This paper investigates model-order reduction methods for geometrically
nonlinear structures. The parametrisation method of invariant manifolds is used
and adapted to the case of mechanical systems expressed in the physical basis,
so that the technique is directly applicable to problems discretised by the
finite element method. Two nonlinear mappings, respectively related to
displacement and velocity, are introduced, and the link between the two is made
explicit at arbitrary order of expansion. The same development is performed on
the reduced-order dynamics which is computed at generic order following the
different styles of parametrisation. More specifically, three different styles
are introduced and commented: the graph style, the complex normal form style
and the real normal form style. These developments allow making better
connections with earlier works using these parametrisation methods. The
technique is then applied to three different examples. A clamped-clamped arch
with increasing curvature is first used to show an example of a system with a
softening behaviour turning to hardening at larger amplitudes, which can be
replicated with a single mode reduction. Secondly, the case of a cantilever
beam is investigated. It is shown that the invariant manifold of the first mode
shows a folding point at large amplitudes which is not connected to an internal
resonance. This exemplifies the failure of the graph style due to the folding
point, whereas the normal form style is able to pass over the folding. Finally,
A MEMS micromirror undergoing large rotations is used to show the importance of
using high-order expansions on an industrial example.