Structured Pseudospectra in Structural Engineering

This paper presents a new method for computing the pseudospectra of a matrix that respects a prescribed sparsity structure. The pseudospectrum is defined as the set of points in the complex plane to which an eigenvalue of the matrix can be shifted by a perturbation of a certain size. A canonical form for sparsity preserving perturbations is given and a computable formula for the corresponding structured pseudospectra is derived. This formula relates the computation of structured pseudospectra to the computation of the structured singular value (ssv) of an associated matrix. Although the computation of the ssv in general is an NP-hard problem, algorithms for its approximation are available and demonstrate good performance when applied to the computation of structured pseudospectra of medium-sized or highly sparse matrices. The method is applied to a wing vibration problem, where it is compared with the matrix polynomial approach, and to the stability analysis of truss structures. New measures for the vulnerability of a truss structure are proposed, which are related to the ‘distance to singularity’ of the associated stiffness matrix.