψ1ðξÞ5212ξ ð12 ξÞ

ψ2ðξÞ5 ð12 ξÞð11 ξÞ

ψ3ðξÞ512ξ ð11 ξÞ

ð5:65Þ

The functions defined in Eq. (5.65) are the shape functions of the para-

bolic or quadratic element.


The mapping of the parabolic element from the xy-plane onto the interval

21# ξ# 1 of the ξη-plane is accomplished through the transformation

xðξÞ5 b0 1 b1ξ1 b2ξ2

yðξÞ5 c0 1 c1ξ1 c2ξ2ð5:66Þ

We can readily conclude from Eqs. (5.59) and (5.66) that the parabolic

element at hand is isoparametric, since both the geometry and the boundary

quantity are approximated by polynomials of the same degree. The coeffi-

cients bk and ck ðk5 0; 1; 2Þ in Eq. (5.66) are evaluated from the requirement

that the element arc should pass through the points ðx1; y1Þ, ðx2; y2Þ, and

ðx3; y3Þ for ξ521; 0; 1, respectively. These conditions are expressed mathe-

matically as

xð21Þ5 x1; xð0Þ5 x2; xð1Þ5 x3

yð21Þ5 y1; yð0Þ5 y2; yð1Þ5 y3

It is apparent that the above conditions yield expressions for xðξÞ and

yðξÞ, which are similar to those of Eq. (5.64). More specifically,

xðξÞ5X3α51

ψαðξÞxα

yðξÞ5X3α51

ψαðξÞyα

ð5:67Þ

where the shape functions ψαðξÞ are given in Eq. (5.65).

Equation (5.67) may be employed to express the distance r , as well as the

kernels vðrÞ and vnðrÞ of the integrals representing the influence coefficients,

as functions of the variable ξ. Finally, the Jacobian of Eq. (5.58) is evaluated

from the expression

ds5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidx2 1 dy2

p5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½x 0ðξÞ�2 1 ½y0ðξÞ�2

qdξ

Hence

J ðξÞ�� 5 ½x 0ðξÞ�21½y0ðξÞ�2� �1=2

5 ðb112b2ξÞ21ðc112c2ξÞ2� �1=2 ð5:68Þ


where, on the basis of Eq. (5.61), it is

b1 5x3 2 x1

2; b2 5

x1 2 2x2 1 x32

c1 5y3 2 y1

2; c2 5

y1 2 2y2 1 y32

ð5:69Þ

After discretizing the boundary into N parabolic elements, the boundary

integral equation (3.31) may be written as

εi ui 52XNj51

ðΓj

v un ds1XNj51

ðΓj

u vn ds ð5:70Þ

Taking into account the parabolic variation of the boundary quantities on

the elements, the second line integral in the right-hand side of Eq. (5.70)

yields

ðΓj

uðqÞ vnðpi; qÞ dsq 5ðΓj

ðψ1 u1 1ψ2 u

2 1ψ3 u3Þ vn ds

5 hij1 u1 1 hij2 u

2 1 hij3 u3

ð5:71Þ

where it has been set

hij1 5

ðΓj

ψ1 vn ds

hij2 5

ðΓj

ψ2 vn ds

hij3 5

ðΓj

ψ3 vn ds

ð5:72Þ

In order to evaluate the above integrals, their integrands are expressed in

terms of the variable ξ

hijα 5

ðΓj

ψα vn ds5ð121

ψαðξÞcosφðξÞ2πrðξÞ J ðξÞ

�� dξ ðα5 1; 2; 3Þ ð5:73Þ

Similarly, we obtain

ðΓj

un v ds5ðΓj

ðψ1 u1n 1ψ2 u

2n 1ψ3 u

3nÞ v ds

5 gij1 u1n 1 gij2 u2

n 1 gij3 u3n

ð5:74Þ


where

gijα 5

ðΓj

ψα v ds

5

ð121

ψαðξÞlnrðξÞ2π

J ðξÞ�� dξ ðα5 1; 2; 3Þ

ð5:75Þ

The computation of integrals (5.73) and (5.75) is performed numeri-

cally following a procedure similar to that employed for the linear element.

