  The homogenization of periodic elastic composites is addressed through the
reformulation of the local equations of the mechanical problem in a geometric
functional setting. This relies on the definition of Hilbert spaces of
kinematically and statically admissible tensor fields, whose orthogonality and
duality properties are recalled. These are endowed with specific energetic
scalar products that make use of a reference and uniform elasticity tensor. The
corresponding strain and stress Green's operators are introduced and
interpreted as orthogonal projection operators in the admissibility spaces. In
this context and as an alternative to classical minimum energy principles, two
geometric variational principles are investigated with the introduction of
functionals that aim at measuring the discrepancy of arbitrary test fields to
the kinematic, static or material admissibility conditions of the problem. By
relaxing the corresponding local equations, this study aims in particular at
laying the groundwork for the homogenization of composites whose constitutive
properties are only partially known or uncertain. The local fields in the
composite and their macroscopic responses are computed through the minimization
of the proposed geometric functionals. To do so, their gradients are computed
using the Green's operators and gradient-based optimization schemes are
discussed. A FFT-based implementation of these schemes is proposed and they are
assessed numerically on a canonical example for which analytical solutions are
available.
