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

TL;DR

This review paper discusses nonlinear model order reduction techniques for geometrically nonlinear structures, emphasizing invariant manifold theory, and covers historical development, computational methods, and recent advances with practical examples.

Contribution

It provides a comprehensive overview of invariant manifold-based nonlinear reduction methods, unifying various approaches and highlighting recent computational advancements.

Findings

Invariant manifolds enable nonlinear reduction without basis expansion.

Recent methods allow direct computation of reduced models from invariant manifold theory.

Applications demonstrate effectiveness in complex structural models.

Abstract

This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes (NNMs) and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations (PDE). They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then the specific case of structures discretized with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models (ROMs) relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted.