the dual reciprocity boundary element method · study on the dual reciprocity boundary element...

DRBEM 3/8/09

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University of Hertfordshire

Department of Mathematics

Study on the Dual Reciprocity Boundary Element Method

Wattana Toutip

Technical Report 3 July 1999

Preface

The boundary Element method (BEM) is now recognised as a well-established

numerical technique for solving problems engineering and applied science. The main

advantage of the BEM is its unique ability to provide a complete solution in terms of

boundary value only. An initial restriction of the BEM is that the fundamental

solution to the original partial differential equation is required in order to obtain an

equivalent boundary integral equation. Another is that non-homogeneous terms are

included in the formulation by means of domain integrals, thus making the technique

lose the attraction of its “boundary only” character.The resolution of these problems

has been the subject of considerable research over the past decade and several

methods have been suggested. It is our opinion that the most successful so far is the

dual reciprocity method (DRM) by means of accuracy and programming point of

view.

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1. Introduction

There are three classical methods for solving problems in engineering and applied

science. The first approach is the finite difference method. This technique

approximates the derivatives in the differential equation which govern each problem

using some type of truncated Taylor expansion. The second one is the finite element

method (FEM). This method involves the approximation of the variables over small

parts of the domain, called elements, in term of polynomial interpolation function.

The disadvantages of FEM are that large quantities of data are required to discretize

the full domain. The third one is the boundary element method (BEM). This approach

is developed as a response to that problem. The method requires discretization of the

boundary only thus reducing the quantity of data necessary to run a program.

However, there are some difficulties of extending the technique to several

applications such as non-homogeneous, non-linear and time-dependent problems for

examples. The main drawback in these case is the need to discretize the domain into a

series of internal cells to deal with the terms not taken to the boundary by application

of the fundamental solution. This additional dicretization destroys some of the

attraction of the method in terms of the data required to run the program and the

complexity of the extra operations involved.

It was then realised that a new approach was needed to deal with domain integrals in

boundary elements. Several methods have been proposed by different authors. The

most important of them are:

1. Analytic Integration of the Domain Integrals.

2. The Use of Fourier Expansion.

3. The Galerkin Vector Technique.

4. The Multiple Reciprocity Method.

5. The Dual Reciprocity Method

The Dual Reciprocity Method (DRM) is essentially a generalised way of constructing

particular solutions that can be used to solve non-linear and time-dependent problems

as well as to represent any internal source distribution. This work is intended to study

this method.

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2. The boundary Element Method for the equation 02 u

and bu 2

2.1 Laplace equation

The Laplace equation satisfied by the potential *u in a domain with a boundary

as shown in Figure 1 can be converted into the well known integral equation (Brebbia,

1978)

dquduqcu ** (2.1)

where 21 and we assume that uu on 1 and qqn

u

on 2 . n denotes

the unit outward normal to and *u is the fundamental solution of the Laplace

equation; c is a real number which is described in the following paragraphs.

Let P be a point in the domain and assume that P is an internal point. Then equation

(2.1) becomes (Paris and Cañas, 1997)

duqdquPu **)( (2.2)

where *u is a fundamental solution and n

uq

** .

On the other hand, in the case when P is a boundary point, equation (2.1) is of the

form (Paris and Cañas, 1997)

dquduqPuPc **)()( (2.3)

where

2

)()(

PPc , )(P is the internal angle at the boundary point P.

Figure 2.1 : Laplace equation posed on the domain with the boundary

n

uu

1 2

qq

n

u

02 u

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For a two-dimensional domain, the fundamental solution is (Gilbert and Howard,

1990)

)1

ln(2

1*

ru

(2.4)

On the other hand, in a three-dimensional domain (Gilbert and Howard, 1990)

r

u4

1*

where r is the distance between the source point and the field point, see Figure 2.2.

The boundary integral equation (2.3) is discretized by partitioning the boundary into

N elements. The element j lies between node j and node j+1 as shown in Figure 2.3.

This is equivalent to replacing the boundary curve by a polygon N

r

Field point

Source point

Figure 2.2 : Distance r between the source point and the field point

Figure 2.3 : Discretization of the boundary into N elements

j

1

3

…

2

i

…

1

j+1

N

r

… …

Base node

[j]

Target element

N

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We choose a suitable set of basis function Njsw j ,...,2,1:)( where s is the

distance around the boundary, , and consider the boundary element approximation

j

N

j

j Uswsu )()(~

1

j

N

j

j Qswsq )()(~

1

where U j and Q j are the, approximate, values of u and q at node j.

