  We determine the material parameters in the relaxed micromorphic generalized
continuum model for a given periodic microstructure in this work. This is
achieved through a least squares fitting of the total energy of the relaxed
micromorphic homogeneous continuum to the total energy of the fully-resolved
heterogeneous microstructure, governed by classical linear elasticity. The
relaxed micromorphic model is a generalized continuum that utilizes the $\Curl$
of a micro-distortion field instead of its full gradient as in the classical
micromorphic theory, leading to several advantages and differences. The most
crucial advantage is that it operates between two well-defined scales. These
scales are determined by linear elasticity with microscopic and macroscopic
elasticity tensors, which respectively bound the stiffness of the relaxed
micromorphic continuum from above and below. While the macroscopic elasticity
tensor is established a priori through standard periodic first-order
homogenization, the microscopic elasticity tensor remains to be determined.
Additionally, the characteristic length parameter, associated with curvature
measurement, controls the transition between the micro- and macro-scales. Both
the microscopic elasticity tensor and the characteristic length parameter are
here determined using a computational approach based on the least squares
fitting of energies. This process involves the consideration of an adequate
number of quadratic deformation modes and different specimen sizes. We conduct
a comparative analysis between the least square fitting results of the relaxed
micromorphic model, the fitting of a skew-symmetric micro-distortion field
(Cosserat-micropolar model), and the fitting of the classical micromorphic
model with two different formulations for the curvature...