Thus, for the outside integration ði 6¼ jÞ the conventional Gauss integration

is utilized. However, for the inside integration ði5 jÞ, the analytical

method for the computation of gijα becomes too complicated and it is not

recommended. Instead, the method of extracting the singularity turns out to

be the most suitable. The elements Hii are computed using the indirect

method presented in Section 5.4.3, avoiding, thus, the evaluation of singu-

lar integrals.

The corner points or the points where there is a change in the boundary

conditions are treated using one-sided discontinuous parabolic elements, for

example, κ2 5 1 (Fig. 5.13). For discontinuous parabolic elements, the coeffi-

cients of the polynomial (5.59) are determined from the conditions

f ð2κ1Þ5 f1

f ð0Þ5 f2

f ðκ2Þ5 f3

ð5:76Þ

which yield

α0 2α1 κ1 1α2 κ21 5 f1

α0 5 f2

α0 1α1 κ2 1α2 κ22 5 f3

ð5:77Þ

FIGURE 5.13 Discontinuous parabolic element in global and local coordinate systems.


Equation (5.77) are solved for the unknown coefficients α0, α1, and α2 giving

α0 5 f2

α1 52κ22

Kf1 1

κ22 2κ2

1

Kf2 1

κ21

Kf3

α2 5κ2

Kf1 2

κ1 1 κ2

Kf2 1

κ1

Kf3

ð5:78Þ

where

K 5κ1 κ2ðκ1 1κ2Þ

Introducing the above values of the coefficients αi in Eq. (5.59), the latter

takes the known form

f ðξÞ5X3α51

ψαðξÞ fα ðα5 1; 2; 3Þ ð5:79Þ

where ψαðξÞ are shape functions given as

ψ1ðξÞ5κ2

Kð2κ2 1 ξÞξ

ψ2ðξÞ5κ1 1 κ2

Kκ1 κ2 1 ðκ2 2κ1Þξ2 ξ2� �

ψ3ðξÞ5κ1

Kðκ1 1 ξÞξ

ð5:80Þ

Evidently, Eq. (5.80) reduce to the shape functions of Eq. (5.65) for

κ1 5κ2 5 1. Higher order elements can be derived by choosing interpolating

polynomials of higher degree and following exactly the same procedure as

for the quadratic element.

All continuous elements derived on the basis of the polynomial (5.55) for

n$ 1, produce a continuous variation of the boundary quantity on the whole

boundary, that is, they do not exhibit jumps at the interelement nodes. This

type of continuity is called C 0 continuity. These elements, however, do not

ensure continuity of the derivative at the interelement nodes. This continuity,

referred to as C 1 continuity, can be achieved using shape functions described

by special third order polynomials known as Hermite polynomials or Hermite

interpolation functions. In this case, the element has two nodes which are

placed at its extreme points. The unknowns are the values f1 and f2 of the

boundary quantity, along with the corresponding derivatives θ1 5 ðdf =dξÞ1and θ2 5 ðdf =dξÞ2. Hence, the boundary quantity is expressed as

f ðξÞ5ψ1ðξÞ f1 1ψ2ðξÞ f2 1ψ3ðξÞ θ1 1ψ4ðξÞ θ2 ð5:81Þ


in which the shape functions ψαðξÞ are given by the Hermite polynomials [1]

ψ1ðξÞ514ð12ξÞ2ð21 ξÞ; ψ2ðξÞ5

14ð11ξÞ2ð22 ξÞ

ψ3ðξÞ514ð12ξÞ2ð11 ξÞ; ψ4ðξÞ52

14ð11ξÞ2ð12 ξÞ

ð5:82Þ

5.6 NEAR-SINGULAR INTEGRALS

Once the unknown boundary quantities have been established from the solu-

tion of the boundary integral equations, the values of the potential u and its

derivatives u;x 5 @u=@x and u;y 5 @u=@y at internal points can be computed

using the expressions (4.11), (4.15), and (4.16), or the corresponding ones for

the linear and the parabolic element approximation. The influence coeffi-

cients involved in the aforementioned equations are expressed in terms of line

integrals on the elements Γj whose integrands involve factors of the form

lnr ;1r;