Setting uu ~ and qq ~ in equation (2.3) we collocate at the N nodal points to

obtain a system of algebraic equation (Davies and Crann, 1996)

dsuQswdsqUswuc j

N

j

jj

N

j

jii

*

1

*

1

)()(

(2.5)

which we may write as

j

N

j

ijj

N

j

ijii QGUHuc

11

ˆ (2.6)

where

dsqswdsqswH

N

jjij

** )()(ˆ

(2.7.1)

dsuswdsuswG

N

jjij

** )()(

(2.7.2)

We shall consider linear elements in which w j(s) is the usual „hat‟ function based at

node j as shown in Figure 2.4.

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Consider an arbitrary segment such as the one shown in Figure 2.5,

With the basis function w j(s) being the hat function shown in Figure 2.4, we see that

the values of u and q at any point of an element can be defined in terms of their

nodal values and the linear interpolation functions 1 and 2 such that

121121)(

j

j

jj U

UUUu

121121)(

j

j

jj Q

QQQq

1

j+2 j+1 j j-1 j-2

Figure 2.4 : Hat function based on node

j

w j(s)

Nodal value of u and q

Node j

Node j+1

Figure 2.5 : Relation between local co-ordinate and

dimensionless co-ordinate, )(2

jssl

= -1

= 1

Local co-ordinate

l

js

sj

2

lss j

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where is the dimensionless co-ordinate and 1 and

2 are the usual Lagrange

interpolation polynomials

)1(2

11

)1(2

12

The integral along the element j in equation (5) becomes, for the left hand side,

2

121

2

1*

][

21

*

U

Uhh

U

Udsqdsqw jj

j

j

N

(2.8)

where for each element j we have two components,

][

*

11

j

j dsqh

][

*

22

j

j dsqh

Similarly, for the right hand side we obtain

1

211

*

][

21

*

j

j

jjj

j

j

j Q

Qgg

Q

Qdsqdsuw

N

(2.9)

where for each element j we have two components,

][

*

11

j

j dsug

][

*

22

j

j dsug

Hence, from equation (2.6), for each collocation node i we have

2)1(1ˆ

jjij hhH (2.10.1)

and 2)1(1 jjij ggG (2.10.2)

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In this development of the boundary element method, the equation

j

N

j

ijj

N

j

ijii QGUHUc

11

ˆ (2.11)

has been obtained by collocating at the nodes defining the function values i.e. the

collocation and functional nodes are the same. In equation (2.11), j defines a

functional node and i denotes a collocation node.

Let us define

ijij HH ˆ where ji

iijij cHH ˆ where ji

then we can write equation (11) as

j

N

j

ijj

N

j

ij QGUH

11

(2.12)

The system of linear equations may be written in matrix form

HU = GQ (2.13)

Applying the boundary conditions to identify the Dirichlet and Neumann boundary

regions we can partition the system (2.13) in the form

2

121

2

121 Q

QGG

U

UHH

where U1 and Q2 comprise known boundary values and U2 and Q1 comprise

unknown boundary values. The equations are then rearranged in the form

AX = F

Where 21 HGA ,

2

1

U

QX , and ][ 1122 UHQGF

Once the values of U2 and Q1 on the whole boundary are known we can calculate

the value of u at any internal point Pk by using equation (2.2). We obtain

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j

N

j

kjj

N

j

kjk UHQGu

11

ˆ (2.14)

The LINBEM code (Mushtag, 1995), based on Brebbia (1978), is implemented by

using this mathematical algorithm and works well with smooth domain boundaries

because there are no corners in such boundaries. However, the MULBEM program

(Toutip, 1999), can solve problems caused corners and discontinuous boundary

condithions using the multiple node method (Subia and Ingber, 1995)

.

2.2 Poisson equation

Consider the Poisson equation

bu 2 in (2.15)

as shown in Figure 2.6

where b is at present assumed to be a known function. In a similar way as done in

the Laplace equation, we have

dquuduqqdubu ***2

12

(2.16)

which inegrated by parts twice to produce

dqudquduqduqdubudu ******2

1212

(2.17)

After substituting the fundamental solution *u of the Laplace equation into (2.17)

and grouping all boundary terms together we obtain

21

1

2

uu

qq

n

Figure 2.6 : Geometric Definitions of the Problem

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dqudubduquc ii

*** (2.18)

Notice that although the function b is known and consequently the integral in

does not introduce any new unknowns, the problem has changed in character as we

need now to carry out a domain integral as well as the boundary integral. The constant

ci depends only on the boundary geometry at the point i under consideration.

The simplest way of computing the domain integral term in equation (2.18) is by

subdividing the region into a series of internal cells, on each of which a numerical

integration scheme such as Gauss quadrature can be applied.

However, this technique loses the attraction of its “boundary only” character. It was

then realised that a new approach was needed to deal with domain integrals in

boundary elements. Several methods have been proposed by different authors. The

most important of them are:

1. Analytic Integration of the Domain Integrals. This approach, although

producing very accurate results, is only applicable to a limited number of

cases for which the integrals can be evaluated analytically.

