Embed Size (px)
Citation preview
Lecture Notes in Engineering Edited by C. A. Brebbia and S. A. Orszag
73
K. Hayami
A Projection Transfonnation Method for Nearly Singular Surface Boundary Element Integrals
..--- ---, Springer-Vertag Bertin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Rel
ativ
e j
Err
or
~
1 10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
.--+
<
\ .11
)'.
/_.-
....
,.
"""_
'_'-
~/'
. "
. ,
--' it \/ '''./
",
• __
. \
....... ...
,. .,...
.. q
.<J"
D.
\li '/'
'n\
Gau
ss
IsC
Pijq
*dS
1,&
\ \.
./'.
•
\ \
'~.,
"-"
Iden
tity
\ ~.~
'b."
(Po
lar)
\
" 'n
.. ~
\ ·~
Tell
es
",
S :
'sph
eric
al' q
uadr
ilate
ral
(SP
Q60
) d
= 0.
01
\ --
-.~
bo.
\.
....... ...
..... ,
.. , "
........
........
.. , '
-.~
'~"'"
R,og
-L 2
. ':-
-...
'"..
....
",
..........
.....
10 -7
1,
" \
'."
"..
'. R,
Og-L
1 ,
, I
,
, I
I,
,
2000
N
um
ber
of
Inte
gra
tion
Po
ints
o
500
1000
15
00
, 10
-8
Fig
. 1
0.3
2
Co
nv
erg
en
ce
gra
ph
fo
r Is
CPij
q*d
S
.1:>0 .... Q)
Series Editors C. A. Brebbia . S. A. Orszag
Consulting Editors J. Argyris . K.-J. Bathe' A. S. Cakmak . J. Connor' R. McCrory C. S. Desai· K.-P. Holz . F. A. Leckie' G. Pinder' A. R. S. Pont J. H. Seinfeld . P. Silvester' P. Spanos' W. Wunderlich . S. Yip
Author Dr. Ken Hayami C & C Infonnation Technology Research Laboratories NEC Corporation 4-1-1 Miyazaki Miyamae, Kawasaki Kanagawa 213 JAPAN
ISBN-13: 978-3-540-55000-6 e-ISBN-13: 978-3-642-84698-4 001: 10.1007/978-3-642-84698-4
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the Gennan Copyright Law of September 9, 1965, in its current version. and pennission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the Gennan Copyright Law. © Springer-Verlag Berlin Heidelberg 1992
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Typesetting: Camera ready by author
ACKNOWLEDGEMENTS
I wish to express my sincere appreciation and gratitude to my advisor
Dr. Carlos A. Brebbia, for his valuable guidance and encouragement throughout
the period of this study, which started with my stay at the Wessex Institute of
Technology, U. K.
I am grateful to Prof. W. S. Hall, Prof. M. Tanaka, Prof. J. C. F. Telles,
Dr. B. Adey, Dr. S. M. Niku, Dr. S. Amini, Dr. W. Tang, Dr. M. H. Aliabadi,
Prof. T. Honma, Dr. T. Takeda, Dr. L. Gray, Dr. N. Nishimura, Dr. Y. Iso, Prof.
M. Mori, Dr. M. Sugihara, Dr. K. Kishimoto and Dr. S. Akaboshi, among many
others, for stimulating discussions, valuable remarks and encouragements. I
would also like to thank Dr. W. Blain for his general assistance.
I am indebted to my company, NEC Corporation, in particular, Mr. Y. Kato,
Dr. K. Iinuma, Mr. A. Managaki, Mr. K. Nakamura, Dr. N. Harada, Mr. Y.
Nagai, Dr. E. Okamoto, Dr. S. Doi, Mr. K. Tsutaki and Mr. Y. Yanai, among
many others, for their support and understanding for the present work. I am
grateful to Mr. H. Matsumoto ofNEC Scientific Information System Development
Co. Ltd. for preparing tables for the Gauss-Legendre formula. I would also like to
thank Ms. M. Nakae, Mrs. K. Ishikura and Ms. T. Kitagawa, for their excellent
typing.
I would like to thank my sister, Mrs. Y. Mino for helping me with the proof
reading.
Last, but not the least, I would like to thank my wife Emiko, for her constant
support and encouragement.
TABLE OF CONTENTS
PART I THEORY AND ALGORITHMS
CHAPTER 1 INTRODUCTION 3
CHAPTER 2 BOUNDARY ELEMENT FORMULATION OF 3-D POTENTIAL PROBLEMS
2.1 Boundary Integral Equation 8
2.2 Treatment of the Exterior Problem 14
2.3 Discretization into Boundary Elements 18
2.4 Row Sum Elimination Method 24
CHAPTER 3 NATURE OF INTEGRALS IN 3-D POTENTIAL PROBLEMS 29
3.1 Weakly Singular Integrals 30
3.2 Hyper Singular Integrals 40
3.3 Nearly Singular Integrals 53
CHAPTER 4 SURVEY OF QUADRATURE METHODS FOR 3-D BOUNDARY ELEMENT METHOD 72
4.1 Closed Form Integrals 73
4.2 Gaussian Quadrature Formula 74
4.3 Quadrature Methods for Singular Integrals 76
(1) The weighted Gauss method 76
(2) Singularity subtraction and Taylor expansion method 77
(3) Variable transformation methods 78
(4) Coordinate transformation methods 80
VI
4.4 Quadrature Methods for Nearly Singular Integrals 81
(1) Element subdivision 82
(2) Variable transformation methods 84
(3) Polar coordinates 88
CHAPTER 5 THE PROJECTION AND ANGULAR & RADIAL TRANSFORMATION (PART) METHOD
5.1 Introduction 89
5.2 Source Projection 93
5.3 Approximate Projection of the Curved Element 97
5.4 Polar Coordinates in the Projected Element 104
5.5 Radial Variable Transformation 113
(i) Weakly Singular Integrals 113
(ii) Nearly Singular Integrals 114
(1) . Singularity cancelling radial variable transformation 114
(2) Consideration of exact inverse projection and curvature of the element in the transformation 121
(3) Adaptive logarithmic transformation (log-L2) 123
(4) Adaptive logarithmic transformation (log-Ll) 125
(5) Single and double exponential transformations 131
5.6 Angular Variable Transformation 139
(1) Adaptive logarithmic angular variable transformation 140
(2) Single and double exponential transformations 143
5.7 hnplementation of the PART method 144
(1) The use of Gauss-Legendre formula 145
(2) The use of truncated trapezium rule 150
5.8 Variable Transformation in the Parametric Space 153
VII
CHAPTER 6 ELEMENTARY ERROR ANALYSIS
6.1 The Use of Error Estimate for Gauss-Legendre Quadrature Formula 155
6.2 Case {3=2 160
(Adaptive Logarithmic Transformation: fog-L2)
6.3 Case {3= 1 Transformation 166
6.4 Case {3=3 Transformation 169
6.5 Case {3 = 4 Transformation 172
6.6 Case {3 = 5 Transformation 185
6.7 Summary of Error Estimates for (3=1-5 188
6.8 Error Analysis for Flux Calculations 190
CHAPTER 7 ERROR ANALYSIS USING COMPLEX FUNCTION THEORY
7.1 Basic Theorem 204
7.2 Asymptotic Expression for the Error Characteristic 207 Function <P n(z)
7.3 Use ofthe Elliptic Contour as the Integral Path 208
7.4 The Saddle Point Method 210
7.5 Integration in the Transformed Radial Variable: R 212
7.6 Error Analysis for the Identity Transformation: R(p) = p 214
(1) Estimation of the size (J of the ellipse cq 217
(2) Estimation of max I j(z) I 218 Z E£q
(3) Error estimate En(j) 220
7.7 Error Analysis for the fog-L 2 Transformation 221
(1) Case: 0 = odd 223
(2) Case : 0 = even 227
(i) Contribution from the branch line f +, f- 229
(ii) Contribution from the ellipse cq 234
(iii) Contribution from the small circle CE 238
(iv) Summary 239
VIII
7.8 Error Analysis for the eog-Ll Transformation 241
(1) Error analysis using the saddle point method 244
(2) Error analysis using the elliptic contour: Cq 249 (i) Estimation of max I f(z) I 250
z ECq
(ii) Estimation of a 252
(iii) Error estimate En(j) 253
7.9 Summary of Theoretical Error Estimates 257
PART II APPLICATIONS AND NUMERICAL RESULTS
CHAPTER 8 NUMERICAL EXPERIMENT PROCEDURES AND ELEMENT TYPES
8.1 Notes on Procedures for Numerical Experiments 263 8.2 Geometry of Boundary Elements used for
Numerical Experiments 265
(1) Planar rectangle (PLR) 265
(2) 'Spherical' quadrilateral (SPQ) 267
(3) Hyperbolic quadrilateral (HYQ) 270
CHAPTER 9 APPLICATIONS TO WEAKLY SINGULAR INTEGRALS
9.1 Check with Analytial Integration Formula for Constant Planar Elements 273
9.2 Planar Rectangular Element with Interpolation
Function rp jj 282
9.3 'Spherical' Quadrilateral Element with
Interpolation Function rpij 294
(1) Results for Isrpjju*dS 294
(2) Results for Is rpijq* dS 303
9.4 Hyperbolic Quadrilateral Element with
Interpolation Function rpij 313
(1) Results for Is rpij u* dS 313
(2) Results for Is rp jj q* dS 321
IX
9.5 Summary of Numerical Results for
Weakly Singular Integrals 329
CHAPTER 10 APPLICATIONS TO NEARLY SINGULAR INTEGRALS
10.1 Analytical Integration Formula for Constant Planar Elements 331
10.2 Singularity Cancelling Radial Variable Transformation
for Constant Planar Elements 335
10.3 Application of the Singularity Cancelling
Transformation to Elements with Curvature
and Interpolation Functions
(1) Application to curved elements
(2) Application to integrals including interpolation functions
10.4 The Derivation of the eog-L2 Radial Variable Transformation
(1) Application of radial variable transformations p dp = r'P dR «(3* a) to integrals J s lira dS over
curved elements (2) Difficulty with flux calculation
10.5 The eog-Ll Radial Variable Transformation
10.6 Comparison of Radial Variable transformations
for the Model Radial Integral la.o
(1) Transformation based on the Gauss-Legendre rule
(i) Identity transformation
(ii) eog-L2 transformation
(iii) eog-Ll transformation
(2) Transformation based on the
truncated trapezium rule
(i) Single Exponential (SE) transformation
(ii) Double Exponential (DE) transformation
(3) Summary
353
353
368
377
377 392
395
397
408
408
409
410
411
411
412
412
x
10.7 Comparison of Different Numerical Integration methods on the 'spherical' Element 413
(1) Effect of the source distance d 414 (2) Effect of the position of the source projection Xs 430
10.8 Summary of Numerical Results for Nearly Singular Integrals 438
CHAPTER 11 APPLICATIONS TO HYPERSINGULAR INTEGRALS 441
CHAPTER 12 CONCLUSIONS 445
REFERENCES 451
PART I
THEORY AND ALGORITHMS
CHAPI'ER 1
INTRODUCTION
In three dimensional boundary element analysis, computation of integrals is
an important aspect since it governs the accuracy of the analysis and also because
it usually takes the major part of the CPU time.
The integrals which determine the influence matrices, the internal field and
its gradients contain (nearly) singular kernels of order lIr a (0:= 1,2,3,4,.··) where
r is the distance between the source point and the integration point on the
boundary element.
For planar elements, analytical integration may be possible 1,2,6. However,
it is becoming increasingly important in practical boundary element codes to use
curved elements, such as the isoparametric elements, to model general curved
surfaces. Since analytical integration is not possible for general isoparametric
curved elements, one has to rely on numerical integration.
When the distance d between the source point and the element over which
the integration is performed is sufficiently large compared to the element size
(d> 1), the standard Gauss-Legendre quadrature formula 1,3 works efficiently.
However, when the source is actually on the element (d=O), the kernel 1I~
becomes singular and the straight forward application of the Gauss-Legendre
quadrature formula breaks down. These integrals will be called singular
integrals. Singular integrals occur when calculating the diagonals of the
influence matrices.
When the source is not on the element but very close to the element
(O<d~l), although the kernel lIr a is regular in the mathematical sense, the
value of the kernel changes rapidly in the neighborhood of the source point and
the standard Gauss-Legendre quadrature formula is not practical since it would
require a huge number of integration points to achieve the required accuracy.
4
These integrals will be called nearly singular integrals. Nearly singular
integrals occur in practice when calculating influence matrices for thin
structures, where distances between different elements can be very small
compared to the element size. They also occur when calculating the field or its
derivatives at an internal point very close to the boundary element.
Singular Integrals Nearly Singular Integrals (d=O) (O<d~1)
I. Analytical ( for planar elements only)
II. Numerical
(1) Weighted Gauss (1) Element Subdivision
(2) Singularity Subtraction (2) Variable Transformation (+Taylor Expansion)
(i) Double Exponential Transformation
(3) Variable Transformation (ii) Cubic Transformation
(4) Coordinate Transformation (3) Coordinate Transformation
(i) Triangle to Quadrilateral Polar Coordinates Transformation ( + modification)
(ii)Polar Coordinates
(5) Finite Part Integrals
Present Method:
Projection and Angular & Radial Transformation (PART)
Table 1.1 Classification of quadrature methods for (nearly) singular integrals in three dimensional boundary element method.
Numerous research works have already been published on this subject and
they may be classified as in Table 1.1. These are numerical methods based on the
Gaussian quadrature formula or the truncated trapezium rule with modifications
5
to suit the (nearly) singular kernels which appear in the Boundary Element
Method (BEM).
Let us first briefly review the methods for singular integrals.
The weighted Gauss 4,5,19,20 method uses the kernel 1Ir as the weight
function for generating the Gauss integration points.
The singularity subtraction 21 with Taylor expansion 6 method expands the
singular kernel by the local parametric coordinates. The main terms containing
the singularity is subtracted and integrated analytically and the remaining well
behaved terms are integrated by Gaussian quadrature.
Then there are the coordinate transformation methods. The first type is the
method of transforming a triangular region into a quadrilateral region so that the
node corresponding to the singularity is expanded to an edge of the quadrilateral,
so that the singularity is weakened 7,8,21. The second type is that of using polar
coordinates (p, 8) around the source point in the parameter space 9,15. This
introduces a Jacobian which cancels the singularity 1Ir.
For higher singularities of order 1Ir2 which appear in elastostatics, the
method for calculating finite part integrals 10,11 may be used.
Although a rigorous comparison is not attempted, the use of polar
coordinates seems to be the most natural and effective way. In the present work
this idea is extended to taking polar coordinates around the source point in the
plane tangent to the curved element at the source point. Further, an angular
variable transformation is introduced which considerably reduces the number of
integration points in the angular variable.
Nearly singular integrals turn out to be more difficult and expensive to
calculate compared to singular integrals. They are becoming more and more
important in practical boundary element codes, since the ability and efficiency to
calculate nearly singular integrals governs the code's versatility in treating
objects containing thin structures, which occur in many important problems in
engineering. Examples are the electrostatic analysis of electron guns with
6
complex geometry, calculation of the magnetic flux in thin gaps occurring in
electric motors, to mention a few. The use of discontinuous elements 1 also
increases the chances of encountering nearly singular integrals. The stress of the
present work is on a new quadrature scheme for the accurate and efficient
evaluation of these nearly singular integrals.
The orthodox way to treat the problem is to increase the number of
integration points as the source to element distance d becomes small, and further
to subdivide the element so that the integration points are concentrated near the
source point 7,12. Subdivision tends to be a cumbersome procedure and would be
inefficient when d is very small compared to the element size.
A recent trend is to transform the integration variables so as to weaken the
singular behaviour of the kernel, such as using the double exponential
transformation with trapezium rules 13,14. A more efficient self-adaptive method
using cubic transformation15 has been proposed. However, this method does not
give accurate results when the ratio of the distance d to the typical element size
is smaller than the order of 10 - 2, which is required in practice.
The use of polar coordinates in the parametric space with correction
procedures is reported to be efficient for potential problems 16.
In the present work a new coordinate transformation method is introduced,
in which the curved boundary element is approximately projected to the tangent
plane at the point on the curved element nearest to the source point, and then
polar coordinates are employed in the tangent plane with a further
transformation of the radial variable in order to cancel out the (near) singularity,
after which the standard Gaussian-Legendre quadrature scheme 17 is applied.
Further more, the method is generalized to cope with arbitrary geometry of
the curved element, such as curved triangular as well as quadrilateral elements.
Then an angular variable transformation is introduced to reduce the number of
integration points in the angular variable.
7
Finally, adaptive (with d) logarithmic radial variable transformations are
proposed, which are shown to be robust radial transformation for near
singulari ties of differen t orders.
The method, which will be referred to as the Projection and Angular &
Radial Transformation (PART) method, enables one to calculate nearly singular
integrals accurately and efficiently, even when the distance d to element size
ratio is smaller than 10-2• The method is also applicable to different types of
problems because of the robustness of the adaptive logarithmic radial variable
transformations for different types of singularities.
CHAPI'ER 2
BOUNDARY ELEMENT FORMULATION OF 3-D POTENTIAL PROBLEMS
Although the quadrature methods to be proposed are applicable to general
problems, let us take potential problems to illustrate the nature of the (near)
singularities of integrals and how the quadrature methods can be applied.
2.1 Boundary Integral Equation
The potential problem in a three dimensional domain V with boundary
surface S can be described by the following Laplace equation:
t:. u(x) = 0
where
with the boundary condition;
u(x) = u (x)
{ iJu-q(x)= - = q (x)
iJn
(2.1)
(2.2)
where alan is the derivative along the unit outward normal vector n of the
boundary surface S at point x .
The fundamental solution u*( x, xs) of the Laplace equation:
t:. u· (x, x ) = -6 (x, x ) X B 8
(2.3)
in the infinite domain is given by
• 1 u (x,x)=-
B 471" r (2.4)
where
and
o (x, x.): Dirac's delta function,
x: field point ,
x.: source point,
r = Irl ,
r = x - X •
9
as shown in Fig. 2.1 .
Using Green's identity:
J J ~ ~ (F b.G-Gt:Ji' )dV = (F - - G-)dS
v s an an
taking F=u and G = u*, we obtain
J •. v ( u b.u - u b.u) dV = J ..
s(uq-u q)dS
where
(r, n ) = ---
Substituting equations (2.1) and (2.3) into equation (2.5) gives
where
and
J u (x) 0 (x, x ) dV = J (q u • - u q.) dS v • s
J u (x) 0 (x, x ) dV = u (x ) for v·'
J u (x) 0 (x, x ) dV = 0 V •
for
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
10
n
Fig.2.1 Source point Xs In region V
11
For the case when Xs E S, i.e. when the source point Xs is actually on the
surface S, the property of the Dirac's delta function yields
f u (x) 8 (x, x ) dV = .!:!.... u (x ) V s 4n: s
where w is the solid angle subtended by V at Xs on S as show in Fig. 2.2 .
For instance, w = 21r when the surface S is smooth at Xs •
From equations (2.8-11),
where
f ·· c(x) u(x) = (qu -uq)dS s • S
c(xs )= { :
w/4n:
(x. E V)
(x i V) • (x E S)
s
(2.11)
(2.12)
(2.13)
Instead of using the Dirac's delta function of equation (2.11), equation (2.12)
can be derived for the case when Xs E S as follows:
Consider a part of a sphere S£ of radius E: centered at Xs as in Fig. 2.3 ,
where S=S' +S£' Since now Xs E V, the left hand side of equation (2.8) is
f u (x) 8 (x, x ) dV = u (x ) V • s
(2.14)
Next, the first term of the right hand side of equation (2.8) is
(2.15)
where
(2.16)
, , , , , , , , ,
Xs
12
s
Fig.2.2 Use of Dirac's delta function at Xs E S
13
n
Fig.2.3 Treatment of Xs E S without the use of Dirac's delta function
14
The second term of the right hand side of equation (2.8) is
-t uq· dS = - t. uq· dS - t uq· dS (2.17)
• where
f · f (r, n ) - S uq dS = -- u dS
S 4lr r3 • •
(2.18)
From equations (2.8), (2.14) and (2.15-18),
u(x)= lim f (qu·-uq·)dS + (l-~) u(xs ) s .-0 S' 4lr
(2.19)
Since for potential problems, u*-O(1Ie), q*-O(lle) (cf. equation (3.40) ) and
dS-O(E2) for xES in the neighborhood of xs, where S now indicates the original
smooth surface, we obtain
(2.20)
Hence, equations (2.19) and (2.20) give
W f.. -u(x)= (qu-uq)dS 4lr s s
(2.21)
which corresponds to equation (2.12) for the case when Xs E S.
2.2 Treatment of the Exterior Problem
One advantage of the boundary element method, especially when treating
electromagnetic or acoustic problems is that exterior problems can be treated
without meshing the infinite exterior regions. A brief explanation will be given in
the following for the three dimensioned potential problem.
Let us consider an exterior problem
l::.u(x) = 0 in V (2.22)
with boundary condition
15
u(x) = u (x) on 8 1 c8
q(x)= q (x) 82= 8-81 (2.23)
on
where the region V is an infinite region as shown in Fig. 2.4.
Since the Green's identity of equation (2.5) is also valid for a multi connected
region, let us take a sphere SR of radius R centered at the source point Xs E V,
such that S is included in the sphere SR, as shown in Fig. 2.5. The boundary
integral formulation of equation (2.12) becomes valid for the region VR enclosed
between surface Sand SR, i.e.
(2.24)
• 1 1 u =
• (r, n) 1 q = ---
- 4u3 - - 411" R2
dB = R2 dR sin 8 dB d<{> (2.25)
where (r, e, r/» is the polar coordinate system centered at xs, and R= IRI. Let us take the limit of R-oo. The value ofu, q, on SR can be considered as
a solution of
(2.26)
where Q is the sum of the source term (e.g. electric charge) inside S, since S
may be considered as a point source of finite magnitude Q when observed from a
distant point x E SR as R-oo.
Hence, on SR
(2.27)
(2.28)
16
Fig. . . n V . 2 4 Exterior reglo
17
s
Fig.2.5 Multi connected region VR
18
as R - 00 , so that the third tenn of equation (2.24) tends to zero as R- 00, since
as R-oo.
Hence, equation (2.24) becomes
J · · c(x)u(x)= (qu-uq)dS B B s (2.30)
for the exterior problem. Note that the boundary that has to be discretized is only
S and no mesh discretization in the infinite region is involved. Note also that the
unit outward nonnal n on S is defined in the opposite direction compared to the
interior problem as shown in Fig. 2.5.
2.3 Discretization into Boundary Elements
Now the boundary element fonnulation can be derived from equation (2.12)
by discretizing the surface S into boundary elements Se, as shown in Fig. 2.6.
Each element contains nodes xei (j = 1-ne), where u (or q) is defined from the
boundary condition, and q (or u) is to be solved.
The element is described by the parameters (711' 712) and interpolation
functions ¢ei (7J1' 712)' (j=1-ne), which are defined so that
n e
{(7!1·7!2)= L ~~(7!1'7!2) (~ }=1
(2.31)
where fei is the value of f(7J1' 712) at node xei . f can representthe potential u,
its nonnal derivative q = au/an or coordinates x, i.e.
n • u(7!1'7!2)= L ~~(7J1'7!2) u~
j=l (2.32)
node x j e
19
Fig.2.6 Discretization of S into boundary elements Se
and
n e
q ('71''72)= 2: 1>;('71 .'72) q~ j=1
n • x ('71''72)= 2: 1>~('71·'72) ~
j=1
20
For a curved quadrilateral element shown in Fig. 2.7,
where
Since
J ds=iJdS S .=1 S.
(2.33)
(2.34)
(2.35)
where m is the total number of elements, equation (2.12) can be discretized as
or n
m i (gk 1 ql _ hk 1 u~ ) k k = 2: C e' u e' e' e e e' e
.=1 1=1
(2.37)
where
ck, = c(xk,) 1 1 • e 1>.=1>.('71 .112 )
u l = u(xl ,) • • G=G('71 ·'72 )
q~ = q (x~,) x=x('71''72)
and
(2.38)
21
112
1
-1 1 0
-1
Fig.2.7 Curved quadrilateral element Se and parametric space (111,112)
22
(2.39)
Reordering the nodes throughout S, equation (2.37) can be written as
N N
c. u. = '" Gq. - '" H .. u. II LlJ J L lJJ (2.40)
j=l j=l
where N is the total number of nodes on S. Ifwe define
(2.41)
equation (2.40) gives
N N
L Hiju j = L Gijqj (2.42) j=l j=l
which can be written in matrix form as
(2.43)
0. represents the Dirichlet boundary condition and q the Neumann boundary
condition.
Equation (2.43) can be rewritten as
(2.44)
so that the unknown u and q can be obtained by solving the system of linear
equations (2.44) .
From equation (2.12) the potential u(xs) at an internal point xsEV is given
by
where • 1
U =-4", r '
(2.45)
• • au (r, n)
q = an - 4", r3
23
and the potential gradient at an internal point Xs E V is given by
• • au I au aq -= (q--u-)dS ax, s ax, ax, (2.46)
where au ·(x,x) 1 r 8
= ax 41r r3 , (2.47)
and
aq "(x, x) 1 { ~ _ 3r(r,n) } , =
ax 41r r3 r5 8
(2.48)
Correspondingly, after the discretized equation (2.44) is solved for the
boundary values u and q, the potential u(xs) at an internal point Xs can be
calculated by
where
and
n
U (xs ) = i i. (g:1 q~ - h:1 u~ ) e=1 1=1
n e
X(1J 1,1J2) = L ,s!(1J1,1J2) x! 1=1
(2.49)
(2.50)
(2.51)
(2.52)
Similarly, the potential gradient at an internal point Xs E V can be
calculated by
(2.53)
where
24
I1 I1 ;~IGI r = -- - dTJ dTJ
-1 -1 4/r r3 1 2 (2.54)
and
b sl = a
hsl • ax • •
L: L: •
;1 aq = IGI - dTJ 1 dTJ 2 e ax
B
= L: L: ;~ I GI { ~ 4/r r3
3r (r, n) } --5- dTJ 1 dTJ 2
r (2.55)
where r = X(7]l' 7]2) - Xs ,and n is the unit outward normal vector of the
boundary element Se at x E Se.
2.4. Row Sum Elimination Method
The calculation of c(xs) in equation (2.13) when Xs is a node shared by two
or more elements as shown in Fig. 2.8 involves the calculation of the solid angle
w subtended by the region V at Xs on S. In order to avoid this, one may use the
row sum elimination method in many cases.
For the three dimensional potential problem defined in the interior region,
consider the equipotential solution u(x)== 1 to the original Laplace equation
b. u(x) = O. This implies that q(x) = au(x)lan = 0 on the boundary S.
Hence equation (2.12) gives
C(X)=-I q·dS B s (2.56)
and Uj == 1 , (j = 1-N) and <Ij == 0, (j = 1-N) in equation (2.42) gives
25
Fig.2.8 The solid angle ill at Xs subtending V
Hence,
H .. =II
N
L j=l(j"'i)
26
(2.57)
H .. I) (2.58)
For the exterior problem, a similar technique can be used with some
modification.
In the equation
f "" f " " C (x ) u (x ) = (q u -uq ) dS + (q u -uq ) dS s s S S
R (2.24)
let us assume the equipotential solution u(x)=l in the region VR of Fig. 2.5.
Since
au q= - "" 0 an
c(x)= -f q"dS -f· q"dS S s s
R
Hence,
Equation (2.61) is satisfied in the limit of R-co.
Hence, equation (2.60) gives
c(x)= -f q"dS+l S s
The discretization of this equation as in equation (2.40) gives
n
c;= - L j=l
which gives
H .. + 1 I)
(2.59)
(2.60)
(2.61)
(2.62)
(2.63)
27
n
H .. = H .. + c. = 1 -II U, L H ..
I} (2.64) j=l(j~i)
From the above argument, the diagonal element Hii can be indirectly
calculated by the row sum of Hij (i =F j), so that the calculation of the solid angle
at node Xi given by
w (x.) I
(2.65) c. = I 41f
and the calculation of the singular integral
H .. = '" hkk It Lee
k (2.66) x = x.
e I
hk k = IiI 1 ,/IGI q·(x,xk) dTJ 1dTJ2 e e -1 -1 e e
(2.67)
where • • au (r,n)
q = -=-an 41f r3
become unnecessary. This technique is equivalent to what is known as the use of
rigid body motion in elastostatics. On the other hand, it is a good check to
calculate Hii directly from Ci and Hij. For discontinuous elements 1, Ci = 1 / 2
for i = 1-N, and calculating the singular integrals Hii directly are reported to
give more accurate results 15.
The diagonal element Gii has to be calculated directly by the singular
integral
where u* = 1/(47l"r) and
Gii = L k
x = x. e I
(2.68)
(2.69)
28
One also has to calculate integrals
f 1 f 1 k I I· Ie h = rp IGI u (X,X ) dTf 1 dTf2 e e -1 -1 e •
(2.70)
and
(2.71)
which contribute to the non-diagonal element Hij and Gij. It will be shown in
Chapter 3 that the integrals heek1, geek1 (k:;t: l) are not singular for three
dimensional potential problems in the sense that the integral kernels are of order
0(1) or O(r) , where r = Ix -xekl, since </>e1(xek) = 0 for k:;t: I.
CHAPTER 3
NATURE OF INTEGRALS IN 3-D POTENTIAL PROBLEMS
From the previous chapter, the integrals which appear in three dimensional
potential problems are the following:
f · f q 1 qudS= --dS s s 4~ r (3.1)
f · f u (r, n) uq dS = - - -- dS s s 4~ r3 (3.2)
related to the calculation of the potential u(xs) and the coefficients of the H, G
matrices, and
f au· f q r q-dS= --dS sax, s 4~ r3
(3.3)
related to the calculation of the flux au/axs at Xs.
As mentioned in the introduction (Chapter 1), these integrals may be
classified by the distance d between the source point Xs and the boundary
surface S (or the boundary element Se ) .
When d = 0, they are called singular integrals.
When 0 <d ~ 1, they are called nearly singular integrals.
When d> 1 , the integrals do not cause difficulties since they may be
calculated accurately using the standard Gauss-Legendre formula 1,3 with
relatively few integration points. (Here the distance d is defined relative to the
element size which is set to 1 .)
It is for the singular (d=O), and nearly singular integrals (O<d~l) that
special attention is necessary in order to calculate their values accurately.
30
3.1 Weakly SingularIntegrals
Weakly singular integrals (d = 0) arise when calculating the diagonal terms
of the G and H influence matrices. This corresponds to integrals ge'ekl and he'ekl
in equations (2.38) and (2.39), when e'=e and k=l.
e' = e means that the source point Xs is actually on the element surface Se,
sothat r = Ix-xsl becomes 0 at x = Xs.
Furthermore, when k = 1, one has to calculate
where
and
o . 1 u (x,xi) = -
e 4/rr
o( __ i) (r,n) q x,~ = - -3-
4r
Interpolation functions are generally constructed so that
where
and
In other words, <Pel is the interpolation function corresponding to node xel .
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
31
Hence, the numerator of the kernels of integrals gee ll, heell of equations (3.5)
and (3.6) take a nonzero value at x = xel, whilst the denominator is zero since
r= Ix-xell = O. This means that the kernel of the integrals are singular, and
special care is required for the evaluation of the integrals.
Let us now consider the order of singularity of the kernels of integrals geell
and hee ll •
For gee ll , since </>el(1'/1' 1'/2)= 1 and IGI* 0 at xel= X(1'/ ll, 1'/21), the order of
singulari ty is lIr.
For he/, the order of singulari ty is also lIr, since (r, n)/r3 has a singulari ty
of order lIr when d = 0 , as shown in the following theorem.
Theorem 3.1
(r, n) K n
2r for o < r ~ 1 (3.10)
where Kn().) is the normal curvature of the curved element along a direction
). = d1'/!d1'/l at a source point x~ on the element, and 1'/1' 1'/2 are the parameters
describing the curved element.
Proof:
Let the curved element be expressed by X=X(1'/l' 1'/2) and the source point be
with
r=x-x s
The unit normal is given by
n=G/IGI (3.11)
where ax ax ar ar
G= -x-=-x-a'll a "2 a'll a "2
(3.12)
32
Taking Taylor expansions around Xs one obtains,
- -r = r,ld'll + r,2d'l2
+ 0 ( d'l3 ) (3.13)
where r,i == art a "Ii etc. and the bar - indicates the quantity at xs=x(~l' ~2).
and
(3.14)
denote the first, second and third order terms of dr; I and dr;2' respectively.
Similarly one can express the derivatives of r as,
dr - - - 2 a;; = r'i + r 'il dT)1 + r'i2 d'l2 + O(d'l ) I
(i = 1,2) (3.15)
Hence, the cross product can be expressed by the following expression
(3.16)
and (r, G) can be calculated from equations (3.13) and (3.16) as follows:
0(1) term = 0 (3.17)
- - --o (dT)) term = (r'l dT)l+ r'2dT)2)-(r'lxr'2)= 0 (3.18)
since - - -r 'i - ( r 'I X r '2) = 0 , (i= 1,2) (3.19)
33
The 0 (d7J2) term is given by
(3.20)
where, from the O(d7J) term of equation (3.13) and the O(d7J) term of equation
(3.16),
(3.20)
since
a·(bXc) = c·(aXb) (3.21)
and, from the 0(d7J2) term of equation (3.13) and the 0(1) term of equation (3.16),
(3.22)
so that equation (3.20) becomes
(3.23)
Hence, from equations (3.17), (3.18) and (3.23),
(3.24)
34
From equation (3.16),
- - - - - - - - 2 G = G + ( r'l X r '12 - r'2 X r'l1 ) d7]l + ( r'l X r'22 - r'2 X r '12) d7]2 + 0 (d7] )
and
= 1 G 12 + 0 (d7])
= 1 G 12 { 1 + 0 (d7]) }
1
1 G 1 = 1 G I {I + 0 (d7]) r = IGI + 0 (d7])
Hence, from equations (3.11), (3.24) and (3.28),
= (r,G) IGI- 1
On the other hand, from equation (3.13),
r 2 =(r,r)
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
and
Hence,
,2 04( drh {l + 0 (d7])}
°3(d7]2) = {l + ° (d7])}
°4(d7]2)
35
where
is the normal curvature of the curved element at Xs. (See for instance 24.)
Here, Kn depends only on the direction specified by
i.e. - - 2-
(r '11 + 2 A r '12 + A r '22 ) . n Kn = - - - - - 2- -
r '1· r '1+ 2Ar '1· r '2+ A r '2· r '2
From equation (3.31) and (3.32),
(r, n )
,3
K 1 = n + 0 (1) 2 ,
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
36
2 r for 0< r ~ 1 (3.34)
Q.E.D.
Hence, for potential problems, singular integrals arising from the diagonal
term of the Hand G matrices are both of order
(3.35)
where (r, 8) are the polar coordinates on the plane tangent to the element at Xs,
and
O(!.)dS - O(!.) r dr dO r r
= 0(1) drdO (3.36)
This means that the integrals defining geeU and hee" in equations (3.5) and
(3.6) are only weakly singular. They can be calculated efficiently using polar
coordinates on the plane tangent at Xs, as will be shown by numerical
experiments in Chapter 7.
For the case when e' = e but k:;t: l, it is shown in the following that the
numerators of the integral kernels also become zero at the source point xe' , so
that the integrals gee kl and hee kl have a even weaker singularity compare to gee"
and hee". From equations (2.38) and (2.39) ,
/"= I II 1 ;'IGlu·(x,x')d7J1 d7J2 Ie -1 -1 e e (3.37)
(3.38)
where
• ,1 1 u (x,x )=- - 0(-)
e 411"r r (3.39)
37
.( k) (r,n) -o(!) q x,x =--e 411"r3 r (3.40)
and
Let us take, as an example, a 9-point (quadrilateral) Lagrangian element
where 1 1
(("'1."'2) = L +p("'l) L +q("'2) ((p,q) (3.41) p=-l q=-l
and
(3.42)
and let the source xek be x(O, 0). Then for k* I, <Pe' in equation (3.37) and (3.38)
will be of the order 0(711) or 0('7.2) or 0('71 '7:J in the neighborhood of
xek = x(O, 0) and will not include a constant term. Also, in generallGI - 0(1) in
the neighborhood of xek • Hence, the kernel of the integrals in geekl, heekl (k* l)
are either of the order O( '71' r), O( '72' r) or O( '71 '72' r).
From equation (3.30), for xek = x(O, 0) ,
(3.43)
where
au = ( r , l' r, 1) , a12 = (r , 1 . r, 2)' a22 = (r , 2' r, 2) (3.44)
Here,
(3.45)
If we let '71 =0 and '72- 0,
o = --':"'2 ---:-3 = 0 a22 7]2 + 0 (7] )
Ifwe let TJl =F 0,
and
since
(because r,2 =Fer, 1),
and a22 > 0,
Hence,
so that
Similarly,
Similarly,
f(A) > 0 for all A
1 -= ---- - 0(1) r2 f (A) + 0 (7/)
7Jl --0(1) r
7J2 - - 0(1) r
( 7J~7J2 r =
38
(3.46)
(3.47)
(3.48)
(3.49)
(3.50)
(3.51)
(3.52)
(3.53)
If "11 = 0, "12 - 0 ,
r
If "11 =t= 0,
( ~~~2 Y =
Hence
To sum up,
~1 --0(1), r
39
~2 --0(1), r
(3.54)
(3.55)
(3.56)
Hence, the kernel of the integrals in geekl, heekl (k=t= l) are of the order 0(1) or
O(r) , which means that they have a even weaker singularity compared to O( 1/ r)
for integrals in geell and heell, and will not cause any substantial difficulty when
integrating them numerically.
This will also imply that, for problems where u* - O( 1/ r2), q* - O( 1/ r2)
as in three dimensional elastostatic problems, the nondiagonal terms geekl, heekl
(k=t= l) can be calculated using polar coordinates around the source point xek , since
for these terms
(3.57)
(3.58)
so that the Jacobian r in dS = r dr d8 will cancell the singularity in <Pel / r2
and hence in geekl and heekl, (k=t= l). The diagonal terms heekk can be calculated
by using the row sum elimination (rigid body motion) technique. The calculation
40
of geekk may require the calculation of the finite part of a hyper singular integral
by, for instance, Kutt's method 10.
3.2 Hyper Singular Integrals
For problems including higher order singularity I.e. Is dS I r a (a~ 2 ), the
singular integrals do not exist in the normal sense.
This can be illustrated by taking S as a circular disc of radius a with the
source point Xs at its centre. (Figure 3.1) Consider now taking a smaller
concentric circular disc of radius € away from S and calculating the integral at
the limit as €-O. This gives
I dS = lim I 2n: dO I a ~ r dr S r a .-0 0 • r
= lim 2rr fa .-0 •
dr
{
log a-log £
= ~: 2rr _1_ (_1 ___ 1_) 2-a aa-2 £a-2
(a=2)
(a>2)
(3.59) .
This means that the Cauchy principal value for Is dS/ra does not exist for
a~ 2. Instead, the integral must be defined by its finite part 10.50, which
corresponds to 2rr log a, (a=2) and 27l' a2 -a I (2-a) , (a>2) respectively in
equation (3.59) .
Alternatively, the physical concept equivalent to rigid body motion in
elasticity may be used to calculate the diagonal coefficients of the influence
matrices.
41
s
Fig.3.1 Circular disc S
42
For potential problems, integrals containing higher order singularities arise
when calculating the derivative of the potential at a point on the element by
where
au f (au· aq· ) - = q--u- dS axs s axs axs
• au 1 r -(x,x) = -axs s 41[" r3
(2.46)
(2.47)
(2.48)
instead of interpolating on the boundary element. The integral of equation (2.46)
does have a Cauchy principal value and can be calculated directly using a method
similar to that of Gray 28 and Rudolphi et al. 51, or as a limit of a nearly singular
integral by the method which will be proposed in Chapter II.
Let the source point Xs be on the boundary element surface S. Since q and
u in equation (2.46) generally contain constant terms qo and Uo when expanded
by Taylor series around the source point xs, the order of singularity of the
integrals involved should be, roughly
I .!E f q a u· dS - f 0 ( ~) dS au S axs s r2 (3.60)
f au· I !E u-dS aq" s ax
• - Is {o(~)-o(~) }dS
- to(~)dS (3.61)
since
n 2 3 K ( ) (r,n) - 2"r +0 r (3.62)
from equations (3.31) and (3.32). Hence, the apparent singularities in the integral
in equation (2.46) is of order O(lIr2) and O(lIr3), suggesting that the integral does
not have a finite value, which is contrary to the fact that q == au/an and au/at
43
(alat: tangential derivative at Xs on the boundary S) usually have finite values
on the boundary.
However, it will be shown in the following that the integrals Iau* and Iaq*
do have Cauchy principal values.
Since only the neighborhood of Xs is relevant, so long as the singularity is
concerned, let us assume that the boundary S is smooth at Xs and take a local
tangential planar disc Sa of radius a centered at Xs as shown in Fig. 3.2, and
calculate the integrals Iau. and Iaq* for the planar disc Sa :
a I au· 1 I r I .... q-dS=- q-dS au s ax 411' s 3
a Bar (3.63)
a-I aq* 1 I {n 3r(r,n)}dS I = u-dS=- u ----"--art s ax 411' S r3 r 5
a' a
(3.64)
Cartesian coordinate (x, y, z) are introduced with the x, y, axis lying in the
tangential disc with (0,0,0) at the centre of the disc and the z-axis perpendicular
to the disc towards the inside of the region (opposite to the normal n).
For simplicity, let u and q in equation (2.46) be given by linear
interpolation:
(3.65)
By taking polar coordinates (p, 8) in the x, y plane centered at (0, 0),
u = U o + p (u cosO + u sinO) " Y
q = qo+p(q cosO+q sinO) " Y (3.66)
In order to calculate the hyper singular integrals of equations (3.63), (3.64),
let us assume that Xs = (0, 0, d) and take the limit as d-O, i.e.
44
S
z
n
Fig.3.2 Polar coordinates (p, 8) on a planar disc Sa
and
where
and
Iau. a = lim Iau. a (d) d-+O
45
Ia-oa(d) = ~f2"d8 fa {uo +p(u cos8+u Sin8)} { n 'I 47f 0 0 % Y r3
Noting that
( p COS8]
r = x-x, = P~~nQ ,
r = Jp2+d2
n =
and
(r,n)=d
we obtain
(3.67)
(3.68)
(3.69)
3r(r,n) }
r 5 p dp
(3.70)
(3.71)
46
(3.72)
so that
lim I = [: J d-O 'lo -21r
(3.73)
and
f2n- fa r I = dO "3 p2 cosO dO
q" 0 0 r
= I:~ dO
• [:~] l-J dp o I 2 2 3
Vp+d
47
= I: [ ,(]
dp
J p2+ d2 3
since
f 2" 2 f 2>r 1 + cos 20 cos dO = dO = 7r
o 0 2
f 21[ f21[ sin20 sinO cosO dO = -- dO = 0
o 0 2
Noting that,
r ~~p=3:::;:=a dp = o J /+ d2 3
lim I = d--+O q"
(3.75)
(3.76)
(3.77)
(3.78)
r~ C = dO 0
since
Hence,
48
["="~'] p3 sin 20
_p2d sinO dp
Jp2+d23
dp
1 - oos20
2 dO = 7C
From equations (3.69), (3.73), (3.78) and (3.81),
I aq* I Q == lim I, .Q(d)= lim u- dS
au· d-+O uU d-+O S ax Q 8
7C a q",
1 = 7Caqy =
47C
- 7Caqo
As for Iaq* a, since
a :4 q", a :4 qy
qo -2
3 r (r, n) 1 [:
r 5 =..;;;:;;i2 3 - 1
[PCOSO]
_ 3d 5 psinO
Jp2+d2 -d
(3.79)
(3.80)
(3.81)
(3.82)
(3.83)
49
in equation (3.70),
Io2x dO loa {n 3 r (r, n) } I = "3 - 5 pdp
Uo r r
(3.84)
where
f2" fa = -3d 0 dO 0 p2coso dp = 0 (3.85)
(3.86)
2~ [ 1 d2 I: =
";p2+d2 Jp2+d23
( ";a2~d2 d2
!.+!.) = 2~ Ja2+d23 d d
= 2~ ( 1 d2 )
Ja2+d2 Ja2+d23
(3.87)
Note that the singularities due to nJr 3 and -3r( r, n )/r5 cancel each other out.
Hence,
2~
a (3.88)
and o
o (3.89)
2~
a
50
Next,
I I:rrd0I:{~ 3r(r,nl} 2 - pcosOdp u
r 5 x
= [:I:~ J (3.90)
where
rrr I: p3cos20 I = -3d 0 dO dp u Jp2+d25 xx
r 3 -31Cd p 5
o Jp2+d2 dp
( 3d ~ + 2 )
1C Va2+d2 - la2+d2 3 (3.91)
since
I: 3
I:(~3 d2 ) P - p dp dp = Jp2+d25 Jp2+d25 p +d
[- Vp2~d2 +
d2 1 r = - .J;;2;;j3 0 3
p +d
1 d2 1 2
Va2+d2 +- - (3.92)
3 .,;;;;;;;z 3 3d a +d
Hence,
lim I 21C (3.93) u d-O xx
Noting that,
51
f 2>< fa p3sinO coso I = -3d dO dp = 0
Uxy 0 0 Jp 2+d25
we have
[ 2:1C] lim I =
d-+O Ux
Similarly,
where
f02>< dO foa {n 3 r (r, n)} 2· I = - - p s,nO dp uy r3 r 5
I U :IX
I U yy
I U yz
I = -3d 12>< dO "y% 0
f2K fa p3sinO Iu = -3d dO dp
yy 0 0 Jp 2+d25
= 1C ( 3d _ ~ + 2) Ja2+d2 1"223
Va"+d"
which gives
and
Hence,
3 P
-":""--5 dp
Jp2+ d2
(3.94)
(3.95)
(3.93)
(3.97)
(3.98)
(3.99)
(3.100)
(3.101)
and
52
From equations (3.70), (3.89), (3.96) and (3.102) ,
U
21t U " " 2
1 u I a - 21t U = ..1. art - 41t y 2
21t U 0
-u a 0 2a
To sum up, for the linear interpolation
I a E Is au*
q-dS au· ax s a
aq"
4 aq
= -y
4
qo
2
I a "" u-dS I aq*
art s ax
=
a S
u "
2 u .2-2 u o
2a
1 Is q r
= dS 41t r3
a
= 1 I u { ~ _ 3 r (r, n) } dS 41t S r3 r 5
a
(3.102)
(3.103)
(3.65)
(3.104)
(3.105)
53
Hence, the contribution of the integration in the local disc Sa including xs,
to the singular integral of equation (2.46), which give the potential derivative
au/axs at Xs is
au I aq::c u a a = ::c = Iau· - Ia«r ax s 4 2
• a aq u -2 ...1. (3.106)
4 2
qo u 0
2 2a
which is finite. Since the integral for the boundary surface S excluding Sa is
finite, the integral
au I (au. aq.) -= q--u- dB , ·a x. s ax. ax.
xES 8 (3.107)
gives a finite value (Cauchy principal value) when calculated in the above
manner, although the integral kernel has an apparent hyper singularity of order
o ( lIr2) - 0 ( llr3), as seen in equations (3.60) and (3.61).
This is reasonable since, physically, one would expect finite values for
au/axs from the interpolation of u, au/an, au/at in the boundary S , where d = 0 .
3.3 Nearly Singular Integrals
When the source point is very near the surface, the integrals geSI, hesl of
equations (2.50) and (2.51), and aesl, besl of equations (2.54) and (2.55) have finite
values. But it is difficult to calculate them accurately and efficiently using the
standard Gauss-Legendre product formula, since the value of the kernels vary
very rapidly near the source point Xs. In fact the nearly singular integrals
54
(O<d~l) turn out to be more difficult to calculate than the singular integrals
(d=O) .
The accurate calculation of these nearly singular integrals is of practical
importance in boundary element codes. They may arise when calculating the H
and G matrices in cases where the elements are very close to each other, when
using discontinuous elements, or when it is necessary to calculate the potential
and its gradients at a point very near the boundary. Good examples are the
analysis of electron guns which have complex geometry and thin structures, or
the analysis of electromagnetic fields in thin gaps arising in motors, to name a
few.
To understand the problem, let us examine the nature of the near
singularity (O<d~l) in comparison with singularity ( d=O ), for the integral
kernels u*, q*, au*laxs and aq*laxs, which occur in three dimensional potential
problems.
Let :Ks be the nearest point on the curved boundary element S to the source
point xs , and let the distance be
as shown in Fig. 3.3. :Ks will be called the source projection.
In general, r/r is a unit vector.
Hence,
( ~ n) = cosO r J rn
where 8rn is the angle between rand n as shown in Fig. 3.4 , so that
1 u* =
4nr
1 (r, n) q*= ----
4n r3
1 cosO rn
(3.108)
(3.109)
(3.110)
(3.111)
55
Xs
Xs
Fig.3.3 Source projection 'is and source distance d
56
Xs
n
Fig.3.4 Angle ern between rand n
57
aq* 1 r 1 1 (r) = = ax 411' r3 411' r2 r s
(3.112)
aq* 1 { ~ _ 3 (r, n) r } = ax 411' r 5
8 r
1 { ;a -3 cosO ; } rn = 411' r3
(3.113)
Now let us examine the behaviour of these kernels when the limit of x - xs
is taken.
For the singular case (d = 0) , since Xs = Xs , taking the limit of x - Xs
i.e. r- 0,
r - -+ t r
where t is the unit tangent vector in the direction of r (as r - 0) , as shown in
Fig. 3.5. Since,
cos 0 = (r, n) rn r
from equation (3.10),
• 1 Kn 1 q -+ - 411' 2 r
au* 1 t -+ ax 411' r2 s
K n
-+ - r 2
3 aq* 1 ( n .!.) -+ --K 411' r3 ax 2 n r2 s
For the nearly singular case (0 < d ~ 1 ), taking the limi t of x - Xs,
i.e. r- d,
r - -+ n r
cosO -+ 1 rn
as shown in Fig. 3.6 .
(3.114)
(3.115)
(3.116)
(3.117)
(3.118)
(3.119)
58
Xs x
n
Fig.3.5 ~ - t (singular case)
59
s
Fig.3.6 ~"".n (nearly singular case)
60
Hence,
• 1 1 q -+ --- (3.120) 411" r2
au· 1 n -+ (3.121) ax 411" r2
B
aq· 1 (n 3 ) 1 ( 2n) a;- -+ 411" ? - ? n = 411" -?" (3.122) B
To sum up, the nature of the singular and near singular kernels in th.e limit
of x - Xs is given in Table 3.1. Notice the difference of the nature of singular
and near singular kernels.
Table 3.1 Nature of (near) singularity near source projection Xs
singular near singular d=O O<d~l
411" u* 1/r 1/r
411" q* - Kn / (2r) -l/r
411" au*laxs t/r n/r
411" aq*laxs nlr - 3/2 Kn tlr2 -2 nlr
For the singular case (d=O), u* and q* are of order O(lIr) and the
integrals are weakly singular in the sense that the singularity is cancelled by the
Jacobian of the polar coordinate system centered at xs, i.e. dS - OCr) dr dO.
Integrals containing au*laxs, aq*laxs are hyper singular in the sense that
the order of singularity is O(lIr2) and O(lIr) respectively and the singularity
cannot be cancelled by the Jacobian OCr) of the polar coordinates. However, as
shown in equations (3.104) and (3.105), these hyper singular integrals render
finite values (Cauchy principal value), which can be calculated using polar
coordinates centered at Xs on the plane tangent to S at Xs.
61
For the nearly singular case (O<d~l), the order of near singularity in the
limit as x - Xs is lira, (a=1-3). However, as will be seen from numerical
results in Chapter 7, the nature of the integral kernel at the limit of x - Xs
alone, does not explain the difficulty in integrating these near singular kernels
numerically, using the log-L2 transformation which will be proposed in Chapter 5.
For instance, for a given source distance d, ( 0 < d ~ 1 ) , a given accuracy
requirement E: , the integration of J s au*/axs dS and J s aq*/axs dS related to the
potential derivative aulaxs, requires far more integration points compared to the
integration of J s u* dS and J s q* dS related to the potential u(xs), when the
log-L2 transformation is used. Looking at the nature of near singular kernels in
the limit of x - Xs (Table 3.1), this seems odd, since the order of near singularity
for q* and au*/axs are both O(lIr2), and one would expect that they can be
integrated numerically with more or less the same number of integration points to
obtain the same accuracy (relative error).
In order to explain the behaviour of the near singular kernels in numerical
integration, it is necessary to consider not only the local behaviour of the kernels
in the limit of X-Xs, but also the global behaviour of the kernels around X=Xs
and in the total integration region S.
To do so, consider the case when the boundary element S is planar. Let the
source point be Xs= (0, 0, d), where (x, y, z) are Cartesian coordinates. The
planar element S lies in the xy-plane and the source projection is xs=(O, 0, 0) ,
as shown in Fig. 3.7.
Taking polar coordinates (p, 8) in the xy-plane centred at xs, the near
singular kernels u*, q*, au*/axs, aq*/axs of equations (2.4), (2.7), (2.47) and (2.48)
for three dimensional potential problems can be expressed as follows:
Noting that,
r = [ p ~SO] pstnO
-d
(3.123)
62
z y
Fig.3.7 The planar element S
n =
r =
(r, n l = d
one obtains,
and
u* =
( r, n l q* = --- =
41l"r3
au*
ax s
iJq*
ax s
1 r = =
=
1 d 41l" r3
p cosO
r3
p sinO
r3
d
r3
3r(r,nl}
r 5
63
pcosO o - 3d r 5
o
1
(3.124)
(3.125)
(3.126)
(3.127)
(3.128)
(3.129)
(3.130)
Since the (near) singularity is determined by the radial ( p) component, we
may neglect the angular ( 8 ) component when discussing the nature of near
singularity. Also, since for a planar element ( r, n ) = d is constant, the nature of
64
near singularity of the kernels u*, q* au*laxs, aq*laxs can be summarized as in
Table 3.2.
Table 3.2 Nature of near singularity (O<d ~1) for 3-D potential problems (planar element)
Order of near singularity
u* lIr
q* lira
au*laxs lira, plr
aq*laxs lIr, lIr5 , plr5
Although the above analysis of near singularity was done for planar
elements, it can be considered that the essential nature of near singularity for
general curved elements is the same, since near singularity is dominant in the
neighborhood of xs, where
Cr, n) - d (3.131)
Hence, the relative difficulty of numerically integrating Is au*laxs dS and
Is aq*laxs dS, compared to Isu* dS and Isq* dS, using the log-L2 transformation,
can be explained by the presence of kernels plr3 and plr5 in the first two
integrals.
(cf. Chapter 5-7,10)
For a constant planar element S,
I=fsFdS
f 2>r fPmax (8) = d8 Fpdp
o 0
(3.132)
65
where (p, 8) are the polar coordinates centered a the source projection Xs in S and
pmax(8) is defined as in Fig. 3.8 .
Hence, the near singular integrals in three dimensional potential problems
involving the kernels u*, q*, au*laxs and aq*laxs in Table 3.2 can be expressed as
12" 1PmlJ%(8)/ 1= dO -dp
o 0 ra (3.133)
where a =1,3,5 and 0=1,2 and
r = Jl+d2 (3.134)
since S is a planar element.
Since the near singularity is essentially due to the radial component,
consider the radial component of the integral in equation (3.133) :
rmlJ% J= 0 Fpdp = rmlJ% o fCp) dp (3.135)
where
fCp)= fl (p) i!! e. = p
r v';Jf7
f3(p) p p ... - = r3
J p2+d23
2 2
f3•2 (P) p p
E - = r3 Jp2+~3
f5 (p) P P
iE! - = r5 Jp2+d25
2 2
f5,2(P) p p .. - = r5
J p2+d25 (3.136)
so that, essentially, the nature of near singularity of the radial component of
integrals containing the kernels u*, q*, au*la Xs, aq*la Xs is given by Table 3.3.
66
Fig.3.8 Definition of Pmax (9) for a planar element S
67
Table 3.3 Nature of near singular kernels ofthe radial
component integrals in 3-D potential problems
Order of near singularity
u* plr
q* plr3
a u* I a Xs plr3 , p2/r3
a q* I a Xs plr3, plr5, p2/r5
The graphs of the near singular kernels f1, f3, f3,2, f5 and f5,2 of the radial
component integrals are given in Figures 3.9 to 3.13. The characteristic feature
of the kernels f3,2 and f5,2, which appear in the calculation of the flux aulaxs, is
that
d f3• 2
dp
d f5 2 = -- = 0
dp (3.137)
For planar elements, the radial component integral of equation (6.135) for
the kernel functions f1, f3, f3,2, f5 and f5,2 of equation (6.136) can be expressed
in closed form as follows:
J 1 rmax p = - dp o r
= [ J p2 + d2 (max
= J 2 +d2 Pmax -d (3.138)
Js =
=
=
JS:J. =
=
=
J = 5
68
rn= P -dp o rS
[ 1 rn= - -v;r;;l 0
1 1 - -d J 2 +rr PmDZ
2 rmDZ p -dp o r S
[ wg (p+v'p2+ rr) - vi-:;-;, rmDZ
p + d 0
wg (p +v'p2 + d2) Pmax
.../,2 +(l2 n= n= PmDZ
~ [ ,~3C-3d p +
=
S PmDZ
- wgd (3.140)
(3.141)
(3.142)
These closed form integrals are useful for performing the integration in the
radial variable analytically for planar elements, as will be mentioned in
Chapter 5. They are also useful in checking numerical integration methods for
the radial variable, and will be used in Chapter 10.
