Computational Mechanics 25 (2000) 394±403 Ó Springer-Verlag 2000
Numerical integration of singularities in meshless implementation
of local boundary integral equations
V. Sladek, J. Sladek, S. N. Atluri, R. Van Keer
394
Abstract The necessity of a special treatment of the numerical integration of the boundary integrals with singular
kernels is revealed for meshless implementation of the
local boundary integral equations in linear elasticity.
Combining the direct limit approach for Cauchy principal
value integrals with an optimal transformation of the integration variable, the singular integrands are recasted into
smooth functions, which can be integrated by standard
quadratures of the numerical integration with suf®cient
accuracy. The proposed technique exhibits numerical
stability in contrast to the direct integration by standard
Gauss quadrature.
1
Introduction
A lot of attention has been paid during the past decade to
meshless implementations of both the formulations based
originally on variational principles (weak form) (Belytschko et al., 1996) and/or boundary integral equations
(Zhu et al., 1998; Mukherjee and Mukherjee, 1997).
Recall that, by using an approach based on the BIE, the
dimension of the integration region is reduced by one as
compared with the dimension of the domain in which a
boundary value problem is solved. Beside this most evident attractive property of the BIE formulations one could
bring other advantages, such as good conditioning and
high accuracy, resulting from the use of singular kernels.
Sometimes the appearance of singular integrals has been
considered as a handicap of the BIE formulations because
of the relative complexity of accurate numerical integration. The problem of singularities has been resolved successfully in boundary element implementations of the BIE
formulations (see e.g., Sladek and Sladek, 1998) when the
boundary densities are approximated within ®nite size
Received 12 August 1999
V. Sladek (&), J. Sladek
Institute of Construction and Architecture,
Slovak Academy of Sciences, 842 20 Bratislava,
Slovak Republic
elements polynomially. Having known the boundary
densities in a closed form, one can regularize the integrands involving singular kernels before utilizing quadratures for numerical integration (Tanaka et al., 1994).
Nevertheless, the question of singularities is to be reconsidered in meshless implementations of the BIE.
Now, instead of the de®nition of ®nite size elements by
grouping nodal points on the boundary, the nodal points
are spread throughout the whole domain including its
boundary. When the coupling among the nodal points is
satis®ed via the moving least-squares (MLS) approximation of physical ®elds (such as potential, displacements),
the boundary densities are not known in a closed form any
more, because the shape functions are evaluated only
digitally at any required point. Thus, the peak-like factors
in singular kernels cannot be smoothed by cancellation of
divergent terms with vanishing ones in boundary densities
before the numerical integration. The proposed method
consists in the use of direct limit approach and utilization
of an optimal transformation of the integration variable.
The smoothed integrands can be integrated with suf®cient
accuracy even by using standard quadratures of numerical
integration.
Section 2 summarizes the important equations of the
local BIE formulation for solution of boundary value
problems of linear elasticity. Section 3 deals with the
derivation of nonsingular integrands in the meshless implementation of the LBIE endowed with the MLS approximation of displacements. Finally, in Sect. 4, the proposed
technique of the numerical integration is tested in a numerical example with comparison of numerical results
with those obtained by using standard Gauss quadrature
without any elaboration of the integrand.
2
The local boundary integral equations for linear elasticity
Let us consider a linear elastostatical problem on the domain X bounded by the boundary C. Then, the displacements are governed by the Navier equations (Balas et al.,
1989)
cijkl uk;jl ‡ bi ˆ 0
1†
S. N. Atluri
Center for Aerospace Research & Education, 48-121,
Engineering IV, University of California at Los Angeles,
Los Angeles, CA 90024, USA
in which bi is the body force and cijkl is the tensor of
material parameters, which are reduced to two constants
in the case of isotropic and homogeneous elastic continua
with
R. Van Keer
Department of Mathematical Analysis,
University of Gent, Galglaan 2, B ± 9000 Gent, Belgium
cijkl ˆ l
2v
dij dkl ‡ dik djl ‡ dil djk
1 ÿ 2v
;
2†
X0s is a circle centered at y. In general, the boundary
oXs ˆ Ls [ Cs , in which Cs ˆ X0s \ C and Ls is a circular
part of oXs , i.e. Ls ˆ oX0s \ X.
2-d problems under plane stress conditions
Then, it is easy to ®nd the companion solution de®ned
otherwise
by
where l is the shear modulus and v is expressed in
terms of Poisson's ratio v as
vˆ
v
1‡v ;
v;
The traction vector is expressed in terms of the displacement gradients as
ti …x† ˆ rij …x†nj …x† ˆ cijkl uk;l …x†nj …x†
3†
cijkl u~km;lj ˆ 0 on X0s ;
u~ij …x; y† ˆ uij …x; y† on oX0s :
8†
Note that the companion solution is non-singular and the
modi®ed test functions are given by
where nj is the unit outward normal to the boundary.
