Hyper-Reduction over Nonlinear Manifolds for Large Nonlinear Mechanical Systems

Shobhit Jain*, Paolo Tiso

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

11 Citations (Scopus)


Common trends in model reduction of large nonlinear finite element (FE)-discretized systems involve Galerkin projection of the governing equations onto a low-dimensional linear subspace. Though this reduces the number of unknowns in the system, the computational cost for obtaining the reduced solution could still be high due to the prohibitive computational costs involved in the evaluation of nonlinear terms. Hyper-reduction methods are then used for fast approximation of these nonlinear terms. In the finite element context, the energy conserving sampling and weighing (ECSW) method has emerged as an effective tool for hyper-reduction of Galerkin-projection-based reduced-order models (ROMs). More recent trends in model reduction involve the use of nonlinear manifolds, which involves projection onto the tangent space of the manifold. While there are many methods to identify such nonlinear manifolds, hyper-reduction techniques to accelerate computation in such ROMs are rare. In this work, we propose an extension to ECSW to allow for hyper-reduction using nonlinear mappings, while retaining its desirable stability and structure-preserving properties. As a proof of concept, the proposed hyper-reduction technique is demonstrated over models of a flat plate and a realistic wing structure, whose dynamics have been shown to evolve over a nonlinear (quadratic) manifold. An online speed-up of over one thousand times relative to the full system has been obtained for the wing structure using the proposed method, which is higher than its linear counterpart using the ECSW.

Original languageEnglish
Article number081008
JournalJournal of Computational and Nonlinear Dynamics
Issue number8
Publication statusPublished - 2019
Externally publishedYes


We thank the anonymous reviewers of this work for several astute observations, which improved the content and quality of the present manuscript. We also thank Ludovic Renson for pointing us to the MATLAB-based tensor product toolbox TPROD. The authors acknowledge the support of the Air Force Office of Scientific Research, Air Force Material Command, USAF under Award No. FA9550-16-1-0096.


  • ECSW
  • Galerkin projection
  • hyper-reduction
  • model order reduction
  • nonlinear dynamics
  • nonlinear manifolds
  • structure-preserving


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