  We apply the Proper Orthogonal Decomposition (POD) method for the efficient
simulation of several scenarios undergone by Micro-Electro-Mechanical-Systems,
involving nonlinearites of geometric and electrostatic nature. The former type
of nonlinearity, associated to the large displacements of the devices, leads to
polynomial terms up to cubic order that are reduced through exact projection
onto a low-dimensional subspace spanned by the Proper Orthogonal Modes (POMs).
On the contrary, electrostatic nonlinearities are modeled resorting to
precomputed manifolds in terms of the amplitudes of the electrically active
POMs. We extensively test the reliability of the assumed linear trial space in
challenging applications focusing on resonators, micromirrors and arches also
displaying internal resonances. We discuss several options to generate the
matrix of snapshots using both classical time marching schemes and more
advanced Harmonic Balance (HB) approaches. Furthermore, we propose a comparison
between the periodic orbits computed with POD and the invariant manifold
approximated with Direct Parametrization approaches, further stressing the
reliability of the technique and its remarkable predictive capabilities, e.g.,
in terms of estimation of the frequency response function of selected output
quantities of interest.
