Computational Mechanics (1991) 8, 57 70
Computational
Mechanics
9 Springer-Verlag 1991
On the evaluation of hyper-singular integrals arising
in the boundary element method for linear elasticity
C. C. Chien, H. Rajiyah and S. N. Atluri
Center for Computational Mechanics, Georgia Institute of Technology, Atlanta, GA 30332-0356, USA
Abstract. The boundary element method (BEM) for linear elasticity in its curent usage is based on the boundary integral
equation for displacements. The stress field in the interior of the body is computed by differentiating the displacement field
at the source point in the BEM formulation, via the strain field. However, at the boundary, this method gives rise to a hypersingular integral relation which becomes numerically intractable. A novel approach is presented here, where hyper-singular
kernels for stresses on the boundary are made numerically tractable through the imposition of certain equilibrated displacement
modes. Numerical results are also presented for benchmark problems, to illustrate the efficacy of the present approach.
Solutions are compared to the commonly used boundary stress algorithm wherein the boundary stresses are computed from
known boundary tractions, and derivatives of known displacements tangential to the boundary. An extension of this approach
to solve linear elasticity problems using the traction boundary integral equation (TBIE) is also discussed.
1 Introduction
It is well known that the boundary element method is a powerful tool in the solution of practical
problems in engineering. Although such formulations were originally thought to be of theoretical
interest, during the past two decades, they have been applied to a wide range of linear and nonlinear problems in solid mechanics (Okada et al. [1]).
The currently popular boundary element method in linear elasticity is based on an integral
representation for displacements which is given in terms of tractions and displacements on the
boundary (Rizzo 1967; Cruse 1969). The stress field in the BEM formulation is computed by differentiating the displacement field at the source point, to obtain the strain field and then applying
Hooke's law to obtain the stress field. Unfortunately, this method works well in the interior of
the body, but at the boundary it gives rise to hyper-singular integrals which are intractable from
a numerical point of view. Hence, to circumvent this predicament at the boundary, the following
techniques have been adopted so far: (i) compute tangential gradients of displacements through
a finite difference approximation of boundary displacements or differentiation of shape functions
associated with nodal displacements (Cruse 1969; Lachat and Watson 1975) (ii) recasting the
integral equation into one that involves displacement derivatives rather than displacements on
the boundary which gave rise to numerically tractable singularities (Ghosh et al. 1986; Okada
et al. 1988). All the above methods for the boundary stress algorithm, however, deviate considerably from the internal stress algorithm of the standard BEM formulation.
The hyper-singular kernels in the integral equation for displacement gradients, in the
traditional BEM, continue to be of concern to researchers in this discipline. A novel approach
is presented here in which the highly singular kernels in the integral equation for stresses on the
boundary are made numerically tractable with a minor modification to the original integral equation. Such a modification is carried out by considering certain properties of the above mentioned
kernel functions and through the imposition of certain equilibrated displacement modes. The
efficacy of the present approach is demonstrated with quadratic isoparametric boundary elements
for several carefully selected two dimensional problems. Comparisons are also made with other
(above mentioned) formulations. An extension of the present methodology to handle the traction
boundary integral equation (TBIE) is also discussed.
Computational Mechanics 8 (1991)
58
2 Present approach for boundary stresses
Let ~ j be the Cartesian components of the Cauchy stress tensor and let fj be the body forces
per unit volume. The equations of linear and angular momentum balance are
aij,i + f; = 0; aij = ~1i"
(1)
The weak form of the linear momentum balance equation, when the displacement field fit due
to the Kelvin's singular solution is taken as the test function, takes the following form:
f (ffij,i-4- fj)(tjdg"2 = 0
(2)
a
Here, ~ and 0~ refer to the domain and the boundary, respectively, of the problem under consideration. Integrating Eq. (2) by parts and applying the divergence theorem twice, the standard BEM
formulation (in the absence of body forces) is obained,
Cijuj(~,,) = S [U*(Xm, ~,,)tj(x,,) -- T*(X,,, ~m)Uj(Xm)]d(~2).
o~
(3)
U*.
Here, Cij = 6i~in the interior and ~6ij at a smooth boundary point.
.
, and
. T*.j are the displacement
. .
and traction fields, respectively, of the fundamental or Kelvin's solution. The above equation IS
discretized and applied at the boundary to determine all the unknown displacements and tractions
on the boundary. The strain field (and hence the stress field) in the interior of the body is obtained
by differentiating Eq. (3) with respect to the load point ~z- Thus,
~,.)uj(x,.)d(a~).
(4)
Such an approach give rise to T* z type kernels which are hyper-singular when ~,, tends to a
boundary point. These hyper-sing~]'lar kernels in its present form are numerically intractable. To
avoid this predicament, various methodologies such as (i) finite difference, or differentiation of
shape functions associated with tangential displacements to obtain tangential displacement gradients, or (ii) recasting the integral equation to obtain a non-hyper-singular form, or evaluating
the finite parts of the hyper-singular integral equation, have been attempted.
The present approach for the boundary stresses, employs the integral equation for displacement
gradients, as in the interior stress algorithm. Subjecting ~ to a rigid body translation, we have
Uj(~m) = Uj(X,,) = Const.; t;(~,,) = tj(x,,) = 0.
(5)
Employing the above relation in Eq. (3), we obtain
C,; + ~ T*(x,,,, ~m)d(~12)= 0.
(6)
at2
Multiplying Eq. (6) with arbitrary U;(~m)and taking ~ ,
we have
It should be noted here that the non-integrable singularity in the first term on the left hand side
of Eq. (7) is of the same form and order as in the second term on the right hand side of Eq. (4) as
Xm~ ~,,. By using the identity of Eq. (7), the second term on the right hand side of Eq. (4) can
be written as,
~ z on
T*(Xm, ~m)Uj(X..)d(~.Q)= -,,~,~a [T*(xm, ~)uj(x,.)-- T*(x m, ~,.)uj({m)] d(~f2) - Cij c~uj(~'')
0r
(8)
C. C. Chien et al.: O n the evaluation of hyper-singular integrals
59
The integrand in the first term on the right hand side of Eq. (8) is only weakly-singular (as
x,,--* ~m) and the derivative O/~,,z could now be taken inside the integral. Hence, from Eq. (8),
one has
Ugl~JD r ~ (Xm, ~m)Uj (Xm)d(r
= S [r~.' (Xrn' ~m)Uj (Xm) -- T?j,l(Xm' ~m)Uj(r
d(~12)
Og2
C3Uj(~m) ~ T*(Xm, ~m)d(Ol2)- Cu ~uj(~m)
(9)
Substituting Eq. (6) into Eq. (9), we have
~-~l~$2T~(ym' ~m)uj(Xm)d(~ff~)~- 0.(2
~ T*ij,l(Xm, ~m)[Uj(Xm)--Uj(r
(10)
Making use of Eq. (10) in Eq. (4), we obtain the modified integral representation for displacement
~radients,
Cijuj,~(~m) = ~ U*,(Xm, r
j" T~,~(Xm,~m)[Uj(Xm)- Uj(~m)]d(OO).
3g2
(11)
~?.O
It is well known that the stresses a u and ui,j can be evaluated in linear elasticity through
a j = EijklUk.t;
ti = aijnj;
aui
Os -- u~'isJ"
(12a-c)
Here, Eok z is the elasticity tensor; tg are the traction components on the boundary (which are known
or have been determined) and nj, sj are the components of the unit normal and tangent vectors at
a boundary point, respectively. Equation (11) is used here (instead of a finite difference algorithm)
to evaluate the tangential derivative at a boundary point in Eq. (12c). It should be noted here that
the integrand in the first integral of Eq. (11) has a principal value singularity, and such integrals
have been successfully evaluated numerically in the past (Okada et al. 1988; Guiggiani and Casalini
1987).
The kernel function T*..
in the second integral is hyper-singular and a sufficiencv
condition
1
tj t
.,
of C '" is required on the density function uj, for such integrals to remain bounded (Hadamard
1923). It has been assumed here that this smoothness requirement is satisfied a priori (this would
then guarantee the boundedness of the integral) in the actual behavior of the function. One may
then discretize the integral equation with any plausible approximation of the density function such
that a numerical solution to the already assumed C 1'" behavior can be obtained.
