Model reduction for dynamical systems with quadratic output

• Published in: International journal for numerical methods in engineering 2012-07, Vol.91 (3), p.229-248
• Main Authors: Van Beeumen, R., Van Nimmen, K., Lombaert, G., Meerbergen, K.
• Format: Article
• Language: English
• Subjects: Applied sciences; Arnoldi method; Bridges; Buildings. Public works; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); modal superposition; model reduction; Numerical analysis. Scientific computation; Physics; quadratic output; recycling; Sciences and techniques of general use; Solid mechanics; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.4255

SUMMARY 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 are appreciated projection‐based model reduction techniques for linear dynamical systems with linear output. This paper extends the framework of the 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 the 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 particularly important case of structural damping. We also give numerical examples for large‐scale systems corresponding to the forced vibration of a simply supported plate and of an existing footbridge. In this case, a block form of the Padé via Krylov method is used. Copyright © 2012 John Wiley & Sons, Ltd.