4-3 Bandwidth

The stiffness matrix K in a typical finite problem can be characterized as symmetric, sparse, and band structured. A banded matrix is one in which the non-zero terms of the matrix are clustered about the main diagonal. A banded matrix is shown in the illustration below; the nonzero terms are denoted by ‘x’.

In the hypothetical case above, all nonzero values lie within a band three columns from the main diagonal. The factorization methods discussed in section 4-2 do not require storage of terms that lie outside the band. Thus it is desirable to have the smallest possible bandwidth when using solution methods of this type.

The maximum bandwidth for a given matrix can be found by considering each element in the mesh and finding the element with the largest difference in node. In the example show, truss elements are joined to form a mesh that is four nodes wide and two nodes deep. The stiffness matrix for this node arrangement is expressed symbolically by the matrix K_a below.

We note that the band of nonzero elements in K_a extends five columns beyond the main diagonal. It may be possible to reduce the bandwidth of this matrix by numbering the nodes differently. If, for example, we renumber the nodes according the scheme shown, we find the following form for the global stiffness matrix, K_b.

With the alternative numbering, the bandwidth is reduced to three columns beyond the main diagonal. Node renumbering software is often employed to reduce matrix bandwidth, computer storage requirements, and solution time. In many commercial FEM codes, its operation is automatic and requires no action on the part of the user.

In the examples above we note that a number of zero terms occur within the matrix band. For large three-dimensional problems, the number of such terms can be quite high, and it is common to use sparse matrix storage methods and iterative solution techniques for such problems, particularly if many nonzero terms fall within the band. In these cases only the nonzero terms and their locations within the array are stored. The bandwidth of the matrix then looses its importance for storage and solution efficiency and is no longer a consideration in the computational process.