  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.