1r2

ð5:83Þ

where r 5 P2 qj j is the distance between the points PAΩ and qAΓj . Clearly,

because point P lies inside the domain Ω, while point q lies on the boundary,

it is always r 6¼ 0. Therefore, these integrals, at least theoretically, are regularsince the value of their integrands is always finite. When point P lies far from

the boundary, the functions (5.83) have a smooth variation and consequently

the conventional Gauss integration gives accurate results. However, when

point P lies near the boundary, the functions (5.83) may take a very large,

though finite, value and, thus, their variation is not smooth any more.

In order to illustrate this behavior, we consider the element of Fig. 5.14.

The internal point P lies near the boundary. Applying Eq. (5.3), one obtains

the global coordinates in terms of the local coordinate ξ,

FIGURE 5.14 Linear element and internal point P near the boundary,

d5minðrÞ5 0:02{1.


xðξÞ5 22 ξ

yðξÞ5 21 ξ

and Eq. (5.12) gives the relative distance between points P and q as

rðξÞ5 ½xðξÞ22:48�21½yðξÞ21:48�2� �1=2

5 ð0:481ξÞ21ð0:521ξÞ2� �1=2 ð5:84Þ

The variation of the functions 1=rðξÞ and 1=rðξÞ2 is shown in Figs. (5.15)

and (5.16), respectively. We notice the large values of these functions at

ξ52 0:5, that is at point A, which is the normal projection of point P on the

element.

Consequently, the integrals of the functions (5.83) for points P near the

boundary behave like singular integrals, although they are not. In the litera-

ture, these integrals are known as near-singular integrals. Their evaluation

faces considerable difficulties, because neither the conventional Gauss inte-

gration nor those methods suitable for singular integrals can be employed.

Nevertheless, other special techniques have been developed for their

evaluation. Among them, the most popular are: the method of the element

subdivision, and the method of the coordinate transformation. The first one

has been discussed in detail by Lachat and Watson [13] and Kane et al.

[14], while the second one has been presented by Telles [15]. Only the first

method is presented below, while the other one can be found in the relevant

literature.

FIGURE 5.15 Variation of the function 1=rðξÞ along the element.

5.6 Near-Singular Integrals 145

5.6.1 The method of element subdivision

This method has been successfully employed to achieve a uniform accuracy

for the values of the line integrals. The success of this technique is based on

the user’s experience. This technique is illustrated by evaluating the follow-

ing integral

ð121

1rðξÞ dξ ð5:85Þ

The plot of its integrand, which is depicted in Fig. 5.15, suggests subdivi-

sion of the interval ½21; 11� in four subintervals:

Subinterval 1: ½2 1; 2 0:6�Subinterval 2: ½2 0:6; 2 0:5�Subinterval 3: ½2 0:5; 2 0:4�Subinterval 4: ½2 0:4; 1 1�In each subinterval, the integral can be evaluated using any integration

rule such as the trapezoidal rule, Simpson’s rule, Newton-Cotes integration

formulae, etc. The Gauss integration is also recommended for this case. Its

application requires first transformation of each subinterval onto the interval

½21; 11�. If ξk and ξk11 represent the end points of the k-th subinterval, then

the linear transformation

ξ5ξk11 1 ξk

21

ξk11 2 ξk2

η ð21# η# 1Þ ð5:86Þ

FIGURE 5.16 Variation of the function 1=rðξÞ2 along the element.


serves the purpose. The value of the integral over the k-th subinterval is

given as

Ik 5ξk11 2 ξk

2

Xni51

1

rξk11 1 ξk

21

ξk11 2 ξk2

ηi

� wi ð5:87Þ

where ηi and wi ði5 1; 2; . . .;nÞ are the abscissas and weights of the n-thorder Gauss integration.