69
---------------------------------1 -------
-1
--~--~~--------P Pmax o
F' 3 9 Graph of Ig, .
f3 (P)
f3 ' (0) = ~
d Pmax 0 ..J2
Fig.3.10 Graph of
1 - p2
P
f3 = P
3 ~P2+d2
70
2...J3 1 --9 d
o -{2d
Fig.3.11 Graph of
16Vs 1 125 d4 - -
o .!L 2
Pmax
f 3,2
Pmax
Fig.3.12 Graph of
1 - p4
71
f 5,2
f5,2' (0) = 0
o Pmax
Fig.3.13 Graph of
CHAPTER 4
SURVEY OF QUADRATURE METHODS FOR
3-D BOUNDARY ELEMENT INTEGRALS
In this chapter we shall review some of the quadrature methods related to
the three dimensional boundary element method in the literature according to
Table 1.1.
As shown in Chapter 2, in three dimensional boundary element analysis,
integrals of the form
g~'el = f _: f _: ~~ jGj u* (x,x~,) d7J 1 d7J2
h~'el = f _: f _: ~~ jGj q* (x.x~,) d7J1d7J2
(4.1)
(4.2)
are necessary for the calculation of Hand G matrices and potential u at an
internal point. For the potential gradient aulaxs at an internal point xs, integrals
of the form
(4.3)
(4.4)
become necessary.
Here tel(r;l' 712) is the interpolation function corresponding to node xel , and
is a polynomial of (711' 712). The field point x on the boundary element e is given
by
(4.5)
where xe l , (l = 1-ne) are the coordinates of the nodes belonging to the element Se .
The Jacobian of the transformation x(r;l' 712) is given by
73
For potential problems,
and
where
1 u* (x, x ) = -
8 411' r
1 (r,n) q* (x x ) = --
, 8 411' i
au* 1 r - (x x) = axs ' 8 411' r3
aq* 1 {n 3r (r , n) } - (x x) = - -ax, ' s 411' r3 - r5
n e
r= (r1,r2 ,ra)T= X-X8 = L;! ('11''12)x~ - X8
1=1
4.1 Closed Form Integrals
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
Closed form or analytical integration formulas for equations (4.1)-(4.4) are
available for constant planar triangular elements 1, planar parallelograms 6 and
for planar triangular elements with higher order interpolation functions 2,29, in
the case of potential problems.
However, for general curved elements with constant or higher order
interpolation functions, it seems impossible to derive closed form integrals for
equation (4.1)-(4.4), let alone for potential problems. The main reason for this is
that the integrands include lira, (a=1-5) and IGI, where r and IGI are given by
the square root of a general polynomial of ('71' '72)' the order of which is higher
74
than 4 for "11 and "12 respectively, for general curved elements which are expressed
by quadratic or higher order polynomials <Pel ("11' "12) in equation (4.5).
4.2 Gaussian Quadrature Formula
Since closed form integrals are not available, numerical quadrature schemes
must be employed for general curved elements.
Let us consider a one dimensional integral
I = I: {(x) dx
Numerical integration formulas for (4.14) is given in the form
n
1- IA '" L wi ((Xi) i'" 1
(4.14)
(4.15)
where Xi and Wi are the position and weight of the i-th integration point,
respectively. The Gaussian quadrature formula 3 is known to give optimum
results, so long as f(x) does not include any singularity or near singularity. The
formula is derived in the following manner. Let a= -1, b=1 and
f In
E (f)= {(x) dx - L w. {(x.) n -1 i'" l' ,
(4.16)
The weights {wJ and positions {xJ are determined to make the error EnW = 0 for
as high a degree polynomial f(x) as possible. Since,
(4.17)
EnW = 0 for every polynomial of degree ~ m if and only if En(xi ) = 0 ; i = 0, 1, ... ,
m. Since the formula has 2n free parameters {Xi} and {Wi}, the equation
E (xi)=o )·=01'" 2n-l n ' t , (4.18)
is solved to obtain {Xi} and {Wi}' i = 1-n.
75
It turns out that equation (4.18) is satisfied if xl' ... , xn are taken as the zero
points
(4.19)
of the Legendre polynomial:
(4.20)
and
(4.21)
The table for the Gauss-Legendre quadrature formula is given in 3, and an
efficient algorithm for generating the table has been proposed by Golub and
Welsch 38.
The error for the Gauss-Legendre quadrature formula when applied to a
function f(x) defined in the interval x E [-1, 1] is given by
22n+1 ( 1)4 En (f) = n. l2n) ( 11 )
(2n+ 1) [(2n)!]3 (4.22)
for some -1 < 1J < 1 . 3,33
For integration over the interval- [a, b] , equation (4.22) can be applied to
fb b-a f 1 {a+b+x(b-a)} f( t) dt = - f dx
a 2 -1 2
as will be shown in Chapter 6.
For two dimensional integrals, the product formula
can be employed.
n n
L Wi L Wj f(xi,x j ) i=l j=l
(4.23)
(4.24)
The Gauss-Legendre quadrature formula is optimal in the sense that it gives
exact results for polynomials of up to order 2n -1 with n integration points.
However, the formula does not give exact results when the integrand f(x) is
singular or nearly singular.
76
Hence, the Gauss-Legendre quadrature formula itself may be used to
calculate the 3D boundary element integrals of equation (4.1)-(4.4) so long as the
distance d between the source point Xs and the boundary element S is sufficiently
large compared to the element size.
4.3 Quadrature Methods for Singular Integrals
Singular integrals arise when the source point Xs lies in the boundary
element S over which the integration is performed. The straight forward
application of the Gauss-Legendre quadrature formula fails, since the integrand
goes to infinity when x coincides with xs, i.e. when r= Ix-xsl = 0
Various methods have been proposed to overcome this difficulty. Some of
them will be explained briefly in the following.
(1) The weighted Gauss method 4,5,11,19,20
In the one dimensional Gaussian formula, one can obtain { Xi } and { Wi } so
that they would give optimum results for a particular weight function w(x), i.e.
f 1 n
1= w(x){(x)dx = L w/(xi ) -1 i=1
(4.25)
For instance, Kutt 10 has obtained quadrature points {Xi} and weights {wil for the
calculation of the finite part of the singular integral
f 1 {(x) n -- dx = " w. ((x.)
A .L." -1 (x+1) 1=1
(4.26)
where w(x) = 1/ (x + l)A is singular at x = -1.
This can be applied to three dimensional boundary element integral, for
instan.ce to equation (4.1) in the form
(4.27)
77
when the source point is at Xs = x( -1, -I} , since the term in { } is well
behaved, because the singularity due to lIr is cancelled by (1/1 + 1}(1/2 + I). This method was improved 4,5,19 by using a two dimensional weight function
(4.28)
where x(~l' ~2} is the source point. Equation (4.28) approximates the distance r
so that the singularity cancellation is improved.
A further modification was introduced 20 where the first term of the Taylor
approximation for the distance r is employed. This method is reported to give
good results for planar parallelograms but relatively poor results on a spherical
patch.
(2) Singularity subtraction and Taylor expansion method 6,21
A classical way of dealing with kernel singularities is to subtract them out
so that F(x, xs}, an integrand containing a singularity, would be dealt with
using
f F(x,x)dS = f {F(X,X)-F·(X,X)} dS+ f F·(x,x)dS s 8 s 8 S S 8 (4.29)
where F*(x, xs} is a function which has the same singularity as F(x, xs} but is of
simpler form which can be integrated exactly. Then, F(x, xs} - F*(x, xs} is non
singular and can be integrated accurately using, for example, the Gauss-Legendre
quadrature formula.
This method was introduced in three dimensional boundary elements in 21 ,
where the exact integration of the subtracted singularity F*(x, xs} for planar
elements was calculated.
78
Aliabadi, Hall and Phemister 6 introduced the idea of using Taylor series
expansion of the complete singular integrand to provide subtracted terms which
can be exactly integrated, though these integrations can be very laborious. They
also recognized that subtraction of only the first term containing the actual
singularity was not sufficient to produce a well behaved remainder integral and
that there was an advantage in subtracting further terms of the series. The
method is reported 6 to give an error of 6X 10-7 with only 64 integration points for
planar parallelograms. However, for a spherical patch, 24X24=576 integration
points are required to achieve an error of 10-6•
(3) Variable transformation methods
Variable transformation is a well-known technique for the evaluation of
improper integrals 36,13.
For a one dimensional integral
I = I 1 f (x) dx -1
the variable x is transformed by
so that
where
and
x = ,s (u)
1= 1: f(,s (u» ,s'(u) du
,s(a)=-l,
d,s ,s'(u)=
du
,s(b)=l
The transformation x= ¢(u) is chosen so that
g(u) = f(,s (u»,s'(u)
(4.30)
(4.31)
(4.32)
(4.33)
(4.33)
(4.34)
79
is no longer an improper integrand. Hence, one can apply a standard quadrature
rule to
1= I:g(U)dU (4.35)
to calculate the original integral.
One such transformation is the error function transformation 13,30
2 fU 2 x = er{(u) = - e-Y dy
VTro (4.36)
which transforms the integral of equate (4.30) to
2 fco 2 1= ...;;- -co f( erf(u» e -u du (4.37)
The integrand is now dominated by e - u2 and may be approximated accurately by
a truncated trapezium rule 31. This method has been applied in the boundary
element analysis of a three dimensional acoustic problem 22 for weakly singular
integrals by using the transformation of equation (4.36) in each direction of the
two dimensional rectangular region.
The double exponential transformation 32 was applied 27 in the form
x(u) = (± forx<O)
'fj = ~ sinh { ~ ( _1 __ 1 )} 2 2 1-u 1+u (4.38)
in each direction of the two dimensional element for weakly singular integrals in
a three dimensional electrostatic problem.
Both methods are reported to give accurate results. However these methods
use extra CPU-time in the calculation of the exponential and error functions,
unless they are prepared before hand as a table of integration points and weights.
Hence, as far as weakly singular integrals are concerned, the simple polar
coordinate transformation, which will be explained in the next section, seems
more efficient in cancelling the weak singularity of order lIr. For hyper singular
80
integrals of order lira (a~ 2), the variable transformations mentioned here do not
work. For such cases the finite part of the integral may be calculated by the
method of equation (4.26) proposed by Kutt 10.
(4) Coordinate transformation methods
(i) Triangle to quadrilateral transformation
Lachat and Waston 7 introduced the transformation of a triangular element
to a square in the parameter space (1'/1' 1'/2)' In this transformation the corner at
which the singularity is placed becomes the fourth side of the square on which the
Jacobian of the transformation is zero. For example,
where
(1 +)' 1) )'2 - (1-)'1)
2
(1 +)' 1)
J ('71' '72) = 2
(4.39)
(4.40)
The above Jacobian J(1'/1' 1'/2) regularizes the integration so that the Gauss
Legendre quadrature formula can be applied.
(ii) Polar coordinates
Rizzo and Shippy 9 introduced the method of using the polar coordinate
system (p, 8) centered at the source point (1'/1' 1'/2) in the (1'/1' 1'/2) parameter space, so
that
(4.41)
81
The Jacobian of the transformation: p cancels the singularity of order lIr. The
method is also mentioned in 15, 25 •
As shown in Chapter 2, the order of singularity of both u* and q* are of
order lIr for weakly singular integrals arising in three dimensional potential
problems. This explains the fact that the use of polar coordinates regularizes
these singularities, so that the Gauss-Legendre quadrature formula can be safely
applied to the variables p and 8.
In this thesis, this idea is extended to taking polar coordinates around the
source point in the plane tangent to the curved element at the source. This
enables one to treat near singular integrals and singular integrals in the same
frame work by introducing a variable transformation R(p) of the radial variable
p, which regularizes the (near) singularity. Further, an angular variable
transformation t(8) is introduced, which considerably reduces the number of
integration points in the angular dirl~ction.
4.4 Quadrature Methods for Nearly Singular Integrals
N early singular integrals turn out to be more difficult and expensive to
calculate compared to the (weakly) singular integrals mentioned in the
proceeding section.
N early singular integrals become important when treating thin structures,
where the distance between elements are very small compared to the element size
as shown in Fig. 4.1. The use of discontinuous elements is another source of
nearly singular integrals, since the~ distance between an element node and an
adjacent element can be very small compared to the element size, as shown in Fig.
4.2. Another important source of nearly singular integrals is the calculation of
the potential or potential derivatives at an internal point very near the boundary.
This arises for instance in the simulation of electron guns in cathode ray tubes,
82
where the accurate value of the electric flux near the cathode is required in order
to calculate the trajectory of electrons.
The stress of this thesis is on the calculation of these near singular integrals,
although singular integrals can be treated efficiently in the same frame work.
Previous methods will be briefly reviewed in the following.
(1) Element subdivision
The orthodox way to treat nearly singular integrals is to increase the
number of integration points as the source to element distance d diminishes
(compared to the element size) 7,12. However, the number of necessary integration
points with the standard Gauss-Legendre quadrature increases rapidly as d
decreases, as will be shown in Chapter 10.
The next thing to do is to subdivide the element so that the integration
points are concentrated near the source point 7,12.
Element subdivision tends to be a cumbersome procedure and would still be
inefficient when d is very small compared to the element size. Another
disadvantage of element subdivision is that it suffers from the fact that the
highest polynomial degree which can be integrated exactly by the Gauss
Legendre quadrature depends on the local number of integration points selected
for each sub-element. For instance, a one-dimensional quadrature which is
subdivided into three sub elements and uses 2, 3 and 4 Gauss points,
respectively, can integrate exactly polynomial integrands of degrees 3,5 and 7
over each part (2n-1), whereas if 9 points are used to integrate over the
complete element, a 17th-degree polynomial is allowable 15.
83
Ie I I •
Fig.4.1 Boundary elements for thin structures
•
•
Fig.4.2 The use of discontinuous elements
84
. (2) Variable transformation methods
A recent trend is to transform the integration variables so as to weaken the
near singularity by the Jacobian of the transformation. The transformation also
has the effect of concentrating the integration points near the source point.
(i) Double exponential transformation
The double exponential transformation was originally proposed by
Takahashi and Mori 13.
The method is efficient for the integration
I = f 1 r(x) dx -1
where f(x) has integrable singularities at x = ± 1 .
The method introduces the variable transformation
x = tanh { ~ sinh(t) }
so that
I = ~ [~ r[tanh { ~ Sinh(t)} I cosht dt
cosh2 { ~ sinh (t)}
(4.42)
(4.43)
(4.44)
The truncated trapezium rule is applied to equation (4.44) to evaluate the integral
numerically.
This method has been applied to nearly singular integrals in boundary
elements in 14,26, 16. However, it will be shown in Chapter 10 that it requires
many integration points when the source distance d is very small, and the
method consumes a lot of CPU-time for the evaluation of exponential functions in
equations (4.43) and (4.44), unless they are prepared before hand in a table.
(ii) Cubic transformation method
A more efficient self-adaptive method using cubic transformation was
proposed by Telles 15. The method is briefly explained in the following.
For a one dimensional integral
1=11 f(")d,, -1
85
(4.45)
let the projection of the source point Xs on to the element r be x(1j), and the
distance d as shown in Fig. 4.3 , where the element is described by
x(7j), (-1~ 7j~ 1).
A cubic transformation
is introduced, such that
where
and
I = f 1 f (" (r) ) J (r) dr -1
d" J(r) = -dr
(4.46)
(4.47)
(4.48)
and J(y) takes a minimum value t(d) at y=y, in order to weaken the near
singular behaviour of f( 7j(y» near 17 = 1j •
f(d) is an optimized function of the distance d given by
r (d) = 0.85 + 0.24 log d (0.05 ;;;; d < 1.3 )
0.893 + 0.832 log d (1.3;;;;d<3.618)
1 (3.618;;;; d) (4.49)
86
r Xs
x (-1) x (11)
x (1)
Fig.4.3 One dimensional integral over r.
87
In order to compare with methods proposed in this thesis for cases where d < 0.05 ,
the function fed) is interpolated between d = 0 and 0.05 to give
r (d) = 2.62 d (0 ;:;; d < 0.05 ) (4.50)
The standard Gauss-Legendre quadrature formula is then applied to equation
(4.47).
For two dimensional integration, equations (4.45) and (4.47) are generalized
to give.
(4.51)
where cubic transformations 7J/Y1)' 7J2(Y2) are applied to each direction 711' 712
respectively. In this case the distance parameters dl, d2 for each direction is
determined by
d = 2d 1 I x(1, TJ 2) - x(-l, TJ z>1 (4.47)
d = 2d 2 I x(TJl'l) - x(TJ 1 ,-1)1
(4.52)
where xs= x(~1' ~2) is the source projection or the point on the curved element S
nearest to the source point Xs. (~1' ~2) can be calculated by the Newton-Raphson
method, as will be explain in Chapter 5.
Telles' method gives good results for d> 10- 2, where d is the relative
distance compared to the element size. However, when 0<d<10-2, the method
does not give accurate enough results even when 32x32 integration points are
used, as will be shown in Chapter 10.
88
(3) Polar coordinates
The use of polar coordinates in the (7J1' 7J2) parameter space, as in the
calculation of weakly singular integrals, alone does not give accurate results for
nearly singular integrals, as will be shown in Chapter 10. A method to improve
the result by correction procedures 16 is reported to be efficient for potential
problems.
However, in this thesis we take a different view and introduce a new method
by taking polar coordinates in the plane tangent to the element at the source
projection. Further, radial and angular variable transformations are introduced
in order to weaken the near singularity before applying the Gauss-Legendre
quadrature formula.
CHAPTER 5
THE PROJECTION AND ANGULAR & RADIAL
TRANSFORMATION (PART) METHOD
5.1 Introduction
Hayami and Brebbia 17 have proposed a new coordinate transformation
method to calculate singular and nearly singular integrals for curved boundary
elements 5.
Here, the method will be generalized to deal with arbitrary curved element
geometries and different types of integral kernels which arise in three
dimensional potential problems.
From Chapter 3 (cf. Tables 3.1 and 3.2), the (nearly) singular integrals
involved in the boundary element analysis of three dimensional potential
problems may be generally expressed as
(5.1)
where
and
a=l for (weakly) singular integrals necessary for the calculation ofH
and G matrices
a = I, 3, 5 for nearly singular integrals necessary for the calculation of
potential and potential gradients at internal (external) points
very near the boundary S.
Here r = Ix - xsl is the distance between the source point Xs and the field point x.
f is a function of xES, which does not have any (near) singularity in r.
Although the method is proposed for three dimensional potential problems,
integrals arising in other areas such as acoustic problems (Helmholtz equation),
90
elastostatics etc., may be regarded to take the form of equation (5.1), so that the
following analysis may also apply to such problems.
As the evaluation of (nearly) singular integrals becomes a difficult problem
for the case of curved elements these are the elements to be treated here in detail.
For planar elements, the problem becomes simpler and closed form integrals such
as those given in Aliabadi, Hall and Phemister 6 or Kuwabara and Takeda 2,29 are
available.
Although the quadrature method to be proposed can be applied to general
curved elements (triangular as well as quadrilateral), we shall use the 9-point
Lagrangian element as an illustration (cf. Fig. 5.1) .
The interpolation in this element can be expressed as
where
1
((1J 1, 1J2)= L j=-1
1
L V>j ("II) V>k(1J.) ((i,k) k =-1
(5.2)
(5.3)
The element is isoparametric in the sense that f can also represent the
coordinates X(7]l' 7]2) • The element S is defined by
where 1 1
and
x ("11' "1 2) = L L V>j(1J 1)V>k(1J.)x j ,k j=-lk=-1
xj,lt. = x(i,k), i,k=-l,O,l (5.4)
for a 9-point Lagrangian element.
91
--1
Fig. 5.1 The 9-point Lagrangian element
92
When calculating boundary element integrals, one needs to know a measure
of the distance d between the source point Xs and the element S over which the
integral is performed. A rough estimate of d can be obtained by the distance d'
between Xs and the centre of element x(O, 0), i.e.
d'=lx(o,O)-x I B
(5.5)
Let us define the size 1 of the element by
l == max ( I x (I, 0) - x (-1,0) I. I x (0,1) - x (0, -1)1 ) (5.6)
If d' ~ 1, the integration can be performed by the standard product Gauss
Legendre quadrature formula 1,3.
It is when d' < 1 that we need to devise a new scheme, since the standard
Gauss-Legendre formula alone does not give accurate estimates of the integral
efficiently. ( From here on, we shall assume that the element has been normalized,
so that 1= 1 and d=d'.)
For such cases, this thesis proposes a new scheme : the Projection and
Angular & Radial transformation (PART) method 18 •
The proposed method consists of the following steps:
(1) Find the point is on the element S nearest to the source point Xs.
(2) Approximately project the curved element S on to a polygon S in the plane
tangent to the element S at is.
(3) Introduce polar coordinates (p, 0) in the projected element S. (4) Transform the radial variable by R(p) in order to weaken the (near)
singularity flra in the integral of equation (5.1) .
(5) Transform the angular variable by t(O) in order to weaken the near
singularity in 0 that arises when is is close to the edges of S.
(6) Apply the Gauss-Legendre quadrature formula to the transformed variables
Rand t in order to calculate the integral of equation (5.1) numerically.
93
5.2 Source Projection
The first step is to find the closest point Xs = x(~1' ~2) on the curved element
S to the source point Xs. This point Xs will be called the 'source projection'.
(~1' ~2) can be obtained by solving the set of nonlinear equations for (7]1' 7]2):
( x-x ~)=o s ' aTJ. ,
(i=1,2) (5.7)
since x - Xs 1. ax/a7] i (i = 1, 2) is satisfied at Xs = x( ~ l' ~ 2) , as shown in Fig. 5.2.
Let r =X-Xs. Then, ar/a7]i = ax/a7]i ' (i=1, 2) so that equations (5.7) is
equivalent to
Here,
where
since
r = x - X B
(i=1,2)
= L¢/TJl' TJ2) xi -xs i
L¢j(TJ 1 , TJ2)=1 J
(5.8)
(5.9)
for complete interpolation functions. For instance for the 9-point Lagrangian
element of equations (5.2)-(5.4) ,
94
Fig. 5.2 The source projection Xs
since
95
I I I
L9I j = L 9I k (TjI) L 91 1 (Tj2) = L 9I k (TjI) == 1 j k=-I 1=-1 k=-I
I
L 9I k (Tj) = Tj(Tj-1)/2 + I_Tj2+ Tj(Tj+1)12 == 1 k=-I
From equation (5.9) ,
(i = 1,2)
Hence, equation (5.9) is equivalent to
where
Let
Since
{fl (TJ I .Tj2):O
f2(TjI·TJ2 )-O
f (II + t.ll ) = f (11) +
afl
aTJ I
af2
aTjI
afl
aTJ2 t.ll+ o (t.TJ 2 )
af2
aTj2
if we have an initial solution 110 which is in the neighborhood of ii : f(ii) = 0,
i.e. I f(ll) I ~ 1, the iteration
where
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
(5.16)
96
afl afl -1
a'll a'l2 ~I}i = f (I}i )
af2 af2
(5.17)
a'll a'l2
should converge to the solution, i.e.
This is what is known as the Newton-Raphson method, and I}i will
converge to the true solution ii. very quickly, provided that the initial solution iio
is in the neighborhood of ii.. In fact the method has the property of second order
convergence, i.e.
(5.18)
where 1 . 1 is a suitable norm for lll}.
For our problem of solving equation (5.7) to find the source projection :Ks , the
Newton-Raphson method converges within few (3-4) iterations to give a relative
error of 10-6, and consumes only about 1% of the total integration time. The
convergence is very good, so long as the solution :Ks lies inside the element S.
Then, the initial solution can be set to I}o T = (77 l' 772) = (0,0) .
However, when the source projection lies far out side the element, i.e.
I~ll~ 1 and/ or 1~21~ 1, the method may not converge to the true solution (~l' ~2).
This is because the interpolation function ¢j (e.g. 9-point Lagrangian element)
diverges as 177il ~ 1 and / or 17721 ~ 1 , i.e. the interpolation function ¢j gives a good
approximation of the function only when 177il~ 1, (i = 1,2).
Hence, it may be safer to start with the initial solution
corresponding to the closest node Xi,k, ( j, k = -1, 0, 1) of the element, to the
source point xs, particularly when:Ks lies near or outside the edge of the element.
97
In the actual application of the proposed integration method (PART), it
turns out that it is only when the source distance d=lxs-xsl is relatively small
compared to the element (d ~ 1), that the integral becomes (nearly) singular. This
means that the method should be applied when the source point Xs is very close to
the element. In this case, the source projection Xs either lies inside the element S,
or very close to the edges of S even when it lies outside the element. Thisjustifies
the use of the Newton-Raphson method to obtain the source projection xs, since
the convergence of the method is guaranteed in such cases due to the nature of the
interpolation functions <po. J
5.3 Approximate Projection ofthe Curved Element
Next, the curved element S is approximately projected on to the plane
tangent to the curved element S at the source projection xs.
First the unit normal vector ns to the element S at xs=x (~l' ~2) is
calculated by
(5.19) where
(5.20)
Denote the four corner nodes of the element by
Xl = x(l,l)
X2 = x(-l,l)
Xa = x(-l, -1)
X4 = x(l,-l) (5.21)
98
Then the perpendicular projection Xj of node Xj onto the plane tangent to the
curved element S at Xs is given by
~ . = x. - { (x.-;- ). n } n J J J B B B
(j= 1-4) (5.22)
The curved element S is then mapped onto the planar quadrilateral S defined by xl' x2' xa and x,, as shown in Fig. 5.3. This is an approximate
projection, since the exact projection of a curved edge of S would, in general, be a
curve in the tangent plane.
The consideration of exact projection will be given in a later section.
However, the approximate projection is shown to be sufficient and more efficient
for introducing the source distance d into the radial variable transformation
R(p), considering that it is the local behaviour of lira, r=lx-xsl, xES, that
dominates the (near) singularity ofthe integrand.
Next, one divides the projected quadrilateral S into four triangular regions
lj , (j = 1-4) centered at Xs as shown in Fig. 5.4.
In each triangular region ti:j defined by xs, Xj and Xj + 1, one defines the following,
geometric quantities:
where X5 == Xl' and
edge-j : edge corresponding to Xj + l -Xj
fj : foot of perpendicular from Xs to edge-j
hj : length of the perpendicular ( hj = Ifj - xsl ),
a j : angle between vectors fj - Xs and Xj - xs,
(5.23)
99
Xs • , ns'~
Xs
-s
Fig. 5.3 ~ppro)(imate projection 01 the curved element
I I I I I I 100 I I I I I I I edge-1 I I I I I I I
edge-4 I I I
edge-2
edge-3
Fig. 5.4 Division of the projected quadrilateral -S into four triangu\ar regions
101
and
(5.24)
as shown in Fig. 5.5 .
Then,
sin /::"8. = sgn (j) -J 1 - cos2 /::"8 . J J
h.=a.cosa. J J J (5.26)
where, as in Fig. 5.6 ,
sgn (j) = 1 when :Ks is on the semi-plane described by the line defined
by the edge - j , which includes the quadrilateral S.
o when :Ks is on the line defined by edge - j .
-1 when :Ks is on the semi-plane described by the line defined
by the edge - j, which does not include the quadrilateral S.
(5.27)
sgn(j) can be easily computed by comparing the values 7Jl and 7J2 with -1
and 1 , since the square: -1 ~ 7J 1 ~ 1 , -1 ~ 7J 2 ~ 1 , which represents the curved
element S is mapped continuously on to the planar quadrilateral S. A certain
accuracy threshold (e.g. ~ = 10-6 ) is necessary in the actual computation to judge
whether :Ks is on the edge or not.
The procedure of approximate projection explained so far, can also be applied
to curved triangular elements. In this case S is divided in to three triangular
regions ~1. tl2 and ~3.
102
Fig. 5.5 TriangUlar region -
Llj
103
sgn( j ) = 0
sgn( j ) = -1
Fig. 5.6 Definition of sgn( j )
104
5.4 Polar Coordinates in the Projected Element
As mentioned before, the singular and nearly singular integrals occurring in
three dimensional boundary element method can be generally expressed as
I=J LdS S rtt
(a> 0) (5.1)
where f is a well behaved function ofx on S. Since the singularity is related to
the integration in the radial direction, it seems natural to separate the integral of
equation (5.1) into its radial and angular components. In this way, one can tackle
the problem of (near) singularity by considering only the radial component of the
integral.
Hence, first consider integrating equation (5.1) using polar coordinates in
the projected quadrilateral S, centered at the source projection xs, where Xs is
situated in the projected quadrilateral S as defined in the previous section.
Polar coordinates in the (1'/1' 712) parameter space have been previously used
to deal with singular integrals 9.15, but here they are generalized by using polar
coordinates in the projected quadrilateral and by introducing further variable
transformations in the angular and radial directions.
Using the (1'/1' 712) parameters defined in equations (5.2)-(5.4) and Fig. 5.1 ,
equation (5.1) becomes
(5.28)
where
(5.29)
Next, divide the square region of the integration in equation (5.28) into four
triangular regions Llj (j=1-4) centered at (~1' ~2) and situated in the (1'/1' 712)
space as shown in Fig. 5.7. Notice that (~1' ~2) corresponds to the source
projection.
105
Fig. 5.7 Four triangular regions in (111,112) space
106
Thus,
4
1= L I. J j=l
where
I. = I I f - I G I dT/ l dT/2 J r" (5.30)
6. J
Let
(5.31)
In order to map the projected quadrilateral § onto the curved element S,
linearly map each triangular region t.j in S (Fig. 5.4) onto each corresponding
triangular region ~j in the (7J1' 7J2) space (Fig. 5.7). This is done by defining
local Cartesian coordinates (~1' ~2) for each triangular region 6j in "S" (Fig. 5.5) as
shown in Fig. 5.8, and then mapping 6j onto ~j ( Fig. 5.9 ) .
The linear map from llj to ~j which maps (0,0) to (~1' ~2)' (aj, 0) to
(7Jlj ,7J2j), and (aj+lcos~8j, aj+1sin~8j) to (7Jr1,7J2j+1) is given by
where
and
(5.32)
(5.33)
j -
J' "12 - "12 l = 21 a,
J
107
j+l -"1 2 - "1 2
+ a j+ 1 sin!::.8 j
From equations (5.32) and (5.33), one obtains
0"1 1 llf
a "1 1
l1~ - = . - = a~l a~2
a "1 2 l2f
a "1 2 l j - = =
a~l a~2 22
so that the Jacobian of the linear mapping from (';1' ';2) to (771' 772) is
Hence, the integral of equation (5.30) becomes
6. J
IL J
(5.34)
(5.35)
(5.36)
(5.37)
The above mentioned procedure for mapping the curved element S onto the
planar polygon S via the parametric space (771' 772) is illustrated in Fig. 5.10.
108
~2
Xs ~ --~~--------------~.-~ ~1 o aj
Fig. 5.8 (~1' ~2) coordinates in region ~ j in S.
109
112
( 'I"I 1j+1, '1"1 2j+1) ( j '1"1 j ) 'I 'I 111 ,'12 ___ ---+--I--~'"
--------- --------~
Fig. 5.9 Region ~ j
Linear map:
L' J
112 1
110
f 1 J 1 f IG I I = -1 -1 ra d1l1 d1l2
4
=LI. j=1 J
s
I. = I L.I f f IG I dJ: dJ: J J _ ra ~1 ~2
aj
Fig. 5.10 Mapping from S to S via the parametric space (111 ' 112 )
111
Now one can introduce polar coordinates (p,O) in each triangular region ~j
of the projected quadrilateral S, as shown in Fig.5.11.
~1 = P cosO
~2 = p sinO (5.38)
Thus, the integral of equation (5.37) becomes
pdpdO (5.39)
and from equations (5.34) and (5.38) one can write
(5.40)
The upper limit Pj(O) ofthe integral in the radial direction plO) is given by
h. p. (0) = J
J cos(O-a.) J
(5.41)
where hj, aj are defined in Fig. 5.5 ofthe previous section.
112
Fig. 5.11 Polar coordinates (P, e) In ~j
113
5.5 Radial Variable Transformation
This section deals with two cases;
(i) when the source point Xs is on the element and produces weak singularity
(a=l)
and
(ii) when the source point Xs is very near the element and results in a near
singularity.
These types of singularities or near singularities can be investigated by
looking at the integral in the radial direction in equation (5.39), i.e.
(5.42)
In order to cope with the near singularity, a transformation of the radial
variable p will be introduced.
(i) Weakly SingularIntegrals
As shown in section 3.1 , the singular integrals (d=O) arising in the
calculation of Hand G matrices in three dimensional potential problems are only
weakly singular and have kernels of order at most O(l/r) ,i.e. a = 1 in equation
(5.42).
Since r - p in the proximity of Xs ,where p is the radial length along the
tangent plane at Xs , the singularity O(l/r) is cancelled by p to give a regular
kernel of order 0(1). Hence, no extra transformation in the radial variable is
necessary for the weakly singular integrals. It will be shown in the numerical
results in Chapter 10 that the use of polar coordinates with the angular variable
114
transformation enables one to calculate the weakly singular integrals very
efficiently, irrespective of the position of the source point Xs on the element.
It is worth noting that in this method, the same set of integration points can
be used for the calculation of both Hand G matrices since the order of singularity
involved is the same, as demonstrated in Chapter 3.
(ii) Nearly Singular Integrals
It will be shown that nearly singular integrals ( 0<d~1 ) are much more
difficult to calculate efficiently compared to singular integrals (d = 0). Using polar
coordinates alone, for instance, does not give accurate results when 0 < d ~ 1 .
(1) Singularity cancelling radial variable transformation
To overcome this difficulty, Hayami and Brebbia 17 have proposed a
transformation of the radial variable p for the integral of equation (5.42). This is
done by approximating the distance r=lx-xsl by r'=-V p2+d2 as shown in
Fig. 5.12, which is equivalent to approximating the curved element by its
projection on to its tangent plane at Xs •
115
S
Fig. 5.12 Approximate distance
r'= ~p2+ d 2 of r
116
Here it is worth nothing that for general curved elements, r' does not
approximate r very well when x is far from the source projection xs. However,
the (near) singularity lira is only dominant in the neighborhood of xs, where
r-r', so that the transformation R(p) based on this approximation r' - r should
work efficiently for (near) singular integrals (0 ~ d.erg 1), even when the curvature
of the element is relatively large.
Hence, a radial variable transformation R(p) is introduced such that
pdp = r,adR = ..jp2+d2 a dR (5.43)
This transforms the radial integral of equation (5.42) as follows:
(5.44)
This operation has the effect of cancelling the singular behaviour of lira by r' a .
The transformation R(p) is defined by equation (5.43) as
R (P) = f p dp ..jp2+d2 a
(5.45)
which can be integrated analytically as follows:
From the definition of the approximate distance
(5.46)
(5.47)
r'dr' = pdp (5.48)
so that
R (p) = f.L dp r,a
117
I r' = - dr' r,a
= I r' (l-a) dr'
{~ for a =1= 2 = 2-a
(5.49) log r' for a=2
,.. 2-a -(p2+d2) 2
for a =1= 2 = 2-a
1 (5.50) -
log (p2+d2) 2 for a=2
From equation (5.50), the inverse function p(R) of R(p) is given by
2 1
p (R) = [ {(2-a)R }2-a _ d2 ]2
1 (5.51)
( e 2R _d 2 )2 (a = 2)
From equation (5.49) the function r'(R) is given by
,'(R) = { ~:-a)R}'~' (a =1= 2)
(5.52) (a = 2)
R(p), p(R) and r'(R) are given in Table 5.1 for a = 1-5 .
It is found that the radial transformation (5.45) exactly cancels out the
(near) singularity in equation (5.44) when f=constant for the case of planar
parallelogram elements and planar triangular elements ( IGI =constant). This
means that only one integration point is required for the radial integration to give
118
Table 5.1 Radial variable transformation R(p)
and p(R), r'(R) for a = 1-5.
a R(p) peR)
1 Jp2+d2 JR2_d2
2 log Jp2+d2 Je2R _d2
1 ) ~-d' 3 -
Jp2+d2 R2
1 J- 2~ -d' 4 -2 (p2+d2)
1 5 - J( -3R)-t - d2
3 (p2+d2)312
r'(R)
R
eR
1 - -
R
Ri 1
( __ )t
3R
an accurate result, independent of the source distance d. In other words, the
integration in the radial variable was done analytically. In this sense, the
method is semi-analytical.
From this point of view, for a planar element S, (r=r'), with f=constant,
the radial component of the integrals
I = I f pY dS «,Y S r« (5.53)
where r = 0 ; a = 1,3,5 and r = 1 ; a = 3,5 (cf. Table 3.2) which occur in three
dimensional potential problems can be calculated analytically as follows.
I = L IdelL·1 I . a, Y J a,1,} j
(5.54)
I1:>.8. IP.(8) f py+l
1.= J d8 J --dp «, Y,J 0 0 r«
119
= )J .(8)d8 f D.8.
o a, 'Y,) (5.55)
where
J . a, r,)
JP'(O) 8
f) p
o .Jp2+d2 a
dp
= f· [R ,{p.(8)} -R ,(Ol] a,o) a,o (5.56)
where
(5.57)
and 0=1'+1, sothat
8=1 a=1,3,5 and
8 = 2 a = 3,5
For 0=1,
R (P)= a,l 2-a
a=1,3,5 (5.58)
from equation (5.49).
For 0= 2, which occurs in flux calculations
(5.59)
120
and 2
J p dp
Jp 2+ d2 5
p3 = (5.60)
from equations (3.141) and (3.142) respectively.
For general curved elements, one can also use the radial variable
transformations of equations (5.50), (5.58) and (5.59). In this case, the radial
variable transformations no longer cancell the (near) singularity exactly, since
the element is curved. Hence, more than one integration points are required for
the integration in the transformed variable R.
For instance, numerical results in Chapter 10 show that, for a spherical
quadrilateral element S, with f-constant, the approach works efficiently when
the source projection Xs is near the centre of the element, although· it requires
additional radial integration points for the case d ~ 1. However, as Xs moves away
from the centre of the element towards the edge of the element, the method seems
to require more radial integration points and becomes inefficient.
Problems also arise for the case of planar quadrilateral elements which are
not parallelograms. This is caused by the fact that the mapping from the square
in the (7]1' 7]2)-space to an element is linear only when the element is a planar
parallelogram.
Similarly, for curved elements the linear mapping Lj defined for each
region ~j in equation (5.34) does not give the exact point on the curved element
whose projection matches the integration point on the tangent plane, since this
inverse projection is defined by a nonlinear mapping.
At first thought, inaccuracy of the inverse mapping appears to be the main
reason for causing inefficient cancellation of the (near) singularity.
121
(2) Consideration of exact inverse projection and curvature of the element
in the radial variable transformation
In order to overcome these difficulties, the following modifications were
considered.
(i) Exact projection of the curved element S on to the plane tangent at:is .
(ii) Exact inverse projection from the integration point on the tangent plane to
the curved element using the Newton-Raphson method to account for the
nonlinear mapping.
(iii) A more accurate approximation of the distance r by taking the curvature of
the element at :is into account in the approximating distance r' and the
radial variable transformation R(p) based on this r'.
The combination of these modifications resulted in a significant decrease of
the number of necessary radial integration points, which remain almost constant
for very small values of the distance d , for the case when feconstant in the
integration
I f dS S r"
However, this approach has the following draw backs:
(i) It was found that when f;to constant, for instance for integrals containing
the interpolation function:
I ;ij dS
S r"
the simple radial variable transformation
122
with the linear mapping of equation (5.23) based on Lj, requires more than
32 Gauss points to achieve a relative error < 10-6, for a planar square
element, as will be shown in Chapter 10.
For planar square elements, the 'approximate' projection of the
element S to S, the inverse projection using the linear mapping Lj, and the
'approximating' distance r' = V p2 + d2 are all exact. Hence, the failure to
integrate Is ¢ij/ra dS indicates that the above modifications are not sufficient
to calculate the integrals Is flra dS for general curved elements, let alone
planar elements.
In fact numerical experiments on curved elements showed that the
above modifications do not help to decrease the necessary integration points
to calculate I flra dS over curved elements when f;e constant.
(ii) The two variable Newton-Raphson iteration has to be applied for each
integration point in order to find the inverse projection, which results in
excessive CPU time.
(iii) Since an exact projection of the curved element S to the tangent plane at Xs
is performed, the projected quadrilateral S generally has curved edges. This
causes many complexities when performing the integration in S .
Another disadvantage of the singularity cancelling radial transformation
proposed so far is the fact that they require a different transformation for different
order of singularities a = 1-5. This implies that different sets of integration
points and calculation of the quantities like r, IGI, ¢ij for them are required for
each a. For instance, different sets of integration points are required for the
computation of near singular integrals ge'e kl and he'e kl of equation (2,38) and (2.39)
in order to obtain the influence matrices G and H.
Consequently a more simple but robust radial transformation is desired.
123
(3) Adaptive logarithmic radial variable transformation (log-L2)
In order to overcome the above difficulties, one resorts to the approximate
projection and the linear mapping from the tangent plane to the curved element
mentioned in sections 5.2 and 5.3.
However, this time, instead of using the same degree a of the radial
transformation as the (near) singularity lira, as in equations (5.43) and (5.44),
radial transformations with different degrees (3* a were attempted, i.e.
(5.61) and
(5.62)
where a = 1-5 and (3= 1-5 independently.
As a result, it was found (cf. Chapter 10) that the transformation
corresponding to (3 = 2 :
pdp = r,2 dR (5.63)
or
R(P) = log Jp 2+ d2 (5.64)
which gives
p(R) = Je 2R _ d2 (5.65)
r'(R) = e R (5.66)
works most efficiently for different types of near singularities lira, (a = 1,2,3,4,5).
We shall refer to this transformation as the adaptive logarithmic transformation
( log-L2), since R(p) is the logarithm of the L2 - norm in the (p, d)-space.
As will be shown in the numerical results in Chapter 10, this log-L2
transformation works efficiently for general curved elements, with arbitrary
position of the source projection xs. It also enables one to calculate accurately and
124
efficiently nearly singular integrals whose kernels include the interpolation
functions <Pij even when O<d~l, which was not possible with previous methods.
The log-L2 radial variable transformation also has the virtue that only one radial
transformation is necessary for kernels with near singularities of different orders,
such as in ge'e kl and he'e kl in equations (2.38) and (2.39) for the G and H matrices,
or geSI , hesl in equations (2.50) and (2.51) for the potential at an internal point Xs.
The reason why the adaptive logarithm radial transformation (log-L2) works
efficiently for nearly singular integrals of the type J s dS/ra , (a = 1,2,3, ... ) seems to
be the following:
(i) For 0<d~1 r'2 , is sufficient to weaken the near singularity of lira
(a=1-5).
(ii) Since the lower and upper bounds of the radial integration is
R (0) = log d (5.67)
and
(5.68)
respectively, the range of integration does not expand drastically as d becomes
very small. This enable the Gauss-Legendre quadrature formula to work
efficiently in the transformed variable R.
For higher degree transformations this is not the case, for example,
1 (5.69) R (0) = d
({3=3)
1 R (0) = -
2d2 ({3=4 ) (5.70)
1 R (0) = -
3Ji ({3=5 ) (5.71)
However, a more rigorous explanation for the optimality of the f3= 2 (log-L2)
transformation will be given in Chapter 6.
125
(4) Adaptive logarithmic radial variable transformation (log-Ll )
As will be shown by numerical experiments in Chapter 10, and explained by
the error analysis in section 6.2, the adaptive logarithmic transformation (log-L2)
R (p) = log JP 2 + d2 (5.64)
which was proposed in the previous section (3) , works efficiently for nearly
singular integrals arising from the calculation of the potential u(xs) at a point Xs
very near the boundary.
However, the log-L2 transformation of equation (5.64) does not work so
efficiently for nearly singular integrals arising from the calculation of the flux
( ego electrostatic field E, magnetostatic field B) or the potential derivative
aulaxs at a point Xs very near the boundary. This is demonstrated by numerical
experiments in Chapter 10. The reason for the inefficiency of the log-L2
transformation for the potential derivative is given in the error analysis of section
6.S.
In short, the near singular integrals for the potential derivative aulaxs
include integrals of the type
fPO p2 J ... J - dp
a,2 0 ra (<<=3.5) (5.72)
in the radial variable p , where
(5.73)
The integral J a) of equation (5.72) is transformed by the log-L2 transformation
R(P) = log Jp 2+ d2 (5.64)
to give
J J 2-a dR IR(PO)
iii pr a.2 R(O) (5.74)
where
for planar elements,
and
so that
F (R) = p r 2- a
dF
dR
dF dp
dp dR
dFI -+00 dR p=+O
and similarly
( dF )(2n) I = _ 00
dR p=+O
126
(5.75)
(5.76)
(5.77)
(5.78)
(5.79)
which means that the application of the Gauss-Legendre formula to the numerical
integration of F(R) = p r2- a with respect to the variable R in the integral of
equation (5.74) is expected to be inaccurate, or inefficient in that it requires a lot
of integration points.
and
The basic reason for this is that for the 10g-L2 transformation
R (p) = log J p2+d2
dR
dp = =
as shown in Fig. 5.13, so that
dp = l+d2
dR p
which results in
~Ip=+o=+oo in equations (5.77) and (5.78) .
(5.64)
(5.80)
(5.81)
(5.82)
(5.83)
127
R
log "" p2+ d 2 I I I I I P
0 Pj
~
log d t
Fig. 5.13 The log-L2 transformation: R( P) = log ~ p2 + d 2
128
In order to avoid this defect of the log-L2 transformation of equation (5.64),
one would like to have a radial variable transformation which retains the nice
logarithmic character of the log-L2 transformation and yet has the property
dR! - * +0 dp p= +0
(5.84)
so that dpldR has a finite value at p = + 0 .
This can be satisfied by the transformation
R (p) = log(p+d) (5.85)
which we shall refer to as the adaptive logarithmic radial variable transformation
(log-Ll) ,or the log-Ll transformation, since it is the logarithm of the Ll-norm in
the (p, d)-space. Note that
R (0) = logd (5.86)
R (p ,) = log (p, + d) J J (5.87)
From equation (5.85),
dR 1 = (5.88)
dp p+d
which satisfies
dR ! 1 - = - "" 0 dp p=o d
(5.89)
as shown in Fig. 5.14, so that
dp ! - =d*oo dR p=O
(5.90)
As will be shown in section 6.8, the essential nature of the radial component
of the near singular integrals in three dimensional potential problems can be
represented by
129
R
log (p. + d) ------------J I
I I I
--~~~~--~--------~ p o p. J
Fig. 5.14 The log-L1 transformation: R{ P} = log {P +d}
130
(5.91)
where
0=1; a=1,3
for the calculation of the potential u(xs) and
0=1; a = 3,5
0=2 ; a = 3,5
for the potential derivative aulaxs. (cf. Table 3.3 of Chapter 3 )
The integral of equation (5.91) is transformed by the log-L1 transformation
of equation (5.85) as
J - J eRdR fR(P') /
a,li - R(O) r a
since from equations (5.85) and (5.88)
dp R -=p+d=e dR
Let the integrand of equation (5.92) be
where
and
so that
liR P e
F(R) E-
ra
/ (p+d)
efl /-1 [ 2{ } 2 R I dR = 7+2 r (0 + l)p + 0 d - ape dF
(p+d) /-1 { 3 2 2.1} = a+2 (0-a+1)p + (o-a) dp + (o+l)d p + oa-
r
(0=1)
(0=2)
(5.92)
(5.93)
(5.94)
(5.95)
(5.81)
(5.97)
131
and dF/dR remains finite for the interval of R in equation (5.95), in contrast
with equation (5.78) for the log-Lz transformation.
Hence, the adaptive logarithmic radial variable transformation (log-Ll)
stands a good chance for the efficient calculation of near singular integrals for the
potential derivative au/axs as well as the potential u(xs). This will be
demonstrated by numerical experiments in Chapter 10.
In fact, it turns out that the log-Ll transformation is more efficient than the
log-Lz transformation, not only for the flux calculation but also for the potential
calculation when curved elements and/or high order interpolation functions are
used. This can be explained by the presence of terms with o~ 2 in equation (5.91)
for such cases.
The log-Ll transformation is also more robust against the change of the
source distance d and the position of the source projection xs , compared to the
log-Lz transformation.
Thus, the log-Ll radial variable transformation is preferred to the log-Lz
transformation.
(5) Single and double exponential radial variable transformations
In order to weaken the near singularity in the radial variable near p = 0, we
can also use exponential type radial transformations 37. The exponential type
transformations themselves are not original 36, 13,32,14,26,16,30,27, but the fact that
they are applied in the radial variable p in the tangent plane S is new.
There are basically two ways of performing the exponential type radial
transformations in the radial variable p. The first way is to map the radial
variable p: [ 0, p/8) ] on to the half-infinite interval R: [- 00,0] . The second way
is to map p on to the whole infinite interval R : [- 00, 00]. Numerical
experiments in Chapter 10 show that the second procedure gives better results for
different nearly singular integrands F and different source distances d. This is
132
because the numerical integration error near the end point p=Pj(B) in equation
(5.39) and (5.42) is effectively reduced by the second procedure.
Given the transformation to the whole infinite interval R: [- 00, + 00] ,
there is another choice of using the single or double exponential transformation.
(a) Single Exponential radial variable transformation (SE)
First consider the single exponential transformation
where
Pj p(R) = - (1+ tanhR)
2
=
Here P: [ 0, Pj] is mapped on to R: [ - 00, + 00] as shown in Fig. 5.15 .
From equation (5.98) ,
Pj p(O)= -
2
p(R) - p. e- 21R1 J
for R -+ + 00
for R -+ - 00
(5.98)
(5.99)
(5.100)
(5.101)
(5.102)
so that p(R) converges exponentially to Pj as R- + 00, and to 0 as R- - 00 ,
respectively. From equation (5.98),
so that
dp
dR
dp I dR p=O
= Pj 2
(5.103)
(5.104)
133
p
-----------------------~~ -------------------------
~==~~~----~----------------R o
Fig. 5.15 The Single Exponential Transformation p.
P (R) = -t ( 1 + tanh R )
134
The integration in the radial variable P in equation (5.42) is transformed by
equation (5.98) and (5.163) as
J (0) = f Pj riG I p dp o r a
[OX> g(R) dR (5.105)
where
(5.106)
Therefore, the integral J(8) can be approximated by the truncated trapezium rule
for the infinite interval R: [- 00, + 00] ;
n
J (0) - J h, n "" h I g (kh ) (5.107) k=-n
It can be seen from Fig. 5.15 that, since the integration points for the trapezium
rule in the variable R in equation (5.107) is equally spaced, the corresponding
integration points in the variable P are concentrated around the near singularity
at p=O so that, effectively, the near singularity is weakened when transformed
to the variable R. Hence, the numerical integration in equation (5.107) results in
high precision and efficiency.
As mentioned before, this single exponential transformation of equation
(5.90) is superior to the 'half-infinite single exponential transformation'
p(R) = p. (1 +tanhR) J
(5.108)
which maps p: [0, Pi] on to R : [- 00, 0] , since the transformation of equation
(5.108) results in inaccuracy due to the truncation error at the end point P = Pi '
where as for the transformation of equation (5.98), integration points in the
variable P are concentrated, not only around the near singularity at p=O, but
also around the end point P = Pi' so that the truncation error of the numerical
integration near the end point P = Pi is significantly reduced. This will be shown
in the numerical results in Chapter 10.
135
(b) Double Exponential radial variable transformation (DE)
Consider next the double exponential transformation 32 applied to the radial
variable p:
(5.109)
Here again, p: [0, p) is mapped on to R: [- 00, + 00] as shown in Fig. 5.16 .
From equation (5.109) ,
Pj p(O) = -
2
" R --e p (R) - p. (1 _ e 2
J
_~)RI . p (R) - p. e 2
J
(5.110)
for R -++ co (5.111)
for R -+_ co (5.112)
so that p(R) converges to Pj as R- + 00, and p(R) converges to 0 as R- - 00,
double exponentially, which is a faster convergence compared to the single
exponential transformation of equations {5.98), (5.101) and (5.102). From
equation (5.109) ,
(5.113)
so that
(5.114)
Equations (5.105), (5.106) and (5.107) can be similarly applied to calculate the
integral J(8) numerically.
(c) Implementation of the single and double exponential transformations
In the numerical integration using the single exponential or double
exponential transformation, the interval h and the number of integration points
2n in equation (5.107) can be determined for a specific integrand function as
follows:
136
p
p. J ------------------------- -----_.;;--=-.... - ................. -..--..-
~========~~--~------~----------R o
Fig. 5.16 The Double Exponential Transformation p.
P{R) = i { 1 + tanh ( ~ sinh R)}
137
Set h= 1, for example, and calculate Jh,n with increasing n until
(5.115)
is satisfied to obtain J h ;;;;; J h, n .
Repeat the process for h = 112, 114, 1/8, .. · until
(5.116)
is satisfied to obtain the final result.
The threshold value Eh may be set equal to the required accuracy E of the
integration. It is advisable to set En ~ Eh •
In order to speed up calculations, it is recommended to calculate the
exponential type function such as cosh(x) by
ex = exp (x)
ei=l/ex
coshx = (ex + ei )/2 (5.117)
instead of calling the internal function cosh (x), so that ex and ei can be reused
to calculate other required exponential type functions such as sinh(x) or tanh(x) .
For instance, for the single exponential transformation,
p.l p(R)= til} I
+ II
coshR =
(5.118)
(5.119)
should be used, where eR and lIeR are computed only once for a given value for
R.
138
Further CPU-time can be saved by calculating g(R) of equation (5.106) for
R = k and R = - k, (k = 1-n) at the same time, using the fact that
cosh (-R) = coshR
sinh ( - R) = - sinh R
tanh(-R) = -tanhR
etc.
However, in doing so one must make sure that the addition
g (-kh) + g (kh)
does not involve substantial rounding error, since from equation (5.106),
Ig(-kh)1 ~ Ig(kh)1
(5.120)
(5.121)
(5.122)
(5.123)
(5.124)
for relatively large k, since lira becomes dominant when R- - 00 i.e. when p-O .
In order to minimize such sources of error, it is best to perform the addition in
equation (5.107) from k=N down to k= -N consecutively. This will result in
adding small quantities together first, instead of adding small numbers to large
numbers as in equation (5.123), thus minimizing round off errors.
The advantage of using the single or double exponential transformation in
combination with the truncated trapezium rule, is that they do not require any
integration tables (such as that of Gauss-Legendre formula), and also they have
the advantage that the results used for interval h can be reused for the
calculation with interval hl2 etc., where as in the Gauss-Legendre formula, the
position of the integration points are independent for different numbers of
integration points used.
However, the single and double exponential transformations consume
significantly more CPU-time per integration point, compared to the log-Ll and
10g-L2 transformations combined with the Gauss-Legendre formula. This is
139
shown in Table 5.2. Chapter 10. (To further speed up the single/double
exponential transformations, one may prepare a table of values for p and dp/dR
of equations (5.98) and (5.103) before hand.)
Table 5.2 Comparison of computation per integration point
Double Single log-L2 log-Ll
Exponential Exponential
exponential 1 0.5 1 1
square root 1
division 2 1.5
multiplication 2.5 1
addi tionlsubtraction 2 1 1 1
divide/multiply by 2 1.5 1
CPU time per
integration time 7.6 5.4 -0.2 -0.9
(I-'- seconds on SX-2)
Hence, over all, the log-Ll radial variable transformation in combination
with the Gauss-Legendre formula seems to be the best, considering accuracy,
robustness to different types of near singularity and CPU-time.
5.6 Angular Variable Transformation
As will be shown in the numerical results in Chapter 10, when using polar
coordinates, the number of integration points required for the angular variable to
140
produce accurate results increases rapidly as the source projection Xs = x (~1' ~2)
approaches the edge of the projected quadrilateral S. This causes a problem not only for nearly singular integrals but also for
singular integrals resulting from discontinuous elements 1, for which case the
source can be very near the element edges.
This phenomenon may be considered as a near singularity in the angular
variable and can be explained as follows.
When Xs is very close to the edge xj +1 -Xj of the projected quadrilateral S
as shown in Fig. 5.17 , the value of Pj(O), and hence the integrand with respect to
the angular variable 0 in the integral
(5.39)
can vary rapidly as 0 varies from 0 to AOj • Thus, a large number of integration
points are necessary to perform the angular integration numerically.
In order to overcome this difficulty of near singularity in the angular
variable, strategies similar to those taken in the case of near singularity in the
radial variable discussed in section 5.5, are proposed. Namely, a transformation
of the angular variable is introduced in order to weaken the near singularity in
the angular variable.
(1) Adaptive logarithmic angular variable transformation
First we propose a logarithmic angular variable transformation which
weakens the angular near singularity before applying the Gauss-Legendre
quadrature formula.
