  Dimensionality reduction in mechanical vibratory systems poses challenges for
distributed structures including geometric nonlinearities, mainly because of
the lack of invariance of the linear subspaces. A reduction method based on
direct normal form computation for large finite element (FE) models is here
detailed. The main advantage resides in operating directly from the physical
space, hence avoiding the computation of the complete eigenfunctions spectrum.
Explicit solutions are given, thus enabling a fully non-intrusive version of
the reduction method. The reduced dynamics is obtained from the normal form of
the geometrically nonlinear mechanical problem, free of non-resonant monomials,
and truncated to the selected master coordinates, thus making a direct link
with the parametrisation of invariant manifolds. The method is fully expressed
with a complex-valued formalism by detailing the homological equations in a
systematic manner, and the link with real-valued expressions is established. A
special emphasis is put on the treatment of second-order internal resonances
and the specific case of a 1:2 resonance is made explicit. Finally,
applications to large-scale models of Micro-Electro-Mechanical structures
featuring 1:2 and 1:3 resonances are reported, along with considerations on
computational efficiency.