Let us consider Eq. (11) in a local coordinate system, which is aligned with the tangent and
normal to the boundary point under consideration. Limiting attention to a two-dimensional
domain, upon discretizing and introducing shape functions, we have
P
Q
e
Cijuj,t( era)= E E t~jP'a) [, U*,t(Xm,r Ntt''q)dFp=l q=l
Fp
Ms Q
E
Q
~ u~jp'a) ~ T*,t(Xm,~m)N~P'q)dF
p= l(pr q= l
Fp
Ms Q
-- Z E u?'q'
~ (Nt~'q)--7's'q)tl)T*U,,'Cxm, e r a ) d e - Z Z ~s'")u'S'~)
~ tlT*,(x=,r
J
j
s=lq:l
es
s= l q= l
Fs
(13)
Here, Q is the number of nodes in an element, P is the total number of elements on the boundary
and M s is the number of singular elements associated with the source point ~m- As shown in
e
Fig. 2, F = ~ Fp is the whole region involving all the elements along the boundary of body
p=l
Ms
and/--s = ~ F~ is the boundary region which contains the singular elements having source point
s=l
~m as one of its nodes. N ~p'q) are the associated nodal shape functions and N ~s'q) (in Fig. 3) are
60
Computational Mechanics 8 (1991)
the modified shape functions such that
N (`'q) = N(r/)
=N(t/)-I
q 4= r
q---era.
for
for
(14)
Here, ~(~'q)(in Fig. 3) is the constant multiplying the linear portion of N (~'q). The first integral on
the right hand side of Eq. (13) is numerically tractable (Okada et al. 1988; Guiggiani and Casalini
1987). The second integral is non-singular and hence amenable to standard numerical Quadrature.
The term involving (N (~'q)- ~(~'q)r/)in the third integral goes to zero quadratically as x,, ~ ~,, and
hence the integral is amenable to standard numerical Quadrature. The term involving ~ r/T* a d F
r~
in the last integral is not amenable to standard numerical Quadrature. A numerical procedure
for indirectly evaluating these integrals through known equilibrated displacement modes is
presented in one of the latter sections.
3 Direct evaluation of Cauchy principal value (CPV) singularities
As shown in Fig. 1, let B P C be the boundary segment where the source point P lies and let Q
be the field point. The straight line E P D is tangent to the curve B P C at P and let E C and BD
be perpendiculars to the line EPD. The points B and C are the ends of the segment under
consideration. Let r be the distance between the source point P and field point Q. The C P V
singularities in Eq. (13) are of the form,
I, = tY(x,,) ~ U*.,(x,~, ~m)Ned(c~-O).
oa
(15)
Here, N e is the shape function associated with point P, and t~ (x,,) corresponds to the nodal traction
at the source point and Og2is part of the boundary BPC. The singular integral could be written
in the following form,
I, = tY (Xm) j (N e -- 1)U*,,( Xm, r
dO
t~. (x,.) ~ U~j,l(Xra , ~m)d(l~Q)
~t2
(16)
= I2 + t~(Xm)I3.
It should be noted here that I2 is amenable to standard numerical Quadrature. As per Fig. 1,
D
fl~f'
/
E%I
%
[
Q : fietd point (x~)
CPB : boundory
F 5
F $
s
1
C
1
r
a
2
b
Figs. 1 and 2 1 Numerical evaluation of C P V integrals. 2 a and b Segmentation of curve along the boundary of the body
(quadratic elements) i.e. F = ( F - / - ~ ) + F~; a ~,.E Extreme node of an element; b r
Mid node of an element
C. C, C h i e n et al.: O n the e v a l u a t i o n of h y p e r - s i n g u l a r i n t e g r a l s
(i)
(2)
fl
.
"
i
'
0
(3)
......
(~)
r
2
61
(2) ~
1
(3)
-"
-I
r
0
N0)= 89
i
;
a0)=_i
(i)
(2)
(3)
-2
-1
0
N(x) = 89
1)7]
;
;
a(1) = 89
a (2) = - 2
N (1) ~ ( r / - l ) ( r / - 2 ) - i
; u(i)_
3
--2
N ( 2 )== ( 2 - q ) /
; a (2)=2
N (a) = 89
1)r/
; 0~<3) = _ 89
N(2) = (I - r/)(1 + ~7) _ I
~(2) = 0
;
N (a) = 89
1)
;
0:(3) = 89
N~ = 89 + 1)(~+ 2) - 1 ; ~(~ = 3
a
b
e
N(2) = -~1(~7 + 2)
Fig. 3 a-e. Modified shape functions N (''q) in F ~ a n d constant ~("q~ for quadratic elements; a source point at node (1); b source
point at node (2); e source point at node (3)
the integral
would take the following form,
13
13= ~ U*~dS.
