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FIGURE 5.13
A banded stiffness matrix. Coefficients within the band
are usually non-zero.
This row contains non-zero terms, in addition to those on the main diagonal, for the DOF at
the node and for the DOF of the other nodes of the elements connected to the primary node.
All other coefficients in this row of the stiffness matrix are zero. It can be concluded, then,
that each row of the stiffness matrix contains non-zero terms only for the DOF belonging
to the elements meeting at the node containing the force for which this row is written. For
a structure of many nodes and elements, the stiffness matrix may appear to contain mostly
zero terms with relatively few non-zero terms. In such cases, the matrix is said to be
sparse
or
weakly populated
.
Some solution procedures for systems of linear equations can take advantage of the
sparseness of a matrix, particularly if the non-zero terms are clustered close to the diagonal,
i.e., if the matrix is
banded
. Also, storage of the stiffness matrix in a computer is simplified and
efficient for such an arrangement. The non-zero terms can be placed close to the diagonal
by judiciously choosing the numbering system for the DOF. The DOF should be numbered
such that a columnar distance from the main diagonal to the most remote non-zero term in a
particular row is minimized. This is referred to as minimizing the
bandwidth
(Fig. 5.13). The
two numberings of the joints (which, in terms of minimizing the bandwidth, correspond
closely to the DOF) of the frame of Fig. 5.14 illustrate the influence of the numbering scheme
for a structure. Usually, a small bandwidth results if nodes across the shorter dimension of
a structure are numbered consecutively. This certainly holds for the numbering schemes of
Fig. 5.14.
Non-zero elements of the stiffness matrix of Fig. 5.13 are contained in the
NB
coefficients
of the semi-band. Since, in practice,
N
is much greater then
B
, it can be important to avoid
storing
N
2
coefficients by retaining only the
NB
essential coefficients. In terms of equation-
solving efficiency, it has been shown [Cook, et al., 1989] that the savings achieved by
utilizing
NB
coefficients rather than all coefficients in the upper triangle is proportional to
(
2
.
A format for the storage of a banded matrix is shown in Fig. 5.15. Rows of the upper
semi-band are simply shifted to the left. The first row remains in place, the second row is
shifted 1 space, the third row 2 spaces, and the
k
th row
k
N
/
B
)
1 spaces. This places the diagonal
coefficients of the matrix of Fig. 5.13 in the first column of the array of Fig. 5.15.
There is considerable literature (see, for example, Cook, et al., (1989)) on economical
methods of retaining only the essential information of a stiffness matrix. Although the band
format is an efficient method, other techniques have been designed which are even more
efficient (e.g., see Everstine (1979)). Also, schemes have been devised (e.g., Everstine (1979)
and Gibbs, et al., (1976)) for the automatic renumbering of nodes, so that a criterion such
−
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