2. The Use of Fourier Expansion. The Fourier expansion method is not

straightforward to apply in many cases as the calculation of the coefficients

can be computationally cumbersome, although the method has been applied

with some success to relatively simple cases.

3. The Galerkin Vector Technique. This approach uses a primitive, higher-

order fundamental solution and Green‟s identity to transform certain types

of domain integrals into equivalent boundary integrals. The main difficulty

of the approach is that it can only solve comparatively simple cases. It has

been extended to deal with other applications giving origin to the technique

discussed in the paragraph.

4. The Multiple Reciprocity Method. This is an extension of the Galerkin

vector technique which utilises as many higher-order fundamental solutions

as required rather than using just one. The main difficulty is that the

method cannot be easily applied to general non-linear problems although it

has been successfully used to solve some time-dependent problems.

5. The Dual Reciprocity Method. This is the subject of this work and

constituted the only general technique other than cell integration.

The Dual Reciprocity Method (DRM) is essentially a generalised way of constructing

particular solutions that can be used to solve non-linear and time-dependent problems

as well as to represent any internal source distribution. The method can be applied to

define sources over the whole domain or only on part of it. The approach will be

described in the next section.

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3. The Dual Reciprocity Method for Equation of the Type

yxbu ,2

3.1 Mathematical Development of the DRM for the Poisson Equation


bu 2 (3.1)

where yxbb , , that is, b is considered to be a known function of position.

The solution to equation (3.1) can be expressed as the sum of the solution of a

homogenous and a particular solution as

uuu hˆ

where hu is the solution of the homogeneous equation and u is a particular solution

of the Poisson equation such that

bu ˆ2 (3.2)

If there are N boundary nodes and L internal nodes, as shown in Figure 3.1

there will be LN values of ju and the approximation of b can be written in the

form

Figure 3.1 : Boundary and internal nodes

: Internal node

: Boundary node

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j

LN

j

j fb

1

(3.3)

where the LNjj ,...,2,1: is a set of coefficients and the jf are approximating

functions. The particular solutions ju , and the approximating, jf are linked through

the relation

jj fu ˆ2 (3.4)

The function jf in (3.3) can be compared with the usual interpolation function i in

expansions such as

iiuu (3.5)

which are used on the boundary elements themselves.

Substituting equation (3.4) into (3.3) gives

jLN

j

j ub ˆ2

1

(3.6)

Equation (3.6) can be substituted into the original equation (3.1) to give the following

expression

jLN

j

j uu ˆ2

1

2

(3.7)

Multiplying by the fundamental solution and integrating by parts over the domain ,

we obtain

duuduuLN

j

jj

*

1

2*2 ˆ)( (3.8)

Note that the same result may be obtained from equation

dbuduu **2 )( (3.9)

Integrating the Laplacian terms by parts in (3.8), produces the following integral

equation for each source node i,

dquduqucqduudquc jjiji

LN

j

jiiˆˆˆ **

1

** (3.10)

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The term jq in equation (3.10) is defined as n

uq

j

j

ˆˆ , where n is the unit

outward normal to , and can be expanded to

n

y

y

u

n

x

x

uq

jj

j

ˆˆˆ (3.11)

Note that equation (3.10) involves no domain integrals. The next step is to write

equation (3.10) in discretized form, with summations over the boundary elements

replacing the integrals. This gives for a source node i the expression

N

k

N

k

jjiji

LN

j

j

N

k

N

k

iik kk k

dquduqucqduudquc1 1

**

11 1

** ˆˆˆ (3.12)

After introducing the interpolation function and integrating over each boundary

element, the above equation can be written in terms of nodal values as

kj

N

k

ikkj

N

k

ikiji

LN

j

jk

N

k

ikk

N

k

ikii qGuHucqGuHuc ˆˆˆ11111

(3.13)

The index k is used for the boundary nodes which are the field points. After

application to all boundary nodes using a collocation technique, equation (3.13) can

be expressed in matrix form as

GqHu

LN

j

j

1

jj qGuH ˆˆ (3.14)

If each of the vectors ju and jq is considered to be one column of the matrices U

and Q respectively, then equation (3.14) may be written without the summation to

produce

GqHu jj qGuH ˆˆ (3.15)

Equation (3.15) is the basis for the application of the Dual Reciprocity Boundary

Element Method and involves discretization of the boundary only.

The process described forms the basis of the method and gives a process for extending

to non-linear problems. However, in the linear case ( b = b(x,y) ), we can develop the

method in an manner, such that we can easily adapt the existing MULBEM code, as

follows:

Consider the Poisson equation (3.1) with the boundary conditions as shown in Figure

2.6. From equation (3.3) we have

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j

LN

j

j fb

1

with jj fu ˆ2 where jf and ju are known.

