  The structural properties of mechanical metamaterials are typically studied
with two-scale methods based on computational homogenization. Because such
materials have a complex microstructure, enriched schemes such as second-order
computational homogenization are required to fully capture their non-linear
behavior, which arises from non-local interactions due to the buckling or
patterning of the microstructure. In the two-scale formulation, the effective
behavior of the microstructure is captured with a representative volume element
(RVE), and a homogenized effective continuum is considered on the macroscale.
  Although an effective continuum formulation is introduced, solving such
two-scale models concurrently is still computationally demanding due to the
many repeated solutions for each RVE at the microscale level. In this work, we
propose a reduced-order model for the microscopic problem arising in
second-order computational homogenization, using proper orthogonal
decomposition and a novel hyperreduction method that is specifically tailored
for this problem and inspired by the empirical cubature method. Two numerical
examples are considered, in which the performance of the reduced-order model is
carefully assessed by comparing its solutions with direct numerical simulations
(entirely resolving the underlying microstructure) and the full second-order
computational homogenization model. The reduced-order model is able to
approximate the result of the full computational homogenization well, provided
that the training data is representative for the problem at hand. Any remaining
errors, when compared with the direct numerical simulation, can be attributed
to the inherent approximation errors in the computational homogenization
scheme. Regarding run times for one thread, speed-ups on the order of 100 are
achieved with the reduced-order model as compared to direct numerical
simulations.
