Information Technology Reference
In-Depth Information
the segments. Then Eq. (9.28) becomes
m
m
uq
∗
dS
qu
∗
dS
π
u
(ξ
)
+
=
(9.107)
i
S
S
=
1
=
1
where
S
is the length of the
i
is a Gauss point on the boundary. The vari-
ables at the Gauss integration points are the unknowns of the problem. The first step for
using the Gauss quadrature is to parameterize the integrands of the integrals in Eq. (9.107)
so that the integration limits are from
th segment and
ξ
−
1 to 1. Then Eq. (6.116) of Chapter 6 can be used for
each of the integrals, i.e.,
n
u
−
r
k
a
k
r
2
u
j
r
k
(ξ
)
a
k
(ξ
)
j
j
uq
∗
dS
W
(
n
)
j
=
dS
≈−
(9.108a)
r
(ξ
)
2
S
S
j
j
=
1
and
q
ln
1
r
dS
n
1
W
(
n
)
j
qu
∗
dS
=
≈
q
j
ln
(9.108b)
r
(ξ
)
S
S
j
j
=
1
where
n
is the number of Gauss points used,
j
represents a Gauss point on segment
S
,u
j
and
q
j
are the values of
u
and
q
at the Gauss points,
W
(
n
)
j
ξ
is the weighting coefficient taken
from Table 6.7,
r
(ξ
j
)
is the distance from point
ξ
i
of Eq. (9.107) to the Gauss point
ξ
j
, and
r
k
(ξ
j
)
0, the integrals in
Eqs.(9.108) become singular. The singularity involved in Eq. (9.108a) will be treated later.
For the integral in Eq. (9.108b) involving
q
, the integrand
q
ln
is the component of
r
(ξ
j
)
in the
x
k
direction. Note that when
r
=
(
1
/
r
)
is singular when
r
→
0,
i.e., when
ξ
j
→
ξ
i
in Fig. 9.12, and numerical integration cannot be used. To overcome
this problem, a special technique is employed. Further divide the
k
th segment containing
ξ
i
into
n
subsegments (Fig. 9.13) with the integration points at the center of these subseg-
ments. If the unknowns are assumed to be constant on these subsegments, the integration
of Eq. (9.108b) can be of the form
q
ln
1
r
dS
q
j
ln
1
r
dS
n
qu
∗
dS
=
=
S
S
S
j
j
=
1
where
S
is the
j
th subsegment of the
k
th segment, and
q
j
is the value of
q
on the
j
th
j
FIGURE 9.13
When
ξ
i
fall in the same segment, divide the
segment into subsegments.
ξ
j
and
Search WWH ::
Custom Search