Information Technology Reference
In-Depth Information
subsegment. Thus, the numerical integration becomes
n
q
ln
1
r
dS
q
j
ln
1
r
dS
m
(
n
)
1
W
(
n
)
j
=
q
j
ln
+
.
(9.109)
r
(ξ
j
)
S
S
j
=
1
j
=
1
j
=
1
=
k
When the segments do not
contain
When the segments contain
ξ
i
ξ
i
i
in parenthesis
of the first term at the left-hand side of Eq. (9.107) should refer to a Gauss point. This leads
to
Substitute Eqs. (9.108) and (9.109) into Eq. (9.107) and note that the point
ξ
m
n
r
k
(ξ
j
)
a
k
(ξ
j
)
W
(
n
)
j
u
(ξ
i
)
−
u
j
π
r
(ξ
j
)
2
=
1
j
=
1
n
q
j
ln
1
r
dS
m
n
1
π
1
W
(
n
)
j
=
q
j
ln
+
(9.110)
r
(ξ
j
)
S
j
=
1
j
=
1
j
=
1
=
k
When the segments do not
contain
When the segments contain
ξ
i
ξ
i
For each
ξ
i
, Eq. (9.110) reduces to a linear equation
=
H
i
V
G
i
P
where
V
and
P
, which are
N
1 vectors, with
N
defined as the number of integration points
on the boundary, contain all the values of
u
and
q
at the Gauss points. The elements of
H
i
and
G
i
are the coefficients of
u
j
and
q
j
in Eq. (9.110). When all the integration points are
spanned, the global equation
×
HV
=
GP
(9.111)
is formed. The matrices
H
and
G
are of order
N
×
N
.
Note that the singularity in Eq. (9.108) oc-
curs at the evaluation of the diagonal elements of
H
This singularity can be avoided by using
the same technique as that used to process the
H
matrix in Sections 9.2.2 and 9.3.5., i.e., the di-
agonal elements are not evaluated when the matrix
H
is formed. After all of the non-diagonal
elements of
H
are calculated, the diagonal elements of
H
can be computed using Eq. (9.39).
Since no boundary elements and, hence, no shape functions are involved in the integra-
tion, the direct integration method appears to be more efficient than the boundary element
method.
It should be noted that for the direct integration method, the segments can be quite large
so that the discretization of the boundary of the domain can be very simple. The total
number of integration points is
N
.
n
, where
m
is the number of segments and
n
is the
number of integration points on each segment. Since relatively few segments are needed,
the number
N
can be small and hence the computation tends to be efficient. Also, since the
Gauss integration points are not located at the ends of the segments, no integration point is
located at the sharp convex and concave corners. Thus, the need to determine the normal
derivative with respect to the outer normal at the sharp corners is avoided.
=
m
×
EXAMPLE 9.6 Torsion Problem
The two cross-sectional shapes of Fig. 9.14 will be used to illustrate the accuracy of this
numerical procedure. Suppose a twisting moment of magnitude 100 occurs at the cross-
sections. For these cross-sections, the boundaries are divided into segments as shown in
Fig. 9.14. An equal number of Gauss integration points is employed on each segment.
Search WWH ::
Custom Search