Set

LN

j

jjuuU1

ˆ (3.16)

Taking Laplacian operator both two sides we obtain

)ˆ(2

1

22

j

LN

j

j uuU

(3.17)

Substituting (3.1) and (3.4) into equation (3.17) we have

j

LN

j

j fbU

1

2 (3.18)

Finally, by substituting (3.3) into equation (3.18), we obtain a new Laplace eqaution

02 U (3.19)

with boundary conditions

LN

j

jjuuUU1

ˆ on 1

and

LN

j

jjqqQQ1

ˆ on 2

where Q is the normal derivative of U.

After the Laplace eqution (3.19) has been solved, the values of U and Q are known

and hence we also obtain the solution of the Poisson equation (3.1) by the followings:

j

LN

j

juUu ˆ1

on boundary and inside (3.20)

and Qq on boundary (3.21)

The MULBEM code is modified to the MULDRM by this manner.

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Interior Nodes

The definition of interior nodes is not normally a necessary condition to obtain a

boundary solution, however, the solution will usually be more accurate if a number of

such nodes is used.

When interior nodes are defined, each one is independently placed, and they do not

form part of any element or cell, thus the co-ordinates only are needed as input data.

Hence these nodes may be defined in any order.

The Vector

The vector in equation (3.15) will now be considered. It was seen in equation (3.3)

that

j

LN

j

j fb

1

(3.22)

This may be expressed in matrix form as

Fb (3.23)

where each column of F consists of a vector f j containing the values of the function

f j at the )( LN DRM collocation points. In the case of the problems considered in

this section, the function b in (3.1) and (3.22) is a known function of position. Thus

equation (3.23) may be inverted to obtain , i.e.

bF 1 (3.24)

The right-hand side of equation (3.15) is thus a known vector. Writing (3.15) as

dGqHu (3.25)

where

QGUHd ˆˆ (3.26)

Applying boundary conditions to (3.25), this equation reduces to the form

yAx (3.27)

where x contains N unknown boundary values of u and q.

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Internal Solution

After equation (3.27) is obtained using standard techniques, the values at any internal

node can be calculated from equation (3.13), each one involving a separate

multiplication of known vectors and matrices. In the case of internal nodes, as was

explained in previous section, 1ic and equation (3.13) becomes

kj

N

k

ikkj

N

k

ikij

LN

j

jk

N

k

ikk

N

k

iki qGuHuqGuHu ˆˆˆ11111

(3.28)

3.2 Different f Expansions

The particular solution, ju , its normal derivative, jq , and the corresponding

approximating functions jf used in DRM analysis are not limited by formulation

except that the resulting F matrix, equation (3.23), should be non-singular.

In order to define these functions it is customary to propose an expansion for f and

then compute u and q using equations (3.4) and (3.11), respectively. The

originators of the method have proposed the following types of functions for f

1. Elements of the Pascal triangle

2. Trigonometric series

3. The distance function r use in the definition of the fundamental solution

The r function was adopted first by Nardini and Brebbia and then by most

researchers as the simplest and most accurate alternative.

The definition of r is that

222

yx rrr (3.29)

where xr and yr are the components of r in the direction of the x and y axes.

If rf , it can easily be shown that the corresponding u function is 9

2r, in the two

dimensional case.

The function q will be given by

),cos(),cos(3

ˆ ynrxnrr

q yx (3.30)

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In the above, the direction cosine refer to the outward normal at the boundary with

respect to the x and y axes. Formula (3.30) may be easily obtained suing (3.11) and

remembering that r

r

x

r x

and

r

r

y

r y

.

Furthermore, some recent works suggest that rf is in fact one component of the

series

mrrrf ...1 2 (3.31)

The u and q functions corresponding to (3.31) are :

2

232

)2(...

94ˆ

m

rrru

m

(3.32)

)2(...

32

1ˆ

m

rr

n

yr

n

xrq

m

yx (3.33)

In principal, any combination of terms may be selected from (3.31). To illustrate this,

for Poisson-type equation, one case will be considered:

rf 1

The presence of the constant guarantees the “completeness” of the expansion and

also implies that the leading diagonal of F is no longer zero. Equation (3.23) may be

solved for using standard Gaussian elimination. This is the simplest alternative to

program. It has already produced excellent results for a wide range of engineering

problems.

Note that in this case

94

ˆ32 rr

u

and

32

1ˆ

r

n

yr

n

xrq yx

We implement the DRBEM1 program first based on this manner. Furthermore, the

program include the thin plate spline function for approximation. However this

program is still separated the execution of solving the boundary solution and internal

solution. Finally, the DRBEM2 is implemented to solve the whole system of equation

in order to apply to solve non-linear case in the future work. We discuss

computational results in the late section.

