Civil Engineering Reference
In-Depth Information
e
n
e
n
The integrals
'
R
and
'
S
are evaluated as explained in Chapter 9. The integrals
ˆ
n
'
E
can be evaluated using Gauss Quadrature as explained previously if the point
P
is
not one of the cell nodes. If
P
a
coincides with the nodes of cells, then the integrand tends
to infinity with
o(r
2
)
for 2-D and
o(r
3
)
for 3-D problems and special attention has to be
given to the evaluation of
ˆ
n
'
E
. As explained in Chapter 13 for the case with initial
stresses a small zone of exclusion is assumed around
P
and this results in the “free
ˆ
n
term”
F ı
. Now, however we need to evaluate the strongly singular integral
'
E
P
0
over the cells excluding the spherical region with radiusH.
z
S
c
dr
RS
()
c
dS
c
d
M T
cos
d
T
T
r
P
H
y
M
x
Figure 15.9
Polar coordinates for integration with a spherical region of exclusion
For this the singularity isolation method
9
is used. The singularity is isolated be rewriting
the strongly singular domain integral in the form
³
ˆ
³
ˆ
E
(,) ()
PQ
ı
Q dV
E
(,)
PQ
ª
ı
()
Q
ı
()
P
º
dV
¬
¼
a
0
a
0
0
a
V
V
(15.47)
ˆ
³
ı
(
PPQdV
)
E
(
,
)
0
a
a
V
The first integral on the right had side of Eq. (15.47) is weakly singular and can be
integrated numerically using the cell subdivision technique. The strong singularity has
been moved to the second integral and can be treated semi-analytically. For this the
domain is divided into a singular and a regular domain. The singular domain is bounded
by the faces of the cells that contain point
P
surrounded by the region of exclusion as
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