Model reduction for dynamical systems with quadratic output

Finite element models for structures and vibrations often lead to second order dynamical systems with large sparse matrices. For large-scale finite element models, the computation of the frequency response function and the structural response to dynamic loads may present a considerable computational cost. Padé via Krylov methods are widely used and appreciated projection-based model reduction techniques for linear dynamical systems with linear output. This paper extends the framework of Krylov methods to systems with a quadratic output arising in linear quadratic optimal control or random vibration problems. Three different two-sided model reduction approaches are formulated based on Krylov methods. For all methods, the control (or right) Krylov space is the same. The difference between the approaches lies thus in the choice of the observation (or left) Krylov space. The algorithms and theory are developed for the important particular case of Rayleigh damping. We also give a numerical example for a large-scale system corresponding to the forced vibration of a simply supported plate. In this case a block form of the Padé via Krylov method is used.