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4. Computational Result of the MULDRM program

Example 1: Square plate with internal heat generation (Gibson, 1985)


12 u (4.1)

The problem is to be solved over an isotropic square plate occupying the region

66 x and 66 y . The symmetry of the region means we need consider

only the negative quadrant as shown Figure 4.1.

Initially the boundary is modelled with 12 boundary elements and 4 multiple nodes at

the corners. The calculation of the potential is required at the five internal points

shown in Figure 4.1.Boundary conditions are specified as zero flux on the line 0x ,

0y and as zero temperature on the line 6x , 6y .

The internal solutions obtained using MULDRM program are compared with those of

the Monte Carlo method and the exact solution in Table 4.1.

12 u

(-6,0)

(-6,6) (0,6)

(0,0) 1

: Boundary node : Internal node : Multiple node

4 3 2

7

6

5

15

16

8 13

14

10 9 12 11

Figure 4.1 : Discretization of the boundary into 12 elements with

4 multiple nodes and 5 internal points

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Table 4.1 : Internal solutions of square plate with internal heat generation problem

Percentage errors of the potentials at the internal points are shown in Figure 4.2.

We see from the figure that the percentage errors of the solution using the multiple

node approach are less than the other methods.

Example 2 : The Torsion problem


22 u (4.2)

on the elliptical domain as shown in Figure 4.3.

Percentage errors of potential

0

1

2

3

4

(-2,2) (-4,2) (-3,3) (-2,4) (-4,4)

Err

or

(%)

Monte Calo Standard Multiple node

Figure 4.2 : Percentage errors of potential at the internal

points

Internal Monte Calo method MULDRM Exact

point 500 1000 3000 Standard Multiple solution

(-2,2) 8.985 8.543 8.537 8.418 8.690 8.690

(-4,2) 5.802 5.645 5.736 5.582 5.772 5.748

(-3,3) 6.498 6.362 6.477 6.358 6.547 6.522

(-2,4) 5.718 5.634 5.633 5.584 5.772 5.748

(-4,4) 3.961 3.987 3.981 3.889 3.987 3.928

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The elliptical section shown in Figure 3 has a semi-major axis a = 2 and a semi-

minor axis b = 1. The equation of the ellipse is

12

2

2

2

b

y

a

x (4.3)

The boundary condition is the Dirichlet condition with u = 0 on the boundary.

The exact solution is

18.0

2

2

2

2

b

y

a

xu (4.4)

The normal derivative is

)8(2.0 22 yxq (4.5)

We use the number of multiple nodes as 4, 8 and 16. The four multiple nodes are

shown in Figure 4.3. The solutions are compared with the cell integration method and

the exact solution as shown in Table 4.2.

(-2,0)

(0,1)

(2,0)

(0,-1)

1

7 6

2

3 4

9

8

13 12

11

10 16

15 14

Figure 4.3 : Discretization of the boundary into 16 elements

17 internal points and 4 multiple nodes in elliptical domain

: Boundary node : Internal point : Multiple node

5

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Table 4.2 : Internal solutions of the Torsion problem

Percentage errors of potential at the internal points are shown in Figure 4.4

We see that from Figure 4.4 that the multiple nodes do not seem to help much because

the domain boundary is smooth. The boundary does not contain real corners.

However, the solution using the method is quite better compared with the others.


0

1

2

3

4

5

6

(1.5,0.0) (1.2,-0.35) (0.6,-0.45) (0.0,-0.45) (0.9,0.0) (0.3,0.0) (0.0,0.0)

Err

or

(%)

Cell Integration Standard Multiple node

Figure 4.4 : Comparison percentage errors at the internal point

Internal Cell MULDRM Program (No. of multiple nodes) Exact point Integration 0 4 8 16 solution

(1.5,0.0) 0.331 0.344 0.347 0.347 0.347 0.350 (1.2,-0.35) 0.401 0.420 0.419 0.419 0.418 0.414 (0.6,-0.45) 0.557 0.576 0.574 0.574 0.574 0.566 (0.0,-0.45) 0.629 0.648 0.646 0.646 0.646 0.638 (0.9,0.0) 0.626 0.646 0.644 0.644 0.643 0.638 (0.3,0.0) 0.772 0.793 0.790 0.790 0.790 0.782 (0.0,0.0) 0.791 0.810 0.808 0.808 0.808 0.800

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Results for different functions ),( yxbb

The problem in Example 2 will now be presented for different known function

),( yxb In all applications the same problem geometry will be used, that given in

figure 4.3 with homogeneous condition 0u .