Only half of the vector components fui ; ti gdiˆ1 is prescribed by boundary conditions at each point on the
boundary. Having known both the displacements and
tractions all over the boundary, the displacements at any
point y 2 X [ C can be expressed integrally as
u~ij …x ÿ y† ˆ
cijkl ukm;jl ‡ dim d…x ÿ y† ˆ 0 :
Concluding, u~ij ˆ 0 on Ls and the local BIE (LBIE)
collocated at y 2 Cs can be rewritten as
1
r
5 ÿ 4v
4v ÿ 3† ln ÿ
8pl…1 ÿ v†
r0 2…3 ÿ 4v†
r2
r2
1 ÿ 2 dij ‡ 1 ÿ 2 r;i r;j ;
r0
r0
Z
1
1 or
r k nk
ui …y† ˆ ‰uij …x; y†tj …x† ÿ tij …x; y†uj …x†ŠdC…x†
~tij …x; y† ˆ ÿ
1 ÿ 2v†
ÿ
dij
4p…1 ÿ v
r on …3 ÿ 4v†r02
C
Z
2 or
1 ÿ 2v
r;i r;j ÿ
r;i nj ÿ r;j ni †
‡
4†
‡ uij …x; y†bj …x†dX…x†
r on
r
3ri nj ÿ rj ni
X
9†
‡
3 ÿ 4v†r02
where uij are fundamental displacements in in®nite space
(Kelvin's solution), i.e. obeying
in which r is the radius of X0 .
5†
s
Z
The fundamental tractions are given by
tim
ˆ cijkl ukm;l nj :
0
6†
ui …y† ˆ ÿ lim
D!0
uj …x†~tij …x; z†dC…x†
Cs
Z
Z
If some point y lies on the boundary, Eq. (4) can be used
~
as an integral equation for the computation of unpreuij …x ÿ y†dC…x†
ÿ uj …x†tij …x; y†dC…x† ‡ tj …x†~
scribed boundary quantities. Due to the strong singularity
Ls
Cs
of fundamental tractions, the boundary integral of this
Z
kernel is to be considered in the Cauchy Principal Value
uij …x ÿ y†dX…x† :
10†
‡ bj …x†~
(CPV) sense when the source point lies on the boundary
Xs
over which the integration is carried out. Without introducing the CPV integral, this BIE can be rewritten as
If y is an interior point of X, then Xs ˆ X0s , oXs ˆ Ls
Z
(Cs ˆ f[g† and the LBIE becomes
ui …y† ˆ ÿ lim
D!0
Z
‡
C
Z
uj …x†tij …x; z†dC…x†
C
tj …x†uij …x; y†dC…x† ‡
Z
uj …x†~tij …x; y†dC…x†
ui …y† ˆ ÿ
Ls
bj …x†uij …x; y†dX…x†
Z
‡
X
7†
where z ˆ y ÿ eD, with D being the distance jy ÿ zj, and e
is the unique vector.
Of course, the BIE can be applied to any subdomain
Xs X, but the number of unknowns on the arti®cial
boundary is increased twice, because neither displacements not tractions are prescribed at such boundary
points. This handicap can be removed by special selection
of subdomains, if possible, as it takes place in the local BIE
approach. In order to get rid of the unknown tractions on
the boundary oXs , Atluri et al. (1999) have considered the
so-called ``companion solution'' u~ij associated with the
fundamental solution uij and the modi®ed test functions
being given by u~ij ˆ uij ÿ u~ij and ~tij ˆ tij ÿ ~tij . In two
dimensions, Xs has been taken as Xs ˆ X \ X0s , where
Xs
bj …x†~
uij …x ÿ y†dX…x†
11†
Without going into details, we give the moving leastsquares (MLS) approximation of displacements in an interpolation form (Atluri et al., 1999)
uhi …x† ˆ
n
X
aˆ1
a†
/…a† …x†^
ui ;
x 2 Xx
12†
in which Xx is the domain of de®nition of the MLS ap…a†
proximation, u^i are the ®ctitious nodal values (not the
a†
nodal values of displacements, u^i 6ˆ uhi …x…a† †,
a†
a†
a†
u^i 6ˆ ui …x ††, and / …x† is the shape function of the
MLS approximation corresponding to nodal point y…a† .
Note that the shape functions are not known in a closed
form like interpolation polynomials, but they are evaluated
at any integration point digitally.
395
396
3
Numerical integration of singular integrals
According to the previous paragraph, we have to deal with
the strong singularity …rÿ1 † and weak singularity (ln r)
involved in the kernels ~tij and u~ij , respectively. Furthermore, the singular integrals occur only if the source point
y is located on the boundary Cs .
Note that the geometry of the circular part of the
boundary, Ls , can be parametrized exactly. On the other
hand, the geometry of the boundary C or its part Cs is to
be approximated before a parametrization, in general. The
simplest way seems to be a polynomial interpolation of the
n-th order on a part of the boundary C de®ned by n ‡ 1
nodes on C. One can use either linear or non-linear interpolation. The latter is recommended for a more faithful
approximation of curved boundaries if the nodes are not
scattered extremely dense on C. Without any restriction
on the nature of the elimination of singular integrals we
con®ne ourselves to quadratic interpolation in what
follows.
