Subspace Iteration

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Subspace Iteration

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The subspace iteration algorithm used in Modes to solve Mathematical (Eq. 1) is described at length in Bathe et al. (1976) and will be only briefly discussed here. The method is particularly suited to structural systems since in general only a small number of solution eigenpairs is required in these cases. Subspace iteration is basically a simultaneous inverse iteration procedure. A small base of vectors, whose number depends on the required number of eigenpairs, is first created; this defines the eigenproblem “subspace”, and this “subspace” is progressively transformed, by iteration, into the space comprising the lowest few eigenvectors of the overall system.

This procedure requires the complete solution of a reduced eigenproblem in each iteration, and this is done in Modes using the Householder and Q-R methods, which are very efficient methods for the complete solution of small eigensystems. The great advantage of subspace iteration is that the eigenpair extraction takes place in a reduced space, giving rapid convergence to the required eigensolution (Bathe et al., 1976). The method is stable, efficient and fast, with runtimes typically less than a couple of minutes.

When specifying the required number of natural frequencies it is recommended that you specify twice the actual required number you are interested in. For example, if the first 5 natural frequencies are of interest, 10 eigenpairs should be requested. This ensures that the actual required values are estimated to an acceptable accuracy (Bathe et al., 1976).