Invariant spectral foliations with applications to model order reduction and synthesis

Abstract

The paper introduces a technique that decomposes the dynamics of a nonlinear system about an equilibrium into low-order components, which then can be used to reconstruct the full dynamics. This is a nonlinear analogue of linear modal analysis. The dynamics is decomposed using Invariant Spectral Foliation (ISF), which is defined as the smoothest invariant foliation about an equilibrium and hence unique under general conditions. The conjugate dynamics of an ISF can be used as a reduced order model. An ISF can be fitted to vibration data without carrying out a model identification first. The theory is illustrated on a analytic example and on free-vibration data of a clamped-clamped beam.

Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds

“…One simply needs to replace λ k with log μ k and Reλ k log jμ k j in formulas ( 13)-( 16) 29 . We close by noting that in a neighborhood of an SSM, an invariant family of surfaces resembling the role of coordinate planes in a linear system exists 66 . This invariant spectral foliation (ISF) can, in principle, be used to generate a nonlinear analogue of linear modal superposition in a vicinity of a fixed point.…”

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

“…The use of normal form to select ex nihilo a suitable nonlinear model before experimental identification, especially in the case of internal resonances, is also a powerful tool [106,184,272,274], which has been recently coupled to phase locked loop experimental continuation method [45,70]. The use of SSM has also been reported for model identification in [266], and a recent contribution proposes to enlarge the scope by using spectral foliations in model identification in order to better take into account transients and orthogonal directions to the SSMs [265].…”

Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds

“…We close by noting that in a neighborhood of an SSM, an invariant family of surfaces resembling the role of coordinate planes in a linear system exists [66]. This invariant spectral foliation (ISF) can, in principle, be used to generate a nonlinear analogue of linear modal superposition in a vicinity of a fixed point.…”