Splitting Cs associated with the node y…a† into the left
a†ÿ
and right subsegments, we have C…a†
[ C…a†‡
,
s ˆ Cs
s
a†
,
C
are
approximated
portions
of
the
where C…a†
s
s
boundary. If y…a† is a corner point, C…a†ÿ
and C…a†‡
belong
s
s
to two different approximations de®ned
on
the
segments
determined
thetriples of nodes y…aÿ2† ; y…aÿ1† ; y…a† and
a† …a‡1† by…a‡2†
y ;y
, respectively (see Fig. 1). Otherwise,
;y
the normal vector is continuous at y…a† (C is smooth) and
C…a†
belong to the same approximation
segment
s
determined by the triple of nodes y…aÿ1† ; y…a† ; y…a‡1† . Thus,
n
aÿ2† 1
C…a†ÿ
ˆ
8 x 2 <2 ; xÿ
N …f ‡ 1†
s
i …f† ˆ yi
o
aÿ1† 2
a†
N …f ‡ 1† ‡ yi N 3 …f ‡ 1†; f 2 ‰fÿ ; 0Š
‡ yi
n
C…a†‡
ˆ
8 x 2 <2 ;
s
a†
1
x‡
i …f† ˆ yi N …f ÿ 1†:
a‡1†
a‡2†
N 2 …f ÿ 1† ‡ yi
o
f 2 ‰0; f‡ Š
‡ yi
N 3 …f ÿ 1†;
13†
if y…a† is a corner point, while
n
C…a†
ˆ
8 x 2 <2 ;
s
aÿ1†
x
i …f† ˆ yi
N 1 …f†
ÿ ‰f ; 0Š
a† 2
a‡1† 3
14†
N …f†; f 2
‡ yi N …f† ‡ yi
‰0; f‡ Š
if the boundary is smooth at y…a† . Note that N a …n† are the
Lagrange interpolation polynomials of the second order:
n
N 1 …n† ˆ …n ÿ 1†; N 2 …n† ˆ 1 ÿ n2 ;
2
n
3
N …n† ˆ …n ‡ 1† :
2
a†
Now, the position vector ri ˆ xi ÿ yi is approximated
over Cs as
ÿ
‰f ; 0Š
a†
15†
ri ˆ xi …f† ÿ yi ˆ f…fai ‡ bi †; f 2
‰0; f‡ Š
with
1 …aÿ2†
a†
aÿ1†
yi
‡ yi
;
ÿ yi
2
1 …aÿ2†
a†
aÿ1†
‡ 3yi
;
yi
ÿ 2yi
bÿ
i ˆ
2
16a†
1 …a†
a‡2†
a‡1†
ˆ
‡
y
;
ÿ
y
a‡
y
i
i
i
2 i
1
a†
a‡2†
a‡1†
3yi ‡ yi
;
‡ 2yi
b‡
i ˆÿ
2
in the case of a corner at y…a† , while
1 …aÿ1†
a‡1†
a†
‡
ÿ yi ;
aÿ
‡ yi
yi
i ˆ ai ˆ
2
16b†
1 …a‡1†
aÿ1†
‡
y
;
ˆ
b
ˆ
ÿ
y
bÿ
i
i
i
2 i
if the boundary is smooth on Cs .
a†
a†
Denoting the radius of the circular arc Ls as r0 , we
have to determine the isoparametric coordinates of the
endpoints of C…a†
s as the roots of the non-linear equations
aÿ
i ˆ
P
4 …f† ˆ 0
with
a†
a†
a†
2
P
4 …f† ˆ …xi …f† ÿ yi †…xi …f† ÿ yi † ÿ …r0 †
a†
2
ˆ f2 …f2 a
i ai ‡ 2ai bi f ‡ bi bi † ÿ …r0 † : …17†
The Newton-Raphson iteration scheme is appropriate for
®nding f . Omitting the superscripts , we may write the
a†
a†
a†
approximation for r~i ˆ xi ÿ zi (with zi ˆ yi ÿ Di ,
Di ˆ ei D) as
r~i ˆ f…fai ‡ bi † ‡ Di
Fig. 1. Sketch of nodal points on a part of the boundary C
with a corner at Y(a)
The tangent and outward normal vectors and Jacobian on
the approximated boundary segments are given as
bi bj
si …q† ˆ hi …q†=J…q†; ni …q† ˆ eij3 sj …q†;
p
J…q† ˆ hi …q†hi …q†
Dij …q† ˆ ci …q†cj …q† ÿ
in which
Gij …q† ˆ bi ci …q† ‡ bj ci …q† :
with
aˆ
p
ai ai ;
Z
b Di
D
aˆ i ; bˆ ;
bD
b
p
b ˆ bi bi :
a†
2
r~ ˆ r~i r~i ˆ …q ‡ c †=B…q† ;
†dC…x† ‡
ˆ
r~i ˆ qbi ‡ ci …q†D ‡ …q2 ‡ c2 †ai ;
1
q
D
bi B…q† ‡ 2
c …q†B…q† ‡ ai B…q† ;
r~;i ˆ 2
2
r~
q ‡c
q ‡ c2 i
1
D
1
X…q†B…q†
;
r~;k nk …q† ˆ EB…q† ‡ 2
r~
q ‡ c2
J…q†
1
B2 …q†
~
~
r;i r;j ˆ
ECij …q† ‡ X…q†ai aj D
r;k nk …q†~
r~
J…q†
q
D‰EGij …q† ‡ Fij X…q†Š
‡ 2
q ‡ c2
D
D2
X…q†Eij …q† ‡ ED ‡ 2 X…q† Dij …q†
‡ 2
2c
q ‡ c2
q
2
D X…q†Gij …q†
‡
q2 ‡ c2 †2
0 3
q
D
X…q†D
q†
;
‡ 2
