https://hal.science/hal-03283647v2Touzé, CyrilCyrilTouzéIMSIA - UMR 9219 - Institut des Sciences de la mécanique et Applications industrielles - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - ENSTA Paris - École Nationale Supérieure de Techniques Avancées - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - EDF R&D - EDF R&D - EDF - EDFVizzaccaro, AlessandraAlessandraVizzaccaroThomas, OlivierOlivierThomasModel order reduction methods for geometrically nonlinear structures: a review of nonlinear techniquesHAL CCSD2021Nonlinear normal modes of vibrationReduced order models ROMGeometric nonlinearityThin structuresInvariant manifold parametrisationNonlinear mapping[SPI.MECA.VIBR] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Vibrations [physics.class-ph][MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS][SPI.MECA.STRU] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Structural mechanics [physics.class-ph]Touzé, Cyril2021-07-19 11:25:422023-02-08 17:11:182021-07-22 10:31:09enJournal articleshttps://hal.science/hal-03283647v2/document10.1007/s11071-021-06693-9https://hal.science/hal-03283647v1application/pdf2This 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.