In order to do so, let us assume in equation (5.39) that
IP}D> flGI J (8 ) = - p dp - p. (8)
. 0 ra J (5.125)
since Pj(O) dominates the behaviour of the integral J(O) when Xs is very near an . -edge of S.
141
Fig. 5.17 Near singularity in the angular variable e
142
Consider next an angular transformation t = t(8) such that
dO 1 cos (O-a) _ = __ = __ --.;J'-
dt p/O) hj (5.126)
where t(8) is given for each region t.j by
t (0) = I p /0) dO
h j { 1 + sin (0 - a j) } = - log
2 1 - sin (0 - a . ) J
(5.127)
Using t as the new variable for the angular integration in equation (5.39), we
obtain
Since
IM . I t(M.) dO O
J J (0 ) dO = J J (0) - dt t(O) dt
J(O) -- - 1 p. (0)
J
I t(M} J(O) -- dt
t(O) P . (0) J
(5.128)
the integrand in (5.128) is expected to be a smooth function of t compared to the
original strong variation of J(8) with respect to O. t(O) acts as a transformation
that weakens the near singulari ty in the angular variable 0 .
Here it is interesting to note that both the radial and angular variable
transformations
R (p) = log J p2+ d2
R (p) = log (p + d )
and hj { 1 + sin (0 - a) }
t(O) = - log 2 1 - sin (0 - a .)
J
(5.64)
(5.64)
(5.127)
which work efficiently for the radial and angular near singularities respectively,
are logarithmic functions related to the near singularity so that
R (p) .... ex:> as r'= J/ + d2 .... 0 (5.129) and
7r t(O) .... ±ex:> as O-a. .... ±-
J 2 (5.130)
143
Hence, the logarithmic function efficiently weakens the near singularity by
magnifying the scale of the integration variable in an optimal manner, when
V p2 + d2 ~ 1 and 1 ± sin(8 - aj) ~ 1 , which are the causes for the radial and
angular near singularities respectively. By doing so, enough information is
sampled to accurately integrate the rapidly changing integrand in the near
singular range.
When Xs is right on the edge, the triangular region ~j collapses to a line and
it has no contribution to the integral. Hence, one can simply skip the integration
loop for that particular j when sgnG) = 0 as defined by (5.27). Using this sgn(j)
makes the programming simple, since it defines the position ofxs with respect to
the edge-j.
Numerical experiments in Chapter 10 confirm the accuracy and efficiency of
the use of the logarithmic angular variable transformation of equation (5.127) in
combination with the Gauss-Legendre formula.
(2) Single and double exponential angular variable transformations
Just as in the case of the radial variable, one can also apply the single/double
exponential transformation to the integration in the angular variable, in order to
weaken the angular near singularity. The truncated trapezium rule can then be
applied to integrate in the infinite interval of the transformed angular variable.
For the integration in the angular variable in equation (5.39), one can apply
the single exponential (SE) transformation
M. 8 (t) = _J (1 + tanh t )
2
or the double exponential (DE) transformation
(5.131)
(5.132)
which map 8: [0, ~8j] onto t:[ - co, + co]. Hence, the integration in the angular
. variable is transformed as
where
and
J J(O)= J(O)- dt JI!.B. Jao dO
o _ao dt
JI!.B.CB) flGI
J(O) = J -- pdp
dO - = dt
o r a
t::.B. _J
2
144
for the single exponential transformation of equation (5.131), and
dO 7C cosh t - = - 60. --:-------:-dt 4 J 2 (7C )
cosh 2" sinh t
for the double exponential transformation of equation (5.132).
In both cases,
where
JI!.B. Jao o J J(O) dO = _ao a(t) dt
dO a(t) .. J{O(t)}
dt
(5.133)
(5.42)
(5.134)
(5.135)
(5.136)
(5.137)
Hence, the integral of equation (5.136) can be calculated numerically using the
truncated tapezium rule as
Jao N
_ao a(t) dt - h kf:.N a (k h) (5.138)
where hand N are determined by the required accuracy as in equations (5.115)
and (5.116).
5.7 Implementation of the PART Method
To sum up, the integral of equation (5.1) which we want to calculate, is
transformed as follows,
I=J LdS S r a
145
(5.139)
where
JA8j JP}8) fl GI 1. = IdetL.1 dO - pdp
J J 0 0 r a
Jt<A8} dO JR1Pj{8(t}}] flGI d = IdetL.1 -dt -_P..E..dR J t (0) dt R(O) r a dR
(5.140)
and
(5.141)
R(p) is the radial variable transfonnation which serves to cancel or weaken
the (near) singularity due to lira. t(O) is the angular variable transfonnation
which serves to weaken the angular near singularity which arises when Xs is near
the edges of the approximately projected element S. Now that the radial and angular (near) singularities have been weakened by
the transfonnation R(p) and t(O) , the integral Ij of equation (5.140) can be
calculated accurately and efficiently by applying standard numerical integration
formulas to the integration in the variables Rand t.
(1) The use of Gauss-Legendre fonnula
The best recommended way is to apply the Gauss-Legendre fonnula 3 to
R(p) and t(O).
The radial variable transfonnation R(p) should be chosen as follows:
(i) For weakly singular integrals (d=O, a = 1) ,
e.g. J [dS s r
and
in potential problems,
146
R (p) = p
(ii) For nearly singular integrals ( 0 < d <ail ) ,
e.g.
for potential calculations and
e.g. f au· - {dB
s ax s
and
and f aq· {dS sax,
for flux calculation in potential problems,
R (p) = log (p + d) ( wg-L1 transformation)
(iii) For planar elements with f=constant in equation (5.139),
2-a -(p2+ d2) 2
R(p) = {or a*-2 2-a
1 - log (/+ d2) {or a = 2 2
(5.142)
(5.85)
(5.50)
for integrals of the type f s lira dS, where a = 1, 3 occur in potential calculations
and a = 3, 5 occur in flux calculations in three dimensional potential problems
(cf. Table 3.2 and equation (5.50», and
R(P)=
3 P
{or a=3
{or a = 5 (5.143)
for integrals of the type f spIra dS , (a = 3,5), which occur in flux calculations of
three dimensional potential problems. The transformation of equations (5.50) and
(5.143) correspond to performing the radial integral analytically. Hence, only one
integration point is required in the radial direction to obtain the exact value.
147
The angular variable transformation t(8) is given by
hj { 1 +sin (0 - a j) } t (0) = - log
2 I-sin (0 - a .) (5.127)
J
(logari thmic angular variable transformation) for each region ti:j (j = 1-4) .
Now that the radial and angular (near) singularities have been weakened
(or cancelled) by the variable transformations R(p) and t(8) , the product Gauss
Legendre quadrature formula can be safely applied to the integration in the
variables Rand t of equation (5.140), i. e.
tv
1. - 1. == IdetLI J J J
tv
t(M.)-t(O) J
2
(5.144)
for each region 3:j. Ij need not be computed when sgn(j) = 0, in which case the
source projection Xs lies on the line defined by edge-j so that the area of region 3:j
is zero and Ij = ° . In the transformed angular variable t(8) , the integration points tk,
(k=1-Nt) are given by
where
and
t(M)-t(O)
tk = J 2 g(k,Nt ) + t(M.)+t(O)
J
2
h. {1+sin(60.-a.)} t (60 .) = ..J.. log J J
J 2 1 - sin ([:,,0. - a . ) J J
h. { 1 - sin a . t (0) = ..J.. log J
2 1 + sin a . J
(5.145)
(5.146)
(5.147)
148
sin(l:J.O .. -a.) = sin l:J. 0 . cos a . - cosl:J.O .sina. J J J J J J (5.148)
Note also that
pJ' (tk) E! p. {O (t lo )} = hj = hJ. cos h ( thk. ) J cos (O-a.)
J J
(5.149)
where hj, sin~8j, sin aj etc. are given in equations (5.25) and (5.26). The inverse
transform 8(tk), which maps the integration point tk to the angle 8(tk) defined in
Fig. 5.10 for each triangular region ~j in S, is given by
since
and
where
( til) sina. sinO (tA:) = cos aj tanh h. + J
J COSh( :.)
sin(O-a.) = J
cos a . J
J
- sina. J
1f 1f ros(O-a.) > 0 J since - - < 8-a. < -
J 2 J 2
(5.150)
(5.151)
(5.152)
(5.153)
The tanh(tk/hj) and cosh(tk/hj) in equations (5.150) and (5.151) should be
calculated using exp(tklhj) and its reciprocal to save CPU-time, as was done in
equation (5.117).
149
g(k, Nt) and w(k, Nt) in equations (5.145) and (5.152) are respectively the
position and weight of the k-th integration point of the Nt point Gauss-Legendre
quadrature formula for the interval [-1,1], which in this case is used for the
integration in the transformed angular variable t.
In the transformed radial variable R(p), the integration points Rkl
(l = 1-N l) for a given angular direction tk are given by
(5.154)
gel, NR) and w(l, NR) are respectively the position and weight of the l-th
integration point of the NR point Gauss-Legendre formula for the interval [-1,1],
which in this case is used for the integration in the transformed radial variable R.
In order to calculate f, rand IGI in equation (5.144), we need to know the
point (711k1' 712kl) in region ~j of the (711' 712) parameter space, which is mapped
linearly from the integration point ( B(tk), p(Rkl» in region Lij. To do so, we use
equations (5.40), (5.150) and (5.151) to obtain
where
and
( i) For R(p) = p of equation (5.142),
p (R) = R
dp -=1 dR
(ii) For the log-L2 transformation
(5.155)
(5.156)
(5.157)
R (p) = log v'p2+d2
/2R2 p(R)= ve~'-d<
150
f"""7"22 v r <-d<
where
r' = .Jp 2 + d2 = l
dp ,2 2R PdR=r =e
(iii) For the log-Ll transformation R(p) = log(p + d) of equation (5.85),
p (R) = eR - d
dp = e R = p + d dR
(2) The use of truncated trapezium rule
In this case, equation (5.140) gives
so that
where
I. = IdetL.1 - dt -_P ...£. dR Joo d8 JOO flGI d
J J _00 dt _00 r<X dR
flGlp F(p,8) !2
(5.64)
(5.158)
(5.159)
(5.160)
(5.161)
(5.162)
(5.163)
(5.165)
and ht and hR are the interval for the trapezium rule for the angular variable t
and the radial variable R, respectively.
151
For the single exponential transformation,
6.8 . o (t) = T ( 1 + tanh t)
dO 6.8. 1 = _J __
dt 2 cosh2 t
for the angular variable, and
p.{O(t)} peR) - J 2 (1+tanhR)
dp p .(0) 1 = _J_
dR 2 cosh2R
for the radial variable.
For the double exponential transformation,
dO 7r cosht = - 6.0. --,.....-----,-
dt 4 J cosh2 (~ sinh t )
for the angular variable, and
dp 7r coshR dR = - p.{O(t)} ( ) 4 J 2 7r
cosh - sinhR 2
(5.131)
(5.134)
(5.166)
(5.167)
(5.132)
(5.135)
(5.168)
(5.169)
Numerical experiments in Chapter 10 show that the single exponential
transformation (SE) is more efficient compared to the double exponential
transformation, in the number of integration points and CPU time, particularly
for nearly singular integrals (0 < d ~ 1) in the radial variable.
152
However, it is also shown in Chapter 10 that the log-Ll and log-L2 radial
variable transfonnations with the Gauss-Legendre fonnula is more efficient
compared to the single/double exponential transfonnation with the truncated
trapezium rule for the integration in the radial variable. Further, the log-Ll
radial variable transformation is more robust compared to the log-L2
transfonnation against different types of kernel and element geometry.
The organization of a program to calculate a (nearly) singular integral
J sf Ira dS would be the following.
Given the nodes xi,k ( i,k = -1, 0, 1 ) of the curved quadrilateral element S,
and the source point Xs and a table containing weights and positions of the
Gauss-Legendre quadrature fonnula,
(1) Find the source projection Xs =x(~l' ~2) by the Newton-Raphson method.
(2) Determine sgn(j), (j = 1 .... 4).
(3) Calculate the unit nonnal ns to the curved element S at xs.
(4) Find the projection Xj , (j = 1 .... 4) of the four corner nodes of S on to the plane
tangent to S at xs.
(5) Detennine the geometry of the projected quadrilateral S and the four
triangular regions Aj (j = 1-4) defined by xs, Xj (j = 1-4). Then, detennine
the linear mapping matrices Lj (j = 1-4).
(6) Perfonn the numerical integration for each triangular region Aj , j = 1-4 ( if
sgn(j) *" 0 ) in the angular and radial transformed variables t and R
respectively, according to equation (5.144) .
Similar procedures can be taken for a curved triangular element (and
arbitrary curved polygonal elements). For triangular elements, the method is
even more attractive since there are generally only three regions Aj (j = 1 .... 3) to
perfonn the integration instead off our for quadrilateral elements.
153
5.8 Variable Transformation in the Parametric Space
So far, we have introduced polar coordinates (p, 0) and radial and angular
variable transformations R(p), teO) in the plane S tangent to the curved element
S at the source projection xs.
In a similar manner, we can also introduce polar coordinates (p, 8) and radial
and angular variable transformations in the parametric space (711' 712) which
describes the curved element x(711' 712).
For instance, for a curved quadrilateral element S defined by
{X(711' 712) 1-1~ 711' 712~ 1}, one can proceed as follows:
(1) Find the point Xs=x(7;1' 7;2) on S, closest to the source pointxs.
(2) Introduce polar coordinates (p, 0) centered at (7;1' ~2) in the square defined by
1< - - s; 1 - = 711'712- .
(3) Divide the square in to four triangular regions t-j, (j = 1-4) as shown in
Fig. 5.7.
(4) In each region t-j, introduce radial and angular variable transformations
R(p), teO) similar to those defined in sections 5.5 and 5.6 for 3j in the tangent
plane, in order to weaken the (near) singularity.
(5) For each region t-j , perform numerical integration with respect to the
transformed variables R(p) and t(0).
For radial variable transformations R(p) which make use of the source
distance d=lxs-xsl, the equivalent source distance D can be defined for the
parametric space (71 l' 71 2) by,
II =Ix(l, ~2) -x(-l, ~2l1
l2 =lx(~I,I) -x(~I,-lll
d
(5.170)
(5.171)
(5.172)
154
for example. Using such an equivalent source distance D, a log-Ll transfonnation
R (p) = log (p+D) (5.173)
and a log-L2 transfonnation
R(p) = log .Jp2+D2 (5.174)
can be defined.
Similar transfonnations can be introduced for curved triangular elements in
the parametric space (7]1' 7]2) by defining an equivalent distance D similar to
equation (5.172) if necessary.
This variable transfonnation in the parametric space (7]1' 7]2) may seem
simpler compared to the PART method, which employs variable transfonnations
in the approximately projected element S in the tangent plane. However, the
concept of the equivalent distance D becomes vague when the element has a
large aspect ratio, whereas the source distance d is always clearly defined for all
kinds of element geometry, so that the radial variable transfonnation in S using
the source distance d itselfis preferred, so long as the source distance is employed
in the radial variable transformation, such as in the log-Ll and 10g-L2
transfonnations.
CHAPTER 6
ELEMENTARY ERROR ANALYSIS
6.1 The Use of Error Estimate for Gauss-Legendre Quadrature Formula
In this section we will perform an error analysis of the proposed numerical
integration (PART) method and explain why the log-L2 radial transformation
R (p) = log J p2 + d2 (5.64)
corresponding to f3 = 2 in
pdp = r'P dR (5.46)
leads to accurate numerical integration results with relatively few integration
points for integrals of the type Is lira dS, whilst the radial transformations
corresponding to f3* 2 (cf. Table 5.1) give poor results. (cf. Chapter 10)
Since we are interested in the integration in the radial direction, consider
the integral
JPj fiGi
J= --pdp o r a
(6.1)
where Pj = Pj(8) is the upper limit of the radial variable in equation (5.42).
For simplicity, take the case of constant planar elements (parallelogram or
triangular) so that the radial integration of equation (6.1) reduces to
(6.2)
where
(6.3)
156
Applying the radial variable transformation corresponding to
pdp = r'P dR (5.46)
we obtain
IR(P.)
J= J r P-" dR R(O)
(6.4)
Weare now interested in estimating the error when applying the Gauss
Legendre quadrature formula to equation (6.4) .
The error estimate for the n-point Gauss-Legendre formula when applied to
the integration over the interval [-1, 1] :
II n
((x) dx = L w. f(x.) + e (f) 1 ' , n
- i=1
is given by 3,33
for some -1 < TJ < 1 ,
where
(6.5)
(6.6)
Hence, the error En(F) of applying the Gauss-Legendre quadrature formula
of equation (6.5) to the integration of a function F(t) over a general interval
t E [a,b] is given as follows:
Ib F(t)dt = b-a II F {a+ b + X(b-a)} dx G 2 -1 2
= b-a II g(x) dx 2 -1
= b-a { ± w. g(x.) + e (g)} 2 i=I' , n
(6.7)
where
and
a+b+ x(b-a) t(x)= --~-....:.
2
dt b-a -=-dx 2
157
{ } { a+b+X(b-a)} g(x)=F t(x) =F 2
(6.8)
(6.9)
(6.10)
Here, t(x) maps the interval x E [-1, 1] to t E [a, b]. From equation (6.9) and
(6.10),
dg dF dt b-a dF - = -=--dx dtdx 2 dt
and
where
so that the error En(F) of equation (6.7) is given, from equation (6.12) by
b-a E (F) = - e (g)
n 2 n
for some a«<b, (-1<~<1) ,
where 22n+ 1 (n!)4
£ ...
n (2n+l){(2n)!}3
Recall Stirling's formula
(6.11)
(6.12)
(6.13)
(6.14)
(6.15)
(6.16)
158
or
lim = 1 (6.17) n!
which means that n! is asymptotically equivalent to e-nnnY2Ifn. The
approximation (6.16) is reasonably accurate for small values of n as well, as
shown in Table 6.1.
Table 6.1 Accuracy of Stirling's formula for small n
e-n nn Y2lfn n
n!
1 0.922
2 0.960
3 0.973
4 0.980
5 0.983
6 0.986
7 0.988
8 0.990
9 0.991
10 0.992
In fact equation (6.16) can be expressed by the following asymptotic
expansion
n n_ r;:;--( 1 1 139 ) n! = e- n v 21fn 1 + - + -- - -- + ... 12n 288n2 5180n3
(6.18)
Substituting Stirling's formula (6.16) in equation (6.15), one obtains
E '" n
22n +1 (n!)4
(2n + 1) {(2n!)}3
1 V;(e)2n -2n- 2 - - - n
2 4
Hence, from equation (6.14),
E (F) = p r 2n) (-r) n n
for some a < 1" < b ,
where
( b-a )2n+l p = -- E
n 2 n
_ b-a ~ {(b-a)e }2n 4 n 8n
159
(6.19)
(6.20)
(6.21)
From equation (6.20), the lower bound Ln(F) and upper bound Un(F) for
IEn(F)1 are given by
and
min
= P n min I r 2n) (r) I a ~ r~ b
U (F) = max lEn (F)I
a~ r~ b n
= Pn max Ir2n)(-r)1
a ~ r~ b
respectively, so that,
Ln (F) ~ lEn (F)I ~ Un (F)
Now we are ready to apply the error analysis to the integral
(6.22)
(6.23)
(6.24)
160
JR(P') J= J r ll - a dR (6.4)
R(O)
where,
F (R) • r(R) II - a (6.25)
a = R (0) (6.26)
and
b=R(p.) J
(6.27)
6.2 Case B = 2 (Adaptive Logarithmic Transfonnation: log-L2)
First, the error analysis will be perfonned for the log-L2 radial variable
transfonnation corresponding to p = 2 :
Since p = 2 and
r= r ' • Jp2 + cI- = eR
we have
which is shown in Fig. 6.1.
Also,
a = R (0) = log d
(5.64)
(6.28)
(6.29)
(6.30)
1
1
a=2 1 1
: a = _1 ___ -
161
F (R)
d 11-.-----
R~(~O)~=~I~O-9-d------------O~----=P~j~~~--R
Fig. 6.1 Graph of F(R) = r ~-a = e (2-a)R
for ~ = 2, a = 1 "'5
where
Hence, r.
b-a=log..l. d
From equation (6.29) ,
162
F(2n) (R) = (2_a)2n e<2-a)R ~ 0
which also stands for a = 2, when
F(R) .. 1
r2n)(R) .. 0
For a=l,
is a strictly increasing function of R, so that
and
min I F(2n) (R) I = F(2n) (a )
a:fiOR:fiOb
= r2n) (log d)
=d
max I r2n) (R) I = r2n) (b )
a:fiOR:fiOb
For a=2,
= r2n) (log r . ) J
= r. J
= J""p 2=-. -+-d-=-2 J
where a ~ T ~ b
For a ~ 3,
(6.31)
(6.32)
(6.33)
(6.34)
(6.35)
(6.36)
(6.37)
(6.38)
(6.39)
163
is a strictly decreasing function of R, so that
and
mm I r 2n) (R) I = F(2n) (b )
a;;;;'R;;;;'b
2 (2-a)logr. = (2 _ a) n e J
= (2_a)2n r. 2-a J
max I r2n) (R) I = r2n) (a)
a;;;;'R;;;;'b
= (2_a)2n e(2-a) logd
= (2_a)2n d2- a
To sum up, we have Table 6.2.
Table 6.2 Minimum and maximum value of F(2n)(r)
(a ~ T ~ b) for f3=2, a=1-5
(2n) (2n) a min F (r) max F (r)
a;;;;' ,;;;;, b a;;;;',;;;;'b
1 d rj
2 0 0
3 r.- 1 J
d- 1
4 22n r.- 2 J
22n d-2
5 32n r.-3 J
32n d-3
(6.33)
(6.40)
(6.41)
164
where
From equations (6.21) and (6.32) ,
r. r. log} lo}
d ~(e gd)2" P" - 4 n Sn
If we define
r . ..!.. = 10~ d
r. log..!.. = k log 10
d
- 2.30 k
For nearly singular integrals d ~ Pj
r j ...r;;;:;;; _ P j
d = d d ~ 1
so that k>O.
Hence,
P _ log 10 k ~ ( k e log 10 )211 " 4 n Sn
_ l.S1k (0.7S2k)211 vn n
(6.42)
(6.43)
(6.43)
(6.44)
(6.45)
(6.46)
From Table 6.2 and equations (6.22), (6.23) and (6.46), the lower bound Ln and
the upper bound Un for the error En(F) of applying the n-point Gauss-Legendre
quadrature formula and the radial variable transformation
R (p) = log vi l+ d2 • (P=2)
to the integral
(5.64)
(6.2)
165
is given by Table 6.3.
Table 6.3 Lower bound Ln(F) and upper bound Un(F) of numerical integration error En(F) for p = 2
a Ln(F) Un (F)
1.81 kd ~0.7:2k In 1.81 krj [O.7~kr 1 Yn Yn
2 0 0
1.81k to.7:2k r 1.81 k. [O.7:2k r 3 rjVn dVn
1.81k [1.5:k r 1.81k [l.~k t 4 r.2Vn d2vn J
1.81k [2.3:kr 1.81k (2.~k) 2n 5 r.3 Yn d3 vn J
To sum up, for p = 2, since
equation (6.40), (6.41) and (6.42) gives
. { 0.783 (a-2) k} 2n = -----+ 0 n n~ CD
where
(6.24)
(6.42)
(6.47)
r. log ..1. d r.
k = - 0 434 log..l log 10 . d
166
(6.48)
Table 6.3 and equation (6.47) show that the adaptive logarithmic radial variable
transformation (log-L2)
R (p) = log J l+ d2 ({3=2) (5.64)
in combination with the Gauss-Legendre quadrature formula has the very good
convergence behaviour of order O(n -2n), for the integration in the radial
variable:
(a=1.2.3.···) (6.2)
where n is the number of integration points for the transformed radial variable
R. This explains why the radial variable transformation R(p) corresponding to
f3=2 (log-L2) works well for nearly singular integrals of the type
(a=1.2.3.···)
f21l: fPPl P = d8 -dp
o 0 ra (6.49)
6.3 Case 8=1 Transformation
For the radial variable transformation corresponding to f3= 1 (cf. Table 5.1) :
(6.50)
we have
r = r' = R (6.51)
F{R) = r P- a = R l - a (6.52)
167
and
a = R (0) = d
"" r. J
From equation (6.52), for a = 1 ,
F(R) = 1
r2n)(R) = 0
and for a ~2 ,
r 2n)(R) = (2n+a-2)! R-2n-a+1 > 0 (a-2) !
Since F(2n)(R) is a strictly decreasing function of R E [ d, rj ] for a~ 2,
and
for a ~ 2.
Since
min r 2n) (R) = r 2n) (r . ) J
a;:;;; R;:;;; b
=' (2n+a-2)! r. -2n-a+l > 0 (a-2)! J
max r2n) (R) = r2n) (d)
a;:;;;R;:;;;b
b-a = r.-d J
= (2n+a-2)! d -2n-a+l > 0 (a-2) !
equation (6.21) gives
p -n
and
r. - d _J_
4
(6.53)
(6.54)
(6.55)
(6.56)
(6.57)
(6.58)
(6.59)
(6.60)
where
168
(2n+a-2)! (a _ 2) ! = (2n)! {a (n)
- e -2n (2n)2n v' 411"n {(n) a
(2n+a-2)! (a(n) = (2n)! (a - 2) !
as shown in Table 6.4 .
Table 6.4 fa(n), «(3= 1, a ~ 2)
a fa
2 1
3 2n+l
4 (2n + 1)(n + 1)
5 {2n + 1Hn + 1H2n + 3} 3
From equations (6.22), (6.23), (6.57), (6.58), (6.60) and (6.61), for a~ 2,
d 1--
Ln(F) - ~ (r. _ d) r .1-a {(n) ( __ r_J )2n 2 J J a 4
r . ..l. -1
11" 1 a (d )2n U (F) - - (r. - d) d - ((n) --n 2 J a 4
(6.61)
(6.62)
(6.63)
(6.64)
Hence, to sum up for the radial transformation R(p) corresponding to {3= 1,
169
for a=l
and for a~2 ,
where, for nearly singular integrals, for which
d o < ~ 1 or r. J
r. .1. ~ 1 d
L (F) _ (~)2n ~ 0 n 4 n-+IZI
U (F)_(~.2)2n ~ 00
n 4 d n-+ IZI
(6.65)
(6.24)
(6.66)
(6.67)
(6.68)
This suggests that the {3 = 1 transformation does not work as efficiently as
the {3 = 2 (log-L2), since
(1)2n (1)2n E(F)- - ~ -n 4 n
(6.69)
except for the case when a = 1, for which
This is shown in the numerical results (Chapter 10) where the {3= 1
transformation is out-performed by the {3=2 (log-L2) transformation, except for
the case of weakly singular integrals (d = 0, a = 1) .
6.4 Case 8=3 Transformation
For {3 = 3 (cf. Table 5.1) ,
so that
1 R(p)=- ~
v p-+ d-(6.70)
and
Since
and
1 a = R(O) = -
d
170
1 1 b=R(p.)=- =--
J ~ rj
1 1 b -a =
d r. J
1 r=r'=--
R
F(R) = ,.p-a = (_R)a-3
Hence, for
a$;; 3 ( a-3) n>-• 2
r2n)(R) iE 0
En (F) "" 0
For a = 1,2
.....l2) (2n+2-a)1 2n l" n (R) = . (_R)a-3-
(2 - a)!
Since
1 1 ;:;; -R ;:;;
r. d J
(6.71)
(6.72)
(6.73)
(6.74)
(6.75)
(6.76)
(6.77)
(6.78)
(6.79)
where 0 <d ~ rj for nearly singular integrals, and F(2n)(R) is a strictly increasing
function of
RE[-~.-.!..] d r. J
one obtains
and
Here
where
or
min
171
r2n)(R) = r2n)(--d1 )= (2n+2-a)! d2n+3-" > 0 (2-a)!
max r2n)(R) =r2n)(_~)= (2n+2-a)! r. 2n+3-" > 0 . r. (2-a)! J
J
(2n+2-a)!
(2-a) ! = (2n)! (,,(n)
(,,(n) = (2n+2-a)!
(2-a)!(2n)!
From equation (6.21) ,
1 1 - --d r. p _ __ J
n 4
Thus, fornearlysingularintegrals (0 < d ~ rj ) and a=l,2 ,
and
d 1--
- - - _ - d 3-" {(n) __ J 1r ( 1 1) (r .) 2n 2 d r. " 4
J
(6.80)
(6.81)
(6.82)
(6.83)
(6.84)
(6.85)
(6.86)
(6.87)
172
U (F) = p ~2n)(_~) n n r.
J
(6.88)
where
L (F) ~ E (F) ~ U (F) n n n (6.89)
To sum up for R(p) ; {3=3, good convergence is expected for a= 3,4,5, ...
( for planar elements), while for a = 1, 2, equations (6.87), (6.88) and (6.89) suggest
that the {3 = 3 transformation does not work as well as the {3 = 2 (log-L2)
transformation, for which it was shown in section 6.2 that
(1 )2n E(f) - L (F) - U (F) - -
n n n
6.5 Case (]=4 Transformation
For {3=4 (cf. Table 5.1),
so that
where
1 R(p) = -----
2 (p2 + d2 )
1 a=R(O)=--
2 d 2
1 b=R(p.)=---
J 2 r. 2 J
(6.90)
(6.91)
(6.92)
(6.93)
(6.94)
and
Since
or
1
r = r' = (-2R) 2
a -- 2
F(R) = r ,p-a = (-2R) 2
173
~-2 ~-2 = 2 2 (-R) 2
3 3
F(R) = 2 2 (-R) 2
1 L
F(R)= 2 2 (-R) 2
F(R)= (-R)o = 1
1 1
F(R)=2 2 (-R) 2
etc.
Hence, for a = 4, (n ~ 1 )
F"2n) (R) iE 0
and
En (F)'" 0
For a'* 4,
(6.95)
(6.96)
(6.97)
a = 1 (6.98)
a=2 (6.99)
a=3 (6.100)
a=4 (6.101)
a=5 (6.102)
(6.103)
(6.104)
174
~ -2 { 2n ( ) } p.<2n) (R) = 22 Dl i- 1-i
1
2n+2-~ (-R) 2
Hence I F(2n)(R) I is a strictly increasing function of
so that
min = I p.<2n) ( __ 1 ) I 2d2
and
max I p.<2n) (R) I = I p.<2n) ( - 2~. 2) I J
From equation (6.14),
( b -a )2n+l p.<2n) (R)
E (F) = - e n 2 n (2n)!
for some RE [a, b] , where
e = n
22n+ 1 (n!)4
(2n+1){(2n)!}2
using Stirling's formula of equation (6.16).
Hence,
for some R E [a,b] ,
(6.105)
(6.106)
(6.107)
(6.108)
(6.109)
(6.110)
:so that,
where
175
L (F) IE min IE (F)I n
t ... n
n
max IE (F)I n .
7r rj 4-a { ( d )2} 2n+1 ( rj )4n =---1-- t-
4 d2 r. n 2d J
(2n) !
= n 11+ 2-:a I i=l 2,
For nearly singular integrals
:50 that
and
d o < - ~ 1 r.
J
L (F) - - d 2-a t -7r (1)2n n 4 n 4
4-a
U (F) - ~ -j- t r (2rJd' )4n n 4 d2 n
to can be estimated by the following theorem.
(6.111)
(6.112)
(6.110)
(6.113)
(6.114)
(6.115)
Theorem 6.1
as n - 00
(1) Case a>O
Since
t = n(1+~) n i=l '
let us define
and
where
f(+O)=+oo
f' (x) a = --- < 0
x(x+a)
176
so that f(x) is a strictly decreasing function of x>O, as shown in Fig. 6.2.
Hence we have
where
I log( 1 + ;) dx = (x+a) log (x+a) - x log x
so that
(6.116)
(6.117)
(6.118)
(6.119)
(6.120)
(6.121)
(6.122)
(6.123)
(6.124)
117
f (X)
o 1 2 2n 2n + 1
Fig. 6.2 f (x) = log (1 + ~ ), a > 0
178
f2n+ 1 ( ) §. ~n == 1 log 1 +; dx
= (2n+ 1 +a) log (2n+ 1 +a) - (2n+ 1) log (2n+ 1) - (1 +a) log (1 +a)
and
= (2n+a) log (2n+a) -2nlog2n -aloga
From equations (6.118), (6.123), (6.125) and (6.126), one obtains
where
and
t < t < t -n n n
S2n t == e-1
-n
= (2n+ 1 +a) 2n+l+a
(2n+1)2n+l (l+a)l+a
{e (2n+1 +a)r n _00 (1 +a)l+a
t n
=
(2 e) a
(l+a)l+a
(2n+a)2n+a
(2n)2n aa
(6.125)
(6.126)
(6.127)
(6.128)
n -+00
( 2e)a a - - n
a
179
From equations (6.127), (6.128) and (6.129) ,
for a> 0
(2) Case a=O
(3) Case a<O
Let b=-a. Then,
2n
11 - 71 t = n b > 0 n
i=1
(6.129)
(6.130)
(6.131)
(6.132)
If b is an integer less than 2n, tn = 0 < O(na) , so let us assume that b is not an
integer and let
(6.133)
where
4-1 I b I S~-1 = ~1 ltJg 1- i (6.134)
and
180
2n ( b) S!n = L wg 1--:-i=k '
(6.135)
where k is defined as the integer which satisfies
k-l < b < k (6.136)
Then, if we define
b>O (6.137)
we have
f{b+O} = - 00 (6.138)
f{+ 00) = - 0 (6.139)
and
b f' ( x) = > 0 for x > b
x{x-b) (6.140)
i.e. fix) is a strictly increasing function of x for x > b, as shown in Fig. 6.3 .
Hence,
where
and
where
so that
S 2n < S2n < S 2n -Ie Ie Ie
S2n = II J
2n + 1 ( b) wg 1- - dx Ie x
(6.141)
(6.142)
(6.143)
(6.144)
I I I I I I
181
bl k k + 1 2n 2n + 1 ----~---+~~--~------~~--~--~X
o
f (x)
Fig. 6.3 f (x) = log (1 - ~ ), b > 0
182
§.!n = (2n-a) log (2n-a) - 2n log 2n
-(k-a-l) log (k-a) + (k-l}logk (6.145)
and
s!n = (2n+l-a) log(2n+l-a) - (2n+1) log (2n+1)
-(k-a) log (k-a) + klogk (6.146)
From equations (6.132), (6.133), (6.141), (6.145) and (6.146),
-t < t < t -n n n (6.147)
where
t E1.-1 + S2n
1 -k "" e
(6.148) -n
-t
n
E1.-1 + S 2n Eel k (6.149)
and
E1.-1 k-ll
bl e 1 = n 1--:-
i=l Z (6.150)
(2n_b)2n-b kk-l ----
(2n)2n (k_b)k-b-l
(1-~) 2n 2n (k_b)b (6.151)
= (2n_b)b (l_~)k-l
k
- {e(k-b)}b (l_~)l-k n-+ 00 2n-b k
( b)l-k {e(k-b)}b -b - 1-- -- n
k 2
(2n+ 1-b) 2n+ 1- b
(2n+ 1)2n+1
(1 __ b _)2n+1 2n+l
183
(k_b)k-b
= -----(2n+l-b)b
( ek )b(I_~)b-k 2n+l-b k
Hence,
which imply that
for a < 0
From equations (6.130), (6.131) and (6.156),
(6.152)
(6.153)
(6.154)
(6.155)
(6.156)
184
t - 0 ( n a) for V a E R n (6.157)
Q.E.D.
Returning to equations (6.113), (6.114) and (6.115), for the radial variable
transformation R(p) corresponding to (3=4, Theorem 6.1 implies that
Hence, for a * 4
-'--')" 0 n_ CO
and
for nearly singular integrals i.e.
d o < ~ 1 r.
J
where
(6.158)
(6.159)
(6.160)
(6.161)
To sum up, the radial variable transformation R(p) corresponding to {3=4
is not expected to give good results for the integration of
(6.2)
compared to the (3=2 (log-L2) transformation, except for the case of a=4 for
planar elements, when En(F) = 0 .
185
6.6 Case (3 = 5 Transformation
For f3=5,
R (p) = - 1
3Jp2+tf 3 (6.162)
(cf. Table 5.1), so that
1 a = R (0) = -- (6.163)
3rt
1 b=R(p.)=
J - 3r. 3 (6.164) J
where
rj ... JPj 2+ d2 (6.165)
and
b -a = i(~-r~3) (6.166) J
Since
1
r = r' = (-3R) 3 (6.167)
5-a 5-a
F(R) = ,.p-a =3 3 (-R) 3 (6.168)
for a = 5,
F(R) = 1 (6.169)
and
r2nl (R) = 0 (6.170)
so that
En (F) = 0 (6.171)
186
For a::;: 5 ,
~{ 2n (2 )} ~-2n F 2n)(R) = 3 3 Dl i-a+i (-R) 3 (6.172)
Hence, 1F<2n)(R)1 is a strictly increasing function of
R E [- 3~ • - 3:.3 1 (6.172) J
so that
min IF2n)(R)1 = IF2nl(- 3~)1 a;;;;R;;;;b
(6.173)
and
(6.173)
From equations (6.108) and (6.109),
( b-a )2n+l 7r F2nl(R)
E (F) - - - ---n 2 22n (2n)! (6.175)
for some R E [a,b] ,
where
(6.176)
Hence,
L (F) ;;;; IE (F)I ;;;; U (F) n n n (6.177)
where
Ln (F) 5 min
a~R~b
IE (F)I n
187
-( d)3 2 Tr { ( d )3} {I ;:: } 2n { 2n 3 - a 6~ 1- ~ 4~J d 6n + 5- a Dl 1+ -i-
Since 2
2n 3- a n 1+i
i=l
from Theorem 6.1 ,
L (F) -n
and
2 _ n 3- a
for nearly singular integrals i. e.
d o < <as 1 r.
J
o
(6.178)
(6.179)
(6.180)
(6.181)
Hence, the transformation R(p) corresponding to f3=5 is expected to give
poor results compared to the f3=2 (log-L2) transformation for the integration of
(6.2)
except for the case of a = 5 for planar elements, when En(F) = 0 .
188
6.7 Summary of Error Estimates for 8=1-5 transformations
To sum up, we have been considering the error estimate En when applying
the Gauss-Legendre formula after applying the radial variable transforamtion
pdp = r'P dR ({1=1-5) (5.46)
to the integration in the radial variable
fPj P J= -dp; (a=1-5)
o ra (6.2)
which occurs in the nearly singular integral
I = f dB S ra
f2n: fP' - dO J!!... dp o 0 ra (6.49)
where S is a planar element, so that
r = r ' • ";p2+ d2 (6.3)
The error estimate En for the radial variable transformations: {3 = 1-5 , is
given for different orders of near singularity: a = 1-5, in Table 6.5 .
Table 6.5 Error estimate En of integration using radial variable
transformation {3 = 1-5
~ 1 2 3 4 5
1 0 4-2n < En < (r./4d)2n fV tV J
2 n-2n 0 n-2n
3 4-2n < E < {r'/(4d)}2n rv n", J 0
4 4 -2n ~ En ~ {(r/(2d)}4n 0 4 -2n < En < {r./(2d)}4n '" 'V J
5 4-2n < E < {r./(4t d)}6n '" n '" J
0
189
Here,
(6.31)
and r. ..!.. ~ 1 d
(6.45)
:for nearly singular integrals.
From Table 6.1, the {3=2 transformation:
R(P) = log J p2 + d2 (5.64)
gives good convergence for all near singularities a=1-5 of the type in equations
(6.2) or (6.49), as will be confirmed by numerical results in Chapter 10 .
The {3=3 transformation also seems promising for a~ 3. However, it should
he reminded that the above error analysis was performed for planar constant
,elements. The effect of curvature and high order polynomial interpolation
functions in the integrand are not taken in to account. For instance, although the
,error is En=O for {3=1, a=l for planar constant elements, it is found in the
numerical results in Chapter 10 that, the {3 = 1 radial variable transformation
(6.50)
does not give accurate results compared to the {3 = 2 (log-L2) transformation
(5.64)
for the integration
I = I dS s r
(6.182)
over a curved quadrilateral patch. This seems to indicate that the n -2n
convergence of the {3 = 2 (log-L2) transformation is robust, not only with respect
to the value of a but also with respect to the curvature of the boundary element.
190
6.8 Error Analysis for Flux Calculations
As will be shown in the numerical results in Chapter 10, the adaptive
logarithmic transformation (log-L2) of the type:
(5.64)
works efficiently for nearly singular integrals arising in the calculation of the
potential u(xs) at a point Xs very near the boundary. However, the radial
transformation of equation (5.64) does not work so efficiently for nearly singular
integrals arising from the calculation of the flux or the potential derivative au/axs
at a point Xs very near the boundary.
This can be explained as follows if we recall the nature of near singular
integrals discussed in Chapter 3 37.40. Since in the vicinity of the source point Xs
which is very near to the curved boundary,
u· - (3.127)
(3.128)
and au· 1 p cosO
ax 411" r3 8
p sinO
r3
d
r3 (3.129)
a q. 1 p sinO - 0 -3d ax 411" r 5 8
0 + -3d p sinO
r5
1 3d2
1
r3 r5 (3.130)
191
where the potential u(xs) and the potential derivative auJaxs at an internal point
Xs near the boundary S are given by
u(x,) = Is (qu*-uq*)dS (2.45)
and
au I au* aq* - - (q--u-)dS ax - s ax ax , s, (2.46)
Since the near singularity is essentially related to r and the radial variable
p, the nature of integral kernels in equations (2.45) and (2.46) can be summarized
as in Table 3.2. In other words,
where as
Is u* dS - Is
Is q* dS - Is
1 -dS r
I a u* I 1 -dS- -dS+ sax, S r3
I aq* III -dS- -dS+ sax, s r5 S
regarding the order of near singularity.
Since
I2K IPm=(8) ,= dO
o 0 F pdp
where (p,e) are the polar coordinates in the plane tangent to S at Xs
(6.183)
(6.184)
(6.185)
(6.186)
(6.187)
(cf. equation (5.39) of section 5.4 ) , the radial component of the integrals of
I~quations (6.183) to (6.186) can be summarized as
whereas
Is u* dS -+
Is q* dB -+
f ilu* -dB
s ilx s
J "" fPj 1 0
192
p - dp r
(6.188)
(6.189)
(6.190)
(6.191)
where the extra p compared to quations (6.183) to (6.186) accounts for the
Jacobian introduced by the polar coordinate system in equation (6.187), i.e.
dB = pdp dO (6.192)
and
(6.193)
It was shown in the error analysis in section 6.2 that for the radial
integration:
(a=1-5) (6.2)
the adaptive logarithmic transformation ofthe type
(5.49)
corresponding to {3=2 (log-L2) reduces the error En of the radial numerical
integration by the Gauss-Legendre formula to an order of
(6.194)
193
where n is the number of integration points in the radial variable. This
guarantees the efficiency of the logarithmic transformation (log-L2) of equation
(5.64) for the integration of the radial integration Ja , (a = 1-5) of equation(6.2),
and hence of the integration
I = J dS a S ra
(a=1-5) (6.195)
and
u(xs)= Is (qu*-uq*)dS (2.45)
for potentials near the boundary, which involve radial integrations Jl and J3,
as demonstrated by the numerical results in Chapter 10 .
However, for flux (potential derivative) calculations involving the integral
of equation (2.46), radial integrals of the type
(a = 3,5) (6.196)
as well as
JP . P
J ... J_ dp a 0 r a
(a=3,5) (6.197)
are required, as shown in equations (6.190) and (6.191). Hence, let us perform
an error analysis of the numerical integration of equation (6.196) when using
radial variable transformations of the type
pdp = r'P dR (5.61)
where
(6.197)
for planar elements. The integral of equation (6.196) is transformed by equation
(5.61) to give
J a.2 J
R(P} = p r p-a dR R(O)
(6.198)
194
Let the kernel of the integral of equation (6.198) be notified by
F(R) = p r fJ-a
as a function of the transformed variable R.
Since from equation(5.61)
dp
dR = p
and from equation (6.199)
dF dr - = r fJ-a + p (fJ-a) r fJ-a-l dp dp
= {(fJ-a+l)p2 + ~ }r fJ - a - 2
where
dr d ( ~) p - = - Vp~+d~ = --;:::=~ dp dp ";p2+d 2
p
r
Hence, from equations (6.200) and (6.201) , one obtains
dF dF dp = dR dp dR
Hence,
dF I d 2fJ- a I - - -- - +00 dR p=+O p' p=+O
so long a~ d>O. This results from the fact that, from equation (6.200),
dp I dfJ I _ - _ - +00 dR p=+O P p=+O
while from equation (6.201)
(6.199)
(6.200)
(6.200)
(6.201)
(6.202)
(6.203)
(6.204)
(6.205)
195
dF I = d P- a > 0 dp p= +0
for d>O.
Similarly, for higher derivatives
we obtain the following theorem:
Theorem 6.2
For integers n G: 0,
n "n 1-2k L a 1_ 2k p
k=O
where a1_2k n are constants which do not depend on p or R, and
where
m n bi = 1 for m < 1 i=1
Proof
For n = 0 , equation (6.207) gives
...cO) (R) = p-a 0 1" r a1 p
where, from equations (6.208) and (6.209) ,
a~ = Ln (1-2i)} = 1 .=1
(6.206)
(6.206)
(6.207)
(6.208)
(6.209)
(6.210)
(6.211)
196
so that
rO) (R) = F(R) = r P- a p (6.212)
which corresponds to equation (6.199).
For n = 1, equation (6.207) gives
(6.213)
and equation (6.208) and (6.209) give
° a_; = {.n (1-20} tf = tf ,=1
(6.214)
which agrees with equation (6.203) .
For n~ 2 , we use mathematical induction. Assume that equations (6.207)
and (6.208) hold for n =m, i.e.
m ....Jm)(R) = (m+1)p-a-2m '" m 1-2k 1'"' r L. a1_ 2k p (6.215)
.=0 and
a~_2m = {rr (1-20} d 2m
,=1 (6.216)
Then, for n = m + 1 one obtains,
d = dR (rg) (6.215)
df + fdg = dR a dR (6.217)
where
f .. r(m+1)P-a-2m (6.218)
and
(6.219)
197
Since
r = r 1-= ~ p2 t d~
pdp = r'PdR = rPdR
and from equation (6.220)
so that
rdr = pdp
From equation (6.221),
dp r P - =-dR p
and from equations (6.221) and (6.223)
Hence,
and
dr
dR
df
dR
dg
dR
P-l = r
df = dr
dr
dR
= {(m+l)p-a-2m} r(m+2)p-a-2(m+l)
dg dp = dp dR
r P m = - L a~_2k (1-2k) pl-2k
P k=O
m
= r P L (1-2k)a~2k p -1-2k
k=O
(6.220)
(6.221)
(6.222)
(6.223)
(6.224)
(6.225)
(6.224)
(6.226)
(6.227)
198
From equations (6.217), (6.226) and (6.227) ,
F(m+l)(R) =/m+2),B-«-2(m+1) [f {(m+l).B-a-2m+1-2k} a~_2k pl-2k
k=O
+ f (1-2k) d 2 a~2k p -1-2k 1 k=O
m+l Er(m+2),B-«-2(m+l) '" m+l 1-2k L. al _ 2k p
k=O
where, for k = 0
m+l _ d 2 m a 1 - a 1
and for 1 ~ k ~ m
a ~+2~ = { (m+l).B - a - 2m + 1 - 2k + (3-2k) d 2 a;'-.2k}
and for k = m+1
= {.Ii (1-2i)} d 2 (m+l)
,=1
from equation (6.216) .
(6.228)
(6.229)
(6.230)
(6.230)
(6.231)
Hence, equaions (6.207) and (6.208) hold for n=m+1. Since equations
(6.207) and (6.208) hold for n = 1, by mathematical induction they are satisfied
for all integers n~ 1. Q.E.D.
From Theorem 6.2, the following corollaries are obtained.
Corollary 6.2.1
For integers n~ 0 ,
199
(6.232)
Since r = v' p2 + d2 = d for p = 0 and since the term including pl-2n
is the dominant term in equation (6.207) for p= +0 , equation (6.232) is
asymptotically true for p = + 0 .
Corollary 6.2.2
For n ~ 1,
_ 00
Applying Stirling's formula :
to 2n-l n (1-211 i=l
(4n-3) ! -------(2n_l)!2 2n - 1
one obtains
2n-l n 0-2i) - _2{2e)-2n n2n-2
i= 1
which gives
Corollary 6.2.3
For relatively large integers n~ 1,
F(2n)(R) _ -2 d P- a n -2 -I ( nd)2n p= +0 2 e
1 4n-l p
(6.233)
(6.234)
(6.235)
(6.236)
(6.237)
From equation (6.20) , the error En(F) of applying the Gauss-Legendre
formula to the integral J a,2 in the radial variable R in equation (6.198) is
200
E (F) = p F(2n) (R) n n
for some R E [a,b] and for some P E [0, Pj]' where
a = R (0)
b =R(Pj )
and from equation (5.50),
R (p) = (p2+d 2)
2-{3
l-~ 2
1 -log (p2 + d 2) 2
and
for
for
b-a ~ { (b-a) e } 2n p -n 4 n 8n
{3*2
{3 = 2
(6.238)
(6.239)
(6.240)
(6.241)
(6.242)
Equations (6.238), (6.242) and Corollary 6.2.3 suggest that if the value
P E [ 0, Pj] of equation (6.238) is near 0, i.e. 0 ~ P ~ Pj ,
(6.243)
Nothing that for 0 < d ~ Pj < rj "".jp?+d2,
r. d r. J J
({3=1)
log r. - logd - -logd J
({3=2 )
1 1 1 R(p.)-R(O) = + - -
J r. d d ({3=3)
J
1 1 1 --- + - -2 r. 2 2d 2 2d2
({3=4 )
J
1 1 1 + - -
3 r. 3 3 d 3 3 d 3 ({3=5) ,
(6.244) J
201
equation (6.243) suggests that many integration points n are required for the
accurate integration of Ja.2 of equation (6.198), compared to the integration of
Ja of equation (6.4) by the log-L2 ((3=2) transformation, whre En(F) - n-2n as in
equation (6.47) and Table 6.5.
To sum up, although IEn(F)1 '* 00, the fact that
_ = {_l)m-l oo ( dF )<m) I dR p=+O
(m~ 1) (6.245)
has a bad effect on the efficiency of the use of the Gauss-Legendre formula for the
integration of equation (6.198). Therefor it is clear that the radial variable
transformation R(p) of the type
pdp = r'P dR (5.46)
does not stand a good chance of accurately calculating the integral J a.2, (a = 3,5 )
of equation (6.196) and hence th-e potential gradient at an internal point near the
boundary.
This is true even for the adaptive logarithmic radial variable transformation
(log-L2) corresponding to {3=2 :
R (P) = wgJ p2+ d2 (5.64)
which was so successful in calculating integrals Ja for the potential. This fact is
depicted in Figure 6.4 which shows the kernel function
(6.199)
of equation (6.198) for {3=2; a=3,5, i.e.
(a=3 ) (6.246)
where, from equation (5.64) ,
p (R) = .j e 2R _ d 2
r (R) = eR
202
-2R e (a=5) , (6.247)
(5.65)
(6.248)
Comparing Fig. 6.4 with Fig. 6.1 , it is seen that for the flux related kernel
of equation (6.196), the 10g-L2 transformation ofthe equation (5.64) «(3 = 2) has
a problem, since the transformed kernel F(R) = p r (i-a of equation (6.199) has an
infinite derivative at p=O or R= R(O) = log d , so that the Gauss-Legendre
formula applied to the variable R is not expected to work as efficiently as it did
. for the potential related kernel of equation (6.197) which renders a transformed
kernel F(R) = r (i-a as seen in equation (6.25) . This problem can be overcome by
using the log-Ll radial variable transformation which was proposed in Chapter 5.
I I I I I I I
dF : -=+00 I
203
------------------
F (R)
2
3...[3 d 2
dR '\:
--+---~----------~----~~=----R ! log (#)d
R (0) = log d
o
Fig. 6.4 Graph of F(R) = P r ~-(l for ~ = 2, (l = 3, 5
CHAPI'ER 7
ERROR ANALYSIS USING COMPLEX FUNCTION THEORY
In this chapter, we will give more precise error estimates for the numerical
integration in the radial variable, using complex function theory 42.52. This also
renders a firm theoretical basis for the optimization of the radial variable
transformation.
7.1 Basic Theorem
The error
of the numerical integration formula n
I - In = L Aj f(a j ) j=1
for the integral
I = f 1 [(x) dx -1
(7.1)
(7.2)
(7.3)
over the interval J = ( -1, 1) on the real axis is given by the following
theorem 43-48, considering fez) as a function of the complex variable z .
205
Theorem 7.1
If fez) is regular on K= [ -1, 1],
1= ~ f 'l'(z)f(z)dz 21rl C
I = ...:.. f 'l' (z) fez) dz n 2tri C n
where
'l' (z)
(7.4)
(7.5)
(7.6)
(7.7)
n A. 'l' n (z) = I _J (7.8)
j=1 z-aj
and the path C of the complex integrals encircles the integration points at' a2,"',
an in the positive direction, as shown in Fig.7.1, and there are no singularities of
the function fez) inside the path C .
cI>n(z) is called the error characteristic function of the numerical integration
formula of equation (7.2) .
206
.. .. .. I
Fig. 7. 1 The integration path C
207
For the Gauss-Legendre fonnula, which we are using,
q, (Z) = n
p (x) _n_ dx z-x
1 k+l
Z
i p (x) dx n
(7.10)
Here, Pn(x) is the n-th order Legendre polynomial defined on the interval
J = (-1, 1), and we adopt the zero points of this polynomial as the
integration points a p a2, ••• , an .
7.2 Asymptotic Expressions for the Error Characteristics Function cI>n(Z)
In order to derive theoretical error estimates of the numerical integration
using Theorem 7.1, asymtotic expressions for the error characteristics function
cI>n(z) as n~ 1 and / or I z I ~ 1 becomes necessary. In the following, we give
known asymptotic expressions for cI> n(z) for the Gauss-Legendre fonnula 46,45,47 •
(1) Case I z I ~ 1
where
and
q, (z) = c -2n-l Z n n
b-2 = n
c =
2n3 + 3n2 -n_l
(2n+3) (2n-l)
n (2n) ! (2n+ 1) !
Using Stirling's fonnula :
(7.11)
(7.12)
(7.13)
(7.14)
for n ~ 1 , we obtain
c -n
(2) Case n ~ 1
7r 2; , 2
208
(n~l ) (7.15)
(i) For all z E C except for an arbitrary neighbourhood of the real segment
K=[-1, 1],
w (z) - 27r (z + ~ )-211-1 n
(ii) For all z except for an arbitrary neighbourhood of z= 1 ,
K (2k~ W (z) _ 2e- il( ......;..0 __
where
n 10 (2k~)
z = e i7r cosh 2~
1 k= n+-
2
(7.16)
(7.17)
(7.18)
(7.19)
and Io(z) and Ko(z) are the modified Bessel function of the first and second
kind, resectively.
7.3 Use of the Elliptic Contour as the Integral Path
In the estimation of the numerical integration error En(J) by Theorem 7.1,
it is often useful to take the ellipse el1 :
I z + ~I =11, (11)1) (7.20)
as the path C of the complex contour integral in equation (7.6).
209
The ellipse €q has the foci at z = ± 1 and encircles the interval
K = [-1, 1]. In fact, it collapses to the interval K = [-1, 1] when q = 1 in
equation (7.20). The major axis of €q is
1 1 x=-(q+-) o 2 q
(7.21)
and the minor axis is
1 1 Yo = 2" (q - ;) (7.22)
If the function fez) is regular inside € q, equation (7.6) of Theorem 7.1
and the asymptotic ( n~ 1) expression of cl>n(z) in equation (7.16) gives 48
E (f) - ~ 1 f(z) dz n i J&(z+-v?:l)2n+l
which gives
where e (€q) is the length of €q.
Since
(n~1) ,
1 1 1 e(tq);;:;; 21r·-(q+-)=Ir(q+-) < 2lrq
2 q q
we have
(7.23)
(7.24)
(7.25)
(7.26)
Hence, it becomes important to estimate the maximum value of q (size of
the ellipse €q), such that there are no singularities of fez) inside the ellipse €q.
In order to do so, we derive the expression of q=q( x, y) for the ellipse €q
which passes through the point z = x + iy, in the following.
Equation (7.20) is equivalent to
(7.27)
which gives
x2 l - + = 1 (7.28) s s-1
210
where
{II } 2 s= 2(q+;;) > 1 for q > 1
Solving equation (7.28) for s under the constraint s> 1 gives
x2 + i + 1 + ..J {(x+l)2+i} {(x-l)2+i} s =
2
(7.29)
= {..J(x+1)2 + i ; ..J (x_l)2 + i r (7.30)
Next, solving equation (7.29) under the condition (i> 1, s> 1 gives
q = y + ..J;'2:'1 (7.31)
where
.j(x+ 1 )2 + i + J (x-l )2 + i y=Vs= >1
2 (7.32)
Note that (i(Y) is a strictly increasing function of
y> 1 and a( 1 ) = 1.
