  Flexoelectricity shows promising applications for self-powered devices with
its increased power density. This paper presents a second-order computational
homogenization strategy for flexoelectric composite. The macro-micro scale
transition, Hill-Mandel energy condition, periodic boundary conditions, and
macroscopic constitutive tangents for the two-scale electromechanical coupling
are investigated and considered in the homogenization formulation. The
macrostructure and microstructure are discretized using $C^1$ triangular finite
elements. The second-order multiscale solution scheme is implemented using
ABAQUS with user subroutines. Finally, we present numerical examples including
parametric analysis of a square plate with holes and the design of
piezoelectric materials made of non-piezoelectric materials to demonstrate the
numerical implementation and the size-dependent effects of flexoelectricity.
