  The domain size dependence of piezoelectric properties of ferroelectrics is
investigated using a continuum Ginzburg-Landau model that incorporates the
long-range elastic and electrostatic interactions. Microstructures with desired
domain sizes are created by quenching from the paraelectric phase by biasing
the initial conditions. Three different two-dimensional microstructures with
different sizes of the $90^{o}$ domains are simulated. An electric field is
applied along the polar as well as non-polar directions and the piezoelectric
response is simulated as a function of domain size for both cases. The
simulations show that the piezoelectric coefficients are enhanced by reducing
the domain size, consistent with recent experimental results of Wada and
Tsurumi (Brit. Ceram. Trans. {\bf 103}, 93, 2004) on domain engineered
$BaTiO_{3} $