8pc
(17)
Due to the inherent property of the U*ij,l kernel (for 2-D problems), it could be expressed in the
following form,
U*u.,
f(O).,
r
-
(18)
f ( O ) = - f ( O + lr).
Here, r is the distance between the source point and field point and 0 is the angle made by the
line connecting the source point and field point with the x-axis. Hence, 13 could be written as
(Okada et al. 1988),
13 =
f(O----~)dS
;
BPC
r
f(~z) d S ' } d S + ~{f(O )
r' dS
PC r
= ({f(O)
BP
r
f(fl) dS' }dS+f(cOln(rDI"
r' dS
\ re/
(19)
Here, the angles ~ and fl are constants and 0 varies as the field point moves along the arc (Fig. 1).
r~ and rD are distances between the source point and points E and D, respectively. It should be
noted here that all the integrals along BP and PC in Eq. (19) are regular and hence are amenable
to standard Gaussian Quadrature.
4 Indirect evaluation of
~r, 1 1 T * , l d F
integrals
The kernel T*,, is hyper-singular and hence appropriate modification to the integral equation
has been carr{~t out (Eq. (11)) in order to make it numerically tractable. An indirect procedure
for the evaluation of such integrals is suggested here. This procedure employs known displacement
modes to any arbitrary shaped body under consideration. Such a method would not only obviate
the need for special Quadrature schemes (Kutt 1975; Ioakimidis 1981) but also help in smoothing
out the numerical errors due to discretization.
Rearranging Eq. (13), we have
Ms
Q
p
Q
~ ~(s'q)u(.~'q)~
rlT~.z(Xm,~m) dF= ~ ~, tj(p'q) ~ U*ij,l (Xm,~m)N(P'q)dr
J
s=lq=l
Fs
1'
-
E
q=l
Fp
~_, u~v'q'~ T*,,(x,,,,r N'p'q'dF
p = l ( p C s ) q= l
Ms
p=l
Q
l"~p
Q
- ~ ~-, u~s'q)j ~ (N(S'q)-~'q'rl) T*ij,,(x~, ~ ) d r - C , j u j . , ( ~ 2 .
s=lq=l
Fs
(20)
62
Computational Mechanics 8 (1991)
As discussed previously, all the kernels on the right hand side of Eq. (20) can be evaluated
numeically through standard Gaussian Quadrature. The integrals ~ t/T*z dF, however, on the
F~
left hand side of Eq. (20) would pose a problem. As shown in Fig. 2, the number of integrals to
be evaluated indirectly, would depend on the locations of the source point (~m)"If ~ extreme
node of an element, then the integrals ~ tlT*zdF from both neighboring elements have to be
F~
evaluated separately. On the other hand, if ~mE mid node of an element, only one such type of
integral over the singular element needs to be evaluated. We now adopt a novel procedure wherein
the integrals of the type ~ t/T*t d F are treated as unknowns to be solved for, from Eq. (20). This is
F~
possible, provided u~ 'q) and the corresponding t(.Jp'q) are taken to be known quantities, as evaluated
from known simple equilibrated displacement fields in elasticity problem. The required number
of these displacement fields clearly depends on the numbers of the quantities ~ t/T*g d F to be
rs
evaluated as the unknowns in Eq. (20). The displacement modes for any arbitrary O (for the
case of plane strain problems in the absence of body forces) can be taken as:
(i) uj = xj + x j+ ~,
uj+ 1
2(V -- 1) X2
(ii) u i = x ~ + ]---2V i+1'
u~+l=0etc"
=
0
(j = 1, 2; no sum on j; indices taken cyclically).
(21)
Here, v is the Poisson's ratio. The corresponding equilibrating traction fields on the boundary
~s which could maintain these displacement fields, could be computed through t~ = njEijktuk, t.