ij
q ‡ c2 2c2
1
q
~
r;i nj ÿ r~;j ni † ˆ eji3 2
B…q†bk sk …q†
~
r
q ‡ c2
D
B…q†ck …q†sk …q† ‡ B…q†ak sk …q† ; …19†
‡ 2
q ‡ c2
where
c2 ˆ b2 …1 ÿ a2 †; ei ˆ Di =D;
a
ai b
;
ci …q† ˆ ei ÿ J…q†si …q† ÿ
b
b
A…q† ÿ1
;
B…q† ˆ bÿ2 1 ‡ 2
q ‡ c2
A…q† ˆ bÿ2 …q ÿ ab†2 ‰…q ÿ ab†2 a2 ‡ 2…q ÿ ab†ai bi ‡ 2ai Di Š;
E ˆ ekl3 bk a1 ;
ak b1
2a…q ÿ ab† ‡ b† ;
X…q† ˆ ekl3 ek hl …q† ‡
b
Fij ˆ ai bj ‡ bi aj ;
Eij …q† ˆ bi bj ‡ …ci …q†aj ‡ cj …q†ai †D;
Cij …q† ˆ Eij …q† ‡ qFij ‡ …q2 ‡ c2 †ai aj ;
Zd
†dC…x†
a†‡
Cs
18†
397
Z
Z
a†ÿ
2
20†
f …x†K…x; z…a† †dC…x†
Cs
Hence,
2
1 ÿ a2 †;
Now, the integral of a (nearly) singular kernel over the
singular portion of the boundary contour can be expressed
as
hi …q† ˆ bi ‡ 2ai …q ÿ ab†
q ˆ f ‡ ab;
b2
Cs
ÿ
ˆ
†J ÿ …q†dq ‡
fÿ ‡dÿ
‡
fZ
‡d‡
†J ‡ …q†dq
21†
d‡
in which
xi jC…a† ˆ x
i …f† ˆ xi …q ÿ d †
s
a†
ˆ yi ‡ …q ÿ d † …q ÿ d †a
i ‡ bi ;
d ˆ a b :
Denoting by hat the quantities taken for D ˆ 0, one
obtains
^
B…q†
ˆ 1=d…q†; d…q† ˆ …bi ‡ qai †…bi ‡ qai †;
a
^ci …q† ˆ ei ÿ …bi ‡ 2qai †;
b
2a
^
X…q† ˆ ekl3 ek …bl ‡ 2qal † ‡ ak bk q ;
b
E^ij …q† ˆ bi bj ;
22†
C^ij …q† ˆ bi bj ‡ qFij ‡ q2 ai aj ;
^ij …q† ˆ ^ci …q†^cj …q† ÿ
D
bi bj
2
b
1 ÿ a2 †;
G^ij …q† ˆ bi^cj …q† ‡ bj^ci …q† :
It can be seen that these quantities are bounded for
q 2 ‰fÿ ; 0Š [ ‰0; f‡ Š. Thus, the peak-like factors in the
nearly-singular integrals of the kernels r~;k nk =~
r ; r~;k nk r~;i r~;j =~
r,
and …~
r;i nj ÿ r~;j ni †=~
r are given as follows
D
qD
;
;
q2 ‡ c2
q2 ‡ c2
0
qD2
q
;
D;
q2 ‡ c2
q2 ‡ c2 †2
q2
q
:
‡ c2
23†
Assuming the boundary densities to be Holder continuous,
it can be shown that the nearly-singular integrals involving
the ®rst four factors of Eq. (23) can be evaluated analytically in the limit D ! 0. Without going into details, we
present the results of analytical integration:
Z
lim
f …x†
D!0
a†
Cs
1 o~
r
dC…x†
r~ on
Z0
ˆ …2p ÿ h†f …y…a† † ‡ Eÿ
fÿ
Z
lim
f …x†
D!0
‡
Zf
ÿ
f …x …q††
dq ‡ E‡
dÿ …q†
0
‡
f …x …q††
dq ;
d‡ …q†
1
bÿ
Z
C^ijÿ …q†dq ‡ E‡
Zf
0
a†ÿ
nl
a†ÿ …a†ÿ ÿ
sk J …0†f …y…a† †
el †nk
ˆ0
1
f …x† …~
r;i nj ÿ r~;j ni †dC…x†
r~
lim
a†ÿ
Cs
8
>
< Z 0 aÿ bÿ ‡ 2…aÿ †2 q
k k
f …xÿ …q††dq
ˆ eji3
ÿ …q†
>
d
:
fÿ
Zdÿ
fÿ
‡
2
because of the orthogonality.
Thus, from Eqs. (25) and (26), we have
D!0
1 o~
r
r~;i r~;j dC…x†
r~ on
1
a†ÿ …a†‡
a†ÿ
ˆ ‰…2p ÿ h†dij ‡ …n…a†‡
sj
m sm †…si
2
2
Z0
1
a†ÿ …a†‡
a†
ÿ
ÿ
ÿ ni nj †Š f …y † ‡ E
f …x …q†† ÿ
d …q†
398
R^ÿ …0† ˆ
2
1
‡
C^ij‡ …q†dq ;
f …x …q†† ‡
d …q†
‡ lim
24†
D!0
fÿ ‡dÿ
27†
9
>
=
q
ÿ
Q …q†dq
2
>
q2 ‡ …cÿ †
;
in which
ÿ
ÿ
ÿ
ÿ
Qÿ …q† ˆ Bÿ …q†bÿ
k sk …q†J …q†f …x …q ÿ d †† :
where h is the angle subtended by the tangents to C at y…a† . Recall that
Recall that
a†
x
i …q† ˆ yi ‡ qgi …q†;
C^ij …q† ˆ gi …q†gj …q†;
d …q† ˆ gi …q†gi …q†;
with gi …q† ˆ b
i ‡ qai
ÿ
ÿ ÿ ÿ
a†
Qÿ …0† ˆ Bÿ …0†bÿ
ÿ dÿ g ÿ …ÿdÿ ††
k …bk ÿ 2ak a b †f …y
:
and
Q^ÿ …0† ˆ
1
bÿ
2
ÿ
a†
^‡
bÿ
k bk f …y † ˆ Q …0† :
28†
The last peak-like factor of Eq. (23) gives rise to a strongly
singular integral.