7.4 The Saddle Point Method
Another technique which proves useful in the theoretical estimation of the
numerical integration error using Theorem 7.1, is the saddle point method 43, 44 •
The saddle point s of a complex function fez) is the point z=s at which
j( z) is regular and
f'(s)=O, f"(s)* 0
As long as j( z) *" 0, we can express j( z) as
f(z) = exp g(z)
which gives
f'(z) = g'(z) expg(z)
Since
g'(s)=O ~ f'(s)=o
the saddle points of g(z) and fez) coincide.
(7.33)
(7.34)
(7.35)
(7.36)
211
In the neighbourhood I z-s I ~ 1 of z=s,
f(z) = expg(z)
- expg(s) exp { ~gll(S) (Z_S)2}
Let
gil (s ) = 1 g "( s) 1 eia , a = arg gil (s), - 11: < a ~ 11:
z_s=ri8 , r= Iz-sl, 8=arg(z-s)
Then,
1 g "( s) 1 g (z) - g (s) + r2 e i (a + 28)
2
f (Z) = expg(Z)
{ gll(S) 2 } - exp g (s) exp -2 - r cos (a + 28 )
{ g"(s) } exp i -2- r2 sin(a + 28)
Now, let us consider the complex contour integral
1= Icf(Z)dz
(7.37)
(7.38)
(7.39)
(7.40)
(7.41)
(7.42)
along a path C. Let z=s be the saddle point of fez), where j(z) is expressed
as
f(z) = exp g(z) (7.34)
in the neighbourhood of z=s. Move the integration path C, without crossing
any singular points of fez), so that it passes through the saddle point z=s in
the direction
a 11: 8=--±-
2 2
( ± depends on the direction of the path ), so that
f ( z) - exp g ( s) exp - 1 z - s 1 2 { } { Ig"(s)1 } 2
(7.43)
(7.44)
212
and
in the neighbourhood I z-s I < 8 of z=s.
Then,
where
I = Ie f (z) dz
L: exp { - _I.,:;:,g_'~,;,..( S....;..) ,;",1 r2} dr
L: exp { - ....;..1.,:;:,g_'~,;,..(s....;..)_1 r2} dr
--= e 2
j 2lr
1 g "(S) 1 f(s)
g "( S) = [ :2 {log f (z) } I Z = B •
(7.45)
(7.46)
(7.47)
The contribution to the integral I from I z-s I < 8 is dominant, provided
I g"(s) I is sufficiently large. Note that equation (7.46) gives an evaluation
of the complex integrall by the information of j(z) at its saddle point z=s.
If there are several saddle points of j(z), the sum of the last expression in
equation (7.46) is taken for these saddle points.
7.5 Integration in the Transformed Radial Variable: R
Using Theorem 7.1 and the techniques mentioned above, we will derive
theoretical error estimates for numerical integration using different radial
variable transformations.
The essence of the radial component of the boundary element integrals
for three dimensional potential problems can be expressed as
IPj / 1= - dp
o ra (7.48)
213
where r=vI P 2+d 2 for planar elements. a and 0 are given in Table 7.1,
according to Table 3.3. Pi = Pi (6) is the upper limit of the radial integral
in equation (5.42).
Table 7.1 Nature of nearly singular kernels of the radial component
integrals in 3-D potential problems
a 0
u* 1 1
q* 3 1
au* 3 1 -axs 2
aq* 3 1 axs 5 1
2
The application of the radial variable transformation to equation (7.48)
gives
fR(Pi / dp 1= --dB
R(O) ra dB (7.49)
Further, transforming R to x so that the interval R : [ R( 0 ), R( Pi)] is
mapped onto x: [-1 , 1], we obtain
where
and
or
I = 1 1 / dp dB dx = f 1 {(x) dx -1 ra dB dx -1
2R - { R Cp .) + R ( 0 ) } x = ___ .....:....J ___ _
R(p.)-R(O) J
(7.50)
(7.51)
(7.52)
so that
R=
dR
214
{R(Pj)-R(O)}x +R(pj)+R(O)
2
R(Pj)-R(O) -= --=----dx 2
(7.53)
(7.54)
Now, Theorem 7.1 and the related techniques can be applied to the
numerical integration of j(x) in equation (7.50) using the Gauss-Legendre
rule.
7.6 Error Analysis for the Identity Transformation: R(p)=p
First, theoretical error estimates will be derived for the basic case of
the identity transformation : R(p) = p, which is equivalent to using just
polar coordinates in the projected plane S (cf. Chapter 5) without any
radial variable transformation. This will clarify the nature of the (radial)
near singularity and the difficulty which results from applying the Gauss
Legendre formula directly to the radial variable p.
Since
we have
and
so that
where
p.(x+l) R (p) = P = ....:J'--__
2
dp Pj -=-dx 2
1 1
r = P j (x + 1 + 2 D 02 (x + 1 - 2 D 02 2
(7.55)
(7.56)
(7.57)
(7.58)
(7.59)
215
is equivalent to the source distance relative to the element size. This
relative source distance : D is the parameter which essentially determines
the degree of nE~ar singularity.
Hence, we obtain
where
or
where
I = J ~ dp = f(x) dx fP . lJ f 1
o r a -1
E (Pj )lJ+1-a fez) = -
2 {z- (-1+2Di) }a/2 {z- (-1-2Di) }a/2
a a
lJ 2 2 f(z) = A (z+ 1) (z-z) (z-z1)
(P. ).l"+1-a
AiiE ..l.. z
The singularities of fez) are 2d
z i5 -l+i- = -1+2Di 1 p. .
J
- 2d z = -l-i- = -1-2Di
1 p. J
Z1 and Z1 are also branching points when a is odd.
The estimate for numerical integration error is obtained by
(7.57)
(7.58)
(7.59)
(7.60)
(7.61)
E (f) = ~ f 4> (z) f(z)dz (7.7) n 21ft C n
where the path C for the complex contour integral is taken as an ellipse
Co:
I z + Jz 2 - 1 I = (1 , «(1>1) (7.20)
which does not eontain the singular points z1 ' Z1 inside or on the ellipse, as
shown in Fig.7.~L
216
Zt
Fig. 7.2 Singularities of f(z) and integration path Ca for the identity radial variable transformation: R(P)=P
Using the asymptotic expression
<t> (z) - 2lr (z +Ji-l )-2n-l n
we obtain
217
IE (f) I ;;;; 2lr 0'-2n max I f(z) I n zEC 11
(1) Estimation of the size (j of the ellipse Co
(7.16)
(7.24)
First, we will determine the size (j of the ellipse Co which passes
through the point
O<t<l
= x+yi (7.62)
It is clear that this ellipse Co does not contain Zl or Zl inside or on itself.
From equations (7.31) and (7.32), we obtain
y (D,t) = .Jl + (Dt)2 +Dt (7.63)
and,
0' (D ,t) = y + ,Ii - 1
(7.64)
so that
>- o (O<t<l) (7.65)
Hence, (J(D, t) is a strictly increasing function of both D and t for
O<t<1 , with the following properties:
O'(D==O) = 1 (7.66)
0'(O<D<l11) - 1 +V2Dt (7.67)
aO' rt __ - (0 <D<l11) - v~- +00 aD 2D D--+O+
(7.68)
218
(1 (D~l) - 4Dt (7.69)
The graph of (J(D, t) vs D for the case t=0.6 is shown in Fig.7.3.
(2) Estimation of max I fez) I zE t: (1
From equation (7.58) ,
so that
I [(;) I = I [(%) I
a a (7.70)
(7.71)
Hence, we need only consider Im(z)!i;, 0 on the ellipse e(1. Since f(z) has a
singularity of order (z - Zl )-a12 at Zl' it is obvious that I f(z) I takes the
maximum value on e(1 when Z is nearest to Zl •
Let Z=Zl +Az. Then, for I Az I <tS 1 ,
a 8--
a
I [(z) I - A8+1-a 2 8- a D 2 I tu I 2
a 8--
a
= 2- 1p.8+1-a D 2 16% I 2 J
Then, for Zt=-1+2Dti, O<t<l, we have
6% = Zt -zl = -2D O-t) i
and
1 t:.z 1 = 2DO-t) .
(7.72)
(7.73)
(7.74)
Hence, for I Az I = 2D I 1- t I <tS 1, which is the case for nearly singular
integrals, where 0 < D-= d/Pj <tS 1 , I j(z) I is nearly maximum at Z=Zt
for all Z E e(1, and
max I[{%)I-I[{%)I zE eO' t
3a a 8--
_ 2- a- 1 p .8+1-a D 2 Il-t 1 2 J
(7.75)
(J
~ 0'
'''1
+-f2
iD
(D~l)
0'''
' 4 t
D
(D>
l)
, ,
, ,
'D
o
0.01
0.
05
0.1
0.15
0.
2 7
'
Fig.
7.
3 G
raph
of
cr(D
, t =
0.6
) vs
D f
or th
e id
entit
y ra
dial
tra
nsfo
rmat
ion:
R( p
)= P
I'.)
.... CD
220
(3) Error estimate En(f)
From equations (7.24) and (7.75), we obtain the error estimate
8- 3a a
1C 2- a p.8+1-a D 2 (l-t) 2 I E (f) I < __ ...;1 ______ _
n "" 0'2n
where 0 < t < 1 .
Since we are interested in cases a=1,3,5 (cf. Table7.1) , a
(I-t) 2<10
implies t < 0.6 .
(7.76)
(7.77)
For the case t=0.6, equation (7.64) gives values of (J(D) for different
values of relative source distance DE dJPj' as shown in Table 7.2 (cf. Fig.
7.3).
Table 7.2 Values of C1(D) for t=0.6, for R(p)=p
D C1
10-4 1.01
10-3 1.04
3X10-3 1.06
10-2 1.12
3X10-2 1.21
10-1 1.42
3X10-1 1.85
1 3.22
For nearly singular integrals whose relative source distance D is in the range
D: 10-3 _10- 1 •
C1 in equation (7.76) takes the value p : 1.04 -1.42
(7.78)
(7.79)
221
In summary, we have obtained an error estimate :
3a 8--
IE(j)I<D 2 n rv
-2n -2n a -a
for the numerical integration
JPo /
1= J - dp o ra
where a=1.04-1.42 for D:IO-3 -10- 1 (7.80)
(7.45)
in the radial variable, (which characterizes the 3-D potential problem,) using
the identity radial variable transformation R(p) = p .
The above theoretical estimate corresponds well with numerical
experiment results, as will be demonstrated in Chapter 10. Equation (7.80)
explains why the use of polar coordinates in the projected plane S, alone,
does not give efficient or accurate results for nearly singular integrals, thus
indicating the necessity of an efficient radial variable transformation R(p) .
7.7 Error Analysis for the tOg-L2 Transformation
In this section, theoretical error estimates will be derived for the
numerical integration in the radial variable using the Gauss-Legendre rule
after applying the tOg-L2 transformation
R(P)= log ..Jp2+d 2 (5.64)
The analysis will clarify quantitatively, the reason why the tOg-L2
transformation works so efficiently for the integration of potential kernels,
while it fails to do so for flux kernels and kernels including interpolation
functions.
Equation (5.64) gives
R(O)= end (7.81)
(7.82)
222
where
(7.83)
and
pdp=r2dr (7.84)
for planar elements, where
r =r' =Jp2+d2 = eR (7.85)
and
(7.86)
Hence,
IP. 6 IR(P.) 6-1 J P J p
1= -dp= - dR o ra R(O) ra - 2
(7.87)
Equations (7.52) and (7.53) for the linear trasformation which maps
R : [R(O) , R(p)] onto x: [-1,1] gives
and
and
2R - tn(r.d) x= __ --=J:...-
tn(r .Id) J
{tn(r.ld)}x+tn(r.d) R= J J
2
dR tn (r.ld) _=_~J_
2
Thus, the integral of equation (7.87) can be expressed as
1= J 1 f(x)dx -1
where
6+1-a tna 2
b ... -(r.d) >0 2 J
a5!2=~>1 d d
(7.88)
(7.89)
(7.90)
(7.91)
(7.92)
(7.93)
(7.94)
223
(1) Case : 0 = odd
When 0 is a (positive) odd number, as in the integration of u* and q*
for the potential, where 0=1 (cf. Table 7.1), the function fez) of equation
(7.92) is regular in the whole complex plane except for z=OO, since (0-1)/2
is a non-negative integer.
Hence, we can take the integration path as C: {z I I z I = R ~ 1 } in
E (f) = 2.. f ~ (z) f(z) dz (7.6) n 2lri c n
of Theorem 7.1, and apply the asymptotic expression of CI>n(z) for I z I ~1
in equation (7.11), and expand fez) in Taylor series as
for z* 00 , so that we obtain the error estimate
En (f) - en ( a2n + b;2 a2n +2 )
Here, we have used the fact that
2.. f zm dz = {I ; m= '-1 2lri cO; m .. -l
(7.95)
(7.96)
(7.97)
for any integration path C that encircles z=O in the positive direction once.
The at in equation (7.95) can be obtained as follows. In equation (7.92),
8-1
n
=L ( 8-1 ) - -It (tna)z
C 2 ( -1)1t 8-1 II e -a
(7.98)
It=o -2
where
c = m! m II (m-k)! k! (7.99)
is the coefficient of binomial expansion. Hence,
224
8-1
n
fez) = b I ( 8-a+l ) ---k (tna)z .
C 2 ( -1)/i 8-1 "e -a
k=O 2
8:1 -1 k ... {( 8 ;+1 -k )<ena)z r = b I ~Ck (-a ) I e!
k=O 2 t=O
since ... t
1= I ~ t=o e!
Thus, we have 8-1
n
a t= b I 8-1 Ck (_a-I)" k=O -2-
{( T-k )ena r e!
so that in equation (7.96),
where
Since
for
a-I "'2 (d )2n
a =b'C ~ 2n f:o 8,1 (2n)! '
8-1 3 2 n (d) 2n+2
-2 2n +3n -n-1 'C 8,k bn a 2n+2 = b L. L ~~-
(2n+3)(2n-1) 1=0 a ... (2n+2)!
8-1 - 2 2n
In (d8,1e) (d8,k) b --c --
n .. l 8n a,k (2n)! 11=0
Ca,le Ei 8-1 C" (_a-I)"
2
d8,k=( 8+~-a -k )ena
2-a 8+1-a 8+1-a -- S; ----k ~ ---
2 - 2 2
8-1 -il:; kil:;O
2
(7.100)
(7.101)
(7.102)
(7.103)
(7.104)
(7.105)
(7.106)
and
225
8-4 8+I-a 8 -s--s-2 - 2 - 2
for a = 1,3,5 (for 3-D potential problems), we have
which gives
18+ I-a 1 max (3,8) -- -k s -----'-
2 - 2 '
(d81e )2 {max(3,8) en(r.fell}2 _'_ s J
8n - 32 n
Hence, for n ~ 1 ,
b - 2 n a2n+2 <lI: a2n
in equation (7.96), so that
En(f) - cn a 2n {I +O(lIn)}
n~l cn a 2n
8+1-a 2n 4
= (r.d)-2- en (r.fell{Un(r./ell} (n!) J J J (2n)! (2n+l)!
cH1-a I IiC -- { e en(r .ld)}2n -y-':"" (r. d) 2 en(r.ld) J
n~l 4 n J J 4n
8-1
n 1e(8-a+I )2n I 8_1CIe (-dlrj ) ---k Ie-O - 2 - 2
using Stirling's formula.
For 0=1, equation (7.111) gives
{(a -2) e en (r .Id) }2n
E(f)- J n 8n '
(7.107)
(7.108)
(7.109)
(7.110)
(7.111)
(7.112)
which matches with equation (6.42), obtained by elementary error analysis
in Chapter 6.
226
For the nearly singular case, where the source distance d is very small
compared to the element size, i.e. 0< DE dip) ~ 1 ,
r. v'p~+d2 .2= J _D-1 d d
(7.113)
(7.114)
so that equation (7.111) for 0': odd gives
E (f) - 8+1-a D n D<IIl Pj
8+1-a 2 2n (n!)"
(-enD) (2lnD) (2n)!(2n+l)!
8-1
n AI (8-a+l )2n L 8-PAI (-D) -2--k AI=OT
~ 8+1-a 2 !.v'~ p. 8+l- a D-2-(_enD)(e enD) n
4 n J 4n
Further, using equation (7.108) for 1~ a~ 5 ,
and since 8-1
8-1 n _
'" AI 2 £... 8-1 CAl (-D) =(1-D) AI=O -2-
we obtain
8-1
n AI(8-a+l)2n L 8_1CAl(-D) -2--k AI=OT
(7.115)
(7.116)
(7.117)
227
In summary, for 0: odd we obtain the following theoretical error
estimates when using the log-L2 radial variable transformation:
For 0: odd
8+1-a 2 -2- {maX(3,8) e enD} n
IE(f) «-enD)D , (n~I,D~I). n tv 8n
(7.118)
For 0=1 2-a -2- {3 e enD }2n
I En(f) I < (-enD) D -- . N 8n
(7.119)
Roughly speaking, rl ----~----~8-+~1--a------2n--------~
I En(f) I ;S D-2-(_e_:D_) - n-2n (7.120)
for 0: odd.
These estimates correspond well with numerical results in Chapter 10,
and explain why the log-L2 radial variable transformation gives efficient
results for the integration of potential kernels u* .and q* (which do not
include any interpolation functions).
(2) Case: 0 = even
When 0 is a (positive) even number, as in the integration of au*/axs
and aq*/axs for the flux, where 0=2 (cf. Table 7.1), the function f(z) of
equation (7.84) has a branch point (singularity) 2lrm
z = -1 + i m en(r.ld)
J
, (m: integer) (7.121)
as shown in Fig.7.4.
In this case, we can modify the condition of Theorem 7.1 to allow a
singularity of f(z) at the end point z= -1 of the interval K=[-1,1] of
integration 45,47. The path C of the contour integral is taken as
Z3 -----------X
Z2 -----------X
-----N-z:i branch lines
-----~---x Z-2
-----------x Z-3
228
o
Fig. 7. 4 Branching singularities Zm of f(z) and integration path for log-L2
transformation, 0 = even
229
(7.122)
as shown in Fig.7.4. ea is an ellipse
I z+.../z2_1 I = (1 (1)1 (7.20)
which has z= ± 1 as its foci, and the branch points zl' z_l are outside the
ellipse. t+ and t_ are the real segment (-xo' -1-e:) in the positive and
negative directions, respectively. 1 1
%=-«(1+-) o 2 (1 (7.123)
is the major axis of ea. C€ is a circle of radius 0<€~1, with its centre at
z=-1.
In the following, we will estimate the error En{f) according to Theorem
7.1, considering the contribution of each component of the path C.
(i) Contribution from the branch line t +! t
In the equation
E (f) = ~ f ~ (z) f(z) dz (7.6) n 21r' C n
of Theorem 7.1, the contribution from the branch lines t + and t _ is
where
Applying the asymptotic expression Ko (2k~)
~n(z+) - 2e±i... • (n~l) 10 (2k~)
where ±" z± = e '" cosh2~ on
(7.124)
(7.17)
(7.125)
and
we obtain
230
KO (2k() W (z ) = W (z ) = -2 ---
n + n' - 10 (2k()
and equation (7.124) can be written as
1 1-1-. Et +. t -=271'i _% {f(z+)-f(z_)}wn(z±)dz
o Next, equations (7.92-94) can be rewritten as
{ }~ (~tna)z fez) =; e(z+1)tna_ 1 2 e 2
2-« 28-« - en a -2- d 2 > 0 b ""2 rj
r, a • ..l>l
d
If we choose xo> 1 such that
%0-1 ~ 1
so that
O<~~l for
we obtain
±' ±'..2 z = e I" cosh2~ - e 1"(1 +2~ )
Hence, in equation (7.128),
so that
and
±i" 2 e(z+l)tna -1 = ee 2~ tna _ 1
(o<~~ 1),
8-1 ,(8-1)" 8-1
{e(Z+1)tna_ 1 }-2- _ e±I-2 -(2ena)-2- ~8-1
( 2-«) «-2 2 -tna z -tna(l+2~ ) '2 2
e - e
(7.19)
(7.126)
(7.127)
(7.128)
(7.129)
(7.130)
(7.131)
(7.132)
(7.133)
(7.134)
(7.135)
(7.136)
231
a-2
- a 2 {l + O(rn . Thus, equation (7.128) can be expressed as
(6-1)"
f~±) - ~ e±i 2 t 6- 1 , (O<t~l)
6-3 6+1
~ lSi 2 2 (ena) 2 d H1 - a .
Hence, in equation (7.127), (6-ll" (6-1)"
f~+)-f(z_) - ~ {e i - 2-_e- i - 2-} t 6- 1 , (O<t~l)
which gives
f(z ) - f~ ) - - t 6- 1 + - B
6+2 8-1 8+1 - - -- ... i(-l) 2 22 (ena) 2 d H1 - a B
Now, equation (7.127) can be expressed as
where
1 I{ {Ko(2kt) } Et +. t _ - 2lri ~ ; t 6- 1 -2 I (2kt) (-4t>dt
'0 0
C iE!
_ 1'0 8 KO(2k() C J- t IO(2kt) dt
2
8+3
2
1 -1 rx;::I { lSi -cosh x -..; ~--o 2 0 2
and we have assumed
n~l
O<E~l
1 i.e. k= n+ - ~ 1
2
i.e. 0< to ~ 1
(7.137)
(7.138)
(7.139)
(7.140)
(7.141)
(7.142)
(7.143)
(7.144)
(7.145)
(7.146)
(7.147)
For
i. e.
232
if we take E > 0 such that
2k~ ~ (2n+1).£ ~ 1 2
2 E~ ---
(2n+ 1)2
equation (7.189) gives
Ko(2k~) 2 -- - -log(k~) - r + O(k~) Io(2k ~)
Hence, if we take the limit as E - 0 ,
.If K (2k~) H1
f 2 8 0 '/1- 0 ( 2 10 ) 0 ~ I (2k~) ""> - E g E -+
o 0
so that equation (7.143) gives
f{o Ko(2k~)
Et+. t _ - ~ ~8 I (2k~) dt o 0
If we further choose Xo so that it satisfies
2k~o= (2n+1)~o - (2n+l)jxo; 1 ~ 1
we obtain
Since
we have
(7.148)
(7.149)
(7.150)
(7.151)
(7.152:
(7.153)
(7.154)
(7.155)
(7.156)
(7.157)
233
Note also that if we let t == 2 k ~ ,
fIX) K (2k~) 1 fIX) K (t)
~8 _0 __ d~ = t 8 _0_ dt _O(k-8 - 1 )
o Io(2k(J (2k) 8+ 1 0 Io(t)
since
is a constant which is independent of k.
From equations (7.157) and (7.158), we have
f ~o 8 Ko(2k~) fIX) 8 Ko(2k~) fIX) 8 Ko(2k~)
o ~ Io(2k~) d~ = 0 ~ Io(2k~) d~ - ~o ~ Io(2k~) d~
_O(k-8- 1 )
Hence, equation (7.152) gives 8+1
Et+.t _ - o{ (ena)-2- d 8+1-a n-8- 1 }
8+1
D~1 o{ (_enD)-2- D8+1-a n -8-1 }
(7.158)
(7.159)
(7.160)
234
(ii) Contribution from the ellipse c,Q.
For the contribution Eta to the error En(f) , from the ellipse c(J of
equation (7.20), we have
I E I < 2rrO'-2n max I f(z) I a zE e a (7.161)
from equation (7.24).
CD Estimation of the size (J of the ellipse
According to equation (7.92), I f(z) I is boumded as long as z is finite
and o~ 1. Hence, the ellipse C (J can be taken as close as possible to the
branch points: 2rr
z =-I±i-±l ena (7.162)
so long as the points Z±l are just outside the ellipse c(J. Thus, we will
estimate the size (J of the ellipse which has its foci at z= ± 1 and passes
through 2rrt
z =-I+i-t en a •
(O<t<l)
From equations (7.29,30), we obtain
0' = q + VI +l + J2q (q+Vl +l)
where rrt rrt 2rrt
qlE-=--= en a rj en (1 +D-2)
en( d)
and D=d/Pj is the relative souce distance.
Equations (7.164,165) give
dO' q l+q+q2+ (1+q)Vl+q2 -=1+ + >0 dq -J1"+;l v'i+7 v'2 q (q + VI + i )
and dq 4rrt -= dD D(D2+1 ){en(1+D-2)}2
(7.163)
(7.164)
(7.165)
(7.166)
(7.167)
and
235
Hence, a(D) is a strictly increasing function of D>O, where ;z;t
a(D~l) -1 +v~--en D
a (D~l) - 8lrt D2 ,
cIa V2;t -(D~l) - ---dD D (-en D)312
- +00 D--+O
(7.168)
(7.169)
(7.170)
Note also that (j(D ,t) is also a strictly increasing function of 0 < t < 1, so
that
a(D,t) < aeD,l) . (7.171)
The table and graph of (j(D,t) for t=1.0 are given in Table 7.3 and Fig. 7.5,
respectively.
Table 7.3 Values of O'(D) for t= 1.0, for the log-L2 transfonnation (0: even)
D (j
10-4 2.37
10-3 2.74
3X10-3 3.02
10-2 3.50
3X10-2 4.24
10-1 5.93
3XlO- 1 10.4
1 36.3
Since 'o'E>O, 3 0>0;
I a(D,t=l) - a(D,t=1-8) I < € (7.172)
we may conclude that, for the nearly singular case, a in equation (7.161)
takes the value
(7.173)
(j
7.0
6.0
5.0
4.0
3.0
2.0
1.0 0
' ,
, I
, 'D
0.
01
0.05
0.
1 0.
15
0.2
""
Fig
.7.5
G
raph
of
(j(D
, t =
1.0
) vs
D
for
the
log-
L2
tran
sfor
mat
ion
(0 :
eve
n)
II.)
~
237
® Estimation of max I fez) I zE to'
We will estimate max I fez) I using z E EO'
and
- - ena z 8-1 (2-a )
f(z) = ; { e( Z + 1) en a -1 } 2 e 2
2-a 28-a a l - - 1 8--
-=~r 2 d 2 - _p,8+1-a(_lnD)D 2>0 b 2 j D<il 2 J
1 1 I x I ~ x ... -(0'+ -)
020'
1 1 I y I ~y ... -(0'--)
020'
(7.128)
(7.174)
(7.175)
(7.176)
where z=x+iy, and xo and Yo are the major and minor axes of the ellipse
€ (J , respectively.
Since, if c is real,
I I I c I %
I eCz I = e C % ~ e 0
we have
I e (2-2 a en a) Z I (.!2.::.!.!.. en a) % .!2.::.!.!.. %
~e 2 o=a 2 0
Similarly, we obtain
since a>l.
( 1) t ( %0 + 1 ) en a Ie Z+ na_1 I ~ 2 e +1
% +1 =2a o +1
Hence, for o!?; 1 , we have
where
8-1 8-1 I 2-a I - 3-2- (%0+ 1 )-2- + -2-%0
max I f (z) I ~ b a zE eO'
8-1
3 2 __ p,8+1-a(_lnD)D ind (a,8)
D<il 2 J
(7.177)
(7.178)
(7.179)
(7.180)
238
cH1-a- (8-1+ I 2-a I )x ind(a,8) E 0
2 (7.181)
Summing up for the contribution from the ellipse CC1, equations (7.161)
and (7.180) give
where
I E I < a
0-1
7r 3 2 p.8+1-a(_enD) Dind(a,o) q-2n , J
q:2.74-5.93 for D:10- 3 -10- 1 .
Comparing equation (7.182) with (7.160), we have
I E a I ~ I Et+,t_ I
for n~ 1.
(iii) Contribution from the small circle C€
(7.182)
(7.173)
(7.183)
Finally, we will estimate the contribution Ec. from the small circle C€ .
From equations (7.92-94), it can be proved that for z= -1+~ such
that I t:.z I ~ 1 , 0-1 0+1
f(z) - : /::,.z 2 + 0 (/::,.z 2 )
where 6+1 6-a+1
_ 1 -2- --2-",,-(ena) d
A 2
As the asymptotic expression for <I>n(z), we adopt
where
2 -in: Ko (2k~) 4> (z) - e (n~ 1 )
n 10 (2k~)
z = - 1 + /::,.z = ein: cosh 2~ , 1
k=n+-2
Noting that 49
Ko(Z) - -lo(Z) (r+ eog i )+ :2 + 0 (Z4) , (I Z I ~ 1)
Z2 10(Z) - 1+7+0(Z'> , (I Z I ~ 1) ,
(7.184)
(7.185)
(7.17)
(7.186)
(7.187)
(7.188)
239
where r ~ 0.577 is the Euler's constant, we obtain
Ko (Z) Z 2 -- - -tog - - r + 0 (Z ) ( I Z I ~ 1 ) 10 (Z) 2
Thus, if we take O<E~ 1 such that I 2~ I ~ 1 for z E C€, we obtain
ell (z) - 2eogk~+ 2r+O(k~)2 n
Since I ~ I ~ 1 , we have
llz = z+1= 1 -cosh2~ - _2~2 ,
Illzl~l,
and
dz = -4~d~ .
Hence, for n~1 and I 2~ I ~ 1, we obtain
Ec = ~ J ell (z) fez) dz ( 27rl C n
(
8+1
- 0 { € ""2 tog (k€) } -::;; 0
Hence, the contribution from CE, (E-O) is zero.
(iv) Summary
(7.189)
(7.190)
(7.191)
(7.192)
(7.193)
(7.194)
In summary, the contribution Et +.t _ from the branch line t+,t_ is
dominant in En(!>. Hence, from equation (7.160), the theoretical error
estimate for the radial numerical integration using the log-L2 transformation
with the Gauss-Legendre rule for the case: 8=even , is given by
8+1
n:C1 0 [{tn(r/d)}""2 d8+1-"n-8- 1 ]
8+1
D-:'1 O{ (_tnD)-2-D 8+1-" n-8- 1}
_ O(n- 8 - 1 ) (7.195)
240
where n is the number of integration points in the radial variable.
This theoretical estimate corresponds well with numerical results in
Chapter 10, and explains why the log-L2 radial variable transformation is
inefficient and gives inaccurate results for the integration of flux kernels
au*laxs and aq*laxs , which include terms corresponding to 0'=2.
Equation (7.195) also explains why the log-L2 transformation tends to
be inefficient for the integration of kernels including interpolation functions
suchas t/>ij of equation (9.17). Since t/>ij{"Il'''I2) are polynomials of "11 and
"12 ' which in turn include first order terms of p as in equation (5.40), t/>ij
include terms of order p, p2, p3, p4, which correspond to 0'=2,3,4,5 in the
radial integral of equation (7.48). According to equation (7.190), this means
that the numerical integration error drops only at the rate of O{n -3) and
O{n- 5) for the components corresponding to 0'=2 and 4, respectively. This
matches with numerical experiment results for Is t/>ij u* dB and Is t/>ij q* dB
in Chapter 10.
241
7.8 Error Analysis for the tog-LJ Transfonnation
In this section, theoretical error estimates will be derived for the
numerical integration in the radial variable using the Gauss-Legendre rule
after applying the tog-Lit transfonnation :
R(,o)= eog(p+d) (5.85)
The analysis will clarify quantitatively, the reason why the tOg-Ll
transfonnation is a robust transfonnation which works efficiently for the
integration of flux kernels as well as potential kernels and kernels including
interpolation functions.
Equation (5.85) gives
R(O) = end ,
R(p.)=en(p.+d) J J
and
peR) = eR_d
dp R -=e dR
(7.196)
(7.197)
(7.198)
(7.199)
Hence, the radial component integral of equation (7.48) can be expressed as
IPj / 1= - dp
o rtll
= IR(Pi (eR_d)6 eR -;::::;::::;::==;;:~ dR .
R(O) J(eR _d 2 )+;1II (7.200)
Then, equations (7.52,53,54) give
Pj 2 R - en (1 + d ) - Un d (7.201)
x=
p. p. X en (1 + ~) + en (1 + ~) + 2 en d
R = -----------(7.202)
2
242
and
dR -= dx
Hence, the integral of equation (7.200) can be expressed as
1= f 1 f(x)dx -1
where
b' (w-l)'" w fez) = ---------
{w- (l_i)}aI2{w_(1+i)}aI2
p. a'IE 1 +..l. > 1
d
fez) has singularities at
~tncJ w lE e 2 =l±i
which are equivalent to ± in2 (4m±112) x
z=z IE-l+-+i----m ina' ina'
( m : integer) .
(7.203)
(7.204)
(7.205)
(7.206)
(7.207)
(7.208)
(7.209)
(7.210)
The singularities z=z± , (m=O, ±1, ±2, ...... ) are also branch points when a
is odd, as shown in Fig. 7.6
Using the relative source distance:
we have
so that
d D=- (7.211)
(7.212)
+ ------------X %1
-----"<-----" %1
branch lines for 0.= odd
~ + ------------« %0
243
-1 _ 0 1 ------------x %0
+ ------------X %-1
------------x %=1
Fig. 7.6 Singularities z~ and branch lines ( for a = odd) of J(z) for the log-L 1 transformation
244
which gives 7C en 2 ± i - (7.213)
± 2 Zo +1- ---enD
Hence, the distance between the nearest singularities z=z± to the end
point z= -1 is of order
o( -:nD) This is the key to understanding why the log-L1 transformation works far
efficiently compared to the identity transformation, where equation (7.61)
shows that the distance between the nearest singularities Z=Zl'Z'l to z=-1
is of order
O(D) ~ 0(_1_) -enD .
(1) Error analysis using the saddle point method
In the equation
E (f) = _1_ f ~ (z) f(z) dz n 27Ci c n
(7.6)
of Theorem 7.1, we can apply the asymptotic expression c
~ (z) _ _ n_ n z2n+1
(7.214)
of equation (7.11) for z E C; I z I ~1.
If we define
F(z)& ~n(z) f(z)
A' (w-l)'" w ( I z I ~ 1) (7.215) - -- ----------
z2n+1 {w-(1-i)}/l/2 {w-(1+i)}/l/2 '
from equations (7.205) and (7.214), where
(7.206)
ena' "'+1 A' E! C b' = - d -II C n 2 n (7.216)
245
we obtain
g(z) - tog F(z)
- tnA' - (2n+ l}tog z + 6tog(w-l) + tog w
and
dg 2n+l tna'{ 8 a( l-i l+i)} - - --- + - 8+1-a+---- +---dz z 2 w-l 2 w-l+i w-l-i
(i) Case Re(z) ~ 1
In this case, tna' -{Re(z)+l)
Iwl=e 2 ~l
where p.
a' ... 1 +..1. > 1 d
If I I m(z) I - 1, we have
1 z 1 -R e(z) ~ 1
which gives
Iwl~lzl~l
or 1 1
T.;T <C T":;T <C 1
Hence, equation (7.218) gives
(Izl ~l) .
, dg 2n+l tna' g(z) - dz - --z- +-2-(8+1-a) , (Re(z)~l)
Thus, the saddle point of F(z) of equation (7.215), which is given by
dg I -0 dz z=.
is 2(2n+l)
s-(8+1-a}tna'
(7.217)
(7.218)
(7.219)
(7.208)
(7.220)
(7.221)
(7.222)
(7.223)
(7.224)
(7.225)
246
The condition Re( s) ~ 1 implies
o+l-a>O, (7.226)
since en a' >0. From Table 7.1, for the basic integral kernels in three
dimensional potential problems, a,o and 0 + 1-a take the values shown in
Table 7.4.
Table 7.4 Values for 3-D potential problem
kernels a 0 o+l-a
u* 1 1 1
q* 3 1 -1
au* 1 -- 3
axs 2 0
3 1 -1 aq*
-3 -- I axs 5
2 -2
Hence, only the case a = 0 = 1 for the u* kernel gives a saddle point of
equation (7.225) satisfying Re(s) ~ 1 for n ~ 1 .
Equation (7.223) gives 2n+l
g"(z) - -2- , z
(Re(z)~l) ,
so that at the saddle point z=s, 2n+l {(o+1-a)ena,}2
g"(s) - ------- T - 4(2n+l)
Equations (7.38) and (7.45) give
a = arg g" (s) = 0
i(+~) dz= e 2 dr= idr
(7.227)
(7.228)
(7.229)
(7.230)
Applying the saddle point method along the contour C shown in Fig.7.7 ,
247
I
------------x ------------x
-----------.:.-~-x~__4----+-) , ....... -...1 0
----------.::--:;(x-----1r-----.:..-IIr---'
------------x ------------x
Fig.7.7 Integration path C for the saddle point method for the log-L 1 transformation
En(f) = 1 fe F(z)dz 27ri
248
1 ii j 27r - - e - F(s)
27ri I g"(S) I
_ j s2 A' s-2n-l W,Hl-a
27r (2n+ 1)
A' = s-2n w8+1- a V27r (2n+ 1)
for a - a + 1> 0 and n ~ 1 .
Since,
we have
2 (2n+l) s-
(8+1-a}ena'
B + 1 tn Q' _2n_+_l_ + _tn_Q'
w(s)= e 2 = e 8 +1- a 2
8+1-a
w8+ 1- a = e 2n +1 a' 2
From equations (7.216) and (7.15) , we have
7r (en a') d H1 - a
A'=------22n+l
Hence, equation (7.231) renders the following error estimate :
For the tOg-Ll transformation, with O'-a+l>O, n~l,
(7.231)
(7.225)
(7.232)
(7.233)
(7.234)
e ~ {(Hl-a)e en(1+ po ld)}2n En(f)- -v~--{en(1+pold)}(dV1+pold)Hl-a () J
4 n+ 112 J J 4 2n+ 1 .
8+1-a _ -2 (enD)2n D~l (- enD) D -;;-
(7.235)
where D == d / Pj is the relative source distance.
This estimate corresponds well with numerical results for the integration of
u* , (a = a = 1) in Chapter 10.
(m Case Re(z) ~ -1
In this case, lno' -{Re(z)+l}
Iwl=e 2 <Iii I
which implies
249
dg 2n+1 tna'{ a( l-i l+i)} - - --+ - 8-a+I-8- - -- + --dz z 2 2 -l+i -l-i
2n+1 tna' =---+-
Z 2
Hence,
dg I - -0 dz .1='
implies 2(2n+l)
s- > 0 tna'
which contradicts with Re(s)~ -1.
(7,236)
(7.237)
(7.224)
(7.238)
Hence, there are no saddle points of F(z) IE III n(z) f(z) in the region Re(z) ~ -1 .
(2) Error analysis using the elliptic contour: €C1
In order to obtain error estimates for general natural numbers a,8,
using equation (7.6) of Theorem 7.1, we turn to the asymptotic expression
4> (z) - 2lr (z + ~ )-2n-l n (7.16)
which is valid for n~ 1 and z E C except for an arbitrary neighbourhood of
K=[-1,1]. The integration path C is taken as the ellipse €C1 :
(11 > I) (7.20)
which does not contain any singularities of f(z) of equation (7.205), inside.
More specifically, we will choose the ellipse eC1 which passes through the
point tn 2 7rt
z --1+- +i--t tna' 2tna'
(O<t<l) (7.239)
which is located just below the singular point :
250
+ en2 " z E -1+- +i--o ena' 2en a'
as shown in Fig.7.S. Then, we can use the equation
IE (f) I ~ 27['Q-2n maf,x I f(z) I n zE
" to estimate the error of numerical integration.
(i) Estimation of max I fez) I zECo
Since
~tna' ;; = e 2
equation (7.205) gives
f(z)=f(z)
which implies
I f(;) I = I f(z) I
Hence, we need only consider the region Im(z)~ 0 .
(7.240)
(7.24)
(7.241)
(7.242)
(7.243)
It is also clear from equation (7.205) that I fez ) I = +00 , where Zo
is the singular point given by equation (7.210). Thus, we obtain
max I f (z) I - I f (Zt) I for I 1- t I <$i 1 zE e"
If we define
we have
so that
(l-t) 7['
t:;z=z-z+=---i t 0 2 en a'
7['
en2 +-i 2
---+t:;z 2 en a'
Zt+ 1 enti -enti -I:.z
w(zt) = e 2 = (1 + i) e 2
(7.244)
(7.245)
(7.246)
(7.247)
251
o -----------o&~---...--
Zi
Fig. 7.8 Singularities of f(z) and integration path Co for the log-L 1 transformation
Hence, for
we have
2 It:.z I~-
ena'
252
W(Zt) = (1+i) {1+ e~a't:.z+ oC~a't:.zy}
so that equation (7.205) gives
where
Thus, for
where
a a
f(zt) = B' [t:.z -2 + e~a' {1+ (1-00} t:.z1- 2
a a 3 . -1-- 1-- --alf'
B' = (i+l) i J 2 4 d H1 - a (ena') 2 e 8
2 It:.z I~-
ena'
a
I f(zt) I - I B' I I t:.z I 2
1 a 1-~
I B' I = 2 2 4 d cH1 - a (en a') 2
Finally, equation (7.244), (7.248) and (7.252) give the following:
For 1-t ~ 1, O<t<l, a-2 a a
z~a£a I f(z) I - I f(zt) 1- 24 1C -2 {en(1+ ::)} d cH1 - a (l_t)-2
For O<D ~1, equation (7.254) gives a-2 tI a - --
z~aea I f(z) I - I f(zt) 1- 2 4 1C 2 p/+l-a (-enD) D cH1 - a (l-t) 2
(ii) Estimation of (J
(7.248)
(7.249)
(7.250)
(7.251)
(7.252)
(7.253)
(7.254)
(7.255)
Next, we estimate the size (J of the ellipse Ca of equation (7.20) which
passes through the point Z=Zt of equation (7.239). Since,
zt=x+iy (7.256)
where
en2 x= --1
en a'
253
Trt y= 2ena'
equations (7.31) and (7.32) give
)2 )2 j2 c C 2 C 2 C 2 a = -p + -p -pen2+1 + -p -pen2 + -p -pen2+1 ,
2 4 2 4
where
and
O<t<l .
(7.257)
(7.258)
(7.259)
(7.260)
It can be shown that q(D) is a strictly increasing funCtion of D>O , i.e.
For D~1,
and
For D~1 ,
da ->0 for dD
D>O .
a(D) - 1+ v'c-en2 j 1 -D +0 1 -enD ~
da { 1 }_ - -0 dD D (-enD)'lJ3 D~+O
a(D)-2cD
The graph of qeD, t) vs.D for the case t=0.6 is shown in Fig.7.9.
(iii) Error estimate En(f)
From equations (7.24) and (7.254), we obtain a+2 a a
IE(f) 1<24 Trl- 2 {enO+Pj)}dHl-aO_t)-2a-2n
n ~ d
a+2 l-! a
D~l 24 Tr 2 p/H1 - a (_enD)DH1 - a (1-tl 2 -2n a
From equations (3.138-142), the analytical expression for
fPj / I = -dp
a,6 0 r a
is given by Table 7.5.
(7.261)
(7.262)
(7.263)
(7.264)
(7.265)
(7.266)
254
It) o o
. 0)
i.J...
255
Table 7.5. Closed form for IP. / I =;: J -dp
eI ,8 0 rei
a 0 Ia,o (exact) Ia,o (D~ 1)
1 1 Pj (,VI +D2 -D) Pj
1 (1 I D - 1 I .y 1 + D2 ) I Pj 1 I (pjD)
3
2 en {(I +.yl+D2) ID}- 1/.yl +D2 -enD
1 { 1 I D3 - 1 I (1 + D2 )3/2 } I ( 3 Pj 3 ) 1 I {3(pj D )3 }
5
2 1 I {3(pjD)2 (I+D2)3/2} 1 I { 3 (Pj D )2}
Then, equation (7.265) and Table 7.5 give the estimate for the relative
error as in Table 7.6.
Table 7.6. Estimates for relative error En(f) IIa, 8
a 0 En(f) /Ia,o
1 1 3.0 (-enD)D(I-t)-l/2 (J-2n
1 1.3 (-enD) (l_t)-3/2 (J-2n
3
2 1.3 (1_t)-3/2 (J-2n
1 1.8 (-enD) (l_t)-5/2 (J-2n
5
2 1.8 (-enD) (l_t)-5/2 (J-2n
256
To maintain the constant coefficient of the error estimates at a
reasonable size, a
(1-t) 2 <10
gives
{
0.99
o < t < 1_10-aI2 = 0.77
0.60
(a=l)
(a=3)
(a=5)
(7.267)
(7.268)
The values of a(D) for t=0.6 are given in Table 7.7 (cf. graph of
Fig.7.9).
Table 7.7 Values of a(D) for t=0.6, for the log-L1 transformation
D (J D (J
10-10 1.16 10-2 1.40
10-8 1.18 3XlO-2 1.48
10-6 1.21 10-1 1.63
10-4 1.26 3 X 10-1 1.94
3X10-4 1.28 1 3.05
10-3 1.31 3 7.23
3X10-3 1.35 10 23.4
For nearly singular integrals, one typically has relative source
distances of the range
D: 10-3 - 10-1 •
for which the size a of the ellipse € a is
q : l.31 -l.63
(7.269)
(7.270)
In summary, we obtain the following error estimate for the numerical
integration of
257
IPj l I = -dp
a.8 0 r a
using the tog-Ll radial variable transformation
R(P) = eog (p +d)
and the Gauss-Legendre formula, for general natural numbers a, 8 :
~ l-~ a IE(f) 1< 24 7r 2p.Hl-aen(1+D-l)DHl-a(1_t) 2 q -2n
n (Y J
where
q 1.31 -1.63
for
where we have taken t=0.6 .
From Table 7.6, the estimate for the relative error e is given by
for the same values for (J.
(7.266)
(5.85)
(7.271)
(7.272)
The theoretical estimate of equation (7.272) corresponds fairly well with
numerical experiment results in Chapter 10. The estimate also clarifies
quantatively, why the tog-LJ radial variable transformation is, by far, more
efficient and robust compared to the tOg-L2 and identity transformations.
7.9 Summary of Theoretical Error Estimates
In this chapter, we derived theoretical error estimates for numerical
integration by means of the theory of complex functions.
The basic radial component integrals
IP. 8 J p
I = -dp a.8 0 ra
where
258
r - "f;f:;i2 P , a,8 EN, (7.273)
characterize the nearly singular integrals occurring in the boundary element
analysis of three dimensional potential problems.
We obtained the following error estimates En for the numerical
integration of la,8 using the Gauss-Legendre rule after applying each radial
variable transformation R(p) .
Let n be the number of integration points in the radial variable, and d
D= - >0 Pj
be the relative source distance.
Then, for n ~ 1 and D ~ 1 (nearly singular) ,
(1) Identity tranformation : R(p) = p
where
for
8-~ II E < D 2
nrv
- (1 -2n
-2n (1
(1 : 1.04 - 1.42
For 8=odd (potential),
8+1-11 2 En;;' (-enD) D-2 - {ma:t{3~8~ e enD} n
8+1-11
_ D 2 (e:D)2n
(7.80)
259
- n -2n
For 8=even (flux, interpolation functions), 6+1
E - (-enD) 2 D8+1-a n -6-1 n
- n -8-1
(3) eog-LJ transformation: R(e)= eog (e+d)
For 8+1-a > 0 (e.g. a=8=1 for potential),
6+1-a 2n E _ (-enD)D-2- {(8+1-a)eenD}
n 8n
8+1-a -2 (enD)2n -D -
n
- n -2n
For general a.8 EN.
E < (-enD) D8+1-a nN
- (1 -2n
-2n (1
where
(1: 1.31 - 1.63
for
D: 10-3 - 10- 1
(7.118,120)
(7.195)
(7.235)
(7.271)
These theoretical error estimates are compared with numerical
experiment results on one dimensional radial variable integration and
260
boundary element surface integrals with potential and flux kernels, in
Chapter 10. The theoretical estimates bear remarkable resemblance with the
numerical experiment results, demonstrating the validity of the estimates
derived in this chapter.
The theoretical estimates in this chapter also give a clear insight
regarding the optimization of the radial variable transformation R(p) for
nearly singular integrals arising in boundary element analysis in general.
To be more precise, the singularities P± = ±d i E C, inherent in the
near singularity of 1 1
Vp2+; If rlf
(7.274) -=----
are mapped to R± =R(P±) by the radial variable transformation R(p). R±
in turn, are mapped to z± = x(R±) by the transformation 2R-{R(Pj)+R(O)}
x= ---~-----R(p.)-R(O)
J
(7.49)
in the process of mapping the interval R: [ R(O), R(Pj) ] to the interval
x: [-1,1] in order to apply the standard Gauss-Legendre rule.
As shown, for example, by equation (7.80) for R(p)=p and equation
(7.271) for R(p) = tog (p + d), the numerical integration error is governed by
the maximum size (J of the ellipse € (J
I z + ~ I = 11. (11 > 1) (7.20)
in the complex plane, which does not include the singularities z± inside.
Therefore, roughly speaking, the optimum radial variable
transformation R(p) is the transformation which maps the singularities
P± == ± d i, inherent in the near singularity, to z± =x{ R(P±)} which are as
far away as possible from the real interval z: [-1,1], allowing an ellipse €11
of maximum size (J.
PART II
APPLICATIONS AND NUMERICAL RESULTS
CHAPTER 8
NUMERICAL EXPERIMENT PROCEDURES
AND ELEMENT GEOMETRY
In the following chapters, results of numerical experiments performed on the
numerical integration methods proposed in Chapter 5 are given with comparison
with previous methods. Weakly singular integrals, nearly singular integrals and
hyper singular integrals (Cauchy principal value) arising in three dimensional
potential problems are treated. In this chapter the numerical experiment
procedures and types of elements used in the experiments are given.
8.1 Notes on Procedures for Numerical Experiments
All the numerical experiments were done on the NEC supercomputer SX-2
(peak performance 1.3GFLOPS) in double precision (except when stated
otherwise). The results for CPU-time presented in milliseconds (msec) were
consistent within a relative difference of about 1% for most cases and within 5%
for some exceptional cases.
When the Gauss-Legendre formula was used as the basic quadrature rule,
the number of integration points N was increased in the following series:
N= 1,2,3,4,5,6,7,8,9,10,11,12,14,16,20,25,28,32,35,40,45,50,55,
60,64,72,80,90,100,1110,120,128,140,150,160,170,180,190,200,
210,220,230,240,250,256,300,250,400,450,500, (8.1)
until the numerical integration results converged. The integration table in
Stroud and Secrest3 and the IMSL mathematicallibrary39 based on the subroutine
GAUSSQUADRULE by Golub and Welsch38 were used to generate the
integration points and weights for the formula.
264
The relative error en of the numerical integration result In was measured
by comparing it with the true value of the integral 1*, which was obtained either
by analytical integration, when it was possible, or by using the converged result
using the best method with sufficient number of integration points. For instance,
for nearly singular integrals, the result obtained by the PART method with the
angular variable transformation t(8) of equation (5.127) and the log-Ll radial
variable transformation R(p) of equation (5.85), was used to obtain the converged
numerical integration result. Then the relative error en was measured by
(8.2)
When the value of the true (converged) integral 1* itself is zero (or when I 1* I is
less than a certain threshold value e.g. 10 -10), the absolute error
E • 11-1*1 n n (8.3)
was used as the measure of convergence, instead of the relative error.
In most tables, the minimum number of integration points required to
obtain a relative error less than E = 10-6 is given with the CPU-time consumed,
as a measure of the efficiency of the method.
In order to obtain the minimum number of integration points in each
variable required to achieve a relative error less than E, the following procedure
was taken. Taking the example of the PART method with integration in the
angular variable t(8) and the radial variable R(p), where Nt and NR are the
number of integration points in the angular and radial variables, respectively,
(1) increase Nt and NR as Nt=NR=l, 2, 3, ... , according to the sequence of
equation (8.1), until the relative error becomes less than E at
Nt=NR=Nmax (1),
(2) set NR=Nmax(1) and increase Nt=l, 2, 3,'" ,according to the sequence of
equation (8.1), until the relative error becomes less than E at
265
Nt=Nmax (2),
(3) set Nt=Nmax(2) and increase NR=l, 2, 3, ... ,according to the sequence of
equation (8.1), until the relative error becomes less than E at NR=Nmax(3) ,
so that the minimum number of integration points required are
Nt = Nmax(2) and NR=Nmax(3) for the angular and radial variables,
respectively.
When there are more than one component in the integral concerned, such as
in integrals related to the flux e.g. Is iJU*/iJxs dS, Is iJq*/iJxs dS or in integrals
containing interpolation functions e.g. IsfJij u* dS, (i,j = -1, 0, 1), the maximum
relative error among all the components concerned is taken as the (maximum)
relative error.
8.2 Geometry of Boundary Elements used for Numerical Experiments
In all cases, the 9-point Lagrangian element defined by equations (5.2), (5.3)
and (5.4), and shown in Fig. 5.1 is used to model the following (curved)
quadrilateral patches.
(1) Planar rectangle (PLR)
A planar rectangle in the xy-plane defined by [-a, a] X [- b, b] ,or
-a ~ x ~ a
-b;:;;; y ~ b
z = 0
as shown in Fig. 8.1. The source point Xs is given by
X,=(x,y,d)
where d is the source distance. The above planar rectangle will be called
PLR(a, b).
(8.4)
(8.5)
266
y A
b
-a 0 a x
-b
Fig. 8.1 Planar rectangle: PLR (a, b )
267
As a special case, the planar square of size 1 can be defined by PLR (0.5, 0.5).
The planar rectangle PLR (a, b) is modelled by the 9-point Lagrangian element of
equation (5.4) by setting
xi,le= x(j,k)=(ja,kb,O); j,k=-1,O, 1 (8.6)
(2) 'Spherical' quadrilateral (SPQ)
A spherical quadrilateral on the spherical surface of radius a defined by
x = asin'l'cos~
y = a sin 'l' sin ~
z = acos'l' (8.7)
where 'l' determines the latitude and q, is the longitude, as shown in Fig. 8.2,
where 'If and q, are bounded by
'l'(1) ~ 'l' ~ 'l'(-1)
(8.8)
The spherical quadrilateral is approximated by the 9-point Lagrangian
element of equation (5.4) such that parameters '1/1 and '1/2 correspond with angles
q, and 'l', respectively, as
'11 = -1 ++ ~ = ~ (-1)
'11 = 0 ++ ~=~(O)
'11 = 1 ++ ~ =~(1)
and
'12 = -1 ++ 'l' = 'l' (-1)
'12 = 0 ++ 'l' = 'l' (0)
'12 = 1 ++ 'l' = 'l' (1 ) (8.9)
'as shown in Fig. 8.3.
268
z
y
~~-------+--~x
Fig. 8.2 Sphere with radius a
<1>(-1 )
269
z
<1>(1) <1>(0)
'1'(0)
'1'(-1)
Fig. 8.3 Spherical quadrilateral ( SPQ )
270
For instance, if we take
a= 1
<l>(-1) = -30", <l>(-1)= -30", <l>(1)=30" ,
W(-1)=120" , W(0)=90" , W(1)=60", (8.10)
we obtain a spherical quadrilateral subtending 600 in each direction <l> and 'l', and
the size of the element is 1, i.e.
I x (1, 0) - x ( - 1 , 0 ) I = 1
Ix(O,l)-x(O,-l)1 = 1 (8.11)
This element, modelled by the 9-point Lagrangian element as in equation (8.8),
will be called SPQ60.
It should be noted that the maximum relative discrepancy between the real
spherical quadrilateral and the 'spherical' quadrilateral SPQ60 (modelled by the
9 point Lagrangian element), is of the order of 10-3• Hence, in actual applications
a more accurate geometrical modelling becomes necessary, depending on the
required accuracy of the analysis.
(3) Hyperbolic quadrilateral (HYQ)
A hyperbolic quadrilateral is defined as the 9-point Lagrangian element
(cf. equation (5.4) ) whose nine nodes Xj,k; j,k= -1,0, 1 are given in x, y, z
coordinates as
and
:x!,k =ja , (j,k=-l,O,l)
yi,k = kb , (j,k= -1,0,1)
Z-1,1= -h zO,1=0
z-1,0= 0 , zo,o= 0
z-1, -1= h , zO, -1= 0
z1, 1 = h
zO, 1= 0 ,
z1, -1= -h (8.12)
271
h
h
-h
Fig. 8.4 Hyperbolic quadrilateral (HYQ)
272
as shown in Fig. 8.4.
This element HYQ is described by
h z = ab XY (8.13)
In this case, the modelling of the hyperbolic quadrilateral by the 9 point
Lagrangian element is exact.
Setting
a = b = 0.5 h = 0.25 (8.14)
gives the hyperbolic quadrilateral HYQ 1, which is described by
z= XY - 0.5 ;;; x ,Y ;;; 0.5 (8.15)
The size of the element is considered to be 1, since
Ix(1,O)-x(-l,O)I= Ix(O,l)-x(O,-l)l= 1 (8.16)
CHAPI'ER9
APPLICATIONS TO WEAKLY SINGULAR INTEGRALS
In this chapter, results of numerical experiments on weakly singular
integrals arising in three dimensional potential problems are presented.
Although weakly singular integrals are not as difficult to calculate as the
nearly singular integrals, difficulty may arise using polar coordinates, when the
source point is very near the edge of the element, as is the case for discontinuous
elements.
It will be demonstrated in this chapter that the method of using polar
coordinates in the plane S tangent to the element S at the source point xs, with the
angular variable transformation t(O) introduced in Chapter 5 (PART method with
R (p) = p), overcomes the above mentioned difficulty.