The integral (5.85) has been computed numerically using various subdivi-

sions of the element shown in Fig. 5.14 and the results are given in Table 5.2

along with the exact value. These results reveal that the choice of element

subdivision greatly affects the accuracy, even though the total number of

integration points remains the same (20 points). Therefore, special care

should be taken for the computation of near-singular integrals to avoid an

uncontrolled error. A general rule for acceptable results is to choose two

equal subintervals at both sides of point A, whose length is sufficiently

smaller than the distance d (see Fig. 5.14).

TABLE 5.2 Values of the Near-Singular Integral (5.85) for Various Subdivisions of

the Element

Number ofSubintervals

Subintervals Number ofGauss Points

ð121

1rdξ

1 [21.0, 11.0] 20 7.55905

2 [21.0, 20.5] 10 6.34858

[20.5, 11.0] 10

2 [21.0, 20.5] 12 6.29535

[20.5, 11.0] 8

2 [21.0, 20.5] 8 6.35520

[20.5, 11.0] 12

4 [21.0, 20.6] 5 6.30650

[20.6, 20.5] 4

[20.5, 20.4] 5

[20.4, 11.0] 6

4 [21.0, 20.6] 6 6.30934

[20.6, 20.5] 4

[20.5, 20.4] 4

[20.4, 11.0] 6

Exact value 6.309586

5.6 Near-Singular Integrals 147

5.7 REFERENCESFrom the presentation of the BEM up to this point, it becomes evident that its evolution as

a computational method for solving realistic engineering problems is based on its success

in solving singular boundary integral equations. Therefore, the technology of the boundary

element, namely, the construction of various types of elements, the effective integration

over them, especially of the singular kernels, as well as their manipulation in order to treat

discontinuities of the boundary quantities, is among the most important ingredients of the

BEM and has been a field of intense research ever since the BEM appeared as a computa-

tional method. For more information about this subject, the reader is advised to look in

books by Brebbia and Dominguez [7], Kane [16], and Banerjee and Butterfield [12].

Regarding the evaluation of singular integrals, a rich technical literature is available.

Suggestively, we mention the work of Hall [17] and Doblare [18], as well as the book of

Sladek and Sladek [19], who presented methods for the evaluation of singular integrals.

Hayami and Brebbia [20], apart from the singular integrals, treated the near-singular ones.

A method for the computation of line integrals with logarithmic singularity has also been

presented by Katsikadelis and Armenakas [21]. Theocaris [22,23] has published extended

work on the integration of singular integrals. Considerable work, especially on the evalua-

tion of hypersingular integrals, has been published by Guiggiani and his coworkers [24,25].

The reader can also find extended literature related to this subject in the chapters

Computational Aspects of the proceedings of the international conferences on Boundary

Elements (BEM/MRM), which have been organized uninterruptedly by Wessex Institute of

technology since 1978.

[1] Reddy JN. An introduction to the finite element method. New York: McGraw-Hill

International Editions; 1993.

[2] Katsikadelis JT, Sapountzakis E. Torsion of composite bars by the boundary element

method. ASCE J Eng Mech 1985;111:1197�210.

[3] Katsikadelis JT, Kallivokas L. Plates on biparametric elastic foundation. ASCE J Eng

Mech 1988;114:547�875.

[4] Katsikadelis JT, Armenakas AE. A new boundary equation solution to the plate prob-

lem. ASME J Appl Mech 1989;56:264�374.

[5] Katsikadelis JT. Special methods for plate analysis. In: Beskos D, editor. Boundary

element analysis of plates and shells. Berlin: Springer-Verlag; 1991. p. 221�311.

[6] Katsikadelis JT, Kokkinos FT. Static and dynamic analysis of composite shear walls

by the boundary element method. Acta Mech 1987;68:231�50.

[7] Brebbia CA, Dominguez J. Boundary elements: an introductory course. 2nd ed.

Southampton: Computational Mechanics Publications; 2001.

[8] Lefeber D. Solving problems with singularities using boundary elements. In: Brebbia

CA, Connor JJ, editors. Topics in engineering, Vol. 6. Southampton: Computational

Mechanics Publications; 1989.