Now, the last integral in Eq. (27) can be rearranged as
In view of Eq. (19), we have
follows
2 ÿ
Z
Zd
1
q
6
f …x† …~
r;i nj ÿ r~;j ni †dC…x†
lim 4
Qÿ …q† ÿ Qÿ …0†dq
r~
2
ÿ 2
D!0
a†ÿ
Cs
Zd
"
ÿ
ˆ eji3
f …xÿ …q ÿ dÿ ††
fÿ ‡dÿ
q
ÿ
Bÿ …q†bÿ
k sk …q†
q2 ‡ …cÿ †2
2
D
ÿ
ÿ
ÿ ÿ
ÿ
Bÿ …q†cÿ
‡
k …q†sk …q† ‡ B …q†ak sk …q† J …q†dq
2
q ‡ …cÿ †2
6
ˆ lim 4
D!0
ÿ
ÿ
Now, denoting f …xÿ …q ÿ dÿ ††Bÿ …q†cÿ
k …q†sk …q†J …q† as
ÿ
R …q† and assuming the Holder continuity of Rÿ …q†, one
obtains
lim
D!0
fÿ ‡dÿ
ˆ R^ÿ …0† lim
D!0
Zd
fÿ ‡dÿ
fÿ ‡dÿ
Z0
ÿ
fÿ
~2
q2 ‡ D
‡
f
ÿ
q
7
2 dq5
ÿ
‡ …c †
q
Qÿ …q† ÿ Qÿ …0††dq
‡ …cÿ †2
Zdÿ
q
q
dq ÿ
q2 ‡ …cÿ †2
Z0
fÿ
1
q
~2
q2 ‡ D
C
dqA
Qÿ …q† ÿ Qÿ …0††dq
3
Z0
26†
q2
fÿ ‡dÿ
ÿ
D
dq ˆ 0
2
q ‡ …cÿ †2
Zdÿ
B
‡ Qÿ …0†@
ÿ
D
Rÿ …q†dq
2
q ‡ …cÿ †2
q2
fÿ ‡dÿ
0
25†
3
Zdÿ
‡Qÿ …0†
#
Zd
q ‡ …c †
fÿ ‡dÿ
q
q2
~2
‡D
7
Qÿ …q†dq5 ;
29†
~ is a dimensionless quantity de®ned as D
~ ˆ D=b ,
The last equality results from the fact that the limit of the where D
in which b is an arbitrary length parameter.
last integral is ®nite and
Z
Hence, in view of Holder continuity of Qÿ …q†, the limit
lim
of the integrals given by Eq. (29) can be rewritten as
D!0
a†ÿ
Cs
Z0
q
lim
D!0
f
1
r;i nj ÿ r~;j ni †dC…x†
f …x† …~
r~
q2
ÿ
~2
‡D
Qÿ …q†dq
0
Zdÿ
B
‡ Q^ÿ …0† lim @
D!0
fÿ ‡dÿ
q
dq ÿ
q2 ‡ …cÿ †2
1
Z0
fÿ
2
Z0
6
ˆ eji3 4
f
ÿ
ÿ
ÿ 2
aÿ
k bk ‡ 2…a † q
f …xÿ …q††dq
dÿ …q†
q
q2
C
dq :
2 A
~
‡D
3
ÿ
1
‡ lim
2 D!0
32†
Zv0
Qÿ …qÿ …v††dv5
399
vÿ
Furthermore, the last two integrals can be evaluated ana- A similar analysis can be carried out for the integral of the
lytically
same integrand over Cs…a†‡ , with the result
0
ÿ
Zd
B
lim @
D!0
q
dq ÿ
2
q ‡ …cÿ †2
fÿ ‡dÿ
ˆ lim
1
"
ln
D!0 2
ÿ 2
Z0
fÿ
1
Z
lim
q
C
dq
2 A
2
~
q ‡D
~2
D
ÿ 2
d † ‡ …c †
ÿ ln ÿ 2
~2
f ‡ dÿ †2 ‡ …cÿ †2
f † ‡ D
D!0
a†‡
Cs
#
lim
a†ÿ
Cs
"
ˆ eji3
Z0
‡
fÿ
ˆ eji3 f …y…a† † ln
b
f …y † ln ÿ ‡ lim
D!0
b
Z0
fÿ
Zv
Zf
0
‡
‡
‡ 2
a‡
k bk ‡ 2…a † q
f …x‡ …q††dq
d‡ …q†
#
Q‡ …q‡ …v††dv
;
33†
v‡
0
where
p
2 2 ‰0; f‡ Š; v 2 ‰v‡ ; v‡ Š;
ev ÿ D
0
ÿ ‡ 2
2
‡
‡
v0 ˆ 2 ln D; v ˆ ln …f † ‡ D
ˆ D=b0 , with b0 being an arbitrary length
in which D
parameter. It is convenient to make the selection
b0 ˆ b‡ . Then,
Z
1ÿ
lim
f …x† r~;i nj ÿ r~;j ni dC…x†
D!0
r~
q‡ …v† ˆ
q
~2
q2 ‡ D
ÿ
ÿ 2
aÿ
k bk ‡ 2…a † q
f …xÿ …q††dq
dÿ …q†
b
‡
b0
1
‡ lim
2 D!0
1
f …x† …~
r;i nj ÿ r~;j ni †dC…x†
r~
a†
‡
‡
Summarizing, Eq. (27) becomes
D!0
"
ÿ
" !2 #
~
1
D 2
b
D
ˆ ln ÿ
ˆ lim ln ÿ ÿ ÿ ln ÿ
D!0 2
b
b f
f
Z
1
f …x† …~
r;i nj ÿ r~;j ni †dC…x†
r~
Qÿ …q†dq
#
30†
Now, instead of the last integral in Eq. (27), we have to
deal with the ®rst integral on the r.h.s of Eq. (30). The
peak-like character of the kernel of this integral can be
smoothed by an appropriate transformation of the integration variable before a numerical integration (Sladek
and Sladek, 1998). Let us use the transformation q ! v
given by
a†
Cs
2
6
ˆ eji3 4
Z0
fÿ
ÿ
ÿ 2
aÿ
k bk ‡ 2…a † q
f …xÿ …q††dq
dÿ …q†
Zf
‡
0
‡
‡
‡ 2
a‡
k bk ‡ 2…a † q
f …x‡ …q††dq
d‡ …q†
ÿ
~ 2 †;
