Computation of Miura surfaces with gradient Dirichlet boundary conditions

Miura surfaces are the solutions of a constrained nonlinear elliptic system of equations. This system is derived by homogenization from the Miura fold, which is a type of origami fold with multiple applications in engineering. A previous inquiry gave suboptimal conditions for existence of solutions and proposed an $H^{2}$-conformal finite element method to approximate them. In this paper the existence of Miura surfaces is studied using a gradient formulation. It is also proved that, under some hypotheses, the constraints propagate from the boundary to the interior of the domain. Then, a numerical method