9.1 Check with Analytical Integration Formula for Constant Planar Element
First, the analytical integration formula for the integral
I • = 411' I u· dS = I dS u s s r
(9.1)
for a constant planar quadrilateral element is presented, so that the method and
code for numerical integration can be checked.
274
Using polar coordinates (p, B) centered at the source point xs, in each
triangle (j = 1-4), as shown in Fig. 9.1,
where
I = u· f dS
s r
4 fl!.O f p.( 0) =L jd8 J ~dp j=1 0 0 r
for singular integrals (d = 0) over planar elements. Hence,
4 f'O. r·(O)
I = L J d8 J dp u·
j=1 o 0
4 ro. = L J p.(8) dp
j=1 o J
4
f:Oj h.
= L J d8 ros(8-a.) j=1 J
4 h. [ {l + sin (M . - a .)} (1 + sin a .) 1 L Jlo J J J
2" g {l - sin (M . - a . )} (1 - sin a . ) j=1 J J J
For a rectangular planar element PLR (a, b) with the source point at
xs = (xs, Ys, 0), as shown in Fig. 9.2, equation (9.4) can be expressed as
since
where
~ hj lo (aj +1+ hj +1 a j + hj _ 1 ) I u. = L. - g . -=------=-
._1 2 a·+ 1-h·+ 1 a.-h. 1 J- J J J J-
hj + 1 sin (!;"8. - a. ) =
J J
sin a. J
h. 1 = ....l..:::...- , (j=l-4)
a. J
(9.2)
(9.3)
(9.4)
(9.5)
(9.6)
(9.7)
(9.8)
276
Fig. 9.1 Planar quadrilateral element
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
276
y
b
--~a~------~----~~~a~-X
-b
Fig. 9.2 Planar rectangle PLR (a, b) with source point at (xs, Ys)
277
When the source point Xs is on the j-th edge of the element, i.e. sgnG) = 0, the j-th
component of the summation in equations (9.4) and (91.5) is set to zero.
As an example, we take the planar square elem~nt S: PLR(0.5. 0.5), which
is described by
- 0.5 ~ x~ 0.5
- 0.5 ~ y~ 0.5
z = 0 (9.9)
The typical size of the element is 1.
In Table 9.1, the result of calculating
tu* dS (9.10)
is given, where
u* = (9.11)
and the source point XS=X(1]l0 "'/2) is set to )C(O, 0), xU, 0), x(1, 1) and x(O, 1), as
shown in Fig. 9.3. The numerical result obtained by Jsing polar coordinates (p, 0)
around the source point, and further using the angular transformation
h. {1+Sin(O-a.)} t(O) = ..l.. log J
2 l-sin(O-a.) J
(5.127)
introduced in Chapter 5, are compared with the analypcal integration of equation
(9.4). The basic quadrature rule used for the integrktion in p, 0 and teO) is the
Gauss-Legendre rule. The minimum number of int4gration points, Np, No and
Nt(o)' required in each variable to achieve a relative error (compared to the
analytical results) less than 10-6 are given with the C~U-time. Only one integration point is required in the radfal variable. This is because
for planar elements r = p, so that
I u* dS = I dO I 4~r p dp .
= :1£ I dO I dp (9.12)
Tab
le 9
.1
Wea
kly
sin
gu
lar
inte
gra
l Is
u* d
S
over
the
un
it p
lan
ar s
qu
are:
PL
R (
0.5,
0.5)
Po
lar
Co
ord
inat
es
An
gu
lar T
rans
form
atio
n In
teg
ral
(an
aly
tica
l)
Sou
rce
poin
t (1
J1,
1J2)
N
oX
Np
to
tal
CP
U
rela
tiv
e N
t(o
)XN
p to
tal
CP
U
rela
tiv
e 4l
T Is
u* d
S
(mse
c)
erro
r (m
sec)
er
ror
(0,0
) 6
Xl
24
0.69
3
X 1
0-7
lX
l 4
0.20
3
XI0
-16
3.52
5494
3
(1,0
) 6
Xl
18
0.54
3
XI0
-7
lXl
3 0.
17
7X 1
0-1
6 2.
4060
591
(1,1
) 4
Xl
8 0.
28
9X
I0-7
lX
l 2
0.13
6
XI0
-16
1.76
2747
2
(0,1
) 6
xl
18
0.54
3X
1O-7
lX
l 3
0.17
7
XI0
-16
2.40
6059
1
N .....
co
279
1
-1 0 1
-1
Fig. 9.3 Position of source point
280
With the angular transformation t(8), only one integration point is required
in the angular variable. This is because
f 1 4 fl:.8 j f Pj (8) S u· dO = - 2 dO dp
4/C j=l 0 0
1 4 J t(M .) dO = - 2 J p. (0) dt 4/C j=l t(O) J dt
1 4 J t(M .) --2 J dt 4/C j=l t(O)
(9.13)
since dO 1
dt p . (0) J
(5.126) -=--
In Fig. 9.4 the relative error E: using polar coordinates ( p, 8 ) is plotted (in
log-scale) against the number of Gauss-Legendre integration points No in the
angular variable 8. The source point is located at xs=x(O, 0)=(0, 0, 0) at the
centre of the planar square element. From the graph, it is estimated that
(9.14)
so that
(9.15)
(From now on the relative error will be plotted in log-scale in all the convergence
graphs showing the relative error vs. number of integration points.)
Relative Error
1
281
Weakly Si~gular Integral
Is u* d$
S: planar square IPLR ( 0.5 , 0.5 )
Xs = ~( 0 , 0 )
Number of Angular Integration Points
Fig. 9.4 Relative error vs. Number ofl angular integration points using polar coordinat~s (p, e )
282
9.2 Planar Rectangular Element with Interpolation Function Pij
Next, numerical integration is performed for the weakly singular integrals
where
I </> .. u* dS s IJ
1 u* =-
41fr
, (i,j= -1,0,1) (9.16)
(9.11)
and ¢ij' (i,j = -1,0,1) are the 9-point Lagrangian interpolation functions defined
by
(9.17)
where
(5.3)
so that
(9.18)
Again taking the unit planar square S: PLR( -0.5,0.5), Table 9.2 gives the
minimum number of Gauss-Legendre integration points Np, No and Nt(o) in the
variables, p, Band t(B), respectively, to achieve a relative error €: < 10-6 for all
the components i, j = -1, 0, 1 of the integral of equation (9.16), using the polar
coordinates (p, B) and with .the angular transformation t(B) of equation (5.127).
The CPU-time and the maximum relative error for i, j = -1, 0, 1 are also given.
The actual value of the integral corresponding to the ¢ 11 component, calculated
by Nt(o) =Np =128 points is also given. The source point xs=x(77l' 772) was
chosen similar to Table 9.1.
This time, more than one integration point is required in each variable,
because of the interpolation function ¢ij. However, only 3 integration points are
required in the radial variable, and the transformation t(B) decreases the
Tab
le 9
.2
Wea
kly
sin
gu
lar
inte
gra
l Is
1>ij u
* dS
ove
r th
e u
nit
pla
nar
squ
are:
PL
R(0
.5,0
.5)
Po
lar
Coo
rdin
ates
(p,
8)
An
gu
lar T
rans
form
atio
n (p
, t(
8))
Ne
S
ourc
e p
oin
t --
('71.
'72)
C
PU
C
PU
/poi
nt
rela
tiv
e C
PU
C
PU
/poi
nt
rela
tive
N
t(el
Ne
XN
p
tota
l N
t(el
XN
p to
tal
(mse
e)
(use
e)
erro
r (m
see)
(u
see)
er
ror
( 0
,0)
8X
3
96
2.8
29
4X
10
-7
5X
3
60
1.8
30
2X
10
-7
1.6
(1,0
) 9
X3
81
2.
4 30
3
XlO
-7
5X
3
45
1.4
30
6X
10
-8
1.8
(1,1
) 7
X3
42
1.
3 30
5
X1
0-8
4
X3
24
0.
75
31
9X
10
-7
1.8
-(0
,1'
9X
3
n 2.
4 .... "
3 X
10
.7
5X
3
~
1.4
6X
:tO
8
1.8
u ...
u
v
""%
u 0
U
Inte
gra
l
4rr
Is 1>
11 u
*d
S
(128
X12
8pts
., t(
8)X
p)
5.32
8399
8 X
10
-2
8.19
6217
3 X
10
-2
2.76
0298
6 X
10
-1
o . .196217~
'" 00 CAl
284
angular integration points by a factor of 1.6-1.8. The extra CPU-time for the
angular variable transformation is shown to be negligible.
Fig. 9.5 shows the relative error against the number of radial integration
points, and Fig. 9.6 shows the relative error against the number of integration
points in the angular variables Band t(B), for the case of Xs = x(O, 0).
N ext, the effect of the position of the source point xs, on the efficiency of the
numerical integration is investigated. Moving the source point xs=x(~,~) from
~ = 0 to ~ = 1.0 along the diagonal of the planar square element, Table 9.3 shows
the minimum number of integration points Np , No and Nt(O) in each variable to
achieve a relative error less than 10-6, for the integrals Is ~ij u* dS, (i,j=
-1,0,1). Fig. 9.7 shows the number of angular integration points No using the
ordinary angular variable B, and N t(O) using the transformed angular variable
t(B) against ~ , which indicates the position of the source point xs=x(~,~) along
the diagonal of the element.
It is evident that as the source point approaches the corner (or edge) of the
element the number of angular integration points No increases rapidly. This is
due to the angular near singularity mentioned in section 5.6. This problem is
overcome by introducing the angular variable transformation t(B) of equation
(5.127), which weakens the near singularity in the angular variable B. Table 9.3
and Fig. 9.7 show that the transformation t(B) is robust against the change of
position of the source point in the element, and that the number of integration
points (, and hence the CPU-time,) can be reduced by a factor of more than 6 by
integrating in the transformed variable t(B) instead of B, as the source point
approaches the corner of the element (~> 0.9). Hence, this angular
transformation t(B) becomes particularly useful when using discontinuous
elements, which employ source points near the edge or corner of the element.
For the case xs=x(0.9,0.9) (or ~ =0.9 in Table 9.3), the maximum relative
error (for i, j = -1, 0, 1) is plotted against the number of radial integration points
Relative Error
1
Fig. 9.5
285
Weakly Singular Integral
Is <P ij u* dS
S l planar square PLR ( 0.5 , 0.5 )
Xs 1= x( 0 , 0 )
( Ne = 8 )
Number of Radial Integration Points
Relative error vs. Nlumber of radial integration points
Relative Error
1
-1 10
-2 10
-3 10
-4 10
-5 10
-6 10
-7
286
Weakly Singular Integral
f s <P ij u* dS
S: planar square PLR ( 0.5 , 0.5 )
Xs = x( 0 , 0 )
10 ~------~----~--~--~~--~--~----~ 1 234
Fig. 9.6
Number of Angular Integration Points
Relative error vs. Number of angular integration points
287
Table 9.3 Weakly singular integral Is ¢ij u* dS over the Jnit planar square:
PLR(0.5, 0.5) ( effect of position of source ~oint)
I
Polar Coordinates Angular Transformation Integral
~ (p,O) (p, t(O» ~ 471'IS ¢ll u*dS
NoX Np total Nt(o) X Np total it(O) (128 X 128pts., t(O) X p)
0 8X3 96 5X3 60 I 1.6 5.3283998 X 10-2
0.2 9X3 108 5X3 60 I 1.8 9.5485251 X 10-2
0.4 12 X 3 144 6X3 72 I 2.0 1.8719597 X 10- 1
0.6 16 X 3 192 6X3 72 I 2.7 3.2822709 X 10-1
0.8 25 X 3 300 7X3 84 I 3.6 4.6027876 X 10- 1
0.9 40 X 3 480 7X3 84 I 5.7 4.6096971 X 10- 1
0.92 45 X 3 540 8X3 96 I 5.6 4.4780849 X 10- 1
0.94 50 X 3 600 8X3 96 I 6.3 4.2744234 X 10-1
0.96 55 X 3 660 8X3 96 I 6.9 3.9746827 X 10-1
0.98 80 X 3 960 8X3 96 110 3.5329583 X 10- 1
0.99 100 X 3 1,200 8X3 96 113 3.2243822 X 10-1
0.995 (128) X 3 (1,536) 8X3 96 (116) 3.0295234 X 10- 1
e:=2X10-6
0.999 (128) X 3 (1,536) 8X3 96 (~6) 2.8307336 X 10-1
e:=5X10-4
1.0 7X3 42 4X3 24 11.8 2.7602986 X 10-1
Posi tion of source point: Xs = x( ~, ~). Relativeerror e: 1<10-6•
Ang
ular
In
tegr
atio
n P
oint
s .
(rel
ativ
e er
ror
< 1
0-6
)
100 50
Wea
kly
Sin
gula
r In
tegr
al
f s <P
ij u*
dS
S:
plan
ar s
quar
e P
LR (
0.5
, 0
.5 )
Xs
= x
( 11
,11 )
r , " " " " 0, " " " " , ,
, ,
, ,
? '
, , , , <> , , ~
Ne
fOO
J ~
( .......
.. ,-..
. ..od
.1>
--<
1'"
____
__ ~ _
____
_ ~--
~ __ ~ _
____
__ 0
----
-__
~ ---
---~ --
-_
---0
/ N
t (e)
o 0.
2 0.
4 ,..
,... ,..
n ,.
.' !.!
" , I'
'~ I,
..
7'
11
Fig.
9.
7 P
ositi
on
of
Sou
rce
Poi
nt
Num
ber
of
angu
lar
inte
grat
ion
poin
ts
vs.
Pos
ition
of
so
urce
po
int
~
289
in Fig. 9.8, and against the number of angular in~gration points in Fig. 9.9,
respectively.
Comparing Fig. 9.6 (1] =0) with Fig. 9.9 (~=0.9), the angular near
singularity and the effect of the angular variable J.ansformation teO) is more
pronounced in the latter (1j = 0.9).
Next, the effect of the aspect ratio bla of the lanar rectangular element
PLR (a, b), on the efficiency of the numerical integra ·on is investigated. Setting
a = 0.5 constant, b is varied from 0.5 to 5, so that the spect ratio ranges from 1 to
10. In each case, the minimum number of integratio points required to achieve
relative errors less than 10-6 for i,j= -1, 0, 1 for th integral IS;iju*dS by the
method of polar coordinates (p, 0) and the method ing the angular variable
transformation (p, t(8», in Table 9.4. Fig. 9.10 gives e convergence graph of the
maximum relative error for i, j = -1, 0, 1 vs. the n ber of integration points in
the angular variable, for the case when the aspect rati is 5 (a = 0.5, b = 2.5).
It is evident that as the aspect ratio increases, the number of integration
points in the angular variable 0 in the method usi g polar coordinates (p, 0)
increases rapidly, whereas with the method usin the transformed angular
variable teO) of equation (5.127) the number of inte ation points increases very
slowly.
From the above numerical experiments on weak y singular integrals
Is u* dS and Is ;iju* dS over the unit planar squar and planar rectangles, the
robustness of the angular transformation teO) again t the position of the source
point and the aspect ratio of the element is verified.
As for the weakly singular integrals Is q* dS nd Is; ij q* dS , results are
not given for planar elements, since (r, n)=O and *= -(r, n) I (471'r)=O for
r=f:. 0 so that the integrals take the value of zero, excl ding the Cauchy principal
value, which is defined as the limit as the so rce distance d-O of the
corresponding nearly singular integral. This will be entioned in Chapter 10.
Relative Error
1
10- 1
10- 2
10- 3
10- 4
10- 5
10- 6
10- 7
Fig. 9.8
290
Weakly Singular Integral
, J S <Pij U* dS , , , , , , S: planar square , , ,
PLR ( 0.5 , 0.5 ) , , , , , , Xs = x( 0.9 , 0.9 ) , , , , , , , , , , , ,
I , , , , , t (8) , ( Nt (9) = 7 ) , , ,
8, f~ , ,
( N9 = 40 ) \
Number of Radial Integration Points
Relative error vs. Number of radial integration points
Rel
ativ
e E
rror
1 ,.. ...
.. ~.
I ..
....
...
I ..
.. .....
... ........
... 10
-1
10-2
10-3
10-4
10-5
t
(8)
10-6
....... ... .....
... ... , , ,
, , , "
, , ... ,
...
Wea
kly
Sin
gula
r In
tegr
al
, , , ,
Is <l
'ij u
* dS
' ..... ,
, , S
: pl
anar
squ
are
PLR
( 0
.5 ,
0.5
)
Xs=
x(
0.9
, 0.9
)
... ... '.,
9
"'"..,r
, ,
~ , ... , , ' ... ,
, , , ,
, ,
-71
I I
, I
I ,"
, I
I I I,
,I!
,I,
""
J ,
10
r -I
n
-I r
I"
In
I"Ir
~n
~r
'.
7
'
Fig.
9.
9 R
elat
ive
erro
r vs
. N
umbe
r of
an
gula
r
40
Num
ber
of
Ang
ular
In
tegr
atio
n P
oint
s in
tegr
atio
n po
ints
'" co .....
Tab
le 9
.4
Wea
kly
sing
ular
inte
gral
Is ~i
j u*
dS o
ver t
he p
lan
ar re
ctan
gula
r ele
men
t: P
LR
(0.5
, b)
Asp
ect
Pol
ar C
oord
inat
es (p
, 0
) A
ngul
ar T
rans
form
atio
n ( p
, t(O
) )
N8
Inte
gral
b --
4trI
s~11
u*d
S
rati
o N
8X
Np
tota
l N
t(8)
X N
p to
tal
Nt(8
) (1
28 X
128
pts.
, t(O
) X p)
1 0.
5 8
X3
96
5
x3
60
l.
6
5.32
8399
8 X
lO-2
2 l.
0
11 X
3
132
6X
3
72
l.8
6.
8978
324
X 1
0-2
N
, ~
3 1.
5 14
X 3
16
8 6
x3
72
2.
3 7.
4886
094
X 1
0-2
,
5 2.
5 20
X 3
24
0 7
X3
84
2.
9 7.
9307
007
X 1
0-2
10
5 28
X 3
33
6 8
x3
96
3.
5 8.
1984
132X
10-2
-
--
-------
'---------------
--
-----_
.. -
Sou
rce
poin
t x
s=( 0
,0,0
),
rela
tive
err
or <
10
-6•
Rel
ativ
e E
rror
A
1
10
-1
10
-2
10
-3
10-4
--10-
5
10
-6
-7
10
5 10
Wea
kly
Sin
gula
r In
tegr
al
f s <i
'ij u
* d
S
S:
plan
ar
rect
angl
e as
pect
rat
io =
5 P
LR (
0.5
, 2
.5 )
15
20
Num
ber
of
An
gu
lar
Inte
grat
ion
Poi
nts
Fig
. 9.
10
Rel
ativ
e e
rro
r vs
.. N
um
be
r o
f a
ng
ula
r in
teg
ratio
n
po
ints
'" co CAl
294
9.3 'Spherical' Quadrilateral Element with Interpolation Function 1>ij
In the following, results of numerical experiments on weakly singular
integrals over curved boundary elements are given in order to demonstrate the
efficiency and robustness of the method proposed in Chapter 5 using polar
coordinates ( p , 8 ) in the plane S tangent to the curved element at the source
point Xs and the angular variable transformation t(8).
First, numerical results on the element SPQ60 defined in section 8.2 (2) is
given. This element is a 9 point Lagrangian element obtained by interpolating a
spherical quadrilateral subtending 60° in both the latitude and longitude on a
sphere of radi us 1. The typical element size is 1.
(1) Results for fS2..ij u* dS
Numerical results on the weakly singular integral
J rp .. u*dS (i,j=-l,O,l) s I)
(9.16)
over the 'spherical' quadrilateral element SPQ60 are given in Table 9.5, where u*
and the 9 point Lagrangian interpolation functions 1> ij' (i, j = -I, 0, 1) are defined
as in equations (9.11) and (9.17). The source point Xs is set to x(O, 0), x(l, 0),
x(l,l) and x(O,l). The minimum number of Gauss-Legendre integration points to
achieve relative errors less than 10-6 for all the components 1>ij' (i,j= -I, 0, I),
using polar coordinates (p, 8) in the tangential plane S, and using the angular
variable transformation t(8) are given in the table, similar to Table 9.2. This
time, 5-7 integration points are required (compared to 3 for planar elements) in
the radial variable p, due to the curved geometry of the element. The angular
transformation t(8) reduces the number of integration points by a factor of
1.3-1.7.
For the case xs=x(O, 0), the (maximum) relative error is plotted against the
number of radial integration points in Fig. 9.11 and against the number of
angular integration points in Fig. 9.12.
Sou
rce
poin
t
(fJ l'
fJ ,
) N
o X
Np
(0,0
) 8
X5
( 1 ,
0)
8X
6
( 1 ,
1 )
7X
7
--
-
(0,1
) 10
X 6
Tab
le 9
.5
Wea
kly
sin
gu
lar
inte
gra
l Is
¢ij u
* dS
ov
er a
'sph
eric
al' q
uad
rila
tera
l: S
PQ
60
Po
lar
Coo
rdin
ates
(p
, B
) A
ng
ula
r Tra
nsf
orm
atio
n (p
, t(B
» N
o --
CP
U
CP
U/p
oint
re
lati
ve
CP
U
CP
U/p
oint
re
lati
ve
Nt(o
) to
tal
Nt(
O)X
Np
tota
l (m
see)
(u
see)
er
ror
(mse
e)
(use
e)
erro
r
160
4.4
28
1 X
10
-7
6X
5
120
3.4
28
7 X
10
-7
1.3
144
4.0
28
3 X
10
-7
6X
6
108
3.0
28
2 X
10
-7
1.3
98
2.8
29
4 X
10
-7
6X
7
84
2.4
29
2 X
10
-7
1.2
---
180
5.0
-- 28
3
X 1
Q=7
6
X6
10
8 ---
s:o
-28
-r
-s-x
TQ-7
T
.I
Inte
gra
l
47rI
s¢11
u*
dS
(128
X 1
28pt
s., t
(B) X
p)
5.67
5909
0 X
10
-2
8.57
4923
2 X
10
-2
2.88
2730
4 X
10
-1
~3577679 X
JU=
2
I\.)
(0
U
1
Relative Error
1
296
Weakly Singular Integral
f s <J>ij u* dS
, ,
S: 'spherical' quadrilateral (SPQ 60)
Xs = x( 0 ,0)
, , , , , , , '\ ;V t (9), (N t(O) = 6 )
~
)"" 9, (No = 8) ""
10-7~ __ ~ ____ ~ __ ~ ____ ~ __ ~'~ __ ~
Fig. 9.11
Number of Radial Integration Points
Relative error vs. Number of radial
integration points
Relative Error
1
-1 10
-2 10
-3 10
-4 10
-5 10
-6 10
-7
297 1
Weakly Singular I Integral
Is <Pij u* dS
s: 'spherical' quadrilateral ( spa 60 )
1
Xs = x( 0 '10 )
t (8)
10 ~--~--~----~--~--~----~--~----~
Number of Angular Integration Points
Fig. 9.12 Relative error vs. Number of angular integration points 1
298
Next, the effect of the position of the source point Xs is investigated by
moving it along the 'diagonal' of the element i.e. xs=x(~,~), O:;£~:;£ 1.
Table 9.6 gives the number of integration points Np , No and Nt(o) in each
variable to obtain a maximum relative error less than 10-6, for the integrals
Is 1>ij u* dS , (i, j= -1,0, 1). Fig. 9.13 plots the number of angular integration
points No and Nt(o) against the position of the source point ~.
Similar to the case of the planar square element in Table 9.3 and Fig. 9.7,
the number of angular integration points No using just polar coordinates (p, 8)
increases rapidly as the source point approaches the comer of the element (~-1).
The angular transformation t(8) of equation (5.127) has a remarkable effect in
reducing the number of angular integration points. The graph of Fig. 9.13
roughly traces that of Fig. 9.7 for the planar square element except that at the
maximum 14 integration points are required for t(8) in the spherical case,
compared to 8 in the planar case, to obtain a maximum relative error less than
10-6•
Also the number of radial integration points Np increases slightly as the
source point approaches the corner of the element, which was not seen for planar
elements. This is considered to be due to the curvature of the element.
For the case of xs=x(0.9, 0.9) , the maximum relative error is plotted
against the number of radial integration points in Fig. 9.14, and against the
number of angular integration points in Fig. 9.15, respectively.
Compared to the case of xs=x(O, 0) in Fig. 9.11 and Fig. 9.12, the necessary
number of radial integration points (for relative error less than 10-6) has
incr.eased from 5 to 7, and the angular near singularity and the effect of the
angular variable transformation t(8) is more pronounced.
299
Table 9.6 Weakly singular integral Is ~ij u* dS over a '('pherical' quadrilateral:
SPQ60 (effect of position of source point)
Polar Coordinates Angular Transformation Integral (p,O) (p, t(O)) No
~ -- 411" Is ~11 u* dS Nt(o) (128XI28pts., t(O)Xp)
NoXNp total Nt(o) X Np total
0 8X5 160 6X5 120 1.3 5.6759090 X 10-2
0.2 9X5 180 7X5 140 1.1 7.2742430 X 10-2
0.4 11 X 5 220 8X5 160 1.4 1.9162439 X 10- 1
0.6 16 X 6 384 10 X 6 240 1.6 3.3693243 X 10- 1
0.8 20 X 7 560 11 X 6 264 2.1 4.7758224 X 10-1
0.9 40 X 7 1,120 14 X 7 392 2.9 4.8092765 X 10- 1
0.92 50 X 9 1,800 14 X 7 392 3.5 4.6761673 X 10-1
0.94 50 X 7 1,400 14 X 7 392 3.6 4.4668974 X 10-1
0.96 45 X 7 1,260 14 X 7 392 3.2 4.1559238 X 10-1
0.98 64 X 7 1,792 12 X 7 336 5.3 3.6945555 X 10-1
0.99 90 X 7 2,520 14 X 7 392 6.4 3.3709633 X 10-1
0.995 120 X 7 3,360 14 X 7 392 8.6 3.1662071 X 10-1
0.999 (128) X 7 (3,584) 12 X 7 336 (11) 2.9569854 X 10-1
1.0 7X7 98 6X7 84 1.2 2.8827304 X 10-1
Position of source point: X5=X(~;~). Relative error ~<10-6.
Ang
ular
In
tegr
atio
n P
oint
s (r
elat
ive
erro
r <
10
-6 )
100
Wea
kly
Sin
gula
r In
tegr
al
J s <P
ij U
* dS
S:
'sp
he
rica
l' qu
adril
ater
al
(SP
Q 6
0)
Xs =
x( 11
, ~ )
)I: ~ ~ " " " " " " " " " ¢
' , , , , , , , , , 9 , , o
Ne
~pq!
~
:~
b /;)
0 ~~
"
___
o __
_ ~ ___
___ ~---
---_o---
----o
-'>
-/' ~ t (
9)
__
__
__
~"
--0
50 o
'""',,
,,'
? 11
Fig
. 9.
13
Nu
mb
er
of
angu
lar
inte
grat
ion
po
ints
vs
. P
ositi
on
of
sour
ce
poin
t
8
Relative Error
1
301
Weakly Singular I Integral I
J s <P ij U* dS .
s: I spherical I quadrilateral (S~Q 60)
I
Xs = xCO.9 , 0.9 )
Number of Radial Integration Points
Fig. 9.14 Relative error vs. Number of radial integration points
Rel
ativ
e E
rror
10 1 10-1
10-2
10-3
10-4
10-5
10-6
It \ \ "
~ .........
. ... '. , , ... , ... ... ... , , "
, , '. ,
t (e
) , , , ,
, , , ,
--, , , ,
, , W
eakl
y S
ingu
lar
Inte
gral
, , , ,
, ,
f S <!
>ij U
* dS
• \ \
S:
'sp
he
rica
l' qu
adril
ater
al
(SP
Q 6
0)
Xs
= x
( 0.
9 , 0
.9 )
\re
\ \ \ \~-
... _ ... _
.........
.......
' .... '''
'', "''
''''''''
'. 10
-71
I !!
!!!
!!!
!!!
!!!
!!!
!!!!!!!
>
5 1 0
15
20
25
30
35
40
=ig .
. 9.1
5 R
elat
ive
erro
r vs
. N
umbe
r of
an
gula
r in
tegr
atio
n po
ints
N
umbe
r of
Ang
ular
In
tegr
atio
n Po
ints
w
o N
303
(2) Results for fs~j g* dS
For planar elements we had (r, n)EO, r'*O andt.ence q*=O, r'*O. However,
for curved elements, (r, n)-O as shown in Fig. 9.16, so that q*=O. In fact it was
proved in Theorem 3.1 (equation (3.10» that
K K __ _ n = _~ u. (9.19)
8lrT 2
for 0<r4i1, where Kn is the normal curvature at =Xs. Hence, it is expected
that the integration of the weakly singular integra Is ~ij q*dS gives numerical
results similar to that of Is ~ij u* dS , for curved e ements. In the following,
numerical results for the weakly singular integral I ~ij q* dS over the 'spherical'
quadrilateral SPQ60 are given.
In all the numerical experiments with weakly s~ngular integrals over curved
elements, the unit outward normal n is defined by
o n=-
101 (9.20)
where
(9.21)
Table 9.7 gives the number of integration pointlf No, Nt(o) and Np required to
obtain a maximum relative error less than 10-6, for the weakly singular integral
I ;. .. q*dS S 'J
(i,j= -1,0, 1) (9.22)
over the 'spherical' quadrilateral SPQ 60. The source I point Xs is chosen as X(O, 0),
x(l, 0), x(1, 1) and x(O, 1). Results similar to Table 9.5 for IS~ij u* dS are
obtained.
For the case xs=x(O, 0), the maximum relativ~ error is plotted against the
number of radial and angular integration points in Fig. 9.17 and Fig. 9.18,
respectively.
The effect of the position of the source point x~ is shown in Table 9.8 and
Fig. 9.19, similar to Table 9.6 and Fig. 9.13.
304
• r .1 n
Xs x
planar element (r ,n) :: 0
n
curved element (r ,n) =f 0
Fig. 9.16 (r ,n) on planar and curved elements
Tab
le 9
.7
Wea
kly
sin
gu
lar
inte
gra
l f s
1> i
j q*
dS
ov
er th
e 's
pher
ical
' qu
adri
late
ral:
SP
Q60
Pol
ar C
oord
inat
es (
p,
8)
An
gu
lar T
rans
form
atio
n (p
, t(8
» N
o In
teg
ral
Sou
rce
po
int
--
4rr f S
1>11
u*
dS
(7
],,7]
2)
CP
U
CP
U/p
oint
re
lati
ve
CP
U
CP
U/p
oint
re
lati
ve
Nt(O
) N
oX
Np
to
tal
Nt(
O)X
Np
tota
l (1
28X
128p
ts.,
t(8)
Xp)
(m
sec)
(,u
see)
er
ror
(mse
c)
("se
c)
erro
r .
(0,0
) 8
X5
16
0 4.
5 28
6
X 1
0-7
6
X4
96
2.
8 29
5
X 1
0-7
1.
3 -2
.68
07
07
7 X
10
-2
.
~
(1,0
) 7
X6
12
6 3.
5 28
7
X 1
0-7
6
X6
10
8 3.
1 29
9
X 1
0-7
1.
2 -4
.02
01
06
7 X
10
-2
• U
1
(1,1
) 7
X6
84
2.
4 29
6
X 1
0-7
6
X6
72
2.
1 29
4
X 1
0-7
1.
2 -1
.16
08
84
5 X
10
-1
(0,1
) 10
X6
18
0 5.
0 28
1
X 1
0-7
7
X5
10
5 3.
0 29
9
X 1
0-7
1.
4 -4
.40
71
13
5 X
10
-2
Relative Error
1
306
Weakly Singular Integral
Is <Pij q* dS
t (9) ,
S: 'spherical' quadrilateral (SPQ 60)
Xs = x( 0 , 0 )
\ \ \
\ \
\V"'-/ 9, (N 0 = 8 )
( Nt (0) = 6 )
Number of Radial Integration Points
Fig. 9.17 Relative error vs. Number of radial integration points
307
Relative Error Weakly Singular Integral
Is <pij q* dS
1 S: 'spherical' quadrilateral (Spa 60)
Xs = x( 0 ,0)
107~------~ ________ ~ ____ ~ __ ~ ____ ~ __ ~7 1 234 5 6
Number of Angular Integration Points
Fig. 9.18 Relative error vs. Number of angular integration points
308
Table 9.8 Weakly singular integral Is rpij q* dS over the 'spherical' quadrilateral:
SPQ60 ( effect of position of source point)
Polar Coordinates !Angular Transformation Integral (p,O) (p,t(O» No
~ -- 471" Is rp11 q* dS Nt(O)
NoXNp total Nt(o) X Np total (128 X 128pts., t(0) X p)
0 8X5 160 6X4 96 1.3 -2.6807077 X 10-2
0.2 10 X 5 200 7X5 140 1.4 -4.5622416 X 10-2
0.4 12 X 6 288 9X5 180 1.3 -8.6645831 X 10-2
0.6 20 X 6 480 11 X 6 264 1.8 -1.4822399 X 10-1
0.8 25 X 6 600 12 X 6 288 2.1 -2.0071886 X 10-1
0.9 40 X 6 960 14 X 6 336 2.9 -1.9613109 X 10-1
0.92 40 X 6 960 14 X 6 336 2.9 -1.8953344 X 10-1
0.94 45 X 6 1,080 11 X 6 264 4.1 -1.8000174 X 10-1
0.96 55 X 6 1,320 14 X 6 336 3.9 -1.6664471 X 10-1
0.98 72 X 6 1,728 14 X 6 336 5.1 -1.4774935 X 10-1
0.99 90 X 7 2,520 14 X 6 336 6.4 -1.3492079 X 10-1
0.995 110X6 2,640 14 X 6 336 7.9 -1.2694574 X 10-1
0.999 (128) X 6
(3,072) 10 X 6 240 (13) -1.1890655 X 10-1 £=3X10-4
1.0 7X6 84 6X6 72 1.2 -1.1608845 X 10-1
Position of source point: xs=x(~, ~). Relative error £<10-6•
Num
ber
of
Ang
ular
In
tegr
atio
n P
oint
s
100 50
Wea
kly
Sin
gula
r In
tegr
al
Is <!>
Ij q*
dS
S:
'sp
he
rica
l' qu
adril
ater
al
(SP
a 6
0)
xs =
x (11
, 11)
t (9)
o .....
.. ....
. ....
.. .....
.. ...J;.,:"J",,~
... 7
11
Fig.
9.
19
Num
ber
of
angu
lar
inte
grat
ion
poin
ts
vs.
Pos
ition
of
so
urce
po
int
eN g
310
For the case xs=x(0.9, 0.9) ,the relative error is plotted against the number
of radial and angular integration points in Fig. 9.20 and Fig. 9.21, respectively.
Comparing the results obtained for Is 1>ijq* dS with the corresponding
results for Is 1>ij u* dS, it is observed that they show very close resemblances. This
can be explained by the resemblance of the kernels
K q* - -~ u·
2 for 0 < r <ii 1 (9.23)
(cf. Theorem 3.1 and equation (9.19», and the fact that the kernels u* and q* are
dominant in the region O<r~l near the source point Xs , so that the
characteristics in the region O<r~l are reflected in the behaviour of the
integrals Is 1> ij u* dS , I s1> ij q* dS, and the precision of the numerical integration
methods for them.
Relative Error
1
311
Weakly Singular Integral
Is <Pij q* dS
S: 'spherical' quadrilateral (SPQ 60)
Xs = x( 0.9 , 0.9 )
( Ne = 40 )
t (9) ,
( Nt(e) = 14 )
10-7~ __ ~ __ ~ ____ ~ __ ~ ____________ ~
Number of Radial Integration Points
Fig. 9.20 Relative error vs. Number of radial integration points
Rel
ative
Erro
r
10 1 10-1
10-2
10-3
10-4
10-5
10-6
" ........
. ,
........ .... .... ... .... .....
. .... ... ... , .... ,
, , , , , ,
, , " , ,
, ... , ... Wea
kly
Sin
gula
r In
tegr
al
Is <i
>ij q*
dS
, , ... ,
, .......
S:
'sp
he
rica
l' qu
adril
ater
al
(SP
Q 6
0)
Xs=
x(
0.9
~ 0.
9 )
........
. " ........
.......... ~e
.... ... , ,
... "', , ... ... ... ... ... ... ...
10-7
1 .
I I
I I
I ,"
I I
I I
I I
I I
I I
I I
I I
I >""
>
5 1 0
15
20
25
30
35
40
Num
ber
of A
ngul
ar
Fig.
9.
21
Rel
ativ
e er
ror
vs.
Num
ber
of
angu
lar
inte
grat
ion
poin
ts
Inte
grat
ion
Poin
ts
Co)
.... I'J
313
9.4 Hyperbolic Quadrilateral Element with Interpolation Function !J?ij
As another type of curved element, numerical experiment results on the
hyperbolic quadrilateral element HYQ1 defined in equation (8.15) of section 8.2
(3) are given in order to show the efficiency and robustness against element
geometry, of the method of polar coordinates with the angular transformation t(8)
for weakly singular integrals.
(1) Results for (sLj u* dS
Table 9.9 gives the number of integration points No, Nt(o) and Np , required
to obtain a maximum relative error less than 10-6 for the weakly singular
integral Is ¢ ij u* dS , (i, j = -1, 0, 1) over the hyperbolic quadrilateral HYQ1. The
source point Xs is located at x(O, 0), x(1,0), x(1,1) and x(O, 1) .
For the case Xs = x(O, 0) , the maximum relative error for i, j = -1, 0, 1 is
plotted against the number of radial and angular integration points in Fig. 9.22
and Fig. 9.23, respectively.
Table 9.10 and Fig. 9.24 show the effect of the position of the source point
xs=x(~, ~), ~ =0-1, on the numerical integration and compare the use of 8 and
t(8) as the angular variable.
For the case xs=x(0.9, 0.9) , the relative error is plotted against the number
of radial and angular integration points in Fig. 9.25 and Fig. 9.26, respectively.
Compared to the case xs=x(O, 0) , the number of radial integration points Np
(required to achieve a relative error less than 10-6) increases from 4 to 7, and the
angular near singularity and the effect of the angular variable transformation
t(8) becomes more pronounced.
Tab
le 9
.9
Wea
kly
sin
gu
lar
inte
gra
l Is
¢ij u
* dS
ov
er th
e hy
perb
olic
qu
adri
late
ral:
HY
Q1
Po
lar
Coo
rdin
ates
(p
, 8
)
An
gu
lar T
ran
sfo
rmat
ion
(p
, t(
8) )
N
e In
teg
ral
I S
ourc
e p
oin
t --
4rr I S
¢l1
u*
dS
( '7
1 '
'72
) C
PU
C
PU
/poi
nt
rela
tiv
e C
PU
C
PU
/poi
nt
rela
tive
N
t(e)
NeX
Np
to
tal
Nt(
e)X
Np
tota
l (1
28 X
128
pts.
, t(8
) X p
) (m
sec)
(u
see)
er
ror
(mse
c)
(use
e)
erro
r
(0,0
) 8
X5
16
0 4.
4 28
6
X 1
0-7
6
X4
96
2.
8 29
4
X 1
0-7
1.
7 5.
8486
239
X 1
0-2
w
.... ~ (1
,0)
9X
6
162
4.5
28
4 X
10
-7
6X
6
108
3.1
28
1 X
10
-7
1.5
8.68
5018
1 X
10
-2
(1,1
) 6
X7
84
2.
4 28
1
X 1
0-7
5
X7
70
2.
0 29
7
X 1
0-7
1.
2 2.
7591
133
X 1
0-1
(0,1
) 9
X6
16
2 4.
5 28
4
X 1
0-7
6
X6
10
8 3.
1 28
1
X 1
0-7
1.
5 8.
6850
181
X 1
0-2
~ ~-
~ --
--
~ ~-
~-
---------
-----
--~~~
-----
----
Relative Error
1
315
Weakly Singular Integral
Is <Pij u* dS
S: hyperbolic quadrilateral (Hya 1 )
Xs = x{ 0 ,0)
t {9} ,
( Nt (6) = 6 )
Number of Radial Integration Points
Fig. 9.22 Relative error vs. Number of radial integration points
Relative Error
1
316
Weakly Singular Integral
Is <Pij u* dS
S: hyperbolic quadrilateral ( HYQ 1 )
Xs = x( 0 , 0 )
10-7~ __ ~ ____ ~ __ ~ ____ ~ __ ~ ____ ~ __ ~ __ ~~ 1
Number of Angular Integration Points
Fig. 9.23 Relative error vs. Number of angular integration points
317
Table 9.10 Weakly singular integral Is <P ij u* dS over the hyperbolic quadrilateral:
HYQ1 ( effect of position of source point)
Polar Coordinates ~ngular Transformation Integral (p,O) (p , t(O» NO
~ -- 4rr Is <Pll u* dS Nt(O)
NoX Np total Nt(o)XNp total (128X128pts., t(O)Xp)
0 8X5 160 6X4 96 1.3 5.8486239 X 10-2
0.2 10 X 5 200 7X5 140 1.4 1.0198064 X 10-1
0.4 14 X 5 280 7X5 140 2.0 1.9736308 X 10-1
0.6 20 X 6 480 8X6 192 2.5 3.4712015 X 10- 1
0.8 25 X 6 600 9X6 216 2.8 4.9061557 X 10- 1
0.9 40 X 7 1,120 9X7 252 4.4 4.8962891 X 10-1
0.92 50 X 8 1,600 8X7 224 6.3 4.7417784 X 10- 1
0.94 50 X 8 1,600 8X8 256 6.3 4.5040850 X 10- 1
0.96 60 X 7 1,680 9X7 252 6.7 4.1555278 X 10-1
0.98 80 X 7 2,240 9X7 252 8.9 3.6439300 X 10- 1
0.99 110X7 3,080 9X7 252 12 3.2883544 X 10- 1
0.995 (128) X 7
(3,584) 9X7 252 (14) 3.0649529 X 10- 1 €=4X10- 6
0.999 (128) X 7
(3,584) 10 X 7 280 (13) 2.8385477 X 10- 1 €=6X10- 4
1.0 6X7 84 5X7 70 1.2 2.7591133 X 10-1
Position of source point: xs=x(~, ~). Relative error €<10- 6•
Num
ber
of
Ang
ular
In
tegr
atio
n P
oint
s
100
50
Wea
kly
Sin
gula
r In
tegr
al
Is <P
ij u*
dS
S:
hype
rbol
ic q
uadr
ilate
ral
( H
YQ
1 )
Xs
= x
( 11
, 11
)
t (8)
I""I'I
I,!
1'1
o n
. n
n A
'"
,.
..
1""
\ n
7
I,
Fig
. 9.
24
Nu
mb
er
of
an
gu
lar
inte
gra
tion
po
ints
vs
. P
ositi
on
of
sour
ce
po
int
w
0)
Relative Error
1
319
Weakly Singular Integral
Is <f>ij u* dS
S: hyperbolic quadrilateral ( HYQ 1 )
xs = x ( 0.9 , 0.9 )
t (9) ,
10-7~ __ ~ __ ~ ____ ~ __ ~ ____ ~ __ ~ ____ ~~ 1 2 3 4 5 6
Fig. 9.25
Number of Radial Integration Points
Relative error vs. Number of radial integration points
Rel
ativ
e E
rror
10 1 -1
10
10-2
10-3
10-4
10-5
10-6
jt" •• -.
I ..
....
... '. ' .... .....
"
t (9)
"'" '.
, , , ,
, , , ,
.,
Wea
kly
Sin
gula
r In
tegr
al
J s <P
ij u*
dS
S:
hype
rbol
ic q
uadr
ilate
ral
"""'"
X
s =
x( 0
.9 ,
0.9
)
'. ( H
ya
1 )
" , ,
, '. , , , ""' .
... ,r 9
, , "
, , , ,
, , , ,
, , '.
-7.
I ,
>
10
40 N
umbe
r of
Ang
ular
Fig
. 9.
26
. .
. .
Inte
grat
ion
Poin
ts
Rel
ativ
e er
ror
vs.
Num
ber
of a
ngul
ar
inte
grat
ion
pOin
ts
w ~
o
321
(2) Results for fsiyg* dS
Table 9.11 gives the number of integration points required to achieve a
relative error less than 10-6 for the integral Is 1> ijq* dS over the hyperbolic
quadrilateral HYQ1 for the same set of source points as in Table 9.9.
For the case Xs = x(O, 0) , the maximum relative error is plotted aginst the
number of radial and angular integration points in Fig. 9.27 and Fig. 9.28,
respectively. For this case, No happens to be less than Nt(o).
Table 9.12 and Fig. 9.29 show the effect of the position of the source point
xs=x(~, ~), ~ =0-1 , on the numerical integration by the angular variables 0
and teO).
For the case xs=x(0.9, 0.9), the relative error is plotted against the number
of radial and angular integration points in Fig. 9.30 and Fig. 9.31, respectively.
Results on the hyperbolic quadrilateral element HYQ1 for the weakly
singular integrals Is 1> ij u* dS and Is 1> ijq* dS , give similar results a.s those on the
spherical element SPQ60. Namely, the method of using polar coordinates (p, 0)
on the plane tangent to the element at xs, works efficiently. However, as the
source point approaches the corner of the element, the number of integration
points required in 0 increases rapidly. On the other hand, the use of the
transformed angular variable t(O) of equation (5.127) overcomes this problem, i.e.
relatively few angular integration points are required even when the source point
is near the corner.
Compared to the 'spherical' quadrilateral (SPQ60), the hyperbolic
quadrilateral (HYQ1) requires less ( 9 compared to 14 ) angular integration
points in the variable t(O) as ~-1 for the integral Is 1>ij u* dS , and slightly more
(16 compared to 14) for the integral Is 1>ijq* dS.
Sou
rce
poin
t
(711
'712
) N
OX
Np
(0,0
) 6
X4
(1,0
) 8
X5
(1,1
) 6
X6
(0,1
) 8
X5
Tab
le 9
.11
Wea
kly
sin
gu
lar i
nte
gra
l f s
1> ij
q* d
S
over
the
hype
rbol
ic q
uad
rila
tera
l: H
YQ
l
Po
lar
Coo
rdin
ates
( p
, 8 )
A
ng
ula
r Tra
nsfo
rmat
ion
( p,
t(8)
)
No --
tota
l C
PU
C
PU
/poi
nt
rela
tiv
e N
t(O
)XN
p to
tal
CP
U
CP
U/p
oint
re
lati
ve
Nt(o
) (m
sec)
(u
see)
er
ror
(mse
c)
(,use
e)
erro
r
96
2.8
29
7 X
10
-7
8X
4
128
3.7
29
4 X
10
-7
0.75
120
3.4
28
5 X
10
-7
7X
5
105
3.0
29
2 X
10
-7
1.1
72
2.1
29
3 X
10
-7
6X
6
72
2.1
29
4 X
10
-7
1.0
120
3.4
28
5 X
10
-7
7X
5
105
3.0
29
2 X
10
-7
1.1
Inte
gra
l
4Jl'
fS</>l
1 u*
dS
(128
XI2
8pts
., t(
8)X
p)
5.41
8110
4 X
10
-2
-3.8
03
06
61
X 1
0-2
5.54
5164
5 X
10
-2
-3.8
03
06
61
X 1
0-2
w "" ""
Relative Error
1
1
Fig. 9.27
323
Weakly Singular Integral
J s <P ij q* dS
t (8) ,
S: hyperbolic quadrilateral (HYQ 1 )
Xs = x( 0 , 0 )
',~8 , ( N9 = 6 ) , , , , , , '. ( Nt(9) = 8 )
2 3 4 Number of Radial
Integration Points
Relative error vs. Number of radial integration points
Relative Error
1
-1 10
-2 10
-3 10
-4 10
-5 10
-6 10
-7
324
Weakly Singular Integral
Is CPij q* dS
S: hyperbolic quadrilateral ( Hya 1 )
Xs = x( 0 , 0 )
10 ~--~1----~--~--~--~----~--~--~7
Number of Angular Integration Points
Fig. 9.28 Relative error vs. Number of angular integration points
325
Table 9.12 Weakly singular integral Is ~ij q* dS over the hyperbolic quadrilateral:
HYQ1 (effectofposition of source point)
Polar Coordinates ~ngular Transfonnation Integral (p,O) (p, t(O» No
~ -- 411" Is ~11 u* dS Nt(o)
(128 X 128pts., teo) X p) NoXNp total Nt(o) X Np total
0 6X4 96 8X4 128 0.75 5.4181104 X 10-2
0.2 9X5 180 9X5 180 1.0 5.6473892X10-2
0.4 10 X 5 200 10 X 5 200 1.0 3.9303177 X 10-2
0.6 16 X 6 384 12 X 6 288 1.3 5.2697491 X 10-3
0.8 25 X 7 700 12 X 7 336 2.1 -1.6637769X10- 2
0.9 32 X 7 896 16 X 6 384 2.0 -1.1809626X 10-3
0.92 32 X 6 896 16 X 6 384 2.0 5.9762477 X 10-3
0.94 35 X 7 98Q 14 X 6 336 2.5 1.4922905 X 10-2
0.96 40 X 7 1,120 16 X 6 384 2.5 2.5909095 X 10-2
0.98 55 X 6 1,320 14 X 6 336 3.9 3.9253216 X 10-2
0.99 64 X 7 1,792 16 X 6 384 4.0 4.6947305X10-2
0.995 80 X 7 2,240 16 X 6 384 5.0 5.1087330X 10-2
0.999 100 X 7 2,800 16 X 6 384 6.3 5.4558558X 10-2
1.0 6X6 72 6X6 72 1.0 5.5451645 X 10-2
Position ofsource point: xs=x(~, ~). Relative error €<10- 6 •
Num
ber
of
Ang
ular
In
tegr
atio
n P
oint
s
100
50
----
--
Wea
kly
Sin
gula
r In
tegr
al
f s <?
ij q
* dS
S:
hype
rbol
ic q
uadr
ilate
ral
( H
YQ
1 )
Xs
= x
( 11
,11 )
o I I I -" ~ " " " " " " 9' , , 9 , 9
J j
pd
~~~~
~~pd
~~
/o--
-~--
-o~~
/
t (9
) ~~
----
---
------~~--~~-~~
o 0.
2 0.
4 ,..
,.. ,..
n ",':
':'
' ,
" '~
',..
,.
7 11
Fig
. 9.
29
Nu
mb
er
of
an
gu
lar
inte
gra
tion
p
oin
ts
vs.
Pos
ition
of
so
urc
e
po
int
w
r-.)
C
>
Relative Error
10
1
327
Weakly Singular Integral
f s <i>ij q* dS
S: hyperbolic quadrilateral ( HYQ 1 )
Xs = x( 0.9 , 0.9 )
( Ne = 32 )
t (9) ,
( N t(e) = 16 ) 10-7~ __ ~ __ ~ ____ ~ __ ~ ____ ~ __ ~ ____ ~ __ ~
1 234 5 6
Fig. 9.30 Relative error vs. integration points
Number of Radial Integration Points
Number of radial
Rel
ative
Er
ror 10
1 10-1
10-2
10-3
10-4
10-5
10-6
'-.... , '~
. '. , .... , ,
, , , ,
, .... , "
, , , ,
, • , , ,
Wea
kly
Sin
gula
r In
tegr
al
Is <P
ij q*
dS
S:
hype
rbol
ic q
uadr
ilate
ral
( H
YQ
1 )
Xs
= x
( 0.
9 ,
0.9
)
"",r
e , ,
., , , , ,
, 'e
-71
,
I I I,
,! \,
I I
I ,I,
I I
'r '
Num
ber
of
Angu
lar
10
r= 01
n
01 r=\
n
n
nr-
nn
\
7'
Inte
grat
ion
Poin
ts
Fig
. 9.
31
Rel
ativ
e er
ror
vs.
Num
ber
of
angu
lar
inte
grat
ion
poin
ts
W
"l co
329
9.5 Summary of Numerical Results for Weakly Singular Integrals
In this chapter, numerical experiment results were presented to verify the
effectiveness ofthe method of using polar coordinates (p, B) in the plane tangent
to the boundary element at the source point xs, with the angular variable
transformation t(O) proposed in Chapter 5, for weakly singular integrals arising
in three dimensional potential problems.
It was shown that the use of polar coordinates alone 9,15,16 requires excessive
number of integration points in the angular variable B, when the source point Xs
is near the corner or edge of the element as in discontinuous elements, or for
elemen ts wi th high aspect ratio.
It was also shown that the proposed angular variable transformation t(B) of
equation (5.127) overcomes the above mentioned problem by weakening the near
singularity in the angular variable by the transformation. Numerical results for
kernels u* and q* with 9 point Lagrangian interpolation functions ~ij over
planar, 'spherical' and hyperbolic quadrilateral elements showed the robustness
of the transformation teO) against the position of the source point, integral kernel
and element geometry.
The effectiveness of using the radial variable p for weakly singular
integrals with kernels u* and q* with interpolation functions ~ij was also
verified, in accordance with Theorem 3.1, which implies that
q* - -Kn/2 u* - O(lIr) - O(lIp) for O<p~ 1.
Summing up, the use of polar coordinates (p, 0) in the tangent plane with
the angular variable transformation t(B), in combination with the Gauss
Legendre quadrature rule, is a robust and efficient method for the calculation of
weakly singular integrals Is ~ij u* dS and IS~ij q* dS over general curved
elements. Hence, the method can be safely applied to weakly singular integrals
which arise in the calculation of H, G matrices for three dimensional boundary
element analysis (e.g. potential problem ).
330
Other methods for weakly singular integrals 6, 8, 19, 20 have not been
compared with the present method, which gives good enough results, mainly
because the stress of this thesis is on nearly singular integrals, which are more
difficult to calculate, and also because the present method of polar coordinates
with angular variable transformation can be considered as a special case of the
proposed PART ( Projection and Angular & Radial Transformation ) method for
nearly singular integrals, indicating a unified approach to weakly singular and
nearly singular as well as hyper singular integrals, as will be shown in the
following chapters.
CHAPTER 10
APPLICATIONS TO NEARLY SINGULAR INTEGRALS
In this chapter, results of numerical experiments on nearly singular
integrals arising in three dimensional potential problems, which is the main
interest of this thesis, are presented. The PART ( Projection and Angular &
Radial Transformation) method proposed in Chapter 5, with its different types of
radial variable transformations R(p) and the angular transformation t(8), will be
evaluated.
10.1 Analytical Integration Formula for Constant Planar Elements
First, the analytical integration formula for the integral
I • IE 41r f u· dS = f dS u s S r
(10.1)
for a constant planar quadrilateral element is presented, so that the numerical
integration methods / codes can be checked.
Using polar coordinates (p, 8) centered at the source projection xs, in each
triangle t.j, (j = 1-4), as shown in Figures 5.4, 5.5, 5.8 and 5.10,
(10.2)
where
f 118 . J P • (8) 1 •. - J d8 J t:..dp
u .J 0 0 r (10.3)
where
(10.4)
for planar elements, where d is the source distance, or the distance of the source
point Xs from the element S .
Since,
fP'(O)
J •. (0) E J ~ dp U ,J 0 r
where
- J P d JP. (0)
= P o V/+d2
[ 12-=2] Pj(O) = Yp~+d~ o
332
- d (10.5)
P . (0) E hj (10.6) J cos(O-a.)
J
J68.
I • . = J J • . (0) dO U,J 0 U.J
J flO j y'h,....~. 2-+-d~2-co""'s2~(0-_-a -. )
= J J dO _ /:;,0. d o cos (0 - a . ) J
. J .
[COS (/:;.O . - a .) . (h. sin a. +.J h . 2 + d2 cos2 a . ) ]
= h. log J J J J J J
J cos a . . {-h.sin(!:;.8 .-a.) +Vh.2 + d2 cos2 (/:;.0 .-a.)} J J JJ J JJ
- cos (
dsina. -1 J
Vh.2 + d2 J
for sgn(j):;t: O. Note that
since
- < /:;,0 .-a. < -2 J J 2
for sgn(j):;t: O. !:.8j can be calculated from
/:;,0 . = sgn(]) cos -1 (cos /:;,0 . ) J J
(10.7)
(10.8)
(10.9)
(10.10)
333
where cost.8j is given by equation (5.26). For sgn(j) = 0 , is lies on edge-j ofS = S,
so that the area of Llj becomes zero, i.e. Iu*,j =0.
Similarly, for the kernel q* ,
Iq" IE 411" Is q* dS
= -f (r,n)dS s r3
4
= L Iq",j j=l
where
f l>8· fP.(9) d J J -'p
I ... . = dO -- dp ~ ,J 0 0 ?
(10.11)
(10.12)
since (r, n)=d , where the outward unit normal n to the planar element S is
taken so that n points to the other side of S regarding the source point Xs.
Since, r=v'p2+d2 for planar elements,
f1>8 . J P
J .... (0)· -d -3 dp ~ ,J 0 r
_ [ 1 1 Pj (0) - -d - ";p2+d2 0
d = - 1 Yp . (0)2 + d2 J
d cos(O-a.) = J _ 1 Vh.2+ d2cos2(O_a.)
J J
(10.13)
f69.
I .... (0) '" J J .... (0) dO ~,J o~,J
(10.14)
334
for sgnU> * 0, where !::.8j is calculated by equation (10.10).