v ˆ ln…q2 ‡ D
vÿ
0
~
ˆ 2 ln D;
with
v 2 ‰vÿ ; vÿ
0 Š;
~ 2† :
v ˆ ln……f † ‡ D
ÿ 2
ÿ
Hence, we have for q 2 ‰fÿ ; 0Š
p
~ 2;
qÿ …v† ˆ ev ÿ D
2q
q2
~2
‡D
dq ˆ dv :
31†
1
‡ lim
2 D!0
1
‡ lim
2 D!0
Zv0
vÿ
Zv‡
Qÿ …qÿ …v††dv
3
7
Q‡ …q‡ …v††dv5 :
34†
v‡
0
Recall that now all the integrands are smooth functions
Since b is still arbitrary, it is reasonable to take b ˆ bÿ . over the integration intervals. Finally, in view of Eqs. (9),
(24) and (34), we may write
Then, in view of Eqs. (30) and (31) we obtain
Z
ui y…a† ‡ lim
uj …x†~tij x; z…a† dC…x†
D!0
a†
Cs
1
a†‡ …a†ÿ
a†ÿ …a†‡
a†ÿ …a†‡
si sj
ÿ ni nj
dij ÿ
n sl
ˆ uj y
2p
4p…1 ÿ m† l
Z0 n
i
1
1 h
2Eÿ ÿ
ÿ
ÿ
ÿ ^ÿ
1
ÿ
2
m
†E
g
d
‡
q†g
q†
‡
1
ÿ
2
m
†e
a
q†
h
ÿ
ij
ij3
i
j
k
k
dÿ …q†
4p…1 ÿ m†
dÿ …q†
fÿ
o
q
ÿ
ÿ
ÿ
ÿ
ÿ
ÿ
2 ‰ gk …q†ekl3 dij ‡ gj …q†eil3 ÿ 3gi …q†ejl3 Šh^l …q† uj …x …q††dq
a†
3 ÿ 4m† r0
400
ÿ
ÿ
a†
h
‡
Zf 1
4p…1 ÿ m†
0
q
a†
3 ÿ 4m† r0
1
2E‡ ‡
‡
‡
‡ ^‡
1
ÿ
2
m
†E
g
d
‡
q†g
q†
‡
1
ÿ
2
m
†
e
a
q†
h
ij
ij3 k k
j
d‡ …q† i
d‡ …q†
‡
‡
‡
‡
‡
^
2 ‰ gk …q†ekl3 dij ‡ gj …q†eil3 ÿ 3gi …q†ejl3 Šhl …q† uj …x …q††dq
1 ÿ 2m
eij3 lim
ÿ
8p…1 ÿ m† D!0
" Zvÿ0
ÿ
ÿ
Zv‡
q …v†uj …~x …v††dv ‡
vÿ
‡
q …v†uj …~
x …v††dv
;
35†
v‡
0
where
with H…x† being the Heaviside unite step function and with
the unit vectors s , oriented from y…a† to the endpoints of
a†
Ls , being given by
ÿ
ÿ
q …v† ˆ B q …v† b
k ‰bk ‡ 2ak q …v† ÿ d Š ;
ÿ
x~ …v† ˆ x q …v† ÿ d :
#
‡
Recall that the limit D ! 0 in the integrals of the nonsingular terms in ~tij has been performed behind the integral sign and that
gi …f †
q
 :
s
ˆ
i
d …f †
Note that all the integrations in Eqs. (35), (36) can be
performed suf®ciently accurately by standard quadratures
h^
of numerical integration (e.g. standard Gauss quadrature)
k …q† ˆ bk ‡ 2qak :
because of the smooth variation of the integrands.
a†
The integral over the circular part of the boundary Ls in
Finally, the logarithmic singularity of the kernel u~ij is to
Eqs. (10) and/or (11) does not give rise to any dif®culty, be dealt with. Apparently,
a†
because it is non-singular owing to the fact that y…a†2
= Ls . Z
It is convenient to perform the integration with respect to
tj …x†~
uij …x ÿ y…a† †dC…x†
the angular variable of the polar coordinates as
a†
Z
Cs
Z0
uj …x†~tij …x; y…a† †dC…x†
ˆ
a†
Ls
tj …xÿ …f††~
uij …fg ÿ …f††J^ÿ …f†df
fÿ
a†ÿ
ˆ
a†
r0
u
Z
‡
L
uj …x
†~tij …xL ; y…a† †du
;
Zf
36†
‡
u…a†‡
tj …x‡ …f††~
uij …fg ‡ …f††J^‡ …f†df
0
a†
where
a†
a†
xLi ˆ yi ‡ r0 mi …u†; mi …u† ˆ di1 cos u ‡ di2 sin u ;
ÿ
ÿ ÿ s
a†
ˆ arctan 2 ‡ pH ÿs
;
u
1 ‡ 2pH s1 H ÿs2
s1
since ri jC…a† ˆ x
i …f† ÿ yi
s
Hence,
rjC…a† ˆ jfj
p
d …f†;
ˆ fgi …f†.
s
p
r;i jC…a† ˆ sign…f†gi …f†= d …f†:
s
37†
The contour of the integration C ˆ Cÿ [ C‡ will be chosen either as the union of straight lines of the union of a
straight and a curved line, with Cÿ being straight and C‡
being a circular arc. The source point y is put into the
origin of the coordinate system for simplicity but without
any restriction of generality.
