  Exotic behaviour of mechanical metamaterials often relies on an internal
transformation of the underlying microstructure triggered by its local
instabilities, rearrangements, and rotations. Depending on the presence and
magnitude of such a transformation, effective properties of a metamaterial may
change significantly. To capture this phenomenon accurately and efficiently,
homogenization schemes are required that reflect microstructural as well as
macro-structural instabilities, large deformations, and non-local effects. To
this end, a micromorphic computational homogenization scheme has recently been
developed, which employs the particular microstructural transformation as a
non-local mechanism, magnitude of which is governed by an additional coupled
partial differential equation. Upon discretizing the resulting problem it turns
out that the macroscopic stiffness matrix requires integration of macro-element
basis functions as well as their derivatives, thus calling for a higher-order
integration rules. Because evaluation of constitutive law in multiscale schemes
involves an expensive solution of a non-linear boundary value problem,
computational efficiency can be improved by reducing the number of integration
points. Therefore, the goal of this paper is to investigate reduced-order
schemes in computational homogenization, with emphasis on the stability of the
resulting elements. In particular, arguments for lowering the order of
integration from the expensive mass-matrix to a cheaper stiffness-matrix
equivalent are first outlined. An efficient one-point integration quadrilateral
element is then introduced and proper hourglass stabilization discussed.
Performance of the resulting set of elements is finally tested on a benchmark
bending example, showing that we achieve accuracy comparable to the full
quadrature rules.