( a ) The case xu 2

This case and another in this section will be modeled using the element geometry

shown in Figure 4.3. The governing equation is

xu 2 (4.6)

The exact solution is given by

1

47

2 22

yxx

u (4.7)

which satisfies the boundary condition 0u on and produces

22

2

3

14

22

yxx

q (4.8)

Results for both the DRBEM program with varieties of multiple nodes and the cell

integration are given in Table 4.3.

Table 4.3 : Internal solution for the equation xu 2


Internal Cell MULDRM Program (NO. of multiple nodes) Exact

point Integration 0 4 8 16 solution

(1.5, 0.00) 0.176 0.179 0.184 0.184 0.184 0.187

(1.2, 0.35) 0.171 0.180 0.182 0.183 0.183 0.177

(0.6,-0.45) 0.118 0.123 0.125 0.125 0.125 0.121

(0.0,-0.45) 0.000 0.000 0.000 0.000 0.000 0.000

(0.9, 0.00) 0.200 0.205 0.209 0.209 0.209 0.205

(0.3, 0.00) 0.082 0.080 0.085 0.085 0.085 0.083

(0.0, 0.00) 0.000 0.000 0.000 0.000 0.000 0.000

DRBEM 3/8/09

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( b ) The case 22 xu

In this case the governing equation is

22 xu (4.9)


1

46.33850

246

1 22

22 yx

yxu (4.10)

which again satisfies the boundary condition 0u on and produces

yyyyxx

xxyxq 2.833296246

1

22.839650

246

1 3223 (4.11)

Results for the DRBEM program with varieties of multiple nodes and the exact

solution are given in Table 4.4.

Percetage errors of potential

0 . 0 0 0

1 . 0 0 0

2 . 0 0 0

3 . 0 0 0

4 . 0 0 0

5 . 0 0 0

6 . 0 0 0

7 . 0 0 0

( 1 . 5 , 0 . 0 0 ) ( 1 . 2 , 0 . 3 5 ) ( 0 . 6 , - 0 . 4 5 ) ( 0 . 9 , 0 . 0 0 ) ( 0 . 3 , 0 . 0 0 )

err

or

(%)

C e ll in te g r a t io n S ta n d a r d M u lt ip le n o d e s

DRBEM 3/8/09

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Table 4.4 : Internal solution for the equation 22 xu


We see from Figure 4.6 that the multiple nodes do not seem to help much in case of

the smooth boundary as we mentioned in the first problem of Example 2.


0

2

4

6

8

10

12

(1.5,0.0) (1.2,-0.35) (0.6,-0.45) (0.0,-0.45) (0.9,0.0) (0.3,0.0) (0.0,0.0)

Err

or

(%)

Standard Multiple node

Figure 4.6 : Comparison percentage errors at the internal points

Internal MULDRM Program (No. of multiple nodes) Exact point 0 4 8 16 solution

(1.5,0.0) 0.264119 0.265076 0.265104 0.264931 0.259 (1.2,-0.35) 0.218793 0.219662 0.219741 0.219798 0.220 (0.6,-0.45) 0.134292 0.135253 0.135211 0.135345 0.143 (0.0,-0.45) 0.091315 0.092104 0.092096 0.092200 0.103 (0.9,0.0) 0.235153 0.236444 0.236452 0.236486 0.240 (0.3,0.0) 0.140506 0.142129 0.142117 0.142225 0.151 (0.0,0.0) 0.124800 0.126591 0.126580 0.126689 0.136

DRBEM 3/8/09

25

5. Computational results of the DRBEM program

We have implemented the program DRBEM2 to solved the Poisson equations in the

form

y

uyxp

x

uyxpuyxpyxpu

),(),(),(),( 4321

2 (5.1)

using both of the radial basis function rf 1 and rrf log2 with linear term.

The previous DRBEM1 program is separated to solve the boundary solution and use

the result to evaluate the internal solution. On the other hand this program solve the

whole system of equations to obtain the boundary solution and the internal solution in

the same time. The program is tested with some problems in 6 cases:

Case 1: 04321 pppp

Case 2: 0432 ppp , 2

1 ),( xyxp

Case 3: 0432 ppp , yxeyxp 2

1 5),(

Case 4: 0431 ppp , 1),(2 yxp

Case 5: 0421 ppp , 1),(3 yxp

Case 6: 021 pp , 1),(,1),( 43 yxpyxp

We are going to present the result in each case.

Case 1: 04321 pppp

This case is a kind of Laplace equation. We test the program with a previous problem

examined by LINBEM and MULBEM program.

Consider the potential problem

02 u (5.2)

in a square plate 60 x , 60 y

with solution xu 50300 .

We partition the boundary into 12 elements without multiple nodes. The boundary

values of function and normal derivative are shown in Figure 5.1.