It should be noted here that only one set of these solutions are needed when ~m~ mid node of
an element (Fig. 2b) and both sets of solutions are needed when r extreme node of an element
(Fig. 2a). In the latter case, the solution of a system of simultaneous equations is required to
indirectly evaluate the quantity ~ tlT*jdF over both neighboring elements. Such a solution
r,
procedure is analogous to the methodology adopted in Chien et al. 1990, where potential modes
were employed to handle hyper-singular kernels in the 3-D exterior acoustic problem.
5 The traction boundary integral equation (TBIE)
It has been demonstrated in the previous sections that the commonly used displacement boundary
integral equation (DBIE) can be employed to compute the boundary stresses. This was made
possible through suitably modifying the hyper-singular integral such that it was made amenable
to numerical evaluation through employing an indirect approach. Alternatively, one may also
develop the modified form of the TBIE directly from Eq. (11) by using Eqs. (12a) and (12b), to
take the following form,
fltp(~m) = ~ nq(~m)EpquU*,,(Xm, ~m)tj(Xm)d(012)-- ~ nq(~m)Epqi~T*d(Xm, ~m)[Uj(Xm)-- Uj(~m)]d(0O).
a.O
c~t2
(22)
Here, fl takes a value of 1 when em~ in 12 and 89when emE smooth 0~2, nq(~m)are the components
of the normal at ~3S2and Epq,~ is the elasticity tensor.
It should be noted here that the above integral equation could be collocated at smooth
boundary points to yield a system of equations as in the case of the DBIE. The numerical
evaluation of singular integrals in Eq. (22) is carried out in a similar manner to that of the
previous section. The first integral which is CPV singular, can be evaluated directly after
discretization. This is possible due to the fact that nq(r
essentially remains a constant over
the line EPD (Fig. 1), and hence the procedure described in one of the previous sections can be
C. C. Chien et al.: On the evaluation of hyper-singular integrals
63
used here as well. The numerical evaluation of the singular integrals in the second integral
(Eq. 22) can be performed indirectly after discretization. Analogous to the previous section, the
quantity S q nqEpqizT*,~d F is evaluated indirectly through the displacement modes described in
F5
Eq. (21). The number of displacement modes necessary would depend whether ~,,e mid node or
extreme node of an element.
6 Numerical results
6.1 Preliminaries
Numerical results are presented here for the tangential boundary stresses obtained by employing
the DBIE. Three-noded quadratic boundary elements are employed in the present study. The traction discontinuity at the corner is modelled through the use of double nodes. The reader is referred
to Brebbia (1989) or Banarjee (1981) for further details regarding numerical implementation. All
numerical integrations are performed using 4 point Gaussian Quadrature scheme. In the present
paper, the designation of conventional method implies that all the boundary stresses at nodes are
computed from the tangential displacement gradients between the neighboring boundary elements
through a finite difference approximation of boundary displacements. Four example problems
are presented to illustrate the efficacy of the present approach.
6.2 Example problems
6.2.1 A cantilever beam under transverse end loading
The data pertaining to the geometry and material properties of the beam is given in Fig. 4. The
three different boundary element meshes considered are shown in Fig. 5. The tangential stresses
along axx along IJ (Fig. 4b) are considered for the present analysis. Results obtained by the
present method and the conventional boundary stress algorithm are compared with the elasticity
solutions for the same problem. Figures 6, 7 and 8 display the non-dimensionalized tangential
normal stresses axx (along I J) for the meshes given in Fig. 5. The results are observed to converge
to the exact solution for a finer mesh refinement. The agreement of the present method with
that of the exact solution was comparable to that of the conventional method.
L
a
&
r o = 11.25 ks
b
i
I
I
I
I
I
I
I
I
i
J
t
i
I
b
4
e
I
i
I
I
i
I
I
Figs. 4 and 5. 4 a and b Problem of a cantilever beam under transverse end loading; a problem to be solved, L = 12 ft, H = 1.2 It,
E = I . 0 • 106ksf, v=0.3, F = 9 . 0 k ; b boundary conditions, Pt. I(x=4.0ft, y = - 0 . 6 f t ) , Pt. J(x=0.0ft, y = - 0 . 6 f t ) .