‡
‡
Denoting the endpoints of Cÿ ÿ Cÿ
e and C ÿ Ce as
ÿ ÿ
‡ ‡
fx ; xe g and fxe ; x g, respectively, we may write for the
exact values of the considered integrals
Thus,
!2 !2
f
ˆ …a† d …f†;
a†
a†
r0
r0
Cs
"
#
!2
r 1
f
ˆ ln
d …f† ‡ ln s ;
ln …a† …a†
2
r
r0 C…a†
0
s
r
where we have used the parametrization
fˆ
sf‡ ; on C…a†‡
s
;
sfÿ ; on C…a†ÿ
s
ÿ
Yÿex ˆ lim ln…jxÿ
e j=jx j†;
e!0
with s 2 ‰0; 1Š:
e!0
40†
If we take jx
e j ˆ ejx j, the exact values become
Eventually,
Z
Y‡ex ˆ lim ln…jx‡ j=jx‡
e j†:
tj …x†~
uij …x ÿ y…a† †dC…x†
a†
Cs
4m ÿ 3
ˆÿ
8pl…1 ÿ m†
‡
1
8pl…1 ÿ m†
Z1
0
Zf
1 ‡ ^‡ ‡
f J …sf †ti …x‡ …sf‡ ††‡jfÿ jJ^ÿ …sfÿ †ti …xÿ …sfÿ †† ds
ln
s
‡
tj …x‡ …f††
0
h
i
p
2
5 ÿ 4m …a†
a†
1 ÿ f=r0
4m ÿ 3†ln f‡ d‡ …f†=r0 ÿ
d‡ …f† dij
2…3 ÿ 4m†
(
‡
Z0
2
p
gi …f†g‡
1
j …f†
a†
ÿ
‡
‡
ÿ
ÿ …f†=r …a†
^ …f†df ‡
J
‡ 1 ÿ f=r0
d …f†
t
x
f††
4
m
ÿ
3†ln
f
d
j
0
8pl…1 ÿ m†
d‡ …f†
fÿ
)
gÿ …f†gÿ …f†
5 ÿ 4m i
j
a† 2 ÿ
a† 2 ÿ
J^ÿ …f†df ;
d …f† dij ‡ 1 ÿ f=r0
d …f†
38†
1 ÿ f=r0
ÿ
dÿ …f†
2…3 ÿ 4m†
Yÿex ˆ lim e;
with
q
J^ …f† ˆ h^ …f†h^ …f†:
k
Y ex ˆ
k
e!0
Yÿex
Y‡ex ˆ ÿYÿex ;
‡ Y‡ex ˆ 0:
41†
Now, the special Gauss quadrature can be used for the
numerical integration of ln…1=s†, while the standard
quadrature is applicable to other integrals in Eq. (38).
To be consistent with the boundary modelling employed in
the previous section, we parametrize the straight segments
4
Numerical examples
In order to compare the accuracy of the numerical integration of a strongly singular integral, we shall consider
the CPV integral with a constant density
2
C‡ ÿ C‡
e ˆ f8 x 2 < ; x2 ˆ 0; x1 2 ‰er0 ; r0 Šg
Y ˆ Yÿ ‡ Y‡ ;
Z
Y ˆ lim
e!0
C
ÿC
e
o
ln jx ÿ yjdC…x†
os…x†
39†
which can be evaluated analytically too. Recall that
1
1
r ;i nj ÿ r~;j ni † ˆ lim eji3 r~;k sk
lim …~
D!0 r
D!0
~
r~
1
o
ln jx ÿ yj:
ˆ eji3 r;k sk ˆ eji3
r
os…x†
2
Cÿ ÿ Cÿ
e ˆ f8 x 2 < ; x2 ˆ 0; x1 2 ‰ÿr0 ; ÿer0 Šg
as
~ÿ ÿ C
~ ÿ ˆ f8 x 2 <2 ; xi ˆ y…a†
C
e
i
ÿ
‡ f…faÿ
‡
b
†;
f
2 ‰fÿ ; fÿ
i
e Šg
i
~ ‡ ˆ f8 x 2 <2 ; xi ˆ y…a†
~‡ ÿ C
C
e
i
‡
‡
‡ f…fa‡
‡
b
†;
f
2 ‰f‡
i
i
e ; f Šg
in which a
i , bi are de®ned by Eq. (16a) with
a†
yi
ˆ 0;
ÿ
aÿ2†
yi
ÿ1
f ˆ ÿr0 =b ;
a‡1†
ˆ 4r0 ti ;
yi
‡
fe ˆ er0 =b‡ ;
ˆ ÿ4r0 di1 ;
aÿ1†
yi
ÿ
fÿ
e ˆ ÿer0 =b ;
a‡2†
yi
ˆ 8r0 ti ;
‡
f ˆ r0 =b‡ ;
ˆ ÿ2r0 di1 ;
42†
401
Table 1. Accuracy of the numerically computed integrals
over straight segments by the
SG- and tG-approaches
402
e
10)2
10)3
10)4
10)5
10)6
10)7
10)8
10)9
10)10
N ÿ ˆ N ‡ ˆ 12
N ÿ ˆ 11; N ‡ ˆ 12
%err YÿSG = %err Y‡SG
%err YÿtG = %err Y‡tG
)0.80866
)14.27229
)32.95013
)46.11878
)55.07868
)61.49427
)66.30734
)70.05095
)73.04585
)0.17358
)0.11443
)0.86789
)0.67888
)0.59145
)0.48492
)0.46287
)0.37716
)0.35487
´
´
´
´
´
´
´
´
´
10)11
10)11
10)12
10)12
10)12
10)12
10)12
10)12
10)12
%err YÿSG
Y SG ˆ YÿSG ‡ Y‡SG
)1.19420
)16.09886
)34.70849
)47.56227
)56.28471
)62.52828
)67.21211
)70.08552
)73.78968
0.0177882
0.1261751
0.1619517
0.1661875
0.1666187
0.1666619
0.1666662
0.1666666
0.1666667
Table 2. Accuracy of the numerical integration over the circular arc and the total value of the integral over two segments by both the
SG- and tG-approaches for different values of curvature and ®xed e = 10)10
R
%err Y‡SG
Y SG
%err Y‡tG
0.5
1
10
100
)73.56478
)73.21228
)73.04769
)73.04587
)0.1194860
)0.3832114 ´ 10)1
)0.4214525 ´ 10)3
)0.4218708 ´ 10)5
0.2799
0.2065
0.1256
0.1248
where t is an arbitrary unit vector. This parametrization is
exact because of the straight character of Cÿ ÿ Cÿ
e and
.