[9] Stroud AH, Secrest D. Gaussian quadrature formulas. Englewood Cliffs, New Jersey:

Prentice Hall; 1966.

[10] Abramowitz M, Stegun I, editors. Handbook of mathematical functions. 10th ed.

New York: Dover Publications; 1972.

[11] Brebbia CA, Telles JCF, Wrobel LC. Boundary element techniques. Berlin: Springer-

Verlag; 1984.

[12] Banerjee PK, Butterfield R. Boundary element methods in engineering science. U.K:

McGraw-Hill; 1981.

[13] Lachat JC, Watson JO. Effective numerical treatment of boundary integral equations:

a formulation for three-dimensional elastostatics. Int J Numer Methods Eng 1976;10:

991�1005.

[14] Kane JH, Gupta A, Saigal S. Reusable Intrinsic Sample Point (RISP) algorithm for

efficient numerical integration of three dimensional curved boundary elements. Int J

Numer Methods Eng 1989;28:1661�76.

[15] Telles JCF. A self-adaptive coordinate transformation for efficient numerical evaluations

of general boundary element integrals. Int J Numer Methods Eng 1987;24:959�73.


http://refhub.elsevier.com/B978-0-12-804493-3.00005-9/sbref1









































[16] Kane JH. Boundary elements analysis in engineering continuum mechanics.

Englewood Cliffs, New Jersey: Prentice Hall; 1994.

[17] Hall WS. Integration methods for singular boundary element integrands. In: Brebbia

CA, editor. Boundary Elements X, vol. 1, Mathematical and Computational Aspects.

Southampton: Computational Mechanics Publications; 1988. p. 219�36.

[18] Doblare M. Computational aspects of the boundary element method. In: Brebbia CA, editor.

Topics in boundary element research, Vol. 3. Berlin: Springer-Verlag; 1987. p. 51�131.

[19] Sladek V, Sladek J, editors. Singular integrals in boundary element methods.

Southampton: Computational Mechanics Publications; 1998.

[20] Hayami K, Brebbia CA. Quadrature methods for singular and nearly singular integrals in

3-D boundary element method. In: Brebbia CA, editor. Boundary elements X, computa-

tional aspects, vol. 1. Southampton: Computational Mechanics Publications; 1988.

[21] Katsikadelis JT, Armenakas AE. Numerical evaluation of line integrals with a loga-

rithmic singularity. AIAA J 1983;23:1135�7.

[22] Theocaris PS. Numerical solution of singular integral equations: methods. ASCE J

Eng Mech 1981;107:733�51.

[23] Theocaris PS. Numerical solution of singular integral equations: applications. ASCE J

Eng Mech 1981;107:753�71.

[24] Guiggiani M. Direct evaluation of hypersingular integrals in 2D BEM. In: Hackbusch

W, editor. Notes in numerical field mechanics, vol. 33. Braunschweig: Vieweg; 1992.

p. 23�4.

[25] Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ. A general algorithm for the

numerical equations. ASME J Appl Mech 1992;59:604�14.

PROBLEMS

5.1. Derive the shape functions for the cubic element when the interior nodal

points are placed at

i. ξ2 52 1=3 and ξ3 5 1=3,ii. ξ2 52 1=2 and ξ3 5 1=2.

5.2. Given the circular sector of radius R5 3 and angle θ0 5 π=12, compute

its area by approximating the circular arc with (1) linear, (2) quadratic,

and (3) cubic elements. For each case determine the error.

5.3. Compute the near-singular integral using Gauss integration

I 5ð11

21

dx

x20:25ð Þ210:05� �4

5.4. Compute the integral ðΓj

ψ1ðξÞlnr ds

when Γj is a quadratic element passing through the points 1(4.30, 2.50),

2(4.10, 2.90), 3(3.80, 3.20) and the source lies at point P(4.15, 2.65).5.5. Compute the integrals gαij ðα5 1; 2Þ for the linear element with nodal

points 1(1, 2) and 2(1.5, 2.3), when κ1 5κ2 5 0:5 and the source lies

consecutively at point 1 and point 2.

Problems 149































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