For sgn (j)=0, Iq*,j=O
It is interesting to note that, from equations (10.11) and (10.14), the limit
J 1 4 lim q* dS = - - L 68. =
d-+O+ S 411" j=l J
1
2 (10.15)
gives the Cauchy principal value of the weakly singular integral Is q* dS; d=O ,
with the source point Xs just inside the region V regarding S. This corresponds
to equation (2.18) with w = 2rr , where the limit was taken with the source point Xs
inside a small hemisphere of radius (-0). (Refer also to the end of section 9.2.)
335
10.2 Singularity Cancelling Radial Variable Transformation for Constant
Planar Elements
In this section, numerical experimental results are given to show the
effectiveness of the singularity cancelling radial variable transformation
proposed in Chapter 5:
or
R(p)= J p dp
Vp2+d2 a
for calculating the integrals
Is u* ciS = 2... I dS 41[" S r a
(a =1 )
and dS I q* dS = - 2... I (r, n) d I
S 41[" S r3 = - 41[" S r a
over constant planar elements S, for which
'- ~+d2 r = V p~+ d~ = r
From Table 5.1, for a = 1 we have
R (p) = .Jp 2+d2
and for ci=3 1
R (p) = --Jp2+d2
(5.43)
(5.45)
(10.16)
(a =3) (10.17)
(10.18)
(10.19)
(10.20)
As an example, we take the planar square element S: PLR (- 0.5, 0.5) ,
which is defined by
-0.5 ;;;; x ;;;; 0.5, -0.5;;i; y ;;;; 0.5, z = 0 (8.24)
The element size is 1, so that the source distance d is equivalent to the relative
source distance.
The source point Xs = ( 0.25, 0.25, d) will be set so that the source projection
is always
336
x s = - (0.25.0.25.0) = x (0.5.0.5 ) (10.21)
as shown in Fig. 10.1, and the source distance d is varied from 10 to 10-3 in
order to see the effect of the source distance on the difficulty of numerical
integration. Xs is set off the centre in order to represent the general unsymmetric
case where Xs *" x(O, 0) .
Numerical integration results for the integral J su*dS using the singularity
cancelling radial variable transformation R(p)=Yp2+d2 , (a=1) , with and
without the angular variable transformation t(8) of equation (5.127) ; the product
type Gauss-Legendre formula 3,38,39 of equation (4.24) and Telles' cubic
transformation method 15 of equation (4.51) are compared in Table 10.1. The
minimum number of integration points in each variable required to obtain a
relative error less than 10-6 is shown. Fig. 10.2 gives the convergence graph of
relative error (in log scale) vs. the number of integration points, for the case
when the source distance d=0.1.
As shown in Table 9.1, only one radial integration point is required for the
integration
f u* dS = 2- f dS S 41f S r a
when the radial variable transformation
f""2"'.2 R (p) = V p~+d~
corresponding to
pdp = r,a dR (a =1 )
(10.22)
(10.19)
(10.23)
is used, irrespective of the source distance d, whereas with the Gauss and Telles'
methods increasing number of integration points are required as the source
distance decreases. This is because for planar elements, r=r' in equations (5.43)
and (5.44), so that the near singularity due to lIr in equation (5.44) and (10.22) is
exactly cancelled by the r' in the Jacobian due to the transformation of equation
(10.19) and (10.23). In other words, the radial integration is done analytically.
337
z
------~--~~~----~~--~x
s
Fig. 10.1 Unit planar. square element S and source point Xs
Tab
le 1
0.1
Nea
rly
sin
gu
lar i
nte
gra
l Is
u* d
S ov
er th
e u
nit
pla
nar
squ
are:
PL
R(0
.5,O
.5)
R(p
)=V
p2+
d2
, (a
=l)
G
auss
T
elle
s S
ourc
e In
teg
ral
dist
ance
A
ng
ula
r var
iabl
e 8
t(8)
N
8 4
rrI s
u*
dS
--
CP
U
CP
U
d N
8X
NR
to
tal
CP
U
Nt(8
) X
NR
tota
l C
PU
N
t(8)
N7J1
XN
7J2
tota
l (m
sec)
N
Y1
XN
Y2
tota
l (m
sec)
(a
naly
tica
l)
(mse
c)
(mse
c)
10
10 X
l 40
1.
3 4
X1
16
0.
68
2.5
2X
2
4 0.
091
2X
2
4 0.
29
9.98
5468
1 X
10
-2
3 10
X 1
40
1.
3 5
X1
20
0.
81
2.0
3X
3
9 0.
20
3X
3
9 0.
40
3.28
1341
0 X
10
-1
1 10
Xl
40
1.3
4X
1
16
0.68
2.
5 4
X5
20
0.
44
2X
3
6 0.
35
8.89
7447
3 X
10
-1
0.3
9X
1
36
1.2
5X
1
20
0.81
1.
8 9
X9
81
1.
8 6
X6
36
0.
99
1.88
6816
5 CA
l ~
0.1
9 X
1
36
1.2
5X
1
20
0.81
1.
8 25
X 2
5 62
5 13
9
X 1
0 90
2.
2 2.
6127
678
0.03
9
X 1
36
1.
2 6
X1
24
0.
92
1.5
64 X
64
4,09
6 88
1
1X
11
12
1 2.
8 2.
9818
223
0.01
9
X1
36
1.
2 3
X1
12
0.
57
3.0
160
X 1
60
25,6
00
554
20 X
20
400
8.8
3.10
1112
3
0.00
3 9
X1
36
1.
2 3
X1
12
0.
57
3.0
250
X 2
50
62,5
00
1,35
0 35
X 3
5 1,
225
27
3.14
4368
0
0.00
1 9
X1
36
1.
2 3
X1
12
0.
57
3.0
256
X 2
56
65,5
36
1,41
0 (€
=2
X1
0-
4 )
45 X
50
2,25
0 49
3.
1568
704
0 9
X1
36
1.
2 1
X1
4
0.35
9.
0 -
--
--
-3.
1631
456
Sou
rce
poin
t x
s=(0
.25
,O.2
5,d
),
rela
tiv
e er
ror
£<
10
-6
Rel
ativ
e E
rror
Nea
rly
Sin
gula
r In
tegr
al
10-5
\,\
0".
•
\\
'x_
x",
'\
·x
-.x
, \
·'x_.
_x...
.....
Ga
uss
'
. ...
...x_
.-x-..
..
'\ \
--,x_'_
'_'_'--'
' .
\ \ \ '\
' .
~ \
~ ,
~ .~
,
",
' ~.
~
....
....
... .
t....
Tel
les
' ---.
. , ~
'--"-
..
\ (l
=1
,e
i"\..
..-,
J s u*
dS
S :
pla
nar
squa
re
Xs =
x (0
, 0
)
d =
0.1
PLR
(0.
5, 0
.5)
10-1
Co)
~
1 10-3
10-4
10-6
, b
10-7
,
~~~~~
50
100
150
'20
0
Num
ber
of
Inte
grat
ion
Po
ints
Fig.
10.
2 R
elat
ive
erro
r vs
. N
umbe
r o
f in
tegr
atio
n po
ints
340
Table 10.1 and Fig. 10.2 also show that the angular variable transformation
t(8) of equation (5.127) is useful in reducing the number of integration points in
the angular variable. This is shown more clearly in Table 10.2 and Fig. 10.3
where the effect of moving the position of the source projection xs=x(~, ~) from
centre ~ = 0 to the corner ~ = 1 along the diagonal is shown, keeping the source
distance constant at d = 10 -1. Fig. 10.4 shows the convergence graph for the case
when the source projection is near the corner of the element i.e. Xs = x(0.9, 0.9),
wi th the source distance d = 10 -1.
The angular variable transformation t(8) becomes particularly effective
when the source projection Xs is situated near the corner or edge of the element S.
Hence, the transformation t(8) gives a robust numerical integration method for
the angular variable.
For the Gauss and Telles methods, the required number of integration points
decreases as Xs approaches the corner or edge of the element. This is because the
integration points of the Gauss and Telles methods are concentrated near the
corner or edge of the element, so that the near singular integral is integrated more
accurately when the source projection (around which the integral kernel changes
most rapidly) is situated near the corner or edge of the element, compared to when
it is situated in the centre of the element.
Table 10.3 shows the effect of the aspect ratio of the element on the
numerical integration. The source projection is set to xs=x(O, 0) and the source
distance is fixed at d= 10-1• Again, the robustness of the angular variable
transformation t(8) against the aspect ratio of the element is shown, compared to
using 8 as the angular variable or using the Gauss and Telles' methods. With the
product type formulas of Gauss and Telles, the number of integration points
increases approximately proportionally to the aspect ratio.
~
0 0.2
0.4
0.6
0.8
0.9
0.92
0.94
0.96
0.98
0.99
0.99
5
0.99
9
1.0
1.05
1.1
Tab
le 1
0.2
Nea
rly
sin
gu
lar i
nte
gra
l Is
u* d
S o
ver t
he u
nit
pla
nar
squ
are:
PL
R(0
.5,0
.5)
(Eff
ect o
f pos
itio
n of
sour
ce p
roje
ctio
n xs
)
R(p
)=Y
p2+
d2
, (a
=l)
G
auss
T
elle
s A
ng
ula
r v
aria
ble
8
t(8)
N
o C
PU
C
PU
--
CP
u
NoX
NR
to
tal
(mse
c) N
t(o)
XN
R
tota
l (m
sec)
N
t(o)
N1J
IXN
7J2
tota
l (m
sec)
N
y1X
NY
2 to
tal
6X
1
24
0.88
5
X1
20
0.
78
1.2
28
x2
8
784
17
10
X8
8
0
7X
1
28
1.0
5X
1
20
0.82
1.
4 2
8x
28
78
4 17
9
X9
81
8X
1
32
1.1
5x
1
20
0.81
1.
6 25
X25
62
5 13
9
X1
0
90
10
X1
40
1.
3 6
X1
24
0.
91
1.7
25X
25
625
13
8X
9
72
16
X1
64
2.
0 6
X1
24
0.
91
2.7
20
X2
0
400
8.6
9X
10
90
25
X1
10
0 3.
0 8
x1
32
1.
2 3.
1 1
6x
16
25
6 5.
5 9
X9
81
25X
1 10
0 3.
0 8
X1
32
1.
2 3.
1 1
4x
14
19
6 4.
2 7
X8
5
6
28X
1 11
2 3.
3 8
X1
32
1.
2 3.
5 1
4x
14
19
6 4.
2 9
X9
81
32
X1
12
8 3.
7 8
X1
32
1.
2 4.
0 1
2X
12
14
4 3.
1 8
x8
64
45
X1
18
0 5.
2 9
X1
3
6
1.2
5.0
9X
12
10
8 2.
4 6
X8
4
8
60
X1
24
0 6.
9 8
X1
32
1.
2 7.
5 1
2X
12
14
4 3.
1 8
X8
6
4
80
X1
32
0 9.
0 8
X1
32
1.
2 10
1
2X
12
14
4 3.
1 8
x8
64
150X
1 60
0 17
6
X1
24
0.
91
25
12
X1
2
144
3.1
8X
8
64
4x
1
8 0.
43
3x
1
6 0.
38
1.3
12
X1
2
144
3.1
8x
8
64
20X
1 80
2.
4 6
X1
24
0.
91
3.3
10
x1
0
100
2.2
7X
7
49
14
X1
56
1.
8 5
X1
20
0.
80
2.8
9X
10
90
2.
0 6
x6
3
6
Pos
itio
n of
sour
ce p
roje
ctio
n: x
s=x(
O,O
).
Sou
rce
dist
ance
d=
10
-1.
Rel
ativ
e er
ror
<1
0-6
•
Inte
gra
l
cp
u
47r Is
u* d
S
(mse
c)
(ana
lyti
cal)
1.9
2.95
3280
9
2.0
2.89
9330
1
2.2
2.73
6286
4
1.8
2.46
0389
4
2.2
2.06
9063
1
2.0
1.83
9027
2
1.4
1.79
2319
4
2.0
1.74
6007
8
1.6
1.70
0423
3
1.3
1.65
5895
5
1.6
1.63
4123
5
1.6
1.62
3375
3
1.6
1.61
4846
3
1.6
1.61
2723
9
1.3
1.51
2115
5
0.99
1.
4229
312
Num
ber
of I
nteg
ratio
n P
oint
s
100
o ._._
._.-
.-.-
JC
Nea
rly
Sin
gula
r In
tegr
al
". ""
Is
u* d
S,
S:
unit
plan
ar s
quar
e P
LR (
0.5
, 0
.5 )
""
Sou
rce
proj
ectio
n is
= x
(11 ,
11 )
",
Sou
rce
dist
ance
d
= 0
.1
'~_._._._._._.~
Rel
ativ
e er
ror
< 1
0 -
6
r'
9 G
auss
",
"',' "
"
"',\
'\ \
~ \
II
, II
,
II
\..
I I
'I
II
\ 9:
,
II
\ I
I x.~
I I
. 9:
T
elle
s 0
.= 1
, e
\cl ~
,J
a. =
1,
t(e}
,J
.a
p.-.J
:',
_ ••
_
•• _
••
_
•• _
• .
6.-
•• -
•• -
•• -
•• -.~ ••
-••
_
•• _
••
_
•• oo
A •
• ,..
... .
. ::..
.·4
A
I
po
-\,
(.
\',0
-<7
: _
. ••
-•• .,/
!i,-•
• -..
..-
" '\
. I ~ __
_ "
__ --
____
__ --
---0
:r
.:"'
JI.
i -
-'''~~ ....
~o
~
__
__
__
__
__
__
__
-0-
--;
I _
----
----
----
----
y---
----
----
----
-.,.,
....-
• •
• • .,
--
~ -
~ ,~
Y
>if
0.
2 0.
4 0.
6 0.
8 0.
9 1.
0 1.
1
Fig
.10.
3 N
umbe
r of
int
egra
tion
poin
ts
VS
. P
ositi
on
of s
ourc
e pr
ojec
tion
Xs
~
I\)
Rel
ativ
e E
rror
1 10-1
10
-3
10-4
10
-5
10-6
q ,
Ne
arl
y S
ing
ula
r In
tegr
al
f u*
dS
s
S :
pla
na
r sq
uare
P
LR
(0
.5,
0.5
)
.~
.lC"'X~,
. '\0
Xs
= x
(0.9
, 0
.9)
d =
0.1
X' ~\
. ,
\ ~
X
• X
-1\
" \.
Q.
. '0
"
\. \:-
"x-"
.x"
\.~.
\\
'''.
_
x _
_ .
Gau
ss
\ \
'x-'
--
.. x
~ --.-
-'.
,
.--.--
\ '
.~
"-:--
" x-.
...._
SA
,
~.
" '"
-.....-.
... '.
0
.-..
..
Tel
les
~"
a =
1, e
.-
-x
, a
= 1
, t(
e)
\(V
•
\,
10~1
• 5
0
o
100
150
20
0
Num
ber
of
Inte
grat
ion
Po
ints
Fig
. 10
.4
Rel
ativ
e er
ror
vs.
Num
ber
of
inte
grat
ion
po
ints
~
w
Asp
ect
b ra
tio 1
0.5
2 1.
0
3 1.
5
5 2.
5
10
5.0
Tab
le 1
0.3
Nea
rly
sin
gu
lar i
nte
gra
l Is
u*
dS o
ver t
he p
lan
ar re
ctan
gle:
PL
R(0
.5, b
)
(Eff
ect o
f the
asp
ect r
atio
of t
he e
lem
ent)
R(p)=~,
(a=
l)
Gau
ss
Tel
les
Ang
ular
var
iabl
e 8
t(8)
N
O
--
CP
U
CP
U
NoX
NR
to
tal
CP
U
Nt(
o)X
NR
to
tal
CP
U
Nt(O
) N
1JIX
N1J2
to
tal
(mse
c)
NY
IXN
Y2
tota
l (m
sec)
(m
sec)
(m
sec)
6X
1
24
0.89
5
X1
20
0.
80
1.2
28X
28
784
17
10X
8 80
1.
9
8X
1
32
1.1
7X
1
28
0.99
1.
1 28
X55
1,
540
34
8X9
72
1.7
10
x1
40
1.
3 8
X1
32
1.
1 1.
3 25
X72
1,
800
39
8X16
12
8 2.
9
14
X1
56
1.
8 9
X1
36
1.
2 1.
6 25
X12
0 3,
000
65
8X25
20
0 4.
4
20X
1 80
2.
4 1
2x
1
48
1.6
1.7
25X
240
6,00
0 13
0 8X
28
224
4.9
--
------
Sou
rce
proj
ecti
on:
:lts=
x(O
,O).
So
urce
dis
tanc
e d
=1
0-1
. R
elat
ive
erro
r <
10
-6•
Inte
gra
l
41t"
Isu
*d
S
(ana
lyti
cal)
2.95
3280
9 ,
t 4.
2282
421
'
5.01
5173
0
6.02
3988
5 '
7.40
4745
9 -~
345
Next, the effectiveness of the radial variable transformation
(10.20)
corresponding to
pdp = r ' " dR (a=3 ) (10.24)
for the integration
Is q* clS 1
Is (r, n)
=--411' r"
d
Is clS
=--411' r"
(a=3 ) (10.17)
over a constant planar element S, is demonstrated in Table 10.4 .
Again, the unit planar square element PLR(0.5, 0.5) with the source point at
xs=(0.25, 0.25, d) is taken as an example. In this case, the unit outward normal n
is defined as
where
n=- G
IGI
ax ax G= -x
a'll a'l2
as shown in Fig. 10.1, so that
(r,n) = d
(9.25)
(9.26)
(9.27)
Similar to the case of Is u* dS, a=l in Table 10.1, only one integration
point is required for the radial variable, independent of the source distance d. For
the Gauss and Telles' methods, even more integration points are required when
0< d ~ 1, since the order of near singularity is a = 3 for q*, compared to a = 1 for u *.
Fig. 10.5 shows the convergence graph for the case when the source distance is
d= 10-1 and the source projection xs=x(O, 0).
Tab
le 1
0.4
Nea
rly
sin
gu
lar
inte
gra
l Is
q*
dS
ove
r th
e u
nit
pla
nar
squ
are:
PL
R(O
.5,0
.5)
R(p
)= -
11
Yp
2+
d2
, (a
=3
) G
auss
T
elle
s S
ourc
e In
teg
ral
dist
ance
A
ng
ula
r va
riab
le B
t(B
) N
o 4r
r Is
q* d
S
--
CP
U
CP
U
d N
oX
NR
to
tal
CP
U
Nt(
O)X
NR
to
tal
CP
U
Nt(o
) N
7hX
N71
2 to
tal
(rns
ec)
NY
IXN
Y2
tota
l (r
nsec
) (a
naly
tica
l)
(rns
ec)
(rns
ec)
10
10
X1
40
1.
4 4
X1
16
0.
70
2.5
3X
3
9 0.
20
3X
3
9 0.
40
-9.9
56
50
67
X 1
0-3
3 1
0X
1
40
1.4
5X
1
20
0.88
2.
0 3
X4
12
0.
27
3X
4
12
0.47
-1
.06
03
98
1 X
10
-1
1 7
X1
28
1.
0 5
X1
20
0.
81
1.4
5X
6
30
0.68
4
X4
16
0.
58
-7.1
92
05
27
X1
0-1
0.3
6X
1
24
0.91
7
X1
28
1.
0 0.
86
12
X1
2
144
3.1
8X
8
64
1.6
-2.7
34
64
19
~
C)
0.1
5X
1
20
0.80
7
X1
28
1.
2 0.
71
32
X3
2
1024
22
1
4X
14
19
6 4.
5 -4
.76
08
10
4
0.03
3
X1
12
0.
57
7X
1
28
1.0
0.43
1
00
X1
10
11
,000
24
0 25
X25
62
5 14
-5
.80
61
32
7
0.01
3
X1
12
0.
57
7X
1
28
1.0
0.43
25
6X25
6 65
,536
1,4
30
45X
45
2,02
5 45
-6
.12
35
15
4
(e=
2X
10
-6)
0.00
3 3
X1
12
0.
57
3X
1
12
0.58
1.
0 2
56
X2
56
1,
420
72
X7
2
5,18
4 11
3 -6
.23
52
61
9
(e=
2X
10
-3)
65,5
36
,
0.00
1 3
X1
12
0.
57
7X
1
28
1.0
0.43
25
6X25
6 65
,536
1,4
30
100X
100
10,0
00
217
-6.2
67
21
02
(e
=2
X1
0-1
) -----
---
--
-
Sou
rce
poin
t xs
=(O
.25,
0.25
,d).
R
elat
ive
erro
r e<
10
-6•
Rel
ativ
e E
rror
10
1 10
-3
10-4
10
-5
10
-6
10
-7, l
Nea
rly
Sin
gu
lar
Inte
gral
J
q* d
S
s S
: p
lan
ar
squa
re
PLR
(0.
5, 0
.5)
\ ~\
--
(0 0
),
d =
0.1
V
x
s-x
,
\ ·V
'-..
, _.x,
G
au
ss
x ·'
x-.-
x ....
.......
__ ~~
.-x
__
. __
x
-'-'-x
'
~-. _._
._._. __
._._.
l-~\
\. '\ /"
I I I I I I I I I I I I ~
'.'
'-.'.6.
-'y
~,.,
a. =
3,
t(e
)
", ","
""'.
t.. '"
" T
elle
s ,.,,
" , ...
........ ,'
"
=6r~~~_ ~
50
100
15
0
200
Num
ber
of
Fig.
10.
5 R
elat
ive
erro
r V
S.
Num
ber
of
inte
grat
ion
poin
ts
Inte
grat
ion
Poi
nts
~ ...,
348
This time, the angular variable transformation t(8) does not payoff when
d;'£ 0.3. However, Table 10.5 and Fig. 10.6 show that when the source projection
xs=x(~,~) is near the corner of the element (0.96;'£~;'£1.0), the transformation
t(8) helps to reduce the number of integration points in the angular variable.
Hence, the angular variable transformation t(8) ensures the robustness of the
numerical integration method against the position of the source projection xs.
This is demonstrated in the convergence graph in Fig. 10.7 for the case
xs=x(0.99,0.99), d=10-1.
Table 10.6 demonstrates the effect of the aspect ratio of the element on the
numerical integration of f s q* dS. It is shown that the radial variable
transformation method is robust against the aspect ratio, where as the Gauss and
Telles' methods are strongly affected by the aspect ratio.
Summing up, it has been demonstrated that the singularity cancelling
radial variable transformations of equations (10.19) and (10.20) require only one
radial integration point for the exact evaluation of the nearly singular integrals
f s u* dS, (a = 1) and f s q* dS' (a = 3), respectively, when S is a constant planar
element. The effectiveness of the angular variable transformation t(8), for cases
where the source projection Xs is very near the element edge, and for elements
with high aspect ratio, was also demonstrated.
-
~
0 0.2
0.4
0.6
0.8
0.9
0.92
0.94
0.96
0.98
0.99
0.99
5
0.99
9
1.0
1.05
1.1
Ave
rage
C
PU
-tim
e p
er p
oin
t
Tab
le 1
0.5
Nea
rly
sin
gu
lar
inte
gra
l Is
q* d
S o
ver
the
un
it p
lan
ar s
qu
are:
PL
R(0
.5,0
.5)
(Eff
ect o
f pos
itio
n of
sour
ce p
roje
ctio
n :K
s)
R(p
)=_
ltY
p2+
d2,
(a=
3)
Gau
ss
Tel
les
Inte
gra
l iA
ngul
ar v
aria
ble
0
t(O
) N
o 4r
r Is
q*
dS
--
(ana
lyti
cal)
N
oxN
R
tota
l N
t(O
)XN
R
tota
l N
t(o)
N7J
j XN7
J2
tota
l N
YjX
NY
2 to
tal
4X
1
16
5X
1
20
0.80
40
X40
1,
600
14
x1
4
196
-5.1
70
19
80
4X
1
16
5X
1
20
0.80
3
5X
35
1,
225
12
X1
4
168
-5.1
16
42
94
4X
1
16
7X
1
28
0.57
35
X40
1,
400
14
x1
4
196
-4.9
30
51
74
5X
1
20
8X
1
32
0.63
32
X32
1,
024
11
X1
2
132
-4.5
06
32
49
7X
1
28
9X
1
36
0.78
2
5x
28
70
0 1
4X
14
19
6 -3
.49
66
63
1
9X
1
36
10
X1
40
0.
90
25
x2
5
625
14
X1
4
196
-2.5
45
77
65
10X
1,
40
8X
1
32
1.3
20X
20
400
12
X1
1
132
-2.3
19
22
53
11
X1
44
1
1X
1
44
1.0
20X
20
400
12
X1
2
144
-2.0
88
05
65
14
x1
56
1
1X
1
44
2.3
20X
20
400
12
X1
2
144
-1.8
58
30
67
20
X1
80
1
2X
1
48
1.7
20X
20
400
11
X1
1
121
-1.6
36
77
57
28
x1
11
2 1
1X
1
44
2.5
10X
20
200
9X
11
99
-1
.53
11
67
9
35
X1
14
0 1
4X
1
56
2.5
20X
20
400
9x
11
99
-1
.47
99
75
2
60
X1
24
0 1
4X
1
56
4.3
20X
20
400
11
X1
1
121
-1.4
39
86
58
3x
1
6 3
X1
6
1.0
20X
20
400
11
X1
1
121
-1.4
29
96
05
7X
1
28
8X
1
32
0.88
1
4x
16
22
4 1
0X
10
10
0 -1
.00
46
30
6
5X
1
20
7X
1
28
0.71
1
2x
12
14
4 8
X8
64
-7
.15
18
00
8X
10
-1
38,u
sec
38,u
sec
22,u
sec
23,u
sec
~ -~
-----------
_ ...
_-
-------~~-
~
Sou
rce
proj
ecti
on:
:Ks =x(~, ~).
S
ourc
e di
stan
ce d
= 1
0-1
• R
elat
ive
erro
r <
10
-6•
w ~
<0
Num
ber
of
Inte
grat
ion
Poi
nts
400
300
200
100
Nea
rly S
ingu
lar
Inte
gral
Is q*
dS
S
: un
it pl
anar
squ
are
Sou
rce
proj
ectio
n x s
= X
( il,
il)
S
ourc
e di
stan
ce
d =
0.1
R
elat
ive
erro
r <
10
-6
\
\ \ \ ·x. '.
rJ G
auss
'<
.~ i i i i i i l*"_
~ i . \
.1 \
Ii .
i' \
.1 \
\i .
Tel
les
~ \
"-"-
---"
---6
-"-"
----
"-"0
1>..
.. -.. -
_ .. .!!
b-.. -
.. ~\
~ \
........
.......
.~..
I"
'--'
".
\ •. ~
._..
}..~
6! x
u=
3
t(e)
u
=3
, e~
~'''
''''
'''
) ,
p:
'. ~
~ 0
': _
_ 0
_. ~
;-----
-l----
---3C-
------
i-----
--3?--
-----i
------
--~---
-----;
------
-~----
---~~
-->
11
o
0.2
0.4
0.6
0.8
0.9
1.0
1.1
Fig
.10.
6 N
umbe
r of
int
egra
tion
poin
ts
VS
. P
ositi
on
of
sour
ce
proj
ectio
n Xs
w
C11 o
Rel
ativ
e E
rror
1 \
Nea
rly
Sin
gula
r In
tegr
al
f q*
dS
s
~\
10-1~'
~f'x
0. •• 'Q
"o '.
.. ,
,
S :
pla
nar
squa
re
"is
= x
(0.9
9, 0
.99)
d =
0.1
PLR
(0
.5,
0.5
)
ex =
3,
t(9)
x \ 0,
. x,
10-2~
\ ....
.. '\
''x_
.. x
'''x
'.
. \
.~ .
x__
Gau
ss
... _A..
.. \.
/. __
x"-..
C,-'
b~.."
\.
........
.. ......
......
A.....
.."--i
. .
/ .....
...... x
_
\". \.
·_
·-·-·_
·-x
"\'"
" /
'''q,
x
'. "A
T
elle
s \
\<~ "rJ
./
"0
, .. ~
9.>
'" ex
= 3,
0
10-3
10
-4
10
-5
10
-6
I I
~
10
-7 I
I I 0
15
0
20
0
Num
ber
of
50
10
Inte
grat
ion
Poi
nts
Fig
. 10
.7
Rel
ativ
e er
ror
vs.
Num
ber
of
inte
grat
ion
poin
ts
w
~
Asp
ect
b ra
tio 1
0.5
2 1.
0
3 1.
5
5 2.
5
10
5.0
Tab
le 1
0.6
Nea
rly
sin
gu
lar i
nte
gra
l Is
q* d
S o
ver t
he
pla
nar
rect
angl
e: P
LR
(0.5
, b)
(Eff
ect o
f the
asp
ect r
atio
of t
he e
lem
ent)
R(p
) = _
ltv
'p2
+d
2 ,
(a=
3)
Gau
ss
Tel
les
Ang
ular
var
iabl
e 8
t(8)
N
o --
CP
U
NoX
NR
to
tal
CP
U
Nt(O
)XN
R
tota
l C
PU
N
t(o)
N'71
XN
'72
tota
l (m
sec)
N
Y1X
NY
2 to
tal
(mse
c)
(mse
c)
4X
l 16
0.
67
5X
1
20
0.79
0.
80
40
X4
0
1,60
0 35
1
4X
14
19
6
4X
1
16
0.67
6
X1
24
0.
92
0.67
4
0X
80
3,
200
70
14
X2
0
280
5X
l 20
0.
78
9X
1
36
1.3
0.56
3
5X
11
0
3,85
0 84
1
4X
25
35
0
5X
l 20
0.
78
llX
1
44
1.5
0.45
40
X19
0 7,
600
167
16
X3
2
512
5X
l 20
0.
79
14
X1
56
1.
9 0.
36
256X
256
65,5
36
1,42
0 1
4X
40
56
0 (E
=8
X1
0-5
)
Sou
rce
proj
ecti
on:
xs=
(O, 0
). S
ourc
e di
stan
ce d
= 1
0-1
. R
elat
ive
erro
r <
10
-6•
CP
U
(mse
c)
4.5
6.3
7.8
11
12
Inte
gra
l
411" I
sq*
dS
(ana
lyti
cal)
-5.1
70
19
80
-5.3
99
83
35
-5.4
50
46
83
-5.4
77
77
87
-5.4
89
61
42
w
U1
N
353
10.3 Application of the Singularity Cancelling Transfonnation to Elements with
Curvature and Interpolation Functions
The success of the singularity cancelling radial variable transfonnation R(p)
of equations (10.19) and (10.20) for constant planar elements, encourages us to
apply the method to curved elements and integrals including interpolation
functions cP ij ( r; l' r; 2) •
(1) Application to curved elements
Results of applying the singularity cancelling radial variable
transfonnation
R (P) = "';p2 + d2 (a = 1) (10.19)
in combination with the angular variable transfonnation teO) of equation (5.127),
to the nearly singular integral
I u· dB = ..!... I dB S 41r S r
(10.22)
over the 'spherical' quadrilateral element SPQ60 defined in section 8.2, are given
in Tables 10.7, 10.8 and 10.9. In each case, the source projection was set to
x=x(~,~) with ~ =0, 0.5 and 0.9, respectively. The source pointxs was located at
x = x + do 8 B
(10.28)
where the source distance d was varied from 10 to 10-3• The unit outward
nonnal n is defined by
0= G/IGI (10.29)
(10.30)
Sou
rce
dist
ance
d
10
3 1 0.3
0.1
0.03
0.01
0.00
3
0.00
1
Tab
le 1
0.7
Nea
rly
sin
gu
lar
inte
gral
Is u
* dS
ove
r th
e 's
pher
ical
' qua
dril
ater
al:
SP
Q60
( Sou
rce
proj
ecti
on ,x
s = x
(O, 0
) )
R(p
)=V
p2
+d2
, (a
=l)
T
elle
s In
tegr
al
Ang
ular
var
iabl
e: t
(O)
411'
Isu
*d
S
Nt(
8)X
NR
(p)
tota
l C
PU
N
YIX
NY
2 to
tal
CP
U
(log
-Ll,
128
X 1
28)
(mse
c)
(mse
c)
5X
3
60
1.7
4X
5
20
0.62
1.
0519
218X
10
-1
5X
3
60
1.7
4X
5
20
0.62
3.
5505
101
X 1
0-1
5X
3
60
1.7
5X
5
25
0.72
1.
0458
484
5X
4
80
2.2
7X
7
49
1.2
2.39
9997
7
5X
6
120
3.1
10
X1
1
110
2.5
3.17
0799
6
5X
12
24
0 5.
9 1
6X
20
32
0 6.
9 3.
4722
881
4X20
32
0 7.
7 2
8X
28
78
4 17
3.
5598
394
4X35
56
0 13
2
5X
25
62
5 13
3.
5905
652
5X
45
90
0 21
5
0X
50
2,
500
53
3.59
9350
0
Ave
rage
CP
U -t
ime
per p
oint
:
26,u
sec
" 25
,use
c ---
Rel
ativ
e er
ror
£<
10
-6•
Sou
rce
dist
ance
d=
O.O
l, t
owar
ds c
entr
e of
sphe
re.
w ~
Tab
le 1
0.8
Nea
rly
sin
gu
lar i
nte
gra
l Is
u* d
S ov
er th
e 's
pher
ical
' qu
adri
late
ral:
SP
Q60
( Sou
rce
proj
ecti
on X
s = x(0
.5, 0
.5)
)
Sou
rce
R(p)~V p
2+d2
, (a
=l)
; T
elle
s In
teg
ral
An
gu
lar
vari
able
: te
e)
dist
ance
4l
r Is
u* d
S
d N
t(O)X
NR
(p)
tota
l C
PU
N
y! X
NY2
to
tal
CP
U
(log
-Ll,
128
X 1
28)
(mse
c)
(mse
c)
10
6X
20
48
0 12
4
X5
20
0.
68
1.0
57
66
05
XI0
-!
3 6
X2
0
480
12
4X
5
20
0.68
3
.60
25
80
8X
I0-!
1 6
X2
0
480
11
5X
4
20
0.62
1.
0467
642
0.3
5X
20
40
0 9.
6 6
X6
36
1.
0 2.
1748
271
0.1
5X
14
28
0 6.
8 9
X9
81
2.
0 2.
8377
859
0.03
6
X2
0
480
12
llX
16
17
6 4.
0 3.
1353
675
0.01
6
X2
8
672
16
20X
2O
400
8.7
3.22
6382
1
0.00
3 6
X4
0
960
23
35X
35
1,22
5 26
3.
2587
902
0.00
1 6
X6
0
1,44
0 34
45
X50
2,
250
47
3.26
8099
5
Ave
rage
CP
U -t
ime
per
po
int
: 24
,use
c "
27,u
sec
Rel
ativ
e er
ror
<1
0-6
• S
ourc
e po
int
xs,
tow
ards
cen
tre
of sp
here
.
w
U'1
U'1
Tab
le 1
0.9
Nea
rly
sin
gu
lar i
nte
gra
l Is
u*
dS o
ver
the
'sph
eric
al' q
uadr
ilat
eral
: S
PQ
60
( Sou
rce
proj
ecti
on X
s =
x(0.
9, 0
.9)
)
Sou
rce
R(p
)=V
p2+
d2
, (a
=l)
;
An
gu
lar v
aria
ble:
t(8
) T
elle
s In
teg
ral
dist
ance
41
l" Is
u* d
S
d N
t(8)
X N
R(P
) to
tal
CP
U
,NY
1 X
NY
2 to
tal
CP
U
(log
-LI,
128
X 1
28)
(mse
c)
(mse
c)
10
6X
20
48
0 12
4
X5
20
0.
68
1.06
8778
4 X
10
-1
3 5
X2
5
500
12
4X
5
20
0.68
3.
7004
261
X 1
0-1
1 5
X2
0
400
10
4X
4
16
1.1
1.02
9869
0
0.3
6X
25
60
0 14
6
X6
36
1.
0 1.
7293
821
0.1
6X
20
48
0 12
7
X7
49
1.
3 2.
0268
382
0.03
7
X2
5
700
17
9X
12
10
8 2.
6 2.
2105
752
0.01
9
X3
2
1,15
2 27
2
0X
20
40
0 8.
7 2.
2935
015
0.00
3 1
0X
55
2,
200
51
28
X2
8
784
17
2.32
7637
9
0.00
1 1
0X
90
3,
600
85
40
X2
5
1,00
0 21
2.
3379
046
Ave
rage
CP
U -
tim
e p
er p
oin
t:
24,u
sec
" 31
,use
c
Rel
ativ
e er
ror
<1
0-6
• S
ourc
e p
oin
t x
s, t
owar
ds c
entr
e of
sphe
re.
w ~
I
Rel
ativ
e E
rror
1
i kt i ~\ ~ II- x,
t. ~
: \
\ I
. :/ , ~ ..
~. \\ .~ ..
Nea
rly S
ingu
lar
Inte
gra.
Is u
· dS
S :
'sph
eric
al'
quad
rilat
eral
sp
a 60
Xs =
x (
O,O
)
d =
0.
01
_ .... -.
.. _, .....
. ........
.....
rv e
x =
1 ,t
(a),
...
.... ...
.: T
elle
s
(Nt(
e)=
4)
... A.
.... ••
-" -"-"A
-7
I N
umbe
r of
10
>
' In
tegr
atio
n P
oint
s 10
0 20
0 30
0 40
0 50
0 60
0 70
0 80
0
Fig.
10,8
R
elat
ive
erro
r vs
, N
umbe
r of
in
tegr
atio
n po
ints
Co)
~
Rel
ativ
e E
rror
1
-3
10 -4
10
-5
10
10
-6
Nea
rly S
ingu
lar
Inte
gral
\.1.:
A \
, . l\
\
/ \,
\/
", A
••
,
Is u·
dS
S :
'sp
heric
al'
quad
rilat
eral
S
PQ
60
is =
x (
0.5
, 0
.5 )
d =
0.0
1
"'"
Tel
les
'. f"
'-/
"
"~
f"'-/
(l
= 1
, t
(6)
,
(Nt(
e)=
6)
-7
I I
Num
ber o
f 10
;>
' In
tegr
atio
n P
oint
s 1 0
0 20
0 30
0 40
0 50
0 60
0 70
0
Fig
. 10
.9
Rel
ativ
e er
ror
vs.
Num
ber
of i
nteg
ratio
n po
ints
w 81
Rel
ativ
e E
rror
1
i i :t' k: \ if,
1
" '" \. \,
Nea
rly S
ingu
lar
Inte
gral
Is u*
dS
S :
'sph
eric
al'
quad
rilat
eral
spa
60
Xs =
x (
0.9
, 0.9
)
d =
0.0
1
-5
10
'~ .. --
-\..
Tel
les
" ~
/J. =
1,
t(e),
r (N
t (9)
= 9
)
-6
10
" " ", ""-
A
, -7
. >
'
10
800
900
1000
11
00
Num
ber o
f 10
0 20
0 30
0
Fig.
10.
10
400
500
600
700
Rel
ativ
e er
ror
vs.
Num
ber
of i
nteg
ratio
n po
ints
In
tegr
atio
n P
oint
s
w !B
360
Hence the source point Xs is located towards the centre of the sphere (inside the
sphere when d~ 1). Fig. 10.8, 10.9 and 10.10 give convergence graphs for the
cases ~ =0,0.5 and 0.9, respectively, with the source distance fixed atd=O.01.
The results indicate that, unlike for constant planar elements, more than
one integration points are required in the transformed radial variable R and that
the number increases rapidly as the source distance d decreases, and as the
source projection Xs approaches the corner of the curved elements. Compared with
Telles' cubic transformation method of equation (4.51), the radial variable
transformation method of equation (10.19) (a = 1) requires less number of
integration points only for cases d~0.03 for ~=O, and d~0.003 for ~=0.5.
Similarly, results of applying the singularity cancelling radial variable
transformation
1 R (p) = - rrr--J[
Vp- + d-(a = 3) (10.20)
with the angular variable transformation teO), to the nearly singular integral
f 1 f (r, n) q* dB= -- -- dB
S 4lf S r3 (10.31)
over the 'spherical' quadrilateral SPQ60, are given in Tables 10.10, 10.11 and
10.12 for the cases ~ = 0, 0.5 and 0.9, respectively. The corresponding
convergence graphs for d=O.Ol are given in Fig. 10.11, 10.12 and 10.13.
Similar to the case of Is u*dS, more than one integration points are
required in the transformed radial variable R, and the number increases rapidly
as the source distance decreases and the source projection Xs approaches the
corner of the curved element. Compared to the Telles' method, this time the
singularity cancelling radial variable transformation method becomes slightly
more competitive. Namely, the method outperforms Telles' method for the cases
d~O.l (~=O), d~O.Ol (~=0.5) and d~O.OOl (~=0.9).
Tab
le 1
0.10
N
earl
y si
ng
ula
r in
teg
ral
Is q*
dS
ove
r th
e 's
pher
ical
' qu
adri
late
ral:
SP
Q60
( Sou
rce
proj
ecti
on :
Ks =
x(O, 0
) )
Sou
rce
R(p
) =
-lI"v
p2+
d2 ,
(a
=3
);
Tel
les
Inte
gra
l A
ng
ula
r va
riab
le:
t(8)
di
stan
ce
4rr Is
q* d
S
d N
t(8)
XN
R(p
) to
tal
CP
U
NY
l XN
Y2
tota
l C
PU
(l
og- L
l,1
28
X1
28
) (m
sec)
(m
sec)
10
5X
2
40
1.3
3X
4
12
0.45
-9
.86
97
96
0X
10
-3
3 5
X3
60
1.
8 4
X4
16
0.
54
-1.1
61
88
76
X1
0-1
1 6
X3
72
2.
1 5
X5
25
0.
74
-1.0
48
15
17
0.3
5X
5
100
2.7
8X
8
64
1.6
-4.7
85
47
23
0.1
5X
8
160
4.1
14
X1
6
224
5.1
-7.0
23
90
76
0.03
6
X1
6
384
9.5
25
X2
8
700
15
-7.7
89
42
55
0.01
6
X2
8
672
17
40
X4
5
1,80
0 40
-7
.99
74
84
5
0.00
3 6
X5
0
1,20
0 29
6
4X
72
4,
608
100
-8.0
68
94
84
0.00
1 6
x9
0
2,16
0 52
1
l0X
11
0
12,1
00
262
-8.0
89
23
19
Ave
rage
CP
U -
tim
e p
er p
oint
:
27,u
sec
" 26
,use
c -
---
-"------
Rel
ativ
e er
ror
<1
0-6
• S
ourc
e p
oin
t x
s,
tow
ards
cen
tre
of sp
here
.
,
w ~
Tab
le 1
0.11
N
earl
y s
ing
ula
r int
egra
l Is
q* d
S o
ver
the
'sph
eric
al' q
uad
rila
tera
l: S
PQ
60
( Sou
rce
proj
ecti
on X
s = x(
0.5,
0.5
) )
Sou
rce
R(p
) =
_lI
Vp
2+
d2
, (a
=3
);
Tel
les
Inte
gra
l A
ng
ula
r va
riab
le:
t(B
) d
ista
nce
4
nI s
q*
dS
d N
t(8
) X
N R
(p)
tota
l C
PU
N
y! X
NY
2 to
tal
CP
U
(log
-LI,
128
X 1
28)
(mse
c)
(mse
c)
10
5X
20
40
0 9.
8 3
X
4 .1
2 0.
52
-9.4
06
71
28
X 1
0-3
3 5
X2
0
400
9.9
4 X
4
16
0.60
-1
.15
62
05
8 X
1O
-!
1 6
X2
0
480
12
5 X
5
25
0.74
-1
.04
99
29
7
0.3
6X
14
33
6 8.
4 7
X
7 49
1.
3 -4
.08
62
77
0
0.1
6X
7
168
4.4
14
X
14
196
4.5
-6.4
26
57
68
0.03
8
X2
0
640
16
25
X
25
625
14
-7.4
55
59
89
0.01
8
X3
2
1,02
4 25
40
X
45
1,
800
40
-7.7
48
01
49
0.00
3 8
X4
5
1,44
0 35
64
X
72
4,
608
100
-7.8
48
97
97
0.00
1 9
X1
10
3,
960
95
100
X 1
00
10,0
00
217
-7.8
77
66
23
Av
erag
e C
PU
-ti
me
per
po
int:
25
,use
c "
28,u
sec
Rel
ativ
e er
ror
< 1
0-6
• S
ourc
e p
oin
t x
s, to
war
ds c
entr
e of
sphe
re.
I i !
w
OJ
N
Tab
le 1
0.12
N
earl
y s
ing
ula
r in
teg
ral
Is q*
dS
ove
r th
e 's
pher
ical
' qu
adri
late
ral:
SP
Q60
(So
urc
e pr
ojec
tion
xs=
x(0
.9, 0
.9)
)
Sou
rce
R(p)=-lI~
Tel
les
Inte
gra
l A
ng
ula
r v
aria
ble
: t(
8)
dist
ance
4;
r Is
q* d
S
d N
t(8)
XN
R(p
) to
tal
CP
U
NY
I X
NY
2
tota
l C
PU
(l
og-L
I, 1
28
X 1
28)
(mse
c)
(mse
c)
10
6X
32
76
8 19
3
X4
12
0.
52
-8.5
56
59
94
X1
0-3
3 6
X2
8
672
17
4X
4
16
0.60
-1
.14
87
44
9X
10
-1
1 6
X1
6
384
10
5X
5
25
1.3
-1.0
16
24
95
0.3
7X
14
39
2 9.
7 7
X8
56
1.
5 -2
.55
47
82
9
0.1
8X
7
224
5.8
llX
14
15
4 3.
6 -3
.70
59
92
1
0.03
1
0X
25
1,
000
24
14
X1
6
224
5.2
-5.5
99
18
54
0.01
1
0X
40
1,
600
39
28X
32
896
20
-6.7
34
61
03
0.00
3 1
4X
72
4,
032
96
50X
55
2,75
0 60
-7
.18
33
14
2
0.00
1 1
0X
12
8
5,12
0 12
2 80
X72
5,
760
125
-7.3
12
95
00
Ave
rage
CP
U -
tim
e p
er p
oint
:
25,u
sec
" 30
,use
c
Rel
ativ
e er
ror
< 1
0-6
• S
ourc
e p
oin
t x
s, to
war
ds
cent
re o
f sph
ere.
w
O'l
W
Rel
ativ
e E
rro
r
1
-4. ". "
" ~ .....
...
Nea
rly S
ingu
lar
Inte
gral
Is q*
dS
S :
'sp
heric
al'
quad
rilat
eral
sp
a 60
Xs =
x (
0, 0
)
......... , .. ,
.. ~ .. ,
.. ,
d =
0.01
"'A
--..
_ ..
-.6
..
Tel
les
'-"
rv
~ a
. = 3
, t (
e) ,
( N
t (s)
= 6
)
"',
.. , .. ,
•• ~.
-••
-••
-••
-••
-••
..6.
-7'
10 ~~~~~~~~~~ __
~~::~~~ __
~~~ __
~~~~~~Numberof
500
1000
15
00
;>'
Inte
grat
ion
Poi
nts
Fig.
10.
11
Rel
ativ
e er
ror
vs.
Num
ber
of
inte
grat
ion
poin
ts
eN
~
Rel
ativ
e E
rror
1 10 -
2
10
-3
-4
10 -
5
10 -
6
10
.. ~, ''\
,
Nea
rly S
ingu
lar
Inte
gral
fsq
* dS
S :
'sph
eric
al'
quad
rilat
eral
S
PQ
60
Xs =
x (
0.5
, 0
.5 )
d =
0.0
1
'L .. _
.. ~ "',"
''lI...
T
elle
s '.
[V
.....
.. ,
[V
a =
3 ,
t (e)
,
( N
qe
)= 8
)
..........
.... .. , .....
........ .....
. ........ .....
. -
71
,.
"r '
N
umbe
r c;>
f 10
>'
In
tegr
atio
n 50
0 10
00
1500
20
00
Poi
nts
Fig.
10.
12
Rel
ativ
e e
rro
r· v
s.
Num
ber
of
inte
grat
ion
poin
ts .
w
Ol
C1'I
Rel
ativ
e E
rror
1
I 1,~ .0\ , ~-
", "
", ", '. " ", "
~ \
Nea
rly S
ingu
lar
Inte
gral
fsq*
dS
S :
'sp
heric
al'
quad
rilat
eral
sp
a 60
Xs =
x
(0.9
, 0.
9 )
d =
0.01
a =
3,
t (9
) ,
~ rJ
( Nt(
S)=
10)
"-.
,. ) ~-
----
----.eo
... " T
elle
s •. ,
•. ~
10-7~'~--~~~-5~0-0~~--~~1-0·0-0~~~--~15~0-0~>~ N
umbe
r o
f In
tegr
atio
n P
oint
s
Fig.
10.
13
Rel
ativ
e er
ror
vs.
Num
ber
of in
tegr
atio
n po
ints
w ~
367
Summing up, for curved elements, the singularity cancelling radial variable
transformations R(p)=v' p2+d2, (a=1) for u* and R(p) = -Vv' p2+d2, (a = 3)for
q* do not perform as dramatically or efficiently as they did on constant planar
elements. Particularly, these radial variable transformations are not robust
against the position of the source projection xs, although the a = 3 transformation
is slightly more robust than the a = 1 transformation. The other shortcoming of
the singularity cancelling transformation is that different sets of integration
points are required for the kerneles u* and q*, unlike the Telles' method. Hence, a
more robust and efficient radial variable transformation is required.
The angular variable transformation teO) of equation (5.127) is robust
against the position of xs, since the number of angular integration points
increases only slightly as Xs approaches the corner of the element.
At this point, a more accurate cancellation of the near singularity by taking
into consideration the curvature of the element at the so.urce projection xs, in the
radial variable transformation was attempted, as mentioned in Chapter 5. This
approach showed some effect in decreasing the number of radial integration
points. However, it has the following shortcomings.
(a) Exact cancellation of u* or q* does not work efficiently when the
integral kernel includes interpolation functions ~ij' as will be shown in
the next section. (b) Much CPU-time is required in determining the exact projection of the
integration point x on the curved element, to the plane S tangent to S
atxs .
(c) The exact (instead of approximate) projection of the curved element Son
to the tangent plane Sbecomes complicated, since the edge of the
projected element is, in general, a curve in S .
(d) Different sets of integration points are required for the kernels u* and
q*.
368
(2) Application to integrals including interpolation functions
Next, the effect of applying the singularity cancelling radial
transformations of equation (10.19) and (10.20), to nearly singular integrals
including the interpolation functions ¢ij is tested.
For simplicity, the unit planar square PLR(0.5, 0.5) is taken as the element,
and the source point is set at xs=(0.25, 0.25, d) with various source distances
10 -3 ~ d ~ 10. The source projection is fixed at :is = x(0.5, 0.5).
The 9-point Lagrangian interpolation function
i,j= -1,0,1 (10.32)
where
(5.3)
is used as the interpolation function.
First, the effect of the singularity cancelling radial variable transformation
(a = 1) (10.19)
in combination with the angular variable transformation t(8) of equation (5.127)
on the nearly singular integral
J ; .. u· dS s IJ
i,j= -1,0,1 (10.33)
is shown in Table 10.13. The number of integration points required to achieve a
relative error less than 10-6 for all the components ¢ij' (i, j=-l, 0, 1) is
compared with the product type Telles' cubic transformation method of equation
(4.51). (The relative error was estimated by comparing with results obtained by
Nt= 128, NR= 128 points with the log-L1 radial variable transformation method,
mentioned later.)
Tab
le 1
0.13
N
earl
y s
ing
ula
r in
teg
ral
Is <Pi
j u*
dS
ov
er th
e u
nit
pla
nar
squ
are:
PL
R(0
.5,0
.5)
Sou
rce
R(p
)=V
p2
+d
2 ,
(a=
l);
An
gu
lar
var
iab
le:
tee)
T
elle
s In
teg
ral
dist
ance
47
r Is
<Pu
u* d
S
d N
t(8)
XN
R(p
) to
tal
CP
U
NY1
XN
Y2
tota
l C
PU
(l
og-L
l, 1
28
X 1
28pt
s.)
(mse
c)
(mse
c)
10
8X
12
8
4,09
6 11
2 3
X3
9
0.45
2.
7788
157
X 1
0-3
3 7
X1
28
3,
584
98
4X
4
16
0.63
9.
2963
434X
10
-3
1 6
X9
0
2,16
0 60
5
X5
25
0.
85
2.85
2009
9X1O
-2
~
<0
0.3
8X
90
2,
880
80
7X
8
56
1.7
9.27
8779
8 X
10
-2
0.1
7X
80
2,
240
61
10
X1
0
100
2.8
1.76
7045
3 X
10
-1
0.03
9
X1
10
3,
960
109
25
X2
5
625
16
2.2
75
63
48
X1
0-1
I I
0.01
9
X1
50
5,
400
147
35
X4
0
1,40
0 36
2.
4436
723
X 1
0-1
0.00
3 7
X8
0
2,24
0 61
5
5X
55
3,
025
79
2.50
4588
3 X
10
-1
0.00
1 6
X6
4
1,53
6 43
7
2X
72
5,
184
134
2.5
22
17
94
XI0
-1
Sou
rce
poin
t x
s=(0
.25
, 0.2
5, d
),
rela
tiv
e er
ror
<1
0-6
•
Rel
ativ
e E
rror
1 N
earl
y S
ingu
lar
Inte
gral
Is <i>
ij u*
dS
s:
plan
ar
squa
re
PLR
( 0
.5 ,
0.5
)
is =
x (
0.5
, 0.
5 )
d =
0.1
10-7l'
~--~~~
-=50~0
~--~~~
1O~~~~
~~~~--
~~~~~~
~t;~~>
10
00
1500
20
00
Num
ber
of
Inte
grat
ion
Poi
nts
Fig.
10.
14
Rel
ativ
e er
ror
vs.
Num
ber
of
inte
grat
ion
poin
ts
w ..., o
Rel
ativ
e E
rror
1 10
-3
10
-4
X, \ \ \ /--
\ lr-
' \ \
./t
Nea
rly S
ingu
lar
Inte
gral
Is <i>
ij u*
dS
S :
pla
nar
squa
re
PLR
( 0
.5 ,
0.5
)
'is =
x (
0.5
, 0
.5 )
d =
0.0
1
a =
1 ,
t(e)
10
-5
. ./ .
}".,
\ T
elle
s \/"
1../
~
10
-6
.l.. ••
_ .. ~
-7
, ,
10
>
1000
20
00
3000
40
00
Fig.
10.
15
Rel
ativ
e er
ror
vs.
Num
ber
of i
nteg
ratio
n po
ints
5000
N
umbe
r of
In
tegr
atio
n P
oin
ts·
w
..... -
372
The singularity cancelling radial variable transformation of equation
(10.19), a = 1, which was so successful with the integral J s u* dS over constant
planar elements turns out to be inefficient for the integral J s <Pij u* dS , containing
the interpolation function. The convergence graphs for the case d=O.l and
d=O.Ol are shown in Fig. 10.14 and Fig. 10.15, respectively. Only in cases
d=0.003, 0.001 where the source distance d is extremely small compared to the
element, the radial variable transformation R(p) of equation (10.19) is more
efficient compared to Telles' method.
The reason for this is that when applying the Gauss-Legendre rule to the
variable R, obtained by the transformation of equation (10.19), the integration
points are ideally positioned to integrate lIr but they are far from ideal for the
polynomial interpolation function <Pl'h' r;2) •
Similarly, the effect of the singularity cancelling radial variable
transformation
1 R(p)=- ~
v p- + d-(a = 3) (10.20)
in combination with the angular variable transformation t(8), on the nearly
singular integral
I 1> .. q* dS s IJ (i,j=-l,O,l) (10.34)
is shown in Table 10.14, in comparison with Telles' method. The convergence
graphs for the cases d=O.l and 0.01 are shown in Fig. 10.16 and Fig. 10.17,
respecti vely.
Many integration points are required in the transformed radial variable R
because the integration points are not optimally placed to integrate <Pij Ir as a
whole, instead of lIr. Summing up, for nearly singular integrals with kernels including
interpolation functions <Pij' the exactly cancelling radial variable transformations
R (P) = Jp2 + d2
1 R (p) = - rr-:rr
vp- + d-
373
(a = 1 for u*) (10.19)
(a = 3 for q*) (10.20)
do not give efficient numerical integration results. Hence, a more effective and
robust radial variable transformation (against interpolation function <Pij ,
curvature of element and position of source point) is required.
Tab
le 1
0.14
N
earl
y si
ng
ula
r in
teg
ral
Is ¢
ij q
* dS
ov
er th
e u
nit
pla
nar
squ
are:
PL
R(0
.5,0
.5)
R(p
) = -
11,.,
1 p2
+d2
,
(a=
3);
S
ourc
e T
elle
s In
teg
ral
Ang
ular
var
iabl
e: t
(8)
dist
ance
4r
r Is
¢u
q*
dS
d N
t(8)
XN
R(p
) to
tal
CP
U
NY
1X
NY2
to
tal
CP
U
(log
-Ll.
128
X 1
28pt
s.)
(mse
c)
(mse
c)
10
8X12
8 4,
096
114
3X
4
12
0.53
-2
.78
08
84
2X
10
-4
3 7X
120
3,36
0 93
4
X5
20
0.
