Civil Engineering Reference
In-Depth Information
Table 3.3 Global matrix assembly for mesh in Figure 3.5. Superscripts indi-
cate element numbers
k
2
,
2
k
2
,
3
k
2
,
1
k
2
,
4
0
0
0
0
k
3
,
2
k
3
,
3
+
k
2
,
2
k
2
,
3
k
3
,
1
k
3
,
4
+
k
2
,
1
k
2
,
4
0
0
k
3
,
2
k
3
,
3
k
3
,
1
k
3
,
4
0
0
0
0
k
1
,
2
k
1
,
3
k
1
,
1
+
k
2
,
2
k
1
,
4
+
k
2
,
3
k
2
,
1
k
2
,
4
0
0
k
4
,
2
k
4
,
3
+
k
1
,
2
k
1
,
3
k
4
,
1
+
k
3
,
2
k
4
,
4
+
k
1
,
1
+
k
3
,
3
k
1
,
4
k
3
,
1
k
3
,
4
k
4
,
2
k
4
,
3
k
4
,
1
k
4
,
4
0
0
0
0
k
1
,
2
k
1
,
3
k
1
,
1
k
1
,
4
0
0
0
0
k
4
,
2
k
4
,
3
k
4
,
1
k
4
,
4
0
0
0
0
This system or global matrix is symmetrical provided its constituent matrices are sym-
metrical. The matrix also possesses the useful property of “bandedness”, which means that
the non-zero terms are concentrated around the “leading diagonal” which stretches from
the upper left to the lower right of the table. In this example, no term in any row can be
more than four locations removed from the leading diagonal, so the system is said to have
a “semi-bandwidth” of
nband = 4
. This can be obtained by inspection from Figure 3.5
by subtracting the lowest from the highest global freedom number in each element.
The importance of efficient mesh numbering is illustrated for a mesh of line elements
in Figure 3.6 where the scheme in parentheses has
nband = 13
compared to the scheme
using circles with
nband = 2
.
If system symmetry exists it should also be taken into account. Using a constant band-
width storage strategy, the system in Table 3.3 would require 40 storage locations (eight
equations times five terms on each line). Greater efficiency can be achieved through “sky-
line” storage (Bathe, 1996), where the variability of the bandwidth is taken into account,
requiring 27 storage locations in this case. Most of the programs described in this topic
make use of this variable bandwidth or “skyline” storage strategy (see Figure 3.18 for
examples of different storage strategies).
1
2
4
6
8
10
(1)
(2)
(3)
(4)
(5)
(6)
3
(14)
(7)
12
(13)
(12)
(11)
(10)
(9)
(8)
5
7
9
11
13
14
Figure 3.6 Alternative mesh numbering schemes
