How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Shobhit Jain*, George Haller

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

39 Citations (Scopus)


Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

Original languageEnglish
Pages (from-to)1417-1450
Number of pages34
JournalNonlinear Dynamics
Issue number2
Publication statusPublished - 2022
Externally publishedYes


We are thankful to Mingwu Li for help in using coco , for his careful proof-reading of this manuscript and for providing valuable comments. We also thank Harry Dankowicz for helpful suggestions and pointing us to the atlas-k d algorithm in coco.


  • Center manifolds
  • Finite elements
  • Invariant manifolds
  • Lyapunov subcenter manifolds
  • Normal forms
  • Reduced-order modeling
  • Spectral submanifolds


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  • SSMTool

    Jain, S. (Creator), Thurnher, T. (Contributor), Li, M. (Contributor) & Haller, G. (Contributor), GitHub/Zenodo, 18 Oct 2022

    Dataset/Software: Software

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