DRBEM 3/8/09

26

The results are compared with those of using LINBEM and MULBEM program and

shown in Table 5.1.

Table 1: The potential at the internal points

Case 2: 0432 ppp , 2

1 ),( xyxp


22 xu (5.3)

on the elliptical domain as shown in Figure 5.2

x

y

(0,0) (6,0)

(6,6) (0,6)

u = 300 u = 0

q = 0

q = 0

Figure 5.1: Boundary conditions for Problem

1

Point x y LINBEM MULBEM DRBEM 2 Exact

f = 1+r f=TPS solution

1 2.00 2.00 200.862 200.044 200.393 200.393 200.000

2 4.00 4.00 99.909 99.957 99.607 99.607 100.000

3 2.00 4.00 200.980 200.044 200.393 200.393 200.000

4 3.00 3.00 150.382 150.000 150.000 150.000 150.000

DRBEM 3/8/09

27

The elliptical section shown in Figure 1 has a semi-major axis a = 2 and a semi-

minor axis b = 1. The equation of the ellipse is

112 2

2

2

2

yx

(5.4)

The boundary condition is the Dirichlet condition with u = 0 on the boundary.


1

46.33850

246

1 22

22 yx

yxu (5.5)

which again satisfies the boundary condition 0u on and produces

yyyyxx

xxyxq 2.833296246

1

22.839650

246

1 3223 (5.6)

The potential solution at the internal nodes are shown in Table 5.2 and the normal

derivatives at the boundary are shown in Table 5.3.

(-2,0)

(0,1)

(2,0)

(0,-1)

1

7 6

2

3 4

9

8

13 12

11

10 16

15 14

Figure 5.2: Discretization of the boundary into 16 elements

and 17 internal points in elliptical domain

: Boundary node : Internal point

5

DRBEM 3/8/09

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Table 5.2: The potential at the internal nodes of the problem

Table 5.3: The normal derivative at the boundary of the problem

Case 3: 0432 ppp , yxeyxp 2

1 5),(


yxeu 22 5 (5.7)

on the quarter of a unit-circle 122 yx . Discretization and boundary condition are

shown in Figure 5.3.

Point x y DRBEM 1 DRBEM 2 Exact

f = 1+r f=TPS f = 1+r f=TPS solution

1 1.50 0.00 0.262345 0.272802 0.262348 0.272800 0.260

2 1.20 -0.35 0.217888 0.244511 0.218010 0.244460 0.220

3 0.60 -0.45 0.131481 0.176643 0.131574 0.176624 0.144

4 0.00 -0.45 0.087549 0.138110 0.087580 0.138108 0.104

5 0.00 0.00 0.122451 0.170248 0.122445 0.170250 0.137

6 0.30 0.00 0.138187 0.184561 0.138178 0.184566 0.151

7 0.90 0.00 0.233807 0.268124 0.233798 0.268137 0.240

Point x y DRBEM 1 DRBEM 2 Exact


1 -2 0 -0.902 -0.853 -0.902 -0.853 -0.950

2 -0.18476 -0.38268 -0.986 -0.932 -0.986 -0.932 -0.947

3 -1.41421 -0.70711 -0.855 -0.812 -0.855 -0.812 -0.790

4 -0.76537 -923880 -0.435 -0.401 -0.435 -0.401 -0.422

5 0 -1 -0.201 -0.168 -0.201 -0.168 -0.208

DRBEM 3/8/09

29

The potential solutions at the internal nodes are shown in Table 5.4 and the normal

derivatives at the boundary are shown in Table 5.5.


yxeq 2

yxeq 22

yxeu 2

1 3 2

9

5

8

7

6

4

12

11

10

Boundary node Internal node

(0,1)

(0,0) (1,0)

Figure 5.3: Discretization and boundary condition of the

mixed problem

Point x y DRBEM 1 DRBEB 2 Exact


1 0.75 0.25 5.721926 5.741004 5.721920 5.741017 5.754

2 0.50 0.25 3.454453 3.458933 3.454437 3.458954 3.490

3 0.25 0.25 2.074674 2.061553 2.074635 2.061583 2.117

4 0.25 0.50 2.691217 2.685164 2.691175 2.685180 2.718

5 0.50 0.50 4.444861 4.454192 4.444830 4.454207 4.482

6 0.75 0.50 7.350586 7.362017 7.350561 7.362695 7.389

7 0.50 0.75 5.737537 5.736434 5.737672 5.737045 5.755

DRBEM 3/8/09

30

Table 5.5: The normal derivative on the boundary of the problem

Case 4: 0431 ppp , 1),(2 yxp


uu 2 (5.8)

on the boundary as shown in Case 2.

