Using spectral submanifolds for optimal mode selection in nonlinear model reduction

Gergely Buza, Shobhit Jain*, George Haller

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

11 Citations (Scopus)

Abstract

Model reduction of large nonlinear systems often involves the projection of the governing equations onto linear subspaces spanned by carefully selected modes. The criteria to select the modes relevant for reduction are usually problem-specific and heuristic. In this work, we propose a rigorous mode-selection criterion based on the recent theory of spectral submanifolds (SSMs), which facilitates a reliable projection of the governing nonlinear equations onto modal subspaces. SSMs are exact invariant manifolds in the phase space that act as nonlinear continuations of linear normal modes. Our criterion identifies critical linear normal modes whose associated SSMs have locally the largest curvature. These modes should then be included in any projection-based model reduction as they are the most sensitive to nonlinearities. To make this mode selection automatic, we develop explicit formulae for the scalar curvature of an SSM and provide an open-source numerical implementation of our mode-selection procedure. We illustrate the power of this procedure by accurately reproducing the forced-response curves on three examples of varying complexity, including high-dimensional finite-element models.

Original languageEnglish
Article number20200725
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume477
Issue number2246
DOIs
Publication statusPublished - 2021
Externally publishedYes

Keywords

  • forced response
  • model reduction
  • spectral submanifolds

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