  The flexoelectric effect, coupling polarization and strain gradient as well
as strain and electric field gradients, is universal to dielectrics, but, as
compared to piezoelectricity, it is more difficult to harness as it requires
field gradients and it is a small-scale effect. These drawbacks can be overcome
by suitably designing metamaterials made of a non-piezoelectric base material
but exhibiting apparent piezoelectricity. We develop a theoretical and
computational framework to perform topology optimization of the representative
volume element of such metamaterials by accurately modeling the governing
equations of flexoelectricity using a Cartesian B-spline method, describing
geometry with a level set, and resorting to genetic algorithms for
optimization. We consider a multi-objective optimization problem where area
fraction competes with four fundamental piezoelectric functionalities
(stress/strain sensor/ actuator). We computationally obtain Pareto fronts, and
discuss the different geometries depending on the apparent piezoelectric
coefficient being optimized. In general, we find competitive estimations of
apparent piezoelectricity as compared to reference materials such as quartz and
PZT ceramics. This opens the possibility to design devices for sensing,
actuation and energy harvesting from a much wider, cheaper and effective class
of materials.
