  An optimization procedure determining the ideal configuration at the
microstructural level of ferroelectric (FE) materials is applied to maximize
piezoelectricity. Piezoelectricity in ceramic FEs differ significantly from
that of single crystals because of the presence of crystallites (grains)
possessing crystallographic axes aligned imperfectly. The piezoelectric
properties of a polycrystalline (ceramic) FE is inextricably related to the
grain orientation distribution (texture). The set of combination of variables,
known as solution space, which dictates the texture of a ceramic is unlimited
and hence the choice of the optimal solution which maximizes the
piezoelectricity is complicated. Thus a stochastic global optimization combined
with homogenization is employed for the identification of the optimal granular
configuration of the FE ceramic microstructure with optimum piezoelectric
properties. The macroscopic equilibrium piezoelectric properties of
polycrystalline FE is calculated using mathematical homogenization at each
iteration step. The configuration of grains characterised by its orientations
at each iteration is generated using a randomly selected set of orientation
distribution parameters. Apparent enhancement of piezoelectric coefficient
$d_{33}$ is observed in an optimally oriented BaTiO$_3$ single crystal. A
configuration of crystallites, simultaneously constraining the orientation
distribution of the c-axis (polar axis) while incorporating ab-plane
randomness, which would multiply the overall piezoelectricity in ceramic
BaTiO$_{3}$ is also identified. The orientation distribution of the c-axes is
found to be a narrow Gaussian distribution centred around ${45^\circ}$. The
piezoelectric coefficient in such a ceramic is found to be nearly three times
as that of the single crystal.