5 a - c Boundary element models of a cantilever beam; a mesh 1:18 boundary elements, 40 boundary nodes; b mesh 2:24
boundary elements, 52 boundary nodes; e mesh 3:32 boundary elements, 68 boundary nodes
64
Computational Mechanics 8 (1991)
50
5 0 [/
1
I
Mesh 1 ' 18 B E
40 B.N.
A
Results by the present method
o
Results by the conventional method
-Exact solution
h{esh 2 24 B.E.
52 B.N.
z~ Results by the present method
r7 Results by the conventional method
-Exact solution
40
a0
T
\
\
30-
20
30
20
o:s
6
1-x/O.333L
I
J
~-
J
l
7
0:s
l - x ~ 0.333L
50
Mesh 3 : 32 B.E.
68 B,N.
Results by the present method
D
Results by the conventional method
-Exact solution
40
I
\
30-
20
8
0
I
015
l~x/O.333L
Figs. 6-8. Stress axx along the boundary
8 for mesh 3 (in Fig. 5c)
1J
(in Fig. 4b) of a cantilever beam, 6 for mesh 1 (in Fig. 5a); 7 for mesh 2 (in Fig. 5b);
6.2.2 A simply supported beam under uniformly distributed loading
The data pertaining to the geometry and material properties of the beam is given in Fig. 9. The
three different boundary element meshes considered for the half symmetrical beam are shown
in Fig. 10. The tangential stresses trry along AB and axe, along CB (Fig. 9b) are considered for
the present analysis. Results obtained by the present method and the conventional boundary
stress algorithm are compared with the elasticity solutions for the same problem. Figures 11-16
display the non-dimensionalized tangential normal stresses ary (along AB) and oxx (along CB)
for the meshes given in Fig. 10. The results are observed to converge to the exact solution for
a finer mesh refinement. The agreement of the present method with that of the exact solution
was comparable to that of the conventional method, although in certain instances some improvement over the conventional method is observed.
C. C. Chien et al.: O n the evaluation of hyper-singular integrals
65
[
r
i
a
P
Y
C
Y
a -~
1
b
C
I
I
I
I
I
I
I
I
I
o
~l
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
t
I
I
10
Figs. 9 and 10. 9 a and b. P r o b l e m of a simply supported beam under uniformly distributed loading; a problem to be solved,
L = 12 ft, H = 1.2 It, E = 1.0 x 106 ksf, v = 0.3, p = 1.0 ksf; b b o u n d a r y conditions, Pt. A(x = 0.0 It, y = 0.0 ft), Pt. B(x = 0.0 ft,
y = - 0.6 ft), Pt. C(x = 2.0 ft, y = - 0.6 It). 10 a - c Boundary element models of a simply supported beam; a mesh 1:18 b o u n d a r y
elements, 40 b o u n d a r y nodes; b mesh 2 : 2 4 b o u n d a r y elements, 52 b o u n d a r y nodes; e mesh 3 : 3 2 b o u n d a r y elements, 68
boundary nodes
6.2.3 A tension plate with a circular hole
The data pertaining to the geometry and material properties of the plate is given in Fig. 17. The
three progressively refined boundary element meshes considered for the quarter symmetrical
plate are shown in Figs. 18a-c. Attention is focused on the solution for the tangential normal stress
a T along the periphery of the hole and at the boundary segments near the hole, as marked by the
lines E F - FG -- GH in Fig. 17b. Results obtained by the present method and the conventional
boundary stress algorithm are compared with the elasticity solutions of circular hole in an infinite
plate subjected to remote uniform loading. Figures 19, 20 and 21 display the non-dimensionalized
tangential normal stresses from E - H (Fig. 17b) for each of the meshes given in Fig. 18, respectively. The results are observed to converge to the exact solution for a finer mesh refinement. The
agreement of the present method with that of the exact solution was comparable to that of the
conventional method.
6.2.4 A tension plate with a square hole
The data pertaining to the geometry and material properties of the plate is given in Fig. 22. The
three progressively refined boundary element meshes considered for the quarter symmetrical plate
are shown in Figs. 23a-c. Attention is focused on the solution for the tangential normal stress
along the segment QPO (Fig. 22b) of the periphery of the square hole. The closed form elasticity
solution for such a problem does not exist in the literature. Hence, a comparison between the
present method and the conventional boundary stress algorithm is carried out in the present
analysis. It should also be noted here that the stresses become singular at the reentrant corner P.