In
the
numerical
computations,
we
have
used
C ‡ ÿ C‡
e
two approaches. In one of them (denoted by the superscript SG), the standard Gauss quadrature is applied to
numerical integration with respect to f over ‰fÿ ; fÿ
e Š and
‡
‰f‡
e ; f Š. In the second approach (denoted by the superscript tG), the singular point y…a† is moved into z…a† with
a†
a†
zi ˆ yi ‡ Ddi2 , and the integration variable f (or more
precisely q) is transformed into m as described in the
previous section, and ®nally, the numerical integration by
standard Gauss quadrature is performed with
D ˆ e 10ÿ7 , for obeying the limit D ! 0. Table 1 shows
the numerical results when the numbers of the Gaussian
points on the left and right hand sides of the singular point
are either the same (N ÿ ˆ N ‡ ˆ 12) or slightly different
(N ÿ ˆ 11; N ‡ ˆ 12).
Note that the accuracy of the numerical integration of
YSG is not good when e ! 0, but Y SG ˆ YÿSG ‡ Y‡SG ˆ 0 as
long as N ÿ ˆ N ‡ . A small deviation of the symmetry of
the distribution of Gaussian points gives rise to an unacceptable inaccuracy of Y SG . On the other hand, the accuracy of YÿtG is not in¯uenced by the small change of the
number of Gaussian points and the accuracy of YtG are still
reasonable even for e ! 0.
Now, let us consider the smooth connection of the
straight line Cÿ from the previous example with the circular arc
C‡ ˆ f8 x 2 <2 ; x1 ˆ R sin u;
x2 ˆ R…1 ÿ cos u†; u 2 ‰0; UŠg
with U ˆ 2 arcsin…r0 =2R†. On C‡ ÿ C‡
e the angular variable varies within the interval ‰Ue ; UŠ in which
Ue ˆ 2 arcsin…er0 =2R†. In the numerical
computations
~‡ ~‡
C ‡ ÿ C‡
e is approximated as C ÿ Ce given by Eq. (42)
with
´
´
´
´
Y tG
10)4
10)5
10)7
10)9
0.6446717
0.4755348
0.2891415
0.2881606
a‡1†
ˆ di1 R sin 2U ‡ di2 R…1 ÿ cos 2U†;
a‡2†
ˆ di1 R sin 4U ‡ di2 R…1 ÿ cos 4U†;
yi
yi
´
´
´
´
10)5
10)6
10)8
10)10
‡
and f‡ , f‡
ˆ0
e are determined as roots of Eq. P4 …f† …a†
with putting r0 and er0 , respectively, instead of r0 in
Eq. (17).
Now, one can expect failure of the SG-approach in the
accuracy of Y SG due to the broken symmetry in the distribution of integration points with respect to the singular
point, even if the number of Gaussian points is the same
on both the sides of this point (N ÿ ˆ N ‡ ). Numerical
experiments con®rm such a failure. We have performed
numerical computations for several different values of the
curvature of C‡ and e varying from 10ÿ2 to 10ÿ10 . Table 2
shows some of the numerical results.
It can be seen that the accuracy of Y SG is improved
when decreasing the curvature of C‡ , though the accuracy
of the integration over this part of the total boundary is
almost unaffected. On the other hand, tG-approach gives
satisfactory results permanently.
5
Conclusions
Numerical integration of boundary integrals with strongly
singular kernels requires special attention in the meshless
implementations of the LBIE when the boundary densities
are known only digitally (e.g., in the case of MLS-approximation). The proposed method leads to a suf®cient
accuracy of the numerical integration. Moreover, it is
stable with respect to violation of the symmetry in distribution of the integration points around the singular
point.
Without any doubts the proposed technique of numerical integration of singularities is applicable also to the
boundary node method (Mukherjee and Mukherjee, 1997),
where the BIE are considered in a global sense in combination with the MLS-approximation of boundary densities.
4. Mukherjee YX, Mukherjee S (1997) The boundary node
method for potential problems. Int. J. Num. Meth. Eng. 40:
797±815
5. Sladek V, Sladek J (1998) Some computational aspects associated with singular kernels, Chapter 10, In: Sladek V, Sladek J
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