Since homogeneous boundary condition will result in the trivial solution 0 qu at

all nodes, a non-homogeneous condition has to be used, for example

xu sin (5.9)

The solutions of the program are also compared with those of the original program

and the exact solutions and are shown in Table 5.6.


Point x y DRBEM 1 DRBEB 2 Exact


1 1 0 4.863224 4.810020 4.863268 4.809980 14.778

2 0.92388 0.382683 20.747750 20.685510 20.747820 20.685450 20.752

3 0.707107 0.707107 17.858360 17.812670 17.858580 17.812560 17.696

4 0.382683 0.92388 9.069196 9.057966 9.068832 9.057836 9.148

5 0 1 -0.910696 -0.895357 -0.910750 -0.895443 2.718

DRBEM 3/8/09

31

Case 5: : 0421 ppp , 1),(3 yxp


x

uu

2 (5.10)




xeu (5.11)




Point x y Original DRBEM 2 Exact

f = 1+r f=1+r+r^2+r^3 f = 1+r f=TPS solution

1 1.50 0.00 0.994 0.995 0.996 0.997 0.997

2 1.20 -0.35 0.928 0.932 0.928 0.931 0.932

3 0.60 -0.45 0.562 0.566 0.562 0.564 0.565

4 0.00 -0.45 0.000 0.000 0.000 0.000 0.000

5 0.90 0.00 0.780 0.784 0.780 0.782 0.783

6 0.30 0.00 0.294 0.296 0.294 0.295 0.295

7 0.00 0.00 0.000 0.000 0.000 0.000 0.000



1 1.50 0.00 0.229 0.214 0.229 0.225 0.223

2 1.20 -0.35 0.307 0.274 0.307 0.305 0.301

3 0.60 -0.45 0.555 0.523 0.555 0.553 0.549

4 0.00 -0.45 1.003 1.006 1.003 1.005 1.000

5 0.90 0.00 0.411 0.363 0.412 0.410 0.406

6 0.30 0.00 0.745 0.725 0.745 0.745 0.741

7 0.00 0.00 1.002 1.002 1.002 1.005 1.000

DRBEM 3/8/09

32

Case 6: 021 pp , 1),(,1),( 43 yxpyxp


y

u

x

uu

2 (5.12)




yx eeu (5.13)




6. Conclusion

We cannot distinguish the results computed by the standard linear elements and the

multiple node approach in problems containing smooth boundary. The MULDRM

program which transform Poisson equation to Laplace one works well but available

only in case of the right hand side function is a position function. The DRBEM1

program which separates the execution of solving boundary and internal solution

works as well as the DRBEM2 one which solves the whole solution in the same time.

However the DRBEM2 is suitable to modify to solve the non-linear problem which is

the future work. In problem containing corner domain, the normal derivative solution

at the boundary is still poor. It is our purpose to modify the program to resolve this

problem. For approximation of functions, the thin plate spline works better than the

linear function rf 1 for all cases.



1 1.50 0.00 1.231 1.214 1.231 1.225 1.223

2 1.20 -0.35 1.714 1.669 1.714 1.717 1.720

3 0.60 -0.45 2.107 2.057 2.107 2.109 2.117

4 0.00 -0.45 2.557 2.547 2.557 2.560 2.568

5 0.90 0.00 1.400 1.345 1.401 1.404 1.406

6 0.30 0.00 1.731 1.691 1.731 1.737 1.741

7 0.00 0.00 1.989 1.963 1.989 1.997 2.000

33

References

Golberg, M.A. (1994) The theory of radial basis functions applied to the BEM for

inhomogrneous partial differential equations. Boundary Elements

Communications 5, 57-61.

Golberg, M.A. (1996) The method of fundamental solutions for Poisson's equation.

Engineering Analysis with Boundary Elements 16, 205-213.

Golberg, M.A., Chen, C.S., Bowman, H. and Power, H. (1998) Some comments on

the use of radial basis functions in the dual reciprocity method.

Computational Mechanics 22, 61-69.

Karur, S.R. and Ramachandran, P.A. (1994) Radial basis function approximation in

the dual reciprocity method. Mathematical and Computer Modelling 20, 59-

70.

Partridge, P.W. and Brebbia, C.A. (1989) Computer implementation of the BEM dual

reciprocity method for the solution of Poisson type equations. Software for

Engineering Workstations 5, 199-206.

Partridge, P.W. and Brebbia, C.A. (1990) Computer implementation of the BEM dual

reciprocity method for the solution of general field equations.

Communications in Applied Numerical Methods 6, 83-92.

Partridge, P.W., Brebbia, C.A. and Wrobel, L.C. (1992) The dual reciprocity

boundary element method, Southampton: Computational Mechanics

Publications.

Powell, M.J.D. (1994) The uniform covergence of thin plate spline interpolation in

two dimension. Numerical Mathematics 68, 107-128.

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