  We present an enhanced version of the parametric nonlinear reduced order
model for shape imperfections in structural dynamics we studied in a previous
work [1]. The model is computed intrusively and with no training using
information about the nominal geometry of the structure and some user-defined
displacement fields representing shape defects, i.e. small deviations from the
nominal geometry parametrized by their respective amplitudes. The linear
superposition of these artificial displacements describe the defected geometry
and can be embedded in the strain formulation in such a way that, in the end,
nonlinear internal elastic forces can be expressed as a polynomial function of
both these defect fields and the actual displacement field. This way, a
tensorial representation of the internal forces can be obtained and, owning the
reduction in size of the model given by a Galerkin projection, high simulation
speed-ups can be achieved. We show that by adopting a rigorous deformation
framework we are able to achieve better accuracy as compared to the previous
work. In particular, exploiting Neumann expansion in the definition of the
Green-Lagrange strain tensor, we show that our previous model is a lower order
approximation with respect to the one we present now. Two numerical examples of
a clamped beam and a MEMS gyroscope finally demonstrate the benefits of the
method in terms of speed and increased accuracy.