75
-3.1
22
53
19
X 1
0-3
1 8X
100
3,20
0 89
6
X6
36
1.
2 -
2.95
2845
3 X
10
-2
0.3
8X
72
2,
304
64
9X
10
90
2.
6 -2
.56
86
67
0X
10
-1
0.1
8X
25
80
0 23
1
4X
14
19
6 5.
3 -6
.34
41
53
9X
10
-1
0.03
7
X2
8
784
22
25
X2
8
700
19
-8.1
66
95
90
X1
0-1
0.01
8
X4
5
1,44
0 40
4
5X
45
2,
025
53
-8.
6279
332
X 1
0-1
0.00
3 8
X7
2
2,30
4 64
7
2X
72
5,
184
136
-8.7
75
24
22
X1
0-1
0.00
1 8X
120
3,84
0 10
6 10
0X10
0 10
,000
26
2 -8
.81
57
51
7X
10
-1
-------
--------
..
----
--------------
------
----
---
Sou
rce
poin
t xs
=(0
.25,
0.2
5, d
),
rela
tive
err
or
<1
0-6
•
w
...... "'"
Relative Error
1
375
Nearly Singular Integral
Is <l'ij q* dS
S : planar square PLR ( 0.5 , 0.5 )
Xs = x ( 0.5 , 0.5 )
d = 0.1
Number of Integration Points
Fig. 10.16 Relative error vs. Number of integration points
Re
lativ
e E
rro
r
1
'\ 'h...
._ ..
'-."
"\ "
Nea
rly
Sin
gu
lar
Inte
gral
Is <P
ij q*
dS
S :
pla
na
r sq
uare
P
LR
( 0
.5 ,
0.5
)
is =
x (
0.5
, 0.5
)
d =
0.0
1
", .~-
.. ~
rv'T
elie
s
<X =
3,
t (a
)
-7'~ __
~ __
~ __
~ __
~ __
~ __
~ __
.. __
~ __
~ __
~ __
~ __
.. __
~ __
~ __
~ __
~ __
~ __
~ __
~ __
~ __
~ __
~~1o-
__
~~
10
-
>
500
10
00
1
50
0
20
00
Fig.
10.
17
Rel
ativ
e er
ror
VS
. N
umbe
r of
in
tegr
atio
n po
ints
Num
ber
of
Inte
grat
ion
Poi
nts
Co)
.....
m
377
10.4 The Derivation of the log-L2 Radial Variable Transformation
(1) Application of radial variable transformations pdp=r'PdR ((]of:: a) to
integrals [s lira dS over curved elements
In search for a more effective and robust radial variable transformation,
transformations of the type
pdp = r'P dR (5.61)
or
R(p)= I ..L dR r'P
(5.71)
where
(5.46)
were attempted on nearly singular integrals of the type
I dS (10.35) S r a
over curved elements. This time, (3 is not necessarily equal to a, unlike the
exactly cancelling transformation. All the combinations of the transformation: (3
and order of nearly singular integral: a were attempted for (3=1-5, a= 1-5.
Tables 10.15-10.19 give the number of integration points required to
achieve relative error e: < 10-6 for the calculation of the integrals f s lira dS ,
(a = 1-5); by the radial variable transformations of equation (5.61) with (3 = 1-5,
respectively. The transformations are given by
({i=1 )
({i=2 )
({i=3 )
«(3=4 )
378
(f3=5 ) (10.36)
The curved element S was chosen as the 'spherical' quadrilateral SPQ60. The
source point Xs was given such that the source projection is xs=x(0.5,0.5), and
the source distance d = 10, 1, 0.1, 0.01, 0.001 towards the centre of the sphere.
Table 10.20 gives the actual values of the integrals Is lira dS, (a=1-5)
calculated by the PART method with the log-Ll radial variable transformation of
equation (5.85) (, which is the most robust and reliable transformation, as will be
shown later on) and the angular variable transformation t(O) of equation (5.127),
with NR=Nt= 128 integration points.
For the case: source distance d=O.Ol, Table 10.21 gives the number of
integration points NR in the transformed radial variable R required to achieve
relative error € < 10-6 for the calculation of the integrals Is lira dS , (a = 1-5)
with the transformations: .8 = 1-5. Fig. 10.18-10.22 give the convergence
graphs of relative error € vs. the number of integration points to calculate the
integrals
Is lira dS, (a = 1-5) by the radial variable transformations: .8= 1-5, respectively.
The results show that the radial variable transformation corresponding to
.8=2 :
(5.64)
gives good results for all types of near singularity a = 1-5, (particularly for
d;2; 0.1). This log-Lz radial variable transformation requires relatively few
integration points, regardless of the order of near singularity a, curvature of the
element and source distance d.
As an exception, for the case a=4, and 5, the radial variable transformation
corresponding to .8 = 3 :
(10.37)
379
gives better results than the (3=2 (log-L2) transformation. (The required number
of radial integration points is less than a half.)
These results can be explained by the error analysis in Chapter 6 (Table 6.5),
where it was shown for planar elements that the numerical integration error
En(F) in the radial variable is given by
E(F)- n , { -2n
no, (<<=1,3,4,5)
«=2
for the (3 = 2 (log-L2) transformation, and
{ > 4 -2n E (F)
n = 0
for the (3 = 3 transformation, and
{ > 4 -2n E (F)
n = 0
for the (3 = 1,4,5 transformation.
(a=1,2)
(<<=3,4,5 )
(<<*{3)
(<<={3)
(10.38)
(10.39)
(10040)
For curved elements, En(F) = 0 does not hold strictly, even for a = (3.
However, the convergence graphs of Fig. 10.18-10.22 indicate that for a=(3,
either the initial relative error is smaller or the initial rate of convergence is
faster compared to a* (3.
To sum up, the radial variable transformation: (3=2 (log-L2) is the most
robust transformation concerning the order of near singularity a= 1-5. The (3=2
transformation gives best results for a = 1-3, and the (3= 3 transformation gives
best results for a = 4, 5 .
Considering the robustness for a=1-5 and the optimality for a=1-3, the
(3 = 2 (log-L2) radial variable transformation appears to be the most attractive
transforma tion.
Tab
le 1
0.15
N
earl
y s
ing
ula
r in
teg
ral
Is li
ra d
S
over
the
'sph
eric
al' q
uad
rila
tera
l: S
PQ
60
( R
adia
l var
iab
le tr
ansf
orm
atio
n (3
= 1
: R
(p) =
y' p
2 +
d2 )
Sou
rce
a=
l a=
2 a=
3 a=
4 a
=5
dist
ance
d N
t(O
)X
NR
(p)
tota
l N
t(O) X
N R
(p)
tota
l N
t(O
)X
NR
(p)
tota
l N
t(O) X
N R
(p)
tota
l N
t(O
)X
NR
(p)
10
6 X
20
480
6 X
20
48
0 6
X 2
0 48
0 5
X 1
6 32
0 4
X 2
0
1 6
X 2
0 48
0 6
X
20
480
6 X
20
480
6 X
20
480
6 X
20
0.1
5 X
14
280
6 X
25
60
0 7
X 3
2 89
6 7
X 3
5 98
0 7
X 3
5
0.01
6
X 2
8 67
2 7
X
40
1,12
0 9
X 5
0 1,
800
9 X
60
2,16
0 9
X 6
4
0.00
1 6
X 6
0 1,
440
9 X
120
4,
320
9 X
150
5,
400
9 X
170
6,
120
9 X
190
Ave
rage
CP
U-t
ime
24,u
sec
23,u
sec
24,u
sec
23,u
sec
24,u
sec
per
poi
nt
Nu
mb
er o
f in
teg
rati
on
poi
nts
nece
ssar
y fo
r re
lati
ve
erro
r <
10
-6•
Sou
rce
proj
ecti
on:
:Ks =
x(0
.5, 0
.5),
so
urce
poi
nt to
war
ds c
entr
e of
sphe
re.
tota
l
320
480
w ~
980
2,30
4
6,84
0
Sou
rce
dist
ance
d
10 1 0.1
0.01
0.00
1
Ave
rage
CP
U-t
ime
per
poi
nt T
able
10.
16
Nea
rly
sin
gu
lar
inte
gra
l Is
lira
dS
ov
er th
e 's
pher
ical
' qua
dril
ater
al:
SP
Q60
( R
adia
l var
iab
le tr
ansf
orm
atio
n (
3 =
2 :
R(p
) =
log vi
p2 +
d2
)
a=1
a=
2
a=3
a=4
a=5
Nt(
B)X
NR
(p)
tota
l N
t(B
)XN
R(p
) to
tal
N t(
B) X
N R
(p)
tota
l N
t(B
)XN
R(p
) to
tal
N t(B
) X
N R
(p)
6 X
20
480
5 X
16
320
6 X
20
480
5 X
16
320
4 X
20
6 X
20
480
6 X
20
480
6 X
20
480
6 X
20
480
6 X
20
5X
5
100
8 X
12
384
8 X
16
512
7 X
20
560
7 X
20
6X
8
192
7x
8
224
9 X
11
396
9 X
14
504
9 X
14
6 X
10
240
9 X
10
360
9 X
14
504
9 X
16
576
9 X
16
26,u
sec
25,u
sec
25,u
sec
24,u
sec
26,u
sec
Nu
mb
er o
f in
teg
rati
on
poi
nts
nece
ssar
y fo
r re
lati
ve
erro
r <
10
-6•
Sou
rce
proj
ecti
on:
xs=
x(0
.5,0
.5),
so
urce
po
int t
owar
ds c
entr
e o
f sph
ere.
tota
l
320
480
I
560
I
504
57"6
,
Co)
co
~
Sou
rce
dist
ance
d
10 1 0.1
0.01
0.00
1
!Ave
rage
C
PU
-tim
e pe
r poi
nt
Tab
le 1
0.17
N
earl
y s
ing
ula
r in
teg
ral
Is li
ra d
S
over
the
'sph
eric
al' q
uad
rila
tera
l: S
PQ
60
( R
adia
l var
iabl
e tr
ansf
orm
atio
n (3
= 3
: R
(p) =
-1
/ Y
p2 +
d2 )
a=
l a=
2 a=
3 a=
4 a=
5
Nt(B
)XN
R(p
) to
tal
Nt(
B)X
NR
(p)
tota
l N
t(B) X
N R
(p)
tota
l N
t(B) X
N R
(p)
tota
l N
t(B)X
NR
(p)
6 X
20
480
6 X
20
480
6 X
20
480
5 X
16
320
4 X
20
6 X
20
480
6 X
20
480
6 X
20
480
6 X
16
384
6 X
16
6 X
10
240
6X
9
216
7X
8
224
7 X
11
308
7 X
11
6 X
35
840
8 X
25
800
8 X
14
448
9X
6
216
9X
6
6 X
110
2,
640
8 X
80
2,56
0 9
X 3
5 1,
260
9X
5
180
9X
6
24,u
sec
24,u
sec
24,u
sec
25,u
sec
25,u
sec
Num
ber o
f int
egra
tion
poi
nts
nece
ssar
y fo
r re
lati
ve e
rro
r <
10
-6•
Sou
rce
proj
ecti
on:
Xs =
x(0
.5, 0
.5),
so
urce
poi
nt to
war
ds c
entr
e of
sphe
re.
tota
l
320
384
I C
o)
~
308
216
216
Tab
le 1
0.18
N
earl
y s
ing
ula
r in
teg
ral
Is l
ira
dS
over
the
'sph
eric
al' q
uadr
ilat
eral
: S
PQ
60
( R
adia
l var
iab
le t
ran
sfo
rmat
ion
(3 =
4 :
R(p
) =
-1
/ {2
(p2 +
d2)}
)
Sou
rce
a=
l a=
2 a=
3 a=
4 a=
5
dist
ance
d N
t(8)
XN
R(p
) to
tal
Nt(
8)X
NR
(p)
tota
l N
t(8)
XN
R(p
) to
tal
Nt(
8)X
NR
(p)
tota
l N
t(8)
X N
R(p
)
10
6 X
20
480
6 X
20
480
6 X
20
480
5 X
16
320
4 X
20
1 6
X 1
6 38
4 6
X 1
6 38
4 6
X 1
6 38
4 6
X 1
6 38
4 6
X 1
6
0.1
6 X
32
768
6 X
25
600
7 X
20
560
7 X
14
392
8 X
12
0.01
(6
X 2
56
€=5
X1
0-
6 6,
144)
7
X 2
20
6,16
0 9
X 1
50
5,40
0 9
X 5
0 1,
800
8 X
25
0.00
1 (6
X 2
56
6,14
4)
(9 X
256
9,
216)
(9
X 2
56
9,21
6)
160
€=4
X1
0-
1 €=
5X
10
-2
€=3
X1
0-
4 8
X5
10
X 5
0
Ave
rage
CP
U-t
ime
24,u
sec
23,u
sec
24,u
sec
24,u
sec
24,u
sec
per
poi
nt
Nu
mb
er o
f int
egra
tion
poi
nts
nece
ssar
y fo
r re
lati
ve
erro
r <
10
-6 •
Sou
rce
proj
ecti
on:
xs=
x(0
.5, 0
.5),
so
urce
po
int t
owar
ds c
entr
e o
f sph
ere.
tota
l
320
384
Co.
) ~
384
800
2,00
0
Sou
rce
dist
ance
d
10 1 0.1
0.01
0.00
1
~ver
age
CP
U-t
ime
per
po
int
Tab
le 1
0.19
N
earl
y s
ing
ula
r in
teg
ral
Is li
ra d
S
over
the
'sph
eric
al' q
uad
rila
tera
l: S
PQ
60
( R
adia
l var
iabl
e tr
ansf
orm
atio
n !3
= 5
: R
(p)
= -1
/ {3
( p
2 +
d2)
312}
)
a=
l a=
2 a=
3 a=
4 a=
5
N te
o) X
N R
(p)
tota
l N
t(O)X
NR
(p)
tota
l N
t(O)X
NR
(p)
tota
l N
t(8)
X N
R(p
) to
tal
Nt(O
)X N
R(p
)
6 X
20
480
6 X
20
480
6 X
20
480
5 X
16
320
4 X
20
6 X
16
384
6 X
16
384
6 X
16
38
4
6 X
16
384
6 X
16
6 X
80
1,92
0 6
X 7
2 1,
728
7 X
60
1,68
0 7
X 4
0 1,
120
7 X
16
(6 X
256
€=
2X
10
-1
6,14
4)
(8 X
256
8,
192)
€=
3X
10
-2
(9 X
256
€=
lX1
0-
3 9,
216)
9
X 2
56
€=2
X1
0-
5 9,
216
10 X
28
(6X
256
€=9
X1
0-
1 6,
144)
(9
X25
6 €=
3X
10
-1
9,21
6)
(9X
25
6
9,21
6)
9X
25
6
9,21
6 €=
lX1
0-
2 €=
lX1
0-
4 10
X 2
0
27,u
sec
26,u
sec
27,u
sec
26,u
sec
28,u
sec
Nu
mb
er o
f in
teg
rati
on
poi
nts
nece
ssar
y fo
r re
lati
ve
erro
r <
10
-6•
Sou
rce
proj
ecti
on:
Xs
= x(
0.5,
0.5
),
sour
ce p
oin
t tow
ards
cen
tre
of s
pher
e.
tota
l
I
320
3841
I ~
i
4481
11
20
, , 80
0
Tab
le 1
0.20
V
alue
of n
earl
y s
ing
ula
rin
teg
ral
Is lir
a dS
ove
r th
e 's
pher
ical
' qu
adri
late
ral:
SP
Q60
( Sou
rce
proj
ecti
on:
Xs =
x(0
.5, 0
.5)
)
Sou
rce
dist
ance
a
=l
a=2
a=3
a=4
a=5
d i !
10
1.05
7660
5 X
10
-1
1.07
2204
1 X
10
-2
1.08
7073
2 X
10
-3
1.10
2276
7 X
10
-"
1.1
17
82
39
X1
0-5
I
1 1.
0467
642
1.05
0118
7 1.
0534
972
1.05
6900
0 1.
0603
272
0.1
2.83
7785
9 1.
0589
840
X 1
0 5.
2819
788
X 1
0 3.
2411
271
X 1
02
2.26
8925
3 X
103
0.01
3.
2263
821
2.43
2106
3 X
10
6.19
1324
3 X
102
3.
1700
026
X 1
0"
2.11
4756
0 X
106
0.00
1 3.
2680
995
3.86
1520
9 X
10
6.27
4135
6 X
103
3.
1446
057
X 1
06
2.09
6418
2 X
109
Cal
cula
ted
by
the
log-
Ll r
adia
l var
iabl
e tr
ansf
orm
atio
n:
R(p
)=lo
g(p
+d
), a
nd
the
ang
ula
r var
iab
le
tran
sfo
rmat
ion
: t(
8) w
ith
NR
(p) = N
t(8
) = 1
28 p
oint
s.
I ~
U1
Tab
le 1
0.21
N
earl
y si
ng
ula
r int
egra
l Is
lira
dS
ov
er th
e 's
pher
ical
' qu
adri
late
ral:
SP
Q60
(Nu
mb
er o
f rad
ial i
nteg
rati
on p
oint
s N
R b
y th
e tr
ansf
ora
mti
on
: (3
)
a=1
a=2
a=3
a=4
{3=1
28
40
50
60
{3=2
8
8 11
14
{3=3
35
25
14
6
(256
) 22
0 15
0 50
{3
=4
~=5X10-6
(256
) (2
56)
(256
) (2
56)
(3=
5 ~=9X10-1 ~=3X1O-2 ~=1X10-3 ~=2XI0-5
Rel
ativ
e er
ror
~ <
10
-6.
Sou
rce
proj
ecti
on X
s =
x(0.
5, 0
.5).
S
ourc
e di
stan
ce d
=0
.01
, to
war
ds c
entr
e of
sphe
re.
a=5
64
14 6 25
28
w ~
Rel
ativ
e E
rror
1
10
-1
10
-2
10 -4
10
-5
10 -6
Nea
rly S
ingu
lar
Inte
gral
.'---.•. ~
.. ~ .. -
-t •.
q ,
, .....
. Q
., ",
~' ..
.. f s
~~
by r
adia
l tra
nsf
orm
atio
n:
B =
1
Q
.~ ...
.... ~
....
....
. h
'to.
'...
'.
and
angu
lar
tran
sfor
mat
ion
t ( e
) S
:
'sph
eric
al' q
uadr
ilate
ral S
PQ
60
'\
~
"'" ...
. ~
, .. , ..
..... .
Q.
~',
....
. xs=
x(
0.5
,0.5
) 0-0
. ,", •
•••••
o ~,",
......
Q
•
","
~,
, ... ,:
..•.
0,
'"
••••
'0
.,
....
....
... .
.
" ',
', ....
. . ',
"
", ..•.
. '0
, "to
."
••••
••
, "..
".
,
., ~
" ~
",
.... '0
......."'
••••
,
".
.......
".
" ~
.........
.. ... ~'"
.,., '
.. ,.~ ....
..•..
Q
, ..
....
....
. .
, •
___ ..
e. "
".....
<X
= 5
~
.~.
",
......
. c-'
\
,.
'. \
"',
....
\ r-
I'~
-.7
',
".
\ <
x=2
rv
-3
,-
....
....
".
\yoJ
v.-
u=
4
.....
". \
'. \
~
d =
0.01
10
-71
500
1000
\
1500
20
00
>
Fig
.10
.18
R
elat
ive
erro
r vs
. N
umbe
r of
inte
grat
ion
poin
ts
Num
ber
of
Inte
grat
ion
Poi
nts
w
00
-..
.J
Rel
ativ
e E
rro
r
1 t~
.. 10
-. ~
~\~.
10-1~~
~\\
q\ \ \\.
10-
2~ \\
\' \
\\
\ 10-
3~ I'
\ \\
, \ \~
~
. ~
, '\ \
10
-4 ~
IJ\~
\ \ \
\ \ \
\ 10-
5~ ~\
\. \\
Q
, ••
l \\\\
.. 0
.=5
1<
... 2
' ·r
J ~=~\'.
0.=
3
\--,
0.=
4
Nea
rly
Sin
gula
r In
tegr
al
J s ~~
by r
adia
l tra
nsf
orm
atio
n:
B =
2 (
log
-L
2 )
and
angu
lar t
rans
form
atio
n t
( 9 )
S
: 's
pher
ical
' qua
drila
tera
l SP
Q60
is =
x( 0
.5,0
.5)
d =
0.0
1
10
-7'
t I
I I
, >
50
0 10
00
1500
20
00
Fig
. 10
.19
Rel
ativ
e e
rro
r vs
. N
umbe
r o
f int
egra
tion
poin
ts
Num
ber
of
Inte
grat
ion
Poi
nts
~
Rel
ativ
e E
rror
1 Q
l\
I .
0 1
0-1
r'\
\ \.
'0
. ,
+:\
00
\\.
0 10
-2 ~
'\: \
'tl
\. ~
.. '\
~
\'
'Cl.
~\:
~ 10-3~
\\\
'\
;\,
\~
\\ .,
\ \\
"
\,
.l\
'\
't?, \:
\ :~
\
\ \ :.
, ~\
a=
S\
\
10-4
10
-5
Nea
rly S
ingu
lar
Inte
gral
J s ~~
by
rad
ial t
rans
form
atio
n:
B =
3 an
d an
gula
r tr
ansf
orm
atio
n t (
e)
S
: 's
pher
ical
' qua
drila
tera
l SP
Q60
xs=
x(
0.5
,0.5
) d
= 0
.01
10-6~a=4'f'
\,}=
3
\ A
\
a=
2
'y"J
'
1 0 -7
1 sbo
9
1 000
1S~
0 2~
00
>
Fig.
10.
20
Rel
ativ
e er
ror
vs.
Num
ber o
f int
egra
tion
poin
ts
Num
ber o
f In
tegr
atio
n P
oint
s
w ~
Nea
rly
Sin
gula
r In
tegr
al
Rel
ativ
e E
rro
r f s
~~
by
radi
al t
ran
sfo
rma
tion
: B
= 4
1 ~
10 -1
" ~'A
and
angu
lar
tran
sfor
mat
ion
t ( e
) S
:
'sph
eric
al' q
uadr
ilate
ral S
PQ
60
xs=x
( 0
.5,0
.5)
d =
0.01
a=
1
\ 00
000'
''''-<
>--0
.-<>-
<> _
__ -0
_
_ 2
-<>----
<>--0
.----<>
----0_
____
a
_ ..
--
---0
.---
-0--
--0
_._
._.
_ _
,
~ --
----
--'0
----
--___
0
-.-
A..
. • ..t
...., ._
"-.-
....
..
a=
3
• 10
-2~
\!I A \ 1
1 .-
.I>
....
-._
:06
....
_."
'b- .-
.-.-.-.-.-'-.-~~.-.-.-
1'\ "11
1 0 -3
~. \
........
. .
: "
1 .....
.. 1 0
-4~
\ ..•
.•...
: ~
4 \
....... ~ .. -
.. -.. -
.. -a
= '-
" "-
"-""
"',,
--I
, /'
.•••
.._
.. _
.. ...
""
1 0 -5
t \
.. ...
. 5
"_
.. _
f ".
a=
.. _
_ k"
..... ,r
J
...•
10 -6
10 -7
. 50
0 10
00
1500
20
00
>
Fig
. 10
.21
Rel
ativ
e er
ror
vs.
Nu
mb
er
of in
tegr
atio
n po
ints
Nu
mb
er
of
Inte
grat
ion
Poi
nts
CAl
co
o
Rel
ativ
e E
rror
1
10 -1
0000
.-...
u=
1
u=
2
-0
--
-_.r
.:-!. -
-0
-~O~O-o-~---O-----o---O--_~ _
_ o-_
___ ~ __
__ ~ __
_ ~ __
___ O
____
-o _
__ ~ __
____
_
~~
.. .....
..........
... ,,-
3 \.
--.t
>.
-.to
. • -
.t..
.-. -
. ~
N
"' ...
10 -2
...
~.-.e.'-'A--'-,6..-._._.-tr._.-.e.._._._.--.t.._.~._._._
'-.,
.. - "'-"
"'-
10 -3
10
-4
10
-5
10
-6
•... ". '''. ' .. ". ...
... .......
'~"-
4 "-"
-"-"
u=
-...
......... -.
-.. -.-
.. -.. -.. -
.. -.. -.
-.. ~-
.. -...
.. -.. -
.. -
......
....
u= 5
....
.... ...
......
......
......
r:--' ...•
Nea
rly S
ingu
lar
Inte
gral
f dS
.
s --r
et by
rad
ial t
rans
form
atio
n:
B =
5 an
d an
gula
r tr
ansf
orm
atio
n t
( e )
S
: 's
pher
ical
' qua
drila
tera
l SP
Q60
Xs =
x( 0
.5,
0.5)
d =
0.01
10 -7
, 50
0 10
00
1500
20
00
::>"
Fig
. 10
.22
Rel
ativ
e er
ror
vs.
Num
ber
of in
tegr
atio
n pO
ints
Num
ber
of
Inte
grat
ion
Poi
nts
eN
<0 -
392
(2) Difficulty with flux calculation
In the previous section, it was shown that the log-L2 radial variable
transformation:
(5.64)
corresponding to f3 = 2 is the most robust radial variable transformation of the
type
pdp = r'P dR (5.61)
for the calculation of integrals
(a=1-5 ) (10.41)
over a curved element B.
However, the log-L2 transformation of equation (5.64) works very poorly for
nearly singular integrals like
J au· -dS I
S ax • J aq·
-dS s ax I
• (10.42)
which arise in the calculation of the flux aulaxs at a point Xs very near the
boundary using equation (2.46).
In Fig. 10.23, the result of applying the log-L2 radial variable
transformation with the angular variable transformation t(O) of equation (5.127)
is shown for the constant planar square element PLR (0.5, 0.5) :
-0.5 ~ X, Y ~ 0.5 I Z=O (10.43)
of size one (cf. section 8.2 (1», with the source point at xs=(0.25, 0.25, d), where
the source distance is d=O.Ol. The integrals Isu*dS, Isq*dS related to the
calculation of the potential at xs, and the integrals Is au*laxs dB, Is aq*lax dB
related to the flux calculation at Xs , were calculated. The relative error c is
plotted against the (necessary and sufficient) number of integration points
N=NtXNR X4, where Nt is the number of Gauss-Legendre integration points
Rel
ativ
e E
rror
1 10
-1
10
-2
10
-3
10-4
I I I I I ,q \ \ I I 1- "
au
* '.
---
----
'.....
axs
' .... .... .....
.. .... .....
. ...... aq
*
·--0 ..
.. .... 0
0_
-0..
-.. --0-lo
g-L
2 tr
ansf
orm
atio
n
plan
ar s
quar
e el
emen
t, d
= 0
.01
00 _
__
-..
.. _
_
10
-5
10-6
q*
---0 _
__ -
--.... --
-0. __
_ _ --.
..
10
-71
u* ~~~~ >
o 50
0 10
00
1500
N
um
be
r o
f In
tegr
atio
n P
oint
s 10
-8
Fig
. 10
.23
Inte
grat
ion
of p
oten
tial a
nd f
lux
kern
els
usin
g th
e lo
g-L2
tra
nsfo
rmat
ion
~
w
394
for the angular variable t, and N R for the radial variable R. Nt was 3, 7, 8 and 12
for the kernels u*, q*, au*laxs , and aq*laxs, respectively, for relative error :
€<10- 6 •
From Fig. 10.23, it is evident that the log-L2 radial variable transformation,
which is so successful with the integration of nearly singular potential kernels u*
and q* , turns out to be inefficient for the flux kernels au*/axs and aq*/axs.
The reason for this was analysed in sections 5.5 (4), 6.8 and 7.7 (2). In short, the
potential kernels essentially give rise to radial integrals with 0= 1 only, in
JP. /
1= J_ dp , a ,8 0 ra
T22 r= vp~+d~ , (10.44)
where as the flux kernels involve radial integrals with 0=2 as well as 0= 1, which
causes the difficulty with the log-L2 transformation.
In fact, for 0=2, the theoretical estimate of equation (7.190) in section 7.7
(iv) for the log-L2 transformation predicts an error estimate of order n -0-1 =n -3,
where n is the number of integration points in the radial variable. This matches
remarkably well with the convergence graphs of numerical experiment results in
Fig. 10.23, where J s au*laxs dS gives a convergence of order n -3.1 and
J s aq*laxs dS gives a convergence of order n -3.0 for the relative error.
The excellent convergence of the log-L2 transformation for the integration of
the potential kernels u* and q*, where 0 = 1 : odd, is also predicted by the
theoretical error estimate En (f) - n -2n of equation (7.120) in section 7.7 (1) .
395
10.5 The log-Lr Radial Variable Transformation
In order to overcome the difficulty with flux calculation using the log-L2
radial variable transformation, in section 5.5 (4) we introduced the log-Ll radial
variable transformation :
R (,0) = log(p+c!) (5.85)
In Fig. 10.24, the result of calculating the same integrals as in Fig. 10.23 but
with the log-L1 instead of the log-L2 radial variable transformation is given.
Notice the remarkable improvement of the log-L1 over the log-L2 for the error
convergence for the flux integrals Is au*/axs dB and Is aq*/axs dB. Note also
that the log-L1 transformation works reasonably well for the potential integrals
Is u* dB and Is q* dB, as well.
The theoretical error estimate of section 7.8 (2) (iii) predicts the relative
error e to be of order
-2n £ - C1
(7.266)
where Q=1.40 for d=O.Ol from Table 7.7, and n is the number of radial
integration points. This matches fairly well with the convergence graphs in
Fig. 10.24 , which give relative error of orders 1.6 -n and 1.7 -n approximately, for
the integration of au*/axs and aq*/axs ,respectively, for flux calculations.
The theoretical error estimate of equation (7.229) in section 7.8 (1), using the
saddle point method, predicts an error of order n -2n for the case a = 8 = 1, which
corresponds to the integration of u*. This explains the excellent convergence for
the u* kernel in Fig. 10.24.
Rel
ativ
e E
rror
1 10
-1
10
-2
10
-3
10-4
10
-5
10-6
10
-7 ,0
1
0-8
500
1000
log-
L] t
rans
form
atio
n
plan
ar s
quar
e el
emen
t, d=
0.0
1
aq*
1500
N
um
be
r of
In
tegr
atio
n P
oint
s
Fig.
10.
24
Inte
grat
ion
of
pote
ntia
l and
flu
x ke
rnel
s us
ing
the
log-
L] t
rans
form
atio
n
~
Cl
397
10.6 Comparison of Radial Variable Transformations for
the Model Radial Integral la, 8
In section 3.3 (Table 3.2, 3.3) and section 7.5 (Table 7.1), it was shown that
the essential nature of nearly singular integrals occurring in the boundary
element analysis of three dimensional potential problems can be characterized by
the following model integral in the radial variable:
(10.45)
where
(10.46)
for planar elements, and Pj= Pj (8) is the upper limit of the radial variable of the
integral in equation (5.42). The basic integral kernels in three dimensional
boundary element analysis give rise to a, 0 given in Table 7.1, which is reproduced
below.
Table 7.1 Nature of nearly singular kernels of the radial component
integrals in 3-D potential problems
a 0
u* 1 1
q* 3 1
au* 3 1 --axs 2
aq* 3 1 axs 5 1
2
398
In the present section, we will compare the effect of different radial variable
transformations on the numerical integration of la,8 of equation (10.45). The
transformations to be compared are
(1) Transformations based on the Gauss-Legendre rule :
(i) Identity transformation :
R (0) = P
(ii) log-L2 transformation :
R (0) = log Jp2 +d2
(iii) log-L1 transformation:
R (0) = log (0 +d)
and
(2) Transformations based on the truncated trapezium rule:
(i) Single Exponential (SE) transformation
p (R) = Pj (1 + tanh R) 2
(ii) Double Exponential (DE) Transformation
P.{ 1r } P (R)= ~ 1 + tanh (2sinh R)
(10.4 7)
(5.64)
(5.85)
(5.98)
(5.109)
Numerical results will be compared with the following closed form integrals
obtained in section 3.3 ( equations (3.138-142) ) and section 7.8 (2) (iii) (Table
7.5).
I = J p .2+d2 -d = p.(J1+D2 -D) 1,1 J J (10.48)
I =!.. _ 1 = 2..(!.. 1 ) 3,1 d ~ p. D - .Jl+D2
J J
(10.49)
399
~ Pj 132 = log (p ,+V P ,M+dM) -logd - r-r-:?2
, J J V p ,-+d-J
= wg ( 1 + v'};[}) _ ---==1 = D .Ji+1)2
= 1
3p, 2 D2 )1 +D2 3 J
(10.50)
(10.51)
(10.52)
Also, theoretical error estimates for the identity, log-L2 and log-Ll
transformations will be compared with numerical results.
Tables 10.22-28 give the number of (radial) integration points required to
obtain a relative error E < 10-6, for the integration of la,o of equation (10.45),
using each radial variable transformation in conjunction with the Gauss
Legendre rule or the truncated trapezium rule. Pj was set to Pj=l and the source
distance d was varied from 10 to 10-3• The integration error was obtained by
comparing with the analytical integration results of equation (10.48-52).
Figures 10.25-30 give the convergence graphs of the relative error of
numerical integration E (in log scale) vs the number of radial integration points
n for each radial variable transformation, for the case d = 0.01 .
Table 10.22 Identity transformation:
R(pJ=p
a 0 Source distance: d
10 1 10- 1 10-2 10-3
1 1 3 5 12 35 80
1 3 6 16 60 190 3
2 3 5 20 55 170
1 3 6 20 64 210 5
2 3 7 25 60 190
(Number of radial integration points n for
relative error E < 10-6 • )
Table 10.24 log-L1 transformation:
R(p)=log(p+d)
a 0 Source distance: d
10 1 10-1 10-2 10-3
1 1 3 5 8 9 8
1 3 5 12 16 20 3
2 3 6 11 11 16
1 3 6 14 20 25 5
2 3 6 14 20 20
(nforE<1O-6 )
400
Table 10.23 log-L2 transformation:
R(p)=log"';p2+d2
a 0 Source distance: d
10 1 10-1 10-2 10-3
1 1 2 3 4 5 6
1 2 3 4 5 6 3
2 55 55 64 72 80
1 2 3 6 8 10 5
2 55 64 120 170 200
( n for E < 10-6 )
Table 10.25 Single Exponential (SE)
transformation with R(p) : [_ 00, + 00]
a 0 Source distance: d
10 1 10-1 10-2 10-3
1 1 32 66 66 30 30
1 32 66 56 46 54 3
2 70 68 62 58 58
5 1 32 64 40 48 58
2 70 58 50 38 48
(Number of radial integration points n for relative error £<10-6 )
Table 10.27 Single Exponential (SE)
transformation with R(p) : [-00,0]
a 0 Source distance: d
10 1 10-1 10-2 10-3
1 1 1916 1015 64 21 13
1 1916 1185 1260 410 128 3
2 3016 743 1032 1196 752
5 1 1917 2318 202 26 31
2 1370 858 521 84 26
(nfor£<10-6 )
401
Table 10.26 Double Exponential (DE)
transformation with R(p) : [- 00, + 00 ]
a 0 Source distance: d
10 1 10-1 10-2 10-3
1 1 22 22 38 38 38
1 22 22 36 66 70 3
2 22 22 38 36 68
5 1 22 22 32 66 72
2 22 22 34 60 66
( n for £ < 10-6 )
Table 10.28 Double Exponential (DE)
transformation withR(p): [-00,0]
a 0 Source distance: d
10 1 10-1 10-2 10-3
1 1 1688 872 55 17 17
1 1688 883 498 266 72 3
2 1360 717 825 479 528
5 1 1689 1741 126 35 38
2 1360 728 439 32 68
(nfor £ < 10-6 )
1
Rel
ativ
e E
rro
r c
10
-1
10
-2
10~
10-4
10~
Ra
dia
llnte
gra
tio
n
Ia,a
= S~
t: dp
Ide
nti
ty tr
an
sfo
rma
tion
{
r =~
P2 +
d 2
R
(P
) =
P
P j =
1
d =
0.0
1
ex =
B=1
: u*
au*.
..,g:
ex
= 3
, B
=1
: q* 'X
s'
Xs
. au
* ex
= 3
, B
=2
. dX
s ex
= 5
{B
=1
. ili
L ,
B=2
.
axs
erro
r -2
n ex
= 1
, B
= 1
: E
-0
.03
98
x
( 1
.17
)
ex=
3,B
=1
:E-4
.47
x(1
.14
)-2
n
B =
2 :
E -
0.7
08
x
( 1
.14
) -2
n
ex =
5,
B =
1 :
E -
20
.0
x (
1.1
3 )
-2n
B =
2 :
E -
5.9
6
x (
1.1
2 )
-2n
(n~10)
ex=
5
B=
2
,
10~ 10~I~~
lb~~~~
~~~~~~
~~i:Ji
~'~~~~
~~ ::> 10
70
R
adia
l In
teg
ratio
n
Po
ints
n
20
30
40
50
60
10-8
. Fig
. 10
.25
Rad
ial i
nte
gra
l Ia
,~
by
ide
nti
ty t
ran
sfo
rma
tion
"""
o N
Re
lativ
e
Err
or
£ R
adia
l In
teg
ratio
n
la,S
=S~
J ~ d
P
log
-L2
tra
nsf
orm
atio
n
{r =
~ p2
+ d
2
R(p)=log~p2+d2
P J=
1
d =
0.01
1
10
-2
10-3
10--
4
10
-5
\ ~ ~4 . ~,
1 :4
~ .
.. ... \
"A .... .....
.
.. .. )..
..........
.. 1
...... 4 .
... ~
\ ....
<X=
5,
0=
2
..........
.... . ~<
X=5
0=
1
........
........
.....
, .....
..........
. ~ .....
..........
. <X
= 3,
0 =
1
a. =
3,
0 =
2
<x=
0=
1
10
-6
10
-71-
.--.
-~~~~~i
i ~~~~-
-' ,
i
i ::::
:::>"
10
20
30
40
50
60
70
80
90
Rad
ial
Inte
gra
tion
P
oin
ts n
1
0-8
Fig
. 10
.26
Rad
ial
inte
gra
l Ia
,s by
,fog
-L2
tra
nsf
orm
ati
on
8
1 Rel
ativ
e E
rror
E
10
-2
10-3
10-4
10-5
10-6
. Rad
ial I
nteg
ratio
n la
,/) =
S:i t:
dp
, (8
= 2
)
1.0g
-L2
tran
sfor
mat
ion
{ r =~
P2+
d2
R (
P) =1.0g~P2+
d2
P J =
1
d =
0.0
1
a=
3,
8=
2
( E
-0.
266
xn -2
.95
) a=
5,
8=
2
n :
Nu
mb
er
of R
adia
l In
tegr
atio
n P
oint
s
( E
-4.
22 x
n -2
.99
)
10~r
l~~~
-'~~
r-r-
~'-~
-r-r
~~-,
~~r-
~~~~
~~~~
~--
10
,>,
j'c
......
0.5
1.0
(n =
10)
1.5
2.0
l.og
10 n
(n
= 1)
(n
= 10
0)
Fig
. 10
.27
Con
verg
ence
rat
e fo
r .(
og-L
2 tr
an
sfo
rma
tion
§
1
Rel
ativ
e E
rro
r E
10
-1
10-2
10-6
10-6
Ra
dia
llnte
gra
tion
Ia
,s =
It! t:
dP
I. Og-
L1
tra
nsf
orm
atio
n
{ r =~P2+ d
2
P j =
1
R (
p)
=I.og
( P
+ d
) d
= 0
.01
ex=
5,
0 =
1
~ex=5,
0=
2
ex=
3,
0 =
1
ex=
3,
0 =
2
ex=
0=
1
ex =
1,
0 =
1 :
E
...
0.03
98
x (
1.88
) -2
n
ex =
3,
0 =
1 :
E
...
6.31
x
( 1.
64 )
-2n
0=
2:
E
...
0.44
7 x
( 1.
54 )
-2n
ex =
5,
0 =
1 :
E
...
44.7
x
( 1
.59
) -2
n
o = 2
: E
...
15
.8
x (
1.56
) -2
n
10
-71
:::>
R
adia
l Int
egra
tion
Poi
nts
n 10
20
30
10-8
Fig
. 10
.28
Rad
ial i
nte
gra
l Ia
,8
by
£og-
L 1 tr
an
sfo
rma
tion
~
o U1
Rel
ativ
e E
rror
e
1 10
-1
10
-2
10-3
10-4
10
-5
10~
10
-7 I
10-8
fo
30
Rad
ial I
nteg
ratio
n la
,/) =
It] ~ d
p
50 S
ingl
e E
xpon
entia
l ( S
E )
tra
nsf
orm
atio
n
p (
R )
= ~
( 1
+ ta
nh R
)
R :
[ -00
, 00
]
r=~p2+d2,
Pj=
1,
d=O
.01
60
70
Rad
ial
Inte
gra
tion
P
oin
ts n
Fig
. 10
.29
Rad
ial
inte
gra
l I<
X,()
b
y S
ing
le
. Exp
on
en
tial (
SE
) tr
an
sfo
rma
tio
n
~
~
Rel
ativ
e E
rro
r €
1 10
-1
10
-2
10-3
10-4
10--
5 a
= 3
,
10
-e
a=B=
1
10
-71
.,~~~~~i~,~~~~y-~~-r-'---
i ,
, ,
30
10
20
10-8
Ra
dla
llnte
gra
tion
Icx
,s =
S~j -
% dP
Dou
ble
Exp
onen
tial (
DE
) tr
an
sfo
rma
tion
P (R
) =
~j {
1 +
tanh
( ~
sinh
R )
}
R :
[ -0
0,
00
]
r=~p2+d2, p
j=1
,d
=O
.01
a =
5,
B =
2
a =
3,
B =
1 B=
1
- T'~'~~
~r-T-~
~~-r~~
'1~::;
:~~'--
------
---,'
... :::
:>
40
50
a 60
70
Rad
ial
Inte
gra
tion
P
oin
ts n
Fig
. 10
.30
Rad
ial i
nte
gra
l Icx
,B
by
Dou
ble
Exp
on
en
tial (
DE
) tr
an
sfo
rma
tion
~
-...J
408
(1) Transformations based on the Gauss-Legendre rule
(i) Identity transformation
From Table 10.22 and Fig. 10.25, it is clear that the identity transformation
(just polar coordinates) requires a huge number of radial integration points for
nearly singular integrals (D == d / Pj ~ 1 ) .
The theoretical estimate of the error En given in section 7.6 (3), equation
(7.76) implies that
a
E - (1- t) 2 0 -2n n
where, in order to maintain
a
o-t) 2<10
t must satisfy
{
0.99
0< t< 1_10-2Ia = 0.77
0.60
(a=l)
(a=3)
(a=5)
Hence, if we take the maximum values for t, equation (7.64) gives
{
1.15
(J = 1.13
1.12
(a=I)
(a=3)
(a=5)
(10.53)
(7.77)
(7.262)
(10.54)
for D=d=O.OI in equation (10.53). This theoretical estimate matches well with
numerical results in Fig. 10.25, where
409
1.17 (a=o=l)
1.14 (a=3,0=1)
(J = 1.14 (a=3,0=2)
1.13 (a=5,0=1)
1.12 (a=5,0=2) (10.55)
(ii) 10g-Lz transformation ,
Table 10.23 and Fig. 10.26 show that the 10g-L2 transformation is very
efficient for the integration of la, ,8 when 0= 1, but on the other hand very
inefficientfor the integration of I • when 0= 2. This is in accordance with a,.
Fig. 10.23. There, the 10g-L2 radial variable transformation gave remarkably
good convergence for the calculation of the potential integrals Isu* dB and
Is q* dS, which correspond to a=o=l and a=3, 0=1, respectively. On the other
hand, the 10g-L2 transformation gave very slow convergence for the flux integrals
Is au*laxs dS and Is aq*laxs dS , which contain components corresponding to
a=3, 0=2 and a=5, 0=2, respectively (cf. Table 7.1).
The theoretical estimates of the error En in section 7.7 (1), equation (7.120)
gives
(10.56)
for 0= 1. This is in good agreement with the numerical results, which show fast
convergence for a = 1, 3, 5 ; 0 = 1 in Fig. 10.26.
For 0 = 2, the theoretical estimate in section 7.7 (2), equation (7.190) gives
E -3 -n n (10.57)
Fig. 10.27 shows the convergence graphs of relative error e vs loglon, for 0=2.
The graphs give
e - 0.266 X n -2.95 (a=3,6=2) (10.58)
410
€ - 4.22 X n -2.99 (a=5,8=2) (10.59)
which matches very well with the theoretical error estimate of equation (10.57).
(iii) log-Lz transformation
Table 10.24 and Fig. 10.28 show that the log-LI transformation works
efficiently and robustly for all types of model kernels which appear in the three
dimensional potential problem.
The theoretical estimate for the relative numerical integration error £ in
section 7.8, Table 7.6 gives
a
In order to maintain
a
(l-t) 2 < 10
in equation (10.60), t must satisfy
{
0.99
0< t < 1_10-2/a = 0.77
0.60
(a=l)
(a=3)
(a=5)
(10.60)
(7.261)
(7.262)
Hence, if we take the maximum values for t, equations (7.252-254) give
{
1.64
(J = 1.51
1.40
(a=l)
(a=3)
(a=5) (10.61)
for D=d=O.Ol, in equation (10.60). This theoretical estimate for the relative
error corresponds well with the numerical results in Fig. 10.28, where
411
1.88 (a=o=l)
1.64 (a=3,0=1)
a= 1.54 (a=3,0=2)
1.59 (a=5,0=1)
1.56 (a=5,8=2)
(2) Transformations based on the truncated trapezium rule
(i) Single Exponential (SE) transformation
(7.264)
Table 10.25 and Fig. 10'.29 give numerical results for the numerical
integration using the single exponential transformation
p(R) = p j (1 +tanhR) 2
(5.98)
which maps R : [- 00, 00] to P: [ 0, p). The truncated trapezium rule is used for
the integration in the transformed variable R : [- 00, 00] •
The detail of the numerical procedure is given in section 5.5 (5), together
with the double exponential (DE) transformation. Although better than the
identity transformation (l)(i) with the Gauss-Legendre rule for D=d ~ 10-2, the
single exponential transformation requires more than twice as much integration
points, compared to the log-L1 transformation of (1) (iii) to obtain the same
accuracy.
Table 10.27 gives numerical results for the single exponential
transformation
p(R) = p , (1 + tank R ) J
(5.108)
412
which maps R : [- 00, 0] to P: [ 0, Pj ]. This transformation gives poor results
compared to that of equation (5.98), which reduced the integration error by
concentrating the integration points near the end point P = Pj as well as P = 0 .
(ii) Double Exponential (DE) transformation
Table 10.26 and Fig. 10.30 give numerical results for the double exponential
transformation
p. { 7r: } p(R) = ~ 1 + tanh ( 2" sinhR)
(5.109)
which maps R : [- 00, 00] to P: [0, Pj]' This transformation requires even more
integration points for D=d ~ 10-2, compared with the single exponential
transformation of equation (5.98). Compared with the identity transformation,
the DE transformation is more efficientfor D= 10-3 and comparable for D= 10-2•
As with the SE transformation, the DE transformation
p(R)=Pj{l+tanh(~SinhR)} , (10.62)
which maps R : [- 00, 0 ] to P : [0, Pj] , gives poor results compared to the
transformation of equation (5.109), as shown in Table 10.28.
(3) Summary
To sum up, the numerical experiments in this section indicate that the most
efficient and robust method for the numerical integration in the radial variable,
so far, is the log-LJ transformation in combination with the Gauss-Legendre rule.
Also important is the fact that the numerical results match very well with
the theoretical error estimates based on complex function theory, which was given
in Chapter 7.
413
10.7 Comparison of Different Numerical Integration Methods
on the 'Spherical' Element
In this section, different numerical integration methods will be compared
for the integration of nearly singular integrals arising in potential calculations
f ¢ij u* dS s
and flux calculations
f ilu* ¢. -dS
IJ ilx S s
f ¢ij q*dS s
f ilq*
S ¢. -dS
!J ilx s
(10.63)
(10.64)
in three dimensional potential problems. The boundary element S is taken as the
'spherical' quadrilateral element SPQ60, which was defined in section 8.2,
equation (8.9). SPQ60 is a spherical quadrilateral subtending 60° in each
direction on a sphere of radius 1, modelled by the 9- point Lagrangian element of
equations (5.2) and (5.3). Its typical element size is 1 as shown by equation (8.10).
The following methods will be compared:
1. Gauss: The product Gauss-Legendre formula 3, given in section 4.2.
2. Telles: Telles' self-adaptive cubic transformation method 15, given in
section 4.4 (2) (ii) .
3. The PART method with (and without) the angular variable transformation
t(B) of equation (5.127), and with the radial variable transformations R(p) :
(i) Identity: R(p) = p
(ii)log-L2: R(p) = log Yp2 + d 2
(iii) log-Lz: R(p)= log (p+d)
414
(1) Effect of the source distance: d
First, the source point Xs was positioned so that its projection Xs on the
element Sis xs=x(O.5, 0.5), i.e. ~1 =~2=0.5, and Xs is situated towards the
centre of the sphere at a distance d from the element surface S .
Figures 10.31-10.34 are the convergence graphs for the integrals
Is rpij u* dS, Is rpij q* dS , Is rpij au/axs dS and Is rpij aq/axs dS , respectively,
for the source distance d=O.OI. Here, rpij is the 9- point Lagrangian
interpolation function corresponding to the node (7J l' 7J2) = (i,j), i.e.
(10.65)
where rp j( 1J), (i = -1, 0, 1) is defined in equation (5.3). In the graphs, the relative
error is plotted against the number of integration points. The relative error is
taken as the maximum relative error for all the interpolation functions i,j= -1,
0,1. The error was calculated by comparing with the result given by the PART
method with the log-Ll radial variable transformation and the logarithmic type
angular variable transformation t(8) of equation (5.127), with Nt=N R = 128, i.e.
128 Gauss-Legendre integration points for the variables t andR, respectively.
The PART method with the log-L1 radial variable transformation is shown
to be the most efficient and robust method for different integral kernels. Note
also that for the integrals Is rpij u* dS and Is rpij q* dS , related to potential
calculations, the log-L2 radial variable transformation does not show good
convergence as it did for the potential related integrals Is u* dS and Is q* dS
on the constant planar element in Fig. 10.23. The reason for this was explained
using the results of theoretical error estimates in section 7.7 (2), i.e. rpij contain
terms equivalent to p and p3, which give rise to terms corresponding to o=even
in the model radial integral kernels in equation (7.45).
For the flux kernels, Fig 10.33 and 10.34, the standard product Gauss
Legendre formula did not converge within the scope of the graphs.
• q
~ G
au
ss
._ ...
. Rel
atlve~
\, I.
/',
._
._...
......
. ..
.... _
_
Err
or
. '.h
, \V
',
. ./
'-..''-
."-
A
'./
'-..-
1'~
""
\..'
--'
''-,.
.--'
\
l Q
,tJ
10 -1
~ ~
'\
~ ',
.."
Ide
ntit
y 1
0 ·2
~ ! f\.
I b-
"'\
"
(Po
lar)
1/ \.
\ '"
10·
3~ j'
\""""
"
Is CP
ij u*d
S
S :
'sph
eric
al' q
uadr
ilate
ral
(SP
Q60
) d
= 0
.01
./ \ _
......
'b o -4~
1 ;
\,
Tel
les
\.
1 ..
_.~"
\
__
.. ~,
0.
• 0
'"
"
10
-5
"""
,"--
, --
~--~
--.--.--A
'"
-0
'--0
' _
__
-=-1I
b R
,og-
L2
10 -7
1 O'--'-'
-~~-'--"
"""~~~~L
....<~:~
~'-I
.0
_--
'-..
...L
.---
L- >
10 -6
R
,og-
L 1
500
1000
15
00
2000
N
umbe
r o
f 10
-8
Inte
gra
tion
Po
ints
Fig
. 10
. 31
Co
nve
rge
nce
gra
ph
fo
r J s
<Pij
u*d
S
.j::o
. - U1
Rel
ativ
e j
Err
or
~
1 10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
.--+
<
\ .11
)'.
/_.-
....
,.
"""_
'_'-
~/'
. "
. ,
--' it \/ '''./
",
• __
. \
....... ...
,. .,...
.. q
.<J"
D.
\li '/'
'n\
Gau
ss
IsC
Pijq
*dS
1,&
\ \.
./'.
•
\ \
'~.,
"-"
Iden
tity
\ ~.~
'b."
(Po
lar)
\
" 'n
.. ~
\ ·~
Tell
es
",
S :
'sph
eric
al' q
uadr
ilate
ral
(SP
Q60
) d
= 0.
01
\ --
-.~
bo.
\.
....... ...
..... ,
.. , "
........
........
.. , '
-.~
'~"'"
R,og
-L 2
. ':-
-...
'"..
....
",
..........
.....
10 -7
1,
" \
'."
"..
'. R,
Og-L
1 ,
, I
,
, I
I,
,
2000
N
um
ber
of
Inte
gra
tion
Po
ints
o
500
1000
15
00
, 10
-8
Fig
. 1
0.3
2
Co
nv
erg
en
ce
gra
ph
fo
r Is
CPij
q*d
S
.1:>0 .... Q)
Rel
ativ
e E
rro
r
1
10 -1
10
-2
10
-3
10 -4
10
-5
10 -6
"\
"~"
f au
* S
cp .. -
':}-
dS
IJ
uXs
S : '
sphe
rical
' qua
drila
tera
l (S
PQ
60)
d =
0.0
1
;;.
-b
-"-
.."
~".
.....
,
<>--~-
-----
.. '".---
.". lo
g-L
2 ~
~---
---
.~
--._.
~---~-
---~
------
-~----
-~ -,
. . ,
, .. ,.,
Iden
tity)
'.
"'u
(P
ola
r .....
........
. .....
..........
... .... ..
....... _
._.-
..
....."
,.,
Tel
les'
.,.'t
:.
10 -7
1 r: ___ L
....
..L
---1
-...
.L.-
I .0
__
_ L-~~~~~~~~~~L-~~~-
I
__
....r........J.'_~
500
1000
15
00
2000
N
um
ber
of
10 -8
In
teg
rati
on
Po
ints
au*
Fig
. 10
. 33
Co
nv
erg
en
ce
gra
ph
fo
r Is
<i>ij ax
-dS
s
"'" ~ -..J
Rel
ativ
e E
rro
r lB
:. Is
cp"
:\ dS
1
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
IJ uX
s ""
'-.
. \~
'''--
n."
",-
S :
sph
eric
al q
uadr
ilate
ral
\..,~
"~'"
d
= 0
.01
~
" .
,~
, ·'b
----
- c
, .....
... ..
... ~.:.,
·O""
' •• "
"'-.I.
dent
ity (
Pol
ar)
.. ",;:~
'O----_
_ "i
J
~-
n '~
,~-o
-_--
-... _
__ o()
__
)Log
-L 2
.~
----~--
-"
---~
, ""'
- '"",-
, ""'-
'"
Te
lles ''',
'' '''6
10 -7
I :-.
J...
..-L
---1
...-
...L
I
.0
___ ~~~
~L-~
~~~~
;-~~
__ ~::
~~ __ L-
L-___
10-8
'
• 20
00
Num
ber
of
Inte
gra
tion
Po
ints
50
0 10
00
1500
Fig
. 10
. 34
C
on
verg
en
ce g
rap
h f
or
Is <P
ij gi:
dS
.".
00
419
Tables 10.29, 10.30, 10.31 and 10.32 compare the number of integration
points required for each method to achieve a relative error less than lO-6 for the
same integrals Is 9Iij u* dB, Is 9Iij q* dS ,Is 9Iij au*/axs dS and Is 9Iij aq*/axs dB, for
different values of the source distance from d= 10 to lO-3. The source projection
Xs was set again to xs=x(0.5, 0.5) on the curved quadrilateral element: SPQ60.
For the Gauss and Telles methods N1 and N2 represent the necessary and
sufficient number of integration points in the 7J 1 and 7J 2 directions, respectively, so
that the total number of integration points is N=N1 XN2 • For the PART method
(identity, log-L2 and log-L1), Nt andNR (or N p ) are the necessary and sufficient
number of integration points in the transformed angular variable t and the
transformed radial variable R ( or p ), respectively, so that N =Nt X N R X 4 , since
there are generally four triangular regions ;;. j , (j = 1-4) to integrate. (For
triangular elements, this becomes N=NtXNRX3, so that the PART methods
would require only 3/4 of the integration points compared to quadrilateral
elements. On the other hand, the Gauss and Telles methods basically require the
same number of integration points for triangular and quadrilateral elements, so
long as the product type formula is used. )
The actual value of the maximum relative error is shown in brackets when
the method failed to converge even with 256 integration points in each variable.
