A flat plate will bend into a curved shell if it experiences an inhomogeneous
growth field or if constrained appropriately at a boundary. While the forward
problem associated with this process is well studied, the inverse problem of
designing the boundary conditions or growth fields to achieve a particular
shape is much less understood. We use ideas from variational optimization
theory to formulate a well posed version of this inverse problem to determine
the optimal growth field or boundary condition that will give rise to an
arbitrary target shape, optimizing for both closeness to the target shape
and for smoothness of the growth field. We solve the resulting system of
PDE numerically using finite element methods with examples for both the
fully non-symmetric case as well as for simplified one-dimensional and
axisymmetric geometries. We also show that the system can also be solved
semi-analytically by positing an ansatz for the deformation and growth
fields in a circular disk with given thickness profile, leading to paraboloidal,
cylindrical and saddle-shaped target shapes, and show how a soft mode can
arise from a non-axisymmetric deformation of a structure with axisymmetric
material properties.