Figures 24-29 display the non-dimensionalized tangential normal stresses in the regions QP and
OP, respectively, for the meshes shown in Fig. 23. From Figs. 24-26, it is seen that the results,
for avv along QP, for those points which are far away from the corner point P are observed to
be the same between the present and conventional methods. However, the present method appears
to have a smoother, less steeper variation of stresses near the corner P than the conventional
method. It should also be noted that the finite difference algorithm is quite sensitive to the round
of error in this region. The results for o-xxalong OP, between the present and conventional methods
shown in Figs. 27-29, show similar trends as in Figs. 24-26.
Computational Mechanics 8 (1991)
66
\
D 015
-0.5
Mesh 1 :q8 B E.
40B.N
A
Results by the present method
o
Results by the conventional method
k
O.6
-07
~ -
Mesh 2 : 24 B.E,
-o.6t \
Exac,solnt,o~
I
~
~
--
52 B.N,
R~:::,l::~:~:c'::;::t'io=:~~
Exact solution
-0.7-
t
.~0.8
-0.!
-0.9
-1.0
0
11_
-1.0 .!
0
015
-y/0.5 H
A
1~
]
- 0.6 ~
Mesh 3 ' 32 R.E
\
Z~
~
I
!
D
~
--
6% B N.
Results by the present method
Results by the conventional method
Exact solution
--
-70-
-0 . 9 ~
1_8
-72.
-
74
-76
0
A
~
-66
Exact solution
-68-
\
1.s
,
Mesh1: I 8 B E
40BN.
A Results by the present method
m Results by the conventional method
-66 j
b
-
-y/O.5H.
-64
0.5 9 ,
I ~
0.5
A
0'.5
-},/0.5 H
1..4
"
o
c
-64t1
Mesh 2 : 24 B.E
52 B.N.
n
Results by the present method
[] Results by the conventional method
-66
./\~
o'.5
l - x ~ 0A67 L
*-
Mesh 3 : 32 B.E.
68B.N
z~ Results by the present method
Results by the conventional method
-Exact solution
\
-72-
-74
-74 ~
-76
15
-76
o
C
0:5
o'.5
1- x / 0 . 1 6 7 L
8
1-6
C
1-x/O.167L
~
B
C. C. Chien et al.: On the evaluation of hyper-singular integrals
p
1" t 1' I t l't l't:
T
67
F
t
I
2/
E
1' r 1'
21_
i
GH
p
Xy v
x
a,
I
1
i
I
I
18
b
17
Figs. 17 and 18. 17a, b Problem of a tension plate with a circular hole; a problem to be solved, L = 100mm, R / L = 1/10,
E = 68 GPa (7000 Kgf/mm2), v = 0.49, p = 9.8 MPa (1.0 Kgf/mmZ); b boundary conditions, Pt. E(x = 0.0 mm, y = 30.0 mm), Pt.
H ( x = 30.0 mm, y = 0.0 mm). 18 a-e Boundary element models of a tension plate with a circular hole; a mesh 1:12 boundary
elements, 29 boundary nodes; b mesh 2:14 boundary elements, 33 boundary nodes; c mesh 3:19 boundary elements, 43 boundary
nodes
3
3
Mesh 1 . 12 B,E,
29 B N
A Results by the present method
Results by the conventtonM method
- - Exact solution for infinite domain
~2
~~
Mesh2. i4 B.E
33 B N
Results by the present method
c Results by the conventional method
- - Exact solution for mfimte dmnam
~2
c-
._c
E
J
1hi
CO
W
cn
E
0
\F
,
-1
~ F
E
H
E
Y
-1
19
20
,i/
[]
G
H
3
8
Q2
/
c_
~0
21
4
E
Mesh 3 : 19 B,E.
43 B.N.
A Results by the present method
~: Results by the conventional method
- - Exact solutmn for infinite domain
\
D
G
H
Figs. 19-21. Tangential normal stress ar along the boundary
E F - F G - G H (in Fig. 17b) of a tension plate with a ctrcular
hole, 19 for mesh 1 (in Fig. 18a); 20 for mesh 2 (in Fig. 18b);
21 for mesh (in Fig. 18c)
Figs. 11-16. 11-13 Stress ary along the boundary A B (in Fig. 9b, of a simply supported beam, 11 for mesh 1 (in Fig. 10a;
12 for mesh 2 (in Fig. 10b); 13 for mesh 3 (in Fig. 10c). 14-16 Stress axx along the boundary B C (in Fig. 9b) of a simply
supported beam, 14 for mesh 1 (in Fig. 10a); 15 for mesh 2 (in Fig. 10b); 16 for mesh 3 (in Fig. 10c)
68
Computational Mechanics 8 (1991)
ttttttttt'
Y
2L
P
!
