The Dual Reciprocity Boundary Element Method · Study on the Dual Reciprocity Boundary Element Method Wattana Toutip Technical Report 3 ...
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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.
DRBEM 3/8/09
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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
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.
DRBEM 3/8/09
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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
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
Consider 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
DRBEM 3/8/09
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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.
DRBEM 3/8/09
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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)
DRBEM 3/8/09
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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)
Consider the Poisson equation
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
Consider the Poisson equation
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
DRBEM 3/8/09
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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.
Percentage errors of potential
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
DRBEM 3/8/09
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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
Percentage errors of potential at the internal points are shown in Figure 4.5
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)
The exact solution is given by
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
Percentage errors of potential at the internal points are shown in Figure 4.6
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.
Percentage errors of potential
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
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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
Consider the Poisson equation
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
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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.
The exact solution is given by
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),(
Consider the Poisson equation
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
f = 1+r f=TPS f = 1+r f=TPS solution
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
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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.
Table 5.4: The potential at the internal nodes of the problem
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
f = 1+r f=TPS f = 1+r f=TPS solution
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
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Table 5.5: The normal derivative on the boundary of the problem
Case 4: 0431 ppp , 1),(2 yxp
Consider the Poisson equation
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.
Table 5.6: The potential at the internal nodes of the problem
Point x y DRBEM 1 DRBEB 2 Exact
f = 1+r f=TPS f = 1+r f=TPS solution
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
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Case 5: : 0421 ppp , 1),(3 yxp
Consider the Poisson equation
x
uu
2 (5.10)
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
xeu (5.11)
The solutions of the program are also compared with those of the original program
and the exact solutions and are shown in Table 5.7.
Table 5.7: The potential at the internal nodes of the problem
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
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 0.229 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
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Case 6: 021 pp , 1),(,1),( 43 yxpyxp
Consider the Poisson equation
y
u
x
uu
2 (5.12)
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
yx eeu (5.13)
The solutions of the program are also compared with those of the original program
and the exact solutions and are shown in Table 5.8.
Table 5.8: The potential at the internal nodes of the problem
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.
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 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
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the use of radial basis functions in the dual reciprocity method.
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Karur, S.R. and Ramachandran, P.A. (1994) Radial basis function approximation in
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Partridge, P.W. and Brebbia, C.A. (1989) Computer implementation of the BEM dual
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