  The Miura ori is a very classical origami pattern used in numerous
applications in Engineering. A study of the shapes that surfaces using this
pattern can assume is still lacking. A constrained nonlinear partial
differential equation (PDE) that models the possible shapes that a periodic
Miura tessellation can take in the homogenization limit has been established
recently and solved only in specific cases. In this paper, the existence and
uniqueness of a solution to the unconstrained PDE is proved for general
Dirichlet boundary conditions. Then a $H^2$-conforming discretization is
introduced to approximate the solution of the PDE coupled to a Newton method to
solve the associated discrete problem. A convergence proof for the method is
given as well as a convergence rate. Finally, numerical experiments show the
robustness of the method and that non trivial shapes can be achieved using
periodic Miura tessellations.