Ii 2L
9 --- 2at-E~
~t t '1' t 1'
......
I
I
b
&
X
Fig. 22 a and b. Problem of a tension plate with a square hole; a problem to be solved, L = 100 mm, a/L = 1/10, E = 68 GPa
(7000Kgf/mm2), v=0.49, p = 9 . 8 M P a (1.0Kgf/mm2); b boundary conditions, Pt. O(x=0.0mm, y = 10.0mm); Pt.
P(x = 10.0 mm, y = 10.0 mm), Pt. Q(x = 10.0ram, y = 0.0mm)
l: I
Part A
%
Detail of part A
,...,7. . . .
c Part B
:
Detail. of part B
Fig. 23 a-e. Boundary element models of a tension plate with a square hole; a mesh 1:18 boundary elements, 42 boundary
nodes; b mesh 2:22 boundary elements, 50 boundary nodes; e mesh 3:34 boundary elements, 74 boundary nodes
7 Conclusion
In the present method, the integral equation for displacement gradients which was used to
compute stresses in the interior of body is also used to compute the stresses on the boundary.
Such a method has been thought of as unattainable in the past, unless a complete reformulation
is carried out in order to reduce the order of the singularity of the integral equation. In the
present approach, the hyper-singular kernels for stresses on the boundary are made numerically
tractable through the imposition of certain displacement modes. This indirect procedure gwes
rise to better numerical smoothing of coefficients multiplying the nodal variables. The accuracy
of the present method is comparable to that of the conventional boundary stress algorithm.
Acknowledgements
The results presented in this paper were obtained during the course of the investigations supported by the the U.S. Office
of Naval Research, with Dr. Y. Rajapakse as the program official. It is a pleasure to thankfully acknowledge this support.
Figs. 24-29. 24-26 Stress c r along the boundary QP (in Fig. 22b) of a tension plate with a square hole, 24 for mesh 1 (in
Fig. 23a); 25 for mesh 2 (in Fig. 23b); 26 for mesh 3 (in Fig. 23c); 27-29 Stress tr~ along the boundary OP (in Fig. 22b) of
a tension plate with a square hole, 27 for mesh 1 (in Fig. 23a); 28 for mesh 2 (in Fig. 23b); 29 for mesh 3 (in Fig. 23c)
C. C. Chien et al.: On the evaluation of hyper-singular integrals
69
Mesh 1 18 B E
42 B N
&
Results by the present method
o
Results by the conventional method
Mesh2' 2 2 B E
50BN
A Results by the present method
[]
Results by the conventional method
3-
r
t31
c~
\
\
#
I
1
d.s
24
o'.s
0
y/o
25
-'-
h
y/u
a
P
0.5Mesh3: 3 4 B E
74BN
Z~
Results by the present method
[]
Results by the conventional method
M e s h l . 18BE.
42BN,
zh
Results by the present method
El Results by the conventional method
D
3
O-
g
\
ck
Q
\
[]
[]
{~
[]
[]
fl
-0.5 ~
~3
[]
1
-1.0
0
26
Q
0.5
0
y/a
=
27
o
0.5-
0'.5
x/o
P
0.5
Mesh2 22B,E,
50BN.
Lx Results by the present method
[]
Results by the conventional method
Mesh3.34BE.
74BN.
A
Results by th~ present method
[]
Results by the conventional method
O
t
\
0
\
-0.5 -
o
A
-0.5-
[]
-1.0
i
0
28
o
-1.0
0.5
X /Cl
- 9
=
P
29
01.5
o
p
70
Computational Mechanics 8 (1991)
References
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Okada, H.; Rajiyah, H.; Atluri, S.N. (1988): A novel displacement gradient boundary element method for elastic stress
analysis with high accuracy. ASME, J. of Appl. Mech. 55, 786-794
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Communicated by G. Yagawa, February 1, 1990