The average CPU-time per integration point is shown to be more or less the same
for different methods, so that basically one may judge the efficiency of each
method by the number of integration points required. The value of the integral
corresponding to 91 11 ( of the x-component for flux integrals) ,calculated by the
PART method ( log-Ll ,t(8» with Nt=NR=128, is also given for each case. The
best method for each distance d is indicated by * .
sour
ce
Gau
ss
dist
ance
d N
1X
N2
tota
l
10
5X
6
*3
0
3 5
X6
*
30
1 5
X5
*
25
0.3
9 X
10
90
0.1
25 X
28
700
0.03
90
X 1
00
9000
0.01
22
0 X
250
55
000
0.00
3 25
6 X
256
65
536
(5 X
10
-')
0.00
1 25
6 X
256
65
536
(2 X
10
-2)
iAye
rage
C
PU
-tim
e 27
,use
e p
er p
oint
Tab
le 1
0.29
C
ompa
riso
n fo
r di
ffer
ent m
etho
ds fo
r Is
~ij u
* dS
Tel
les
Iden
tity
(P
ola
r)
log-
L 2
log-
L 1
N1X
N2
tota
l N
tXN
p to
tal
NtX
NR
to
tal
NtX
NR
5X
6
30
8X
5
160
8 X
128
40
96
8X
6
5X
6
30
8X
6
192
8 X
120
38
40
8X
6
6X
6
36
8X
6
192
8 X
110
35
20
8X
7
7X
8
*5
6
7X
8
224
8 X
72
2304
7
X8
9 X
10
* 90
8
X 1
4 44
8 8
X 4
5 14
40
8X
9
20 X
25
500
7 X
20
560
7 X
32
896
7 X
12
35 X
40
1400
10
X 4
5 18
00
10 X
40
1600
10
X 1
4
55 X
55
3025
11
X 6
4 28
16
10 X
20
800
10 X
16
72 X
72
5184
10
X 1
0 44
00
10 X
20
*800
10
X 2
0
30,u
see
28,u
see
29,u
see
29,u
see
tota
l
192
192
224
224
288
*3
36
*560
*6
40
*8
00
Inte
gra
l
Is ~llu*dS.
2.36
9067
0 X
10
-'
7.8
51
22
15
X1
0-'
2.36
6559
6 X
10
-3
7.93
4006
3X10
-3
1.49
9020
2 X
10
-2
1.88
1873
5 X
10
-2
2.00
0924
2 X
10
-2
2.04
3229
2 X
10
-2
2.05
5363
7 X
10
-2
.j:o.
N
o
sour
ce
Gau
ss
dist
ance
d N
1X
N2
tota
l
10
4X
5
* 20
3 5
X5
*
25
1 6
X6
*
36
0.3
12 X
14
168
0.1
32 X
35
1120
0.03
11
0 X
110
12
100
0.01
25
6 X
256
65
536
(3 X
10
-6)
0.00
3 25
6 X
256
65
536
(4 X
10
-3)
0.00
1 25
6 X
256
65
536
(3 X
10
-1)
!Ave
rage
C
PU
-tim
e 27
/-ls
ec
per
poi
nt
Tab
le 1
0.30
C
ompa
riso
n fo
r di
ffer
ent m
etho
ds fo
r Is
<Pij q
* dS
Tel
les
Iden
tity
(P
ola
r)
log-
L 2
log-
Ll
N1X
N2
tota
l N
tXN
p
tota
l N
tXN
R
tota
l N
tXN
R
4X
5
20
8X
5
160
8 X
128
40
96
8X
5
5X
5
25
8X
6
192
8 X
128
40
96
8X
6
6X
6
36
8
X6
19
2 8
X 1
10
3520
8
X7
8X
8
* 64
8
X 1
0 32
0 9
X 1
10
3960
8
X8
14 X
14
* 19
6 7
X 1
6
448
9 X
55
1980
7
X 1
1
25 X
25
625
9 X
28
1008
9
X 4
0 14
40
9 X
14
45 X
45
2025
9
X 5
0 18
00
9 X
28
1008
10
X 1
6
72 X
72
5184
9
X 9
0 32
40
9 X
28
900
9 X
20
100
X 1
00
1000
0 9
X 1
50
5400
9
X 2
0 *
72
0
10 X
20
31/-
lsec
28
/-ls
ec
30/-
lsec
29
/-ls
ec
tota
l
160
192
224
256
308
* 50
4
* 64
0
* 72
0
800
Inte
gra
l
Is <P
ll q*
dS
-2.4
45
20
62
X 1
0-5
-2.6
50
96
34
X 1
0-4
-2.3
55
19
83
X 1
0-3
-2.3
43
72
38
X1
0-2
-5.8
70
27
28
X 1
0-2
-7.4
05
88
44
X 1
0-2
-7.7
82
22
66
X1
0-2
-7.9
02
73
56
X1
0-2
-7.9
35
97
58
X1
0-2
~
I\)
~
~ouree
Gau
ss
dist
ance
d N
1X
N2
tota
l
10
5X
6
* 30
3 6
X6
*
36
1 6
X7
*4
2
0.3
14 X
14
196
0.1
40 X
40
1600
0.03
15
0 X
150
22
500
256
X 2
56
6553
6 0.
01
(2 X
10
-3)
256
X 2
56
6553
6 0.
003
(13)
256
X 2
56
6553
6 0.
001
(62)
!Ave
rage
C
PU
-tim
e 31
,use
e p
er p
oin
t
Tab
le 1
0.31
C
ompa
riso
n fo
r di
ffer
ent m
etho
ds fo
r Is
¢ij a
u*la
xs d
S
Tel
les
Iden
tity
(P
ola
r)
log-
L2
log-
L 1
N1X
N2
tota
l N
tXN
p
tota
l N
tXN
R
tota
l N
tXN
R
5X
6
30
8X
5
160
8 X
128
40
96
8X
6
6X
6
36
9X
7
252
9 X
170
61
20
9X
7
7X
8
56
10 X
7
280
10 X
256
10
240
10 X
9
10 X
10
* 10
0 9
X 1
1 39
6 9
X 1
80
6480
9
X 1
0
14 X
16
* 22
4 9
X 2
0 72
0 10
X 2
10
8400
10
X 1
1
14 X
256
14
336
32 X
32
* 10
24
14 X
40
2240
(8
X 1
0-6
) 1
4 X
20
14 X
256
14
336
50 X
50
2500
12
X 6
4 30
72
(2 X
10
-6)
12 X
20
9000
14
X 1
20
14 X
256
14
336
14 X
25
90 X
100
67
20
(5 X
10
-6)
14 X
256
14
336
14 X
25
120
X 1
20
1440
0 16
X 1
70
1088
0 (4
X 1
0-6
)
33.u
see
32.u
see
33.u
see
34.u
see
-_
.-
--_
._
--
-
tota
l
192
252
360
360
440
1120
* 96
0
* 14
00
* 14
00
Inte
gral
Is ¢
llau
*lax
sdS
( x-c
ompo
nent
)
2.17
4021
2 X
10
-5
2.31
5882
0 X
10
-4
1.97
9310
3 X
10
-3
1.62
9241
3 X
10
-2
3.00
4361
4 X
10
-2
2.98
2088
2X10
-2
2.84
2878
7 X
10
-2
2.77
7773
5 X
10
-2
2.75
7571
5 X
10
-2
.j::>
"-
l "-
l
sour
ce
Gau
ss
dist
ance
d N
1X
N2
tota
l
10
5X
5
*2
5
3 5
X6
*"
30
1 7
X7
*
49
0.3
14 X
16
224
0.1
40 X
45
1800
0.03
16
0 X
180
28
800
256
X 2
56
6553
6 0.
01
(3 X
10
-2)
256
X 2
56
6553
6 0.
003
(90)
256
X 2
56
6553
6 0.
001
(590
)
Ave
rage
C
PU
-tim
e 3O
.use
e pe
r poi
nt
Tab
le 1
0.32
C
ompa
riso
n fo
r di
ffer
ent m
etho
ds fo
r Is
¢ij
aq*
/axs
dS
Tel
les
Iden
ti ty
( P
ola
r)
log-
L 2
log-
L 1
N1X
N2
tota
l N
tXN
p
tota
l N
tXN
R
tota
l N
tXN
R
5X
5
25
8X
6
192
8 X
140
44
80
8X
6
5X
6
30
8X
6
192
8 X
128
40
96
8X
7
8x
8
64
10 X
7
286
10 X
220
88
00
10 X
9
9 X
10
*90
9 X
12
432
9 X
150
54
00
9 X
10
20 X
20
*400
10
X 2
0 80
0 11
X 2
10
9240
11
X 1
4
40 X
40
1600
12
X 4
5 21
60
12 X
350
16
800
14 X
20
(8 X
10
-6)
14 X
256
14
336
14 X
25
64 X
64
4096
14
X 8
0 44
80
(7 X
10
-6)
110
X 1
10
1210
0 14
X 1
50
8400
14
X 2
56
1433
6 14
X 2
8 (3
X 1
0-5
)
16 X
256
16
384
16 X
256
16
384
170
X 1
70
2890
0 16
X 3
5 (2
X 1
0-6
) (1
X 1
0-4
)
37.u
sec
32.u
see
38.u
sec
38.u
see
------------
tota
l
192
224
360
360
616
* 11
20
* 14
00
* 15
68
*2
24
0
I
Inte
gral
I
Is ¢
11 a
q*/a
xs d
S
( x-c
ompo
nent
)
-4.7
31
40
67
X1
0-6
-1.6
19
15
72
X1
O-4
-3.9
03
58
21
X1
0-3
-7.5
01
26
85
X1
0-2
-8.9
87
37
60
X1
0-2
-7.7
11
13
43
X 1
0-3
2.77
9438
5 X
10
-2 I
4.12
2683
7X 1
0-2
4.51
4835
9X10
-2
~
....., w
424
Fig.10.35-38 compare in more detail, the effect of the source distance
d on the different methods for the calculation of Is <P ij u* , Is <P ij q* dS ,
Is <P ij iJu*liJxs dS and Is <P ij iJq*liJxs dS. Similar to Table 10.29-32, the number
of integration points required for each method to obtain relative error
e < 10-6 for all i, j = -1,0, 1 (and x, y, z component), was plotted against
the source distance d = 10-4 -10. The element is the same 'spherical'
quadrilateral SPQ60 with the source projection at Xs = x (0.5, 0.5) and the
source distance is measured from the element towards the centre of the
sphere. Table 10.33 summarizes the results.
Table 10.33 Range of source distance d best suited to each method
integral log -L2 log -L1 Telles Gauss kernel
<Pij u* d ~ 0.001 d ~0.04 0.04 < d ~ 0.5 0.5~ d
<Pij q* d ~ 0.002 0.002 ~ d ~ 0.04 0.04 < d ;;;0.7 0.6;;; d
<Pij iJu*liJxs - d <0.03 0.03 ~ d <0.7 0.7 ~ d
<Pij iJq*liJxs - d <0.04 0.04 ~ d <0.8 0.8~ d
(S: SPQ60, xs= x (0.5, 0.5), relative error < 10-6 )
In order to save CPU-time, it is advisable to use the same set of
integration points for the calculation of Is <Pij u* dS and Is<Pij q* dS in order
to calculate the potential u(xs) at X=Xs. Similarly, the same set of
integration points should be used for the calculation of I s<Pij iJu*liJxs dS and
Is <Pij iJq*liJxs dS in order to calculate the flux (potential derivative) iJu/iJxs
at x = Xs . Further, if one wants to calculate both potential and flux at the
same point x = xs , the same set of integration points should be used for all
the kernels: IS<Piju*dS , IS<Pijq*dS , Is<PijiJu*liJXsdS and Is<PijiJq*liJXsdS.
425
The strategy for the choice of numerical integration method with regard to
the source distance d for each situation is given in Table 10.34.
Table 10.34 Strategy for the choice of numerical integration method
calculation log -L1 Telles Gauss item
potential only d < 0.05 0.05 ~ d <0.6 0.6~ d
flux only d < 0.04 0.04 ~ d <0.8 0.8~ d
potential and flux d < 0.04 0.04 ~ d <0.8 0.8~ d
(s: SPQ60, XS = x(0.5, 0.5), relative error < 10-6 )
The above results may vary depending on the position of the source
projection xs , the geometry of the element S. However, they may be
considered as a rough guide for choosing the numerical integration method
according to the (relative) sour~e distance d.
Num
ber o
f Int
egra
tion
Poi
nts
2500
2000
1500
10-4
0----
---0
" 9
00
0'"
, ,
\.
9-1
\ I
\ 1
I ~
, !
\\ · n
\ !
~~~ '\
\ /'
'
'A '
,-1
' \,
'\' :~~
,'-1&
'P
"
'~
\ I
lJ' ,
I '
, I
q "
" : "\
0','
I bb
r,
I
I ..
....
. ,,'
,
: \
"...
. 0-
&
tIcoo
Is 'P
I) u
*dS
P -+
\'"
, ,
iog-
L2
I \
\ i
\ ~ :
,l!\
,:
, G
auss
fl
\ \:
v-.r
{
S:
'sph
eric
al' q
uadr
ilate
ral
(SP
Q6
0 )
is =
X(
0.5,
0.5
) re
lativ
e e
rro
r E
< 1
0-6
I "
~ \
----"j
~ \\ l
· \. _
./ Ide
ntity
( P
ola
r)
~
' .... 'a~ ~-
Tell
es~'
~~ =~~
: -
-'hA
ollo6
. __
_ ~
10-3
10
-2
10-1
1
5 10
Fig
. 10.
35
Num
ber o
f Int
egra
tion
Poi
nts
vs S
ourc
e "D
ista
nce
for
Is 'P
I) u
* dS
S
ourc
e D
ista
nce
d
.1=00
I\,)
C
'l
Num
ber o
f Int
egra
tion
Poi
nts
3500
3000
2500
2000
1500
1000
9 I
~
If!
~
\:
, I
i J l
iJ
" \ ~
i~
: Xi
\ P1
~
,0-'
, ,
})/
\
\", I
de
ntit
y ,
9"-
--0
o
, o
,",
90-&
•
I,
I
I '"
, '.
, I
,_
,
I 'I
,
~ 9
:: :
: I>
I ,
I ,
bd
Is CP
lj q*
dS
{ S
: 's
pher
ical
' qua
drila
tera
l (S
PQ
60
) Xs
= X
( 0.
5, 0
.5)
rela
tive
err
or
£ <
10-
6 ,,\
'-« Po
lar)
\
,-'
\ '\
\
~ G
auss
po
o "\,
''b
--q,
\
})-o
-o-o
J . \
. '.
'\
" ~
\q
p--
--'
,.
'b
' f"
t 'a
----
5001
----
-o--
-o r°o-
oo-o
o
Ii>. ~ l~
::~-
~~~m
H!;:
':"=
=~_~
::::
: R
og-L
2
10-4
Fig
. 10.
36
10-3
10
-2
10-1
1
5 10
S
ourc
e D
ista
nce
d N
umbe
r of I
nteg
ratio
n P
oint
s vs
Sou
rce
Dis
tanc
e fo
r Is
CPlj
q* d
S
.l>
I'.)
-..
.I
Num
ber
of I
nteg
atio
n P
oint
s
3500
1-~
, t'
I
3000
l-II
~ j\ \
\ Id
entit
y
I G
auss
\ 25
00 l-
I \rv
( Pola
r)
I \
\
~ J
au·
\ \"4
S
CPij
ax
s dS
2000
l-I
\ I
log-
L1
~
I
\ \
\ ~
S:
'sph
eric
al' q
uad~
late
ral
\ ,.
/ I
(SP
Q 6
0)
I I
I
1500
f---\/\
n ~
~ \
is =
X (
0.5
,0.5
) ~
"l
\
co
rela
tive
err
or
E <
10
-6
1000
500
Tel
les
10-4
10
-3
10
-2
10
-1
1 5
10
. f
au·
Sou
rce
Dis
tanc
e d
Fig
. 10.
37
Num
ber
of I
nteg
ratio
n P
oint
s vs
Sou
rce
Dis
tanc
e fo
r 5 <
i>ij
axs d
S
Num
ber
of I
nteg
atio
n P
oint
s
3500
3000
2500
2000
1500
1000
500 10
-4
10-3
iOg
-L1
\\.
Ide
ntit
y \ (' (
Po
lar)
'Vi
\ \
D
: I
LJ
I f.
QsL
dS
\ :
I G
auss
S
<i>i
j ax
s .\
I~
I~ 1 ~~\
I
I \
S :
'sp
he
rica
l' q
ua
dri
late
ral
k \
\
· .......
..... ,1
\. \
\ \.
\ X
s = X
( 0.
5, 0
.5 )
rela
tive
err
or
E <
10
-6
~ \"
i 1
~~
J~,
0
Te
"es
~~A ~
......
+
+~~
(SP
Q 6
0)
10-2
1
5 10
Fig
. 10
. 38
Num
ber
of I
nte
gra
tion
Po
ints
vs
Sou
rce
Dis
tanc
e fo
r J
<p ..
.£9.: d
S S
ou
rce
Dis
tanc
e d
S IJ
axs
.j:>.
~
430
(2) Effect of the position of the source projection: Xs
Next, the effect of the position of the source projection Xs on the
accuracy and efficiency of each numerical integration method, is
investigated.
The same 'spherical' quadrilateral S: SPQ60 is taken as the curved
boundary element over which the integration is performed. The source
projection
(10.66)
is moved along the diagonal of the element from the centre (~= 0) to the
corner (~= 1) and outside ( ~ = 1.1). The source point is positioned at
x = x + dn 8 B (10.67)
where n is the unit normal at Xs pointing towards the centre of the sphere.
The source distance was fixed at d=O.01.
The following three methods were compared:
1. The PART method with the log-L1 radial variable transformation and
the angular variable transformation t (8 ) :
(i) Identity: t (8) = 8 ,
(ii) log-type :
hj {l+sin(O-aj )} t(O) = -log
2 1-sin(8-a ,) J
(5.127)
2. Telles' self-adaptive cubic transformation method 15, given in section 4.4
(2)(ii).
Fig 10.39-42 are the graphs of the number of integration points
required to achieve relative error e < 10-6 for all i, j = -1,0,1 (and x,y, z)
components , vs ~ (position of the source projection xs) , for the calculation
of Is r/>ij u* dS, Is r/>ij q* dS, Is r/>ij au*laxs dS and Is r/>ij aq*laxs dS ,
respectively.
431
The figures clearly show the effect of the angular variable
transformation t(8) of equation (5.127), particularly when ~ ~ 0.7, in
reducing the number of integration points in the angular variable compared
to the identity transformation t(8) = 8 ,which requires a huge number of
angular integration points as the source projection Xs approaches the edge
of the element ( ~-1 ).
Telles' cubic transformation method becomes advantageous as the
source projection Xs approaches the edge (corner) of the element. The
results are summarized in Table 10.35. However, the results are only for the
case when the source distance d=O.Ol. The PART method becomes more
advantageous as the source distance d decreases.
432
Table 10.35 Position of source projection xs=x (~ ,~) suited to each method
Type of kernel PART method Telles log-LI ' t ( 8 )
<Pij u* o ~ ~ ~ 0.92 0.92 < ~
<P ij q* o ~ ~ ~ 0.96 0.96 < ~
potential calculation o ~ ~ ~ 0.96 0.96 < ~
<Pij au*laxs o ~ ~ ~ 0.90 0.90 < ~
<P ij aq*laxs o ~ ~ ~ 0.93 0.93 <~
flux calculation o ~ ~ ~ 0.93 0.93 < ~
potential & flux calculation o ~ ~ ~ 0.93 0.93 < ~
(S : SPQ60, d= 0.01)
In the implementation of the PART method and Telles' method in this
experiment, the source distance d was determined as the distance between
Xs and xs, even when the source projection xs=x (~l ' ~2) falls outside the
element S, i.e. when I ~l I >1 or I ~2 I >1 ( I ~ I >1 in the case above).
However, since the integration is performed only inside the element S, the
effective source distance between the source point Xs and the element S is,
generally speaking, larger than the distance I Xs - Xs I. Actually, the
nearest point to Xs lies on the edge of the element S, in this case. Using
this definition of the effective source distance, the PART method ( in
particular) and Telles' method can be improved.
Also, even when the source projection Xs lies on the element S , but
very close to the element edge, results by the PART method may be
improved by artificially moving the source projection to the nearest point on
the edge, provided that the distance between the original source projection
433
:Ks and the closest element edge is smaller than or comparable to the source
distance d.
Num
ber o
f Int
egra
tion
poin
ts
3500
3000
2500
2000
1500
10
00
500
Is % U*
dS
sour
ce d
ista
nce
: d =
0.01
sour
ce p
roJe
ctio
n : X
s= X
(ii , i
i) S
: 's
pher
ical
' qua
drila
tera
l ( S
PQ
60
)
rela
tive
err
or:
e<
10-6
R
I I
I I
I I
I I
I I
I I
~ I~
,
Iy
, "
, "
, "
,: ~
, , , , , lo
g-L
J ,9
1 ,
""I ,
Tel
les
, , ,
" ,~
;
~
, '
~'
\ 1
~_.6
: ,
,.Il
/p
--
o~
\ ,/
,,/
",
.... -........
.... "'"
""
/'"
,,~-----o
~........-A
" ~-
"'V""~
" ~
~~~~
o'
~-
~~~~
~~q~
~~
"
I 1 I
o I I
\ b I I I I I o I I \ q \ boq I I I bo
____
__ o-----~-.o-
01
, ,
, ,
Posi
lion
at
0.1
0.2
0.3
0.4
0 '5
'
, ,
, _
_
sour
ce p
roje
ctio
n .
0.6
0.7
0.8
11
11
11
11
11
11
11
11
11
11
7
" Ti
log-
LJ ,
t (9
)
0.9
1.1
Fig
. 10
. 39
Num
ber
of
inte
gra
tion
po
ints
vs
Pos
ition
of s
ou
rce
pro
ject
ion
fo
r Is
<Ilq
u* d
S
.j::o
~
Num
ber
of i
nteg
ratio
n pO
ints
35
00
30
00
f % q*
dS
S
d =0
.01
Xs=
X (T
\, T1
)
S :
SP
Q 6
0
'i' I I I I I I I I I I I o I I I I
25
00
1'..
........
Tel
les
\. E
< 1
0-6
? ~
I ....
'I
I
/ .......... 4
!
/: ;
20
00
15
00
10
00
50
0
I ...
......
. ,
Q:
: /..........
' I
I
I A-
----
.6,.
--~
" :
'
log-
LJ ,
e
r ~~~------~-----
----
--~~
----
-~--
------
---'"
'........
.. I
:~
...
I I
I ,,_
I
I I
~
I I'
" :'
0
0
, ,
" ,.
.-A
--j-
..It
~~
~~
___ -
-.0-:.
_
,~/
' \
Y
/ ~
~j /l
\ .&~~~~-
/
log-
LJ ,
t (e
)
Q I I I I I I I I I I I q , , , , Q
, , , ~ , , I , , , 00
:~
Pos
ition
of
sour
ce p
roje
ctio
n
O'
I I
I ,
, I
I ,
0.1
0.2
0.
3 0.
4 0.
5 0.
6 0.
7 0.
8 >
1.
0 1.
1 0.
9 1'\
Fig
. 1
0.4
0
Nu
mb
er
of
inte
gra
tio
n p
oin
ts v
s P
osi
tio
n o
f so
urc
e p
roje
ctio
n f
or
Is % q*
dS
"'" w
U1
Num
ber o
f Int
egra
tion
poin
ts
3500
3000
2500
2000
1500
1000
J CR·
au* t
IS
s IJ
ax
s d
=0.0
1
Xs=
X(T
\,rl
) f , ,
r---
""\,
1 S
: SP
Q 6
0 !
' \
0
/ '\
E
< 1
0-6
! '
~ /,
, ,---
-.Ii'
'-c.--
", !
' I'
~ /
Tel
les
\ ,,/
' ~--~'
/ ",
' /
, lo
g-L
l,e
"
" ",
'\",,
, , "
,,0
log-
L 1
, t (
e) sf
"
, "
q , , , ~ \ 1
I "
~ 'I
\ : :
, "
~I , , I , I , 60
Pos
ition
of
sour
ce p
roje
ctio
n 0'
,
I I
, ,
, "
:r
11
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
;;>
Fig
. 10
.41
Num
ber
of
inte
gra
tion
po
ints
vs
Pos
ition
of s
ou
rce
pro
ject
ion
fo
r Is
CPu ~~:
dS
.j:o 1$
Num
ber
of In
tegr
atio
n po
ints
J ~. ~dS
S
IJ aX
s
d =0
.01
---
Xs=
X(l
1,T
l)
25
00
r S
: S
PQ
60
E <
10
-6
20
00
r
1500
1000
~-
----
O---
----
O---
----
O---
----
O-
500
, _--
---0
, , , , , ,
\ 9
, ,
\ , , ,
, ,
\ , , ,
j'
, , T
elle
s \
, , ,
?f
, '" '"
'\
:
11
'X' co
o
log-
Ll
,e
" , , ,
'"'
JS','
,' , , , , , , , , ,
/'p-_
____
_ 'e'
? lo
g-L
1 ,
t (e
)
0.9
q , , , , ~ , , , :
~ ,
" ,
" '
" oe
D , , , , b
e
~
1.0
1.1
Fig
. 10.
42
Num
ber
of
inte
gra
tion
po
ints
vs
Po
sitio
n o
f so
urc
e p
roje
ctio
n f
or
Is % ~~
: dS
"""
w
'-I
438
10.8 Summary of Numerical Results for Nearly Singular Integrals
In this chapter, numerical experiment results were presented to evaluate
the numerical integration methods (PART) proposed for nearly singular
integrals in Chapter 5, in comparison with previous methods.
First, the singularity cancelling radial variable transformation R(p)
corresponding to
pdp = ria dR (5.43)
where a is the order of near singularity of the integral kernel, was shown to
require only one radial integration point for the accurate integration of u* and q*
over constant planar elements. However, for curved elements and kernels
including interpolation functions r/> .. , the method does not work as efficiently as II
Telles' cubic transformation method.
Then the 10g-L2 radial variable transformation
R(p)= log .Jl+d2 (5.64)
which corresponds to the singularity cancelling transformation for a = 2, was
shown to work efficiently for nearly singular integrals
f ~ (a=1-5) (10.35) s
of different orders of near singularity over the 'spherical' quadrilateral element.
SPQ60. However, the method is shown to be inefficient for the integration of the
kernels au*laxs and aq*laxs arising from flux calculation.
Next, the 10g-L1 radial variable transformation
R(p)= log(p+d) (5.85)
was shown to work efficiently for the integration of the flux kernels au*laxs and
aq*laxs, as well as the potential kernels u* and q*, over constant planar
elements.
439
Different radial variable transformations were tested for the model radial
variable integrals
I -fPj / dp a,o - 0 r a
(10.45)
Results showed that the log-LJ radial variable transformation is by far the best
transformation, compared to the identity and log-L2 transformations based on
the Gauss-Legendre rule, and the single and double exponential transformations
based on the truncated trapezium rule. It was also shown that the theoretical
error estimates of Chapter 7 based on complex fluction theory, matches very well
with numerical experiment results.
Finally, different numerical integration methods were tested for the
integration of
f iJu* </> .. -dS vax
s and
over the 'spherical' quadrilateral element SPQ60 of unit size.
f iJq*
</> •• -dS s v ax
8
First, the effect of the source distance d, with the source projection fixed at
xs=x (0.5, 0.5), was tested. As a result, it was shown that for potential and flux
calculations,
(1) For d < 0.04 , the PART method with the log-L J radial variable
transformation and the log-type angular variable transformation
hj {1+sin(8-a j )} t(8)= -log
2 1 - sin ( 8 - a ) (5.127)
J
works most efficiently.
(2) For 0.04;;; d ;;; 0.8 , Telles' cubic transformation method works most
efficien tly.
(3) For d ;:::: 0.8 , the product Gauss-Legendre method works most efficiently.
Next, the effect of the position of the source projection Xs =x(~ , ~), as it is
moved along the diagonal of the element from the centre (~=O) to the corner
440
(~= 1) and outside (~= 1.1) the element S • was tested with the source distance
fixed at d = 0.01.
Results showed that
(1) For 0 ~ ~ ~ 0.93 • the PART method ( with the log-LI radial variable
transformation and the log-type angular variable transformation of equation
(5.127) ) works most efficiently.
(2) For ~ $; 0.93 • Telles' method works most efficiently.
It was also shown that the log-type angular variable transformation teO)
works very effectively and is indispensable when the source projection :is lies near
the element edge i.e. when (0.7 < ~ < 1.1 ) . tv I'V
CHAPTER 11
APPLICATION TO HYPERSINGULAR INTEGRALS
Because of the robustness and efficiency of the PART method with the log-LJ
radial variable transformation, one is tempted to see how close one can let the
source point Xs approach the element surface (d-O) and still obtain the accurate
value of the nearly singular integral.
This is of particular interest for the flux integrals Is au*/axs dS and
Is aq*/axs dS , since in the limit of d-O, or when the source point Xs is on the
element, they become hypersingular, and they exist only in the Cauchy principal
value sense, whereas the potential integrals Is u* dS and Is q* dS are only
weakly singular, i.e. u* - O(lIr) for d=O (cf. section 3.1) and can be easily
calculated just using polar coordinates centered at Xs in the plane tangent to the
element at Xs (PART method with R(p) = p ).
In fact, it turns out that one can calculate these hypersingular integrals by
setting the source distance d sufficiently small and calculating the corresponding
nearly singular integral by the PART (log-LJ) method. To be more specific, let
x'=x +dn s s s
where 0 < d <Ii 1 and ns is the uni t normal to the boundary surface S at Xs E S ,
and calculate the corresponding integral with the source point at xs' •
(11.1)
First, we will calculate the nearly singular integral for the planar square
element PLR (0.5,0.5) ,mentioned before, letting the source point
xs' = (0.25, 0.25, d) approach the element surface (d-O). In order to see how the
nearly singular integral converges to the Cauchy principal value of the
hypersingular integral as d-O, we will compare the former with the latter, which
can be calculated analytically as follows:
442
For a planar rectangular element
s: {X=(X,y, z) I -a~x~a, -b~y~ b, z=O}, with the source point
Xs = (xs , ys , 0) on the element S, the hypersingular integral can be calculated as
the limit of d-O, where xs=(xs,ys, d), to give
where
and
where
f au*/ax dS = (I, " I, " I,")T S aU , % I aU I Y I aU, Z
(11.2) S
I = 1 ± [(SinO 1)[Og{ h; }-(Sino.) [Og{ hj } du",x 41r ;'--1 ;+ cos(O -a.) ; cos(O.-a.)
;+1 ; ; ;
1 I - -du·,z 2
( b-Y ) o = tan -1 __ 8 ,
1 a-x 8
-(rosa.) ;
tan( 0; + 1 - a; + ~) _ (sin a .) log 1 2 4 I]
; tane j : a; + ~)
o
o
4
L {sin ( 0 j + 1 - a j ) - sin ( 0 j - a j ) } / h j ;=1
( b-Y ) o =1r_tan- 1 __ 8 ,
2 x +a 8
( y +b) o = 1r + tan- 1 _8_ 3 x +a
3 (a-x) -1 8 o =-1r+tan --4 2 Y +b
8 s
(11.3)
(11.4)
(11.5)
(11.6)
(11. 7)
and
TC a = -
1 2'
443
3 a = -TC
3 2 ' (11.8)
h = a-x 4 8 (11.9)
For the planar square element PLR (0.5, 0.5), a=b=1I2 and xs=Ys=1I4.
The result of numerical experiments on the planar square element
PLR (0.5, 0.5) is given in Table 11.1, where the source distance d and the number
of integration points for the angular (t) and radial (R) variables using the PART
(log-L1 transformation) method, required to achieve an accuracy of (relative
error<10- 6 compared to the analytical value of the hypersingularintegral (d=O),
are shown. For the x, y, component integrals (*) of Is aq*laxs dS , partial
quadruple precision was necessary, and the absolute error was taken, since the
true values are both zero.
Table 11.1 Convergence of nearly singular integrals to hypersingular integrals ( planar square element: PLR( 0.5, 0.5) )
Integral Component Source Number of Value of integral distance integration points
d NtXNR total (analytical, d=O)
x,y 10-4 8 X 25 800 -1.1122013 X 10-1
Is au*laxs dS 10-7 10 X 32 1280 - 5.0000000 X 10-1 Z
* x ,y 10-8 20 X 80 6400 0 Is aq*laxs dS
10-4 20 X 50 4000 1.2712670 z
Similarly, one can calculate the hypersingular integrals Is au*laxs dS and
Is aq*laxs dS over curved elements, as the limit as d-O of nearly singular
integrals. As an example, we take the 'spherical' quadrilateral element SPQ60
mentioned before with the source point on the element at xs=x (~l' ~2)' where
444
~1 = ~2=0.5. Numerical results are shown in Table 11.2. The value of the source
distance d and the required number of integration points to achieve a relative
error < 10-6 compared to the converged result (d-O) is given. The actual value of
the converged result for the hypersingular integrals (Cauchy principal value) are
also given. In this case, partial quadruple precision was necessary for calculating
the integral f s aq*/axs dS. This is because aq*/axs contains terms like p cos 8/ r5
(cf. equation 3.130), which take very large absolute values near the source point
Xs and cancel each other when integrated in the angular direction, so that
rounding error b~comes significant.
Table 11.2 Convergence of nearly singular integrals to hypersingular integrals ( 'spherical' quadrilateral element: SPQ60 )
Integral Component Source Number of Value of integral distance integration points
d NtXNR total (converged)
x 10-6 16 X 40 2560 4.0224494 X 10-1
f s au*/axs dS y 10-8 16 X 45 2880 -9.9926710X10-2
z 10-8 16 X 45 2880 -6.1706620X10- 1
x 10-7 25 X 80 8000 -1.0959943
f s aq*/axs dS y 10-8 25 X 90 9000 2.221675X10- 1
Z 10-7 25 X 80 8000 -2.3174413 X 10-1
The above numerical experiments demonstrate the robustness of the PART
(log-Ll transformation) method with regard to the source distance d, by applying
it successfully to the calculation of hypersingular integrals as the limit (d-O) of
nearly singular integrals. This may prove useful especially when calculating
hypersingular integrals with the source point at the corner or on the edge of the
element, in which case previous techniques 28 would not work.
CHAPTER 12
CONCLUSIONS
This thesis was primarily concerned with the numerical integration method
for the accurate and efficient calculation of nearly singular integrals over general
curved surfaces, whi<:h plays an important role in three dimensional boundary
element analysis.
Nearly singular integrals frequently arise in engineering problems, when
analysing thin structures or thin gaps, when using boundary elements with high
aspect ratio and when calculating the potential or flux very near the boundary.
The thesis is divided into two parts. In the first part, the theory and
formulation of three dimensional boundary element method (BEM) for potential
problems is presented, and the nature of nearly singular, weakly singular and
hypersingular integral kernels is analysed by focusing on the radial component of
the kernels. Previous work on numerical integration methods for three
dimensional BEM is briefly reviewed before proceeding to describe the new
technique developed to deal with nearly singular integrals. This technique has
been described in detail in Chapter 5 and is referred to as the "Projection and
Angular & Radial Transformation (PART) method". The method consists of the
following stages:
(1) Find the closest point Xs on the curved element S from the source point Xs.
(2) Approximately project the element S on to a polygon S in the plane tangent
toSatxs.
(3) Introduce polar coordinates (p, 8) centred at Xs in S .
(4) Apply a radial variable transformation R(p) in order to weaken the near
singularity inherent in the integral kernel.
(5) Apply the log-type angular variable transformation
446
h. {l+Sin{U-a.)} t{U) = ..1.. log J
2 l-sin{U-a.) J
in order to weaken the angular near singularity, which arises when Xs is very
near the edge of S or when S has a high aspect ratio.
(6) Use the Gauss-Legendre formula in the transformed radial and angular
variables Rand t .
For weakly singular integrals, which appear in the calculation of the
diagonals of Hand G matrices for three dimensional potential problems, the
identity radial variable transformation R(p) = p gives efficient results.
For nearly singular integrals, the singularity cancelling radial variable
transformation gives the exact result with only one integration point in the radial
variable for constant planar elements. For curved elements, the log-L2 radial
variable transformation R(p) = logY p2 + d2 , where d is the source distance, gives
more efficient results for near singularities of different orders. For general
kernels, including kernels arising in flux calculation or kernels with interpolation
functions, the log-Ll transformation R(p) = log(p + d) is the most robust and
efficient transformation.
The possibility of using the single or double exponential transformation with
the truncated trapezium rule for the radial and angular variable integration is
also discussed. Finally, the detailed implementation of the PART method is
discussed. The advantage of the method lies in the fact that the radial and
angular (near) singularities are separated so that they can be analysed and
treated independently from each other.
An elementary error analysis was presented in Chapter 6, explaining why
the log-L2 radial variable transformation ( corresponding to order f3= 2) gives best
results among other singularity cancelling transformations corresponding to
f3* 2. In Chapter 7, complex function theory was used to derive more rigorous
theoretical error estimates for different radial variable transformations. The
447
analysis showed that the PART method using the log-L2 radial variable
transformation converges with error of the order n -2n, where n is the number of
radial integration points, for integrations arising from potential calculations,
whereas the method converges only at a rate of n -3 for integrations arising from
flux calculations. On the other hand, the log-LJ transformation gives convergence
with error of the order cr- 2n for both types of integrals, where cr: 1.3-1.6 for D:
10-3-10-1, where D is the source distance relative to the element size. This
method of using complex function theory for the theoretical error estimate also
gives a clear perspective for the construction of the optimum radial variable
transformation.
The second part of the thesis presented numerical experiment results for the
new numerical integration technique. In Chapter 9, the PART method with the
identity radial variable transformation and the log-type angular variable
transformation was applied to weakly singular integrals and gave accurate and
efficient results for planar and curved elements with interpolation functions. In
particular, the effect of the angular variable transformation t(B) in decreasing the
number of angular integration points when the source point Xs is near the element
edge, as in discontinuous elements and elements with high aspect ratio, was
demonstrated.
Chapter 10 presented numerical results for nearly singular integrals, which
was the primary concern of this thesis. First, results for constant planar
elements were presented, together with the closed form integral. The singularity
cancelling radial variable transformation corresponding to p dp = r a dR , where
r=V p2+d 2 is the source distance and a is the order of near singularity, proved
to be more efficient compared to the Telles and Gauss methods when the relative
source distance D~ 0.1. The log-type angular variable transformation t(B) was
shown to be effective when the source projection :Ks is near the element edge.
The same singularity cancelling radial variable transformation
corresponding to p dp = r' a dR , where r' = V p2 + d 2 is the approximate source
448
distance, was applied to curved elements and kernels with interpolation
functions, and was shown to be inefficient compared to Telles' cubic
transformation method. To overcome this difficulty, the log-L2 radial variable
transformation R(p) = logY p2 + d2 , which corresponds to the singularity
cancelling transformation for a = 2, was introduced and shown to be most efficient
and robust among other transformations corresponding to a* 2, for integrals of
the type Is 1/r a dB with different orders of near singularity a=1-5 over a
curved element B.
However, numerical experiments on a planar element showed that the
log-L2 radial variable transformation is inefficient for integrals arising from flux
calculations. In fact, the convergence was of order n -3, where n is the number of
radial integration points, as predicted in Chapter 7. To solve this difficulty, the
log-LJ radial variable transformation R(p)=log(p+d ) was introduced and was
shown to work efficiently for flux integrals as well as potential integrals over a
constant planar element.
Next, different radial variable transformations were compared for the model
radial component integrals l a•8 = IoPj II r" dp for potential and flux kernels. For
transformations based on the Gauss-Legendre rule, the identity transformation
R(p) = p , the log-L2 transformation and the log-LJ transformation were compared.
Numerical results showed that the log-LJ transformation is the most robust and
efficient transformation with error convergence of order q-2n where q : 1.56-1.88
for a=5 -1 and D=O.Ol . This corresponds well with the theoretical estimate of
Chapter 7 which predicts q: 1.40-1.64. The identity and log-L2 transformations
also gave convergence results which were consistent with the theoretical
estimates. The single and double exponential transformations based on the
truncated trapezium rule were also compared. As a result, they required 2-3
times as many integration points compared to the log-LJ transformation to obtain
the same level of accuracy.
449
Finally, the product type Gauss, Telles' methods and the PART method with
the identity, log-L2 and log-L1 radial variable transformations and identity, log
type angular variable transformations were compared for the nearly singular
integrals Is ~ij u* dS, Is ~ij q* dS arising from potential calculations and
I s ~ ij au*laxs dS, I s ~ ij aq*laxs dS from flux calculations over a curved surface
element S. Results showed that the PART method with the log-Ll radial variable
transformation and the log-type angular variable transformation is the most
robust method with regard to the type of integral kernels, source distance d and
the position ofthe source projection is in S.
To be more precise, the effect of the relative source distance D with the
source projection fixed at i s=x(0.5, 0.5) was investigated for a 'spherical'
quadrilateral element S. As a result, we obtained the following guide for the
choice of numerical integration methods:
For D < 0.04 , use the proposed PART method with the log-L1 radial
variable transformation and the log-type angular variable transformation.
For 0.04 ~ D ~ 0.8 , use Telles' cubic transformation method.
For D > 0.8 , use the product Gauss-Legendre method.
The effect of the position of the source projection is=x (~1' ~2) was also
examined by fixing the source distance at d=D=0.01 and moving is=x (~, ~)
along the diagonal of the element from ~ = 0 to ~ = 1.1 . Results showed that the
above mentioned PART method is most efficient for 0 ~ ~ ~ 0.93, whilst Telles'
method is most efficient for ~ > 0.93, i.e. when the source projection is is very
near the element edge.
The criterion for choosing between the PART and Telles' method much
depends on the relative source distance D. For D ~ 0.01, I~ll, 1~21 - 0.9 is the
rough border for the choice. For D < 0.01 , it is advisable to use the PART
method, irrespective of the position of the source projection is, as long as it is with
in the element i.e. I~ll, 1~21 ~ 1. This is because the log-Ll radial variable
transformation and the log-type angular variable transformation maintains the
450
PART method relatively robust against the change of source distance D and
position of the source projection xs , respectively, whereas the number of necessary
integration points increases rapidly for Telles' method as D decreased beyond
0.01.
The effect of the log-type angular variable transformation in the PART
method was also verified by comparing with results obtained by the identity
angular variable transformation ( t(O) = 0). The effect becomes pronounced when
the source projection approaches the element edge ( ~ ~ 0.7).
Chapter 11 showed that the PART method with the log-L1 radial variable
transformation can be applied to the calculation of hypersingular integrals
( Cauchy principal value) arising from flux calculations on the boundary. This
was done by taking the source distance d sufficiently small to give an
approximation of the limit as d-O. Numerical results matched with analytical
integration results for a planar element. Results on a curved element with
interpolation functions were also presented.
In summary, the thesis proposed a robust and efficient numerical
integration method for nearly singular integrals with arbitrary small source
distance d , which had been an open problem in three dimensional boundary
element analysis. The thesis treated the three dimensional potential problem as
an example, but the method can also be applied to other problems such as
elastostatics. The method can also be applied to two dimensional problems by
using the log -L1 transformation. These topics are left for future research.
REFERENCES
1. BREBBIA, C.A., TELLES, J.C.F. and WROBEL, L.C. (1984) Boundary
Element Techniques: Theory and Applications in Engineering, Springer
Verlag, Berlin.
2. KUWAHARA, T. and TAKEDA, T. (1985) Calculation of Potential and
Potential Gradient for 3-Dimensional Field by B.E.M. using Analytical
Integration, Proc. on Rotary Machines and Stational Apparatus, JECM., SA-
85-47,211-216. (lnJapanese)
3. STROUD, A.H. and SECREST, D. (1966) Gaussian Quadrature Formulas,
Prentice-Hall, New York.
4. CRISTESCU, M. and LOUBIGNAC, G. (1978) Gaussian Quadrature
Formulas for Functions with Singularities in lIr over Triangles and
Quadrangles, in Recent Advances in Boundary Element Methods, Ed. C.
Brebbia, Pentech Press, London.
5. PINA, H.L.G., FERNANDES, J.L.M. and BREBBIA, C.A. (1981) Some
Numerical Integration Formulae over Triangles and Squares with a lIR
Singularity, Appl. Math. Modelling, 5, 209-211.
6. ALIABADI, M.H., HALL, W.S. and PHEMISTER, T.G. (1985) Taylor
Expansion for Singular Kernels in the Boundary Element Method, Int. J.
Num. Meth. Engng., 21, 2221-2236.
7. LACHAT, J.C. and WATSON, J.O. (1976) Effective Numerical Treatment
of Boundary Integral Equations: A Formulation for Three Dimensional
Elastostatics, Int. J. Num. Meth. Engng., 10,273-289.
8. LEAN, M.H. and WEXLER, A. (1985) Accurate Numerical Integration of
Singular Boundary Element Kernels over Boundaries with Curvature, Int.
J. N urn. Meth. Engng., 21, 211-228.
452
9. RIZZO, F.J. and SHIPPY, D.J. (1977) An Advanced Boundary Integral
Equation Method for Three Dimensional Thermoelasticity, Int. J. Num.
Meth. Engng., 11, 1753-1768.
10. KUTT, H.R. (1975) Quadrature Formulae for Finite Part Integrals, Report
WISK 178, The National Institute for Mathematical Sciences, Pretoria.
11. SCHWAB, C.H. and WENDLAND, W.L. (1985) 3-D BEM Numerical
Integration, in C. Brebbia and G. Maier (eds), Boundary Elements VII, Proc.
7th Int. Conf., Como, Italy, Springer Verlag, Berlin & CML Publications,
Southampton, 2, 13-85-101.
12. JUN, L., BEER, G. and MEEK, J.L. (1985) Efficient Evaluation ofIntegrals
of Order lIr, lIr2, lira using Gauss Quadrature, Engineering Analysis, 2,
118-123.
13. TAKAHASHI, M. and MORI, M. (1973) Quadrature Formulas obtained by
Variable Transformation, Numer. Math., 21, 206-219.
14. HIGASHIMACHI, T., OKAMOTO, N., EZAWA, Y., AIZAWA, T. and ITO,
A. (1983) Interactive Structural Analysis System using the Advanced
Boundary Element Method, in Boundary Elements, Proc. 5th Int. Conf.,
Hiroshima, Japan, eds. C. Brebbia, T. Futagami and M. Tanaka, Springer
Verlag, Berlin & CML Publications, Southampton.
15. TELLES, J.C.F. (1987) A Self-Adaptive Co-ordinate Transformation for
Efficient Numerical Evaluations of General Boundary Element Integrals,
Int. J. Num. Meth. Engng., 24, 959-973.
16. KOIZUMI, M. and UTAMURA, M. (1987) A Polar Coordinate Integration
Scheme with a Hierarchical Correction Procedure to Improve Accuracy,
Proc. IUTAM Symp. on Advanced Boundary Element Methods, San Antonio,
Texas, ed. T.A. Cruse, 215-222.
17. HAYAMI, K. and BREBBIA, C.A. (1987) A New Coordinate
Transformation Method for Singular and Nearly Singular Integrals over
General Curved Boundary Elements, in C.A. Brebbia, W.L. Wendland, G.
453
Kuhn (eds), Boundary Elements IX, Proc. 9th Int. Conf., Stuttgart,
Germany, 1,379-399.
18. HA YAMI, K. and BREBBIA, C.A. (1988) Quadrature Methods for Singular
and Nearly Singular Integrals in 3-D Boundary Element Method, (Invited
Paper) in C.A. Brebbia (ed.), Boundary Elements X, Proc. 10th Int. Conf.,
Southampton, U.K., 1,237-264.
19. PEREIRA, M.F.S., SOARES, C.A.M. and FARlA, L.M.O. (1981)
A Comparative Study of Several Boundary Element Methods, Proc. 3rd Int.
Seminar on Boundary Element Methods, Irvine, California, Springer
Verlag.
20. ALIABADI, M.A. and HALL, W.S. (1987) Weighted Gaussian Methods for
Three Dimensional Boundary Element Kernel Integration, Comm. Appl.
Num. Meth., 3, 89-96.
21. JENG, G. and WEXLER, A. (1977) Isoparametric, Finite Element,
Variational Solution of Integral Equations for Three-Dimensional Fields,
Int. J. Num. Meth. Engng., 11, 1455-1471.
22. AMINI, S. and WILTON, D.T. (1986) An Investigation of Boundary
Element Methods for the Exterior Acoustic Problem, Computer Methods in
Applied Mechanics and Engineering, 54, 49-65.
23. HALL, W.S. (1988) Integration Methods for Singular Boundary Element
Integrals, (Invited Paper), in C.A. Brebbia (ed.), Boundary Elements X, Proc.
10th Int. Conf., Southampton, U.K., 1, 219-236.
24. KRAUS, H. (1967) Thin Elastic Shells, John Wiley & Sons.
25. UTAMURA, M. and KOIZUMI, M. (1985) Development of Computer
Program for Analyzing Three-Dimensional Pressure Field in Pressure
Suppression System, J. Nucl. Sci., Technol., 22, 733-741.
26. JUN, L., BEER, G. and MEEK, J.L. (1985) The Application of Double
Exponential Formulas in the Boundary Element Method, in C. Brebbia and
454
G. Maier (eds), Boundary Elements vrr, Proc. 7th Int. Conf., Como, Italy,
Springer-Verlag/CMP, Berlin, 2,13-3-17.
27. KUNO, Y., YAGI, A., MORISHIMA, T. and UCHIKAWA, Y., (1988)
Fundamental Study for High Accuracy Calculation of 3-D Electromagnetic
Field, IEEE Trans. Mag., MAG-24, 1,295-298.
28. GRAY, L.J. and GILES, G.E., (1988) Application of the Thin Cavity Method
in Electroplating, in C.A. Brebbia (ed.), Boundary Element X, Proc. 10th Int.
Conf., Southampton, U.K., 441-452.
29. KUWAHARA, T. and TAKEDA, T. (1987) An Effective Analysis for Three
Dimensional Boundary Element Method Using Analytically Integrated
Higher-Order Elements, The Transactions of the Institute of Electrical
Engineers of Japan, 107-A, 6, 275-282 (in Japanese).
30. AMINI, S. (1986) Efficient Quadrature Rules with A Priori Error Estimates
for Integrals with End Point Singularities, BIT 26,200-208.
31. DAVIS, P.J. and RABINOWITZ, P. (1984) Methods of Numerical
Integration, Academic Press; New York.
32. TAKAHASHI, H. and MORI, M. (1974) Double Exponential Formulas for
Numerical Integration, Publ. RIMS, Kyoto Univ., 9, 721-741.
33. ATKINSON, K.E. (1978) An Introduction to Numerical Analysis, John
Wiley & Sons.
34 HAY AMI, K., and BREBBIA, C.A. Projection and Angular & Radial
Transformation Method for Singular and Nearly Singular Integrals in 3D
Boundary Element Method (to be published).
35. HADAMARD, J. (1923) Lectures on Cauchy's Problem in Linear Partial
Differen tial Equations, Yale U niversi ty Press.
36. IRI, M., MORIGUTI, S., and TAKASAWA, Y. (1970) On a Certain
Quadrature Formula, Kokyuroku of Res. Inst. for Math. Sci., Kyoto Univ.
37. HAY AMI, K. (1990) High Precision Numerical Integration Methods for 3-D
Boundary Element Analysis, 1E3 Trans. Mag., MAG-26, 2, 603-606.
455
38. GOLUB, G.H., and WELSCH, J. H. (1969) Calculation of Gauss
Quadrature Rules, Mathematics of Computation, 23, 221-230.
39. IMSL, (1987) User's Manual Math/Library, Fortran Subroutines for
Mathematical Applications.
40. HAYAMI, K. (1989) Improvement and Error Analysis of the High Precision
Numerical Integration Method: PART, Proc. 6th Japan National
Symposium on Boundary Element Methods, 37-42 (in Japanese).
41. HAYAMI, K. (1990) A Robust Numerical Integration Method for Three
Dimensional Boundary Element Analysis, in M. Tanaka, C.A.Brebbia and
T. Honma (eds), Boundary Elements XII, Proc. 12th Int. Conf., Sapporo,
Japan, 1,33-51.
42. HAY AMI, K. (1991) High Precision Numerical Integration Method for the
Boundary Element Method and its Error Analysis, Kokyuroku of Res. Inst.
for Math. Sci., Kyoto Univ., 744, 188-206 (in Japanese).
43. MORI, M. (1975) Numerical Analysis and Complex Function Theory,
Chikuma-shobo, (in Japanese).
44. TAKAHASm, H. and MORI, M. (1970) Error Estimation in the Numerical
Integration of Analytic Functions, Report of the Computer Centre, Univ. of
Tokyo, 3, 41-108.
45. BARRET, W., (1960) Convergence Properties of Gaussian Quadrature
Formulae, Comput. J., 3, 272-277.
46. MCNAMEE, J. (1964) Error-Bounds for the Evaluation ofIntegrals by the
Euler-Maclaurin Formula and by Gauss-Type Formulae, Math, Comp., 18,
368-38l.
47. DONALDSON, J.D. and ELLIOT, D., (1972) A Unified Approach to
Quadrature Rules with Asymptotic Estimates of their Remainders, SIAM J.
Num. Anal., 9 No.4, 573-602.
48. DAVIS, P.J. and RABINOWITZ P. (1984) Methods of Numerical
Integration, Second Edition, Academic Press.
456
49. MORIGUTI, S., UDAGAWA, K. and HITOTSUMATSU, S. (1960)
Mathematical Formulae, Iwanami-shoten, (in Japanese).
50. HADAMARD, J. (1923) Lectures on Cauchy's Problem in Linear Partial
Differential Equations, Yale University Press.
51. RUDOLPll, T. J., KRISHNASAMY, G., SCHMERR, L. W. and RIZZO, F. J.
(1988) On the Use' of Strongly Singular Integral Equations for Crack
Problems, (Invited Paper), in C. A. Brebbia (ed.), Boundary Elements X,
Proc. 10th Int. Conf., Southampton, U. K., 3, 249-263.
52. HAY AMI, K. (1991/92) A Robust Numerical Integration Method for 3-D
Boundary Element Analysis and its Error Analysis using Complex Function
Theory, NATO ASI Series, Numerical Integration: Recent Developments,
Software and Applications, (to appear).
Lecture Notes in Engineering
Edited by C.A. Brebbia and S.A. Orszag
Vol. 40: R. Borghi, S. N. B. Murhty (Eds.) Turbulent Reactive Flows VIII, 950 pages. 1989
Vol. 41: W. J. Lick Difference Equations from Differential Equations X, 282 pages. 1989
Vol.42: H. A. Eschenauer, G. Thierauf (Eds.) Discretization Methods and Structural Optimization -Procedures and Applications Proceedings of a GAMM-Seminar October 5-7,1988, Siegen, FRG XV, 360 pages. 1989
Vol. 43: C. C. Chao, S. A. Orszag, W. Shyy (Eds.) Recent Advances in Computational Fluid Dynamics Proceedings of the US/ROC (Taiwan) Joint Workshop in Recent Advances in Computational Fluid Dynamics V, 529 pages. 1989
Vol. 44: R. S. Edgar Field Analysis and Potential Theory XII, 696 pages. 1989
Vol. 45: M. Gad-el-Hak (Ed.) Advances in Fluid Mechanics Measurements VII, 606 pages. 1989
Vol.46: M. Gad-el-Hak (Ed.) Frontiers in Experimental Fluid Mechanics VI, 532 pages. 1989
Vol. 47: H. W. Bergmann (Ed.) Optimization: Methods and Applications, Possibilities and Limitations Proceedings of an International Seminar Organized by Deutsche Forschungsanstalt fUr Luft- und Raumfahrt (DLR), Bonn, June 1989 IV, 155 pages. 1989
Vol. 48: P. Thoft-Christensen (Ed.) Reliability and Optimization of Structural Systems '88 Proceedings of the 2nd IFIP WG 7.5 Conference London, UK, September 26-28, 1988 VII, 434 pages. 1989
Vol. 49: J. P. Boyd Chebyshev & Fourier Spectral Methods XVI, 798 pages. 1989
Vol. 50: L. Chibani Optimum Design of Structures VIII, 154 pages. 1989
Vol. 51: G. Karami A Boundary Element Method for Two-Dimensional Contact Problems VII, 243 pages. 1989
Vol. 52: Y. S. Jiang Slope Analysis Using Boundary Elements IV, 176 pages. 1989
Vol. 53: A. S. Jovanovic, K. F. Kussmaul, A. C. Lucia, P. P. Bonissone (Eds.) Expert Systems in Structural Safety Assessment X, 493 pages. 1989
Vol. 54: T. J. Mueller (Ed.) Low Reynolds Number Aerodynamics V, 446 pages. 1989
Vol. 55: K. Kitagawa Boundary Element Analysis of Viscous Flow VII, 136 pages. 1990
Vol. 56: A. A. Aldama Filtering Techniques for Turbulent Flow Simulation VIII, 397 pages. 1990
Vol. 57: M. G. Donley, P. D. Spanos Dynamic Analysis of Non-Linear Structures by the Method of Statistical Quadratization VII, 186 pages. 1990
Vol. 58: S. Naomis, P. C. M. Lau Computational Tensor Analysis of Shell Structures XII,304 pages. 1990
![ROUGH BILINEAR SINGULAR INTEGRALSfaculty.missouri.edu/~grafakosl/preprints/Rough Bilinear Singular Integrals 29.pdfSeeger [28] in all dimensions and was later extended by Tao [30]](https://img.pdfslide.net/doc/110x75/5f4869d25a9b145ee16f767c/rough-bilinear-singular-grafakoslpreprintsrough-bilinear-singular-integrals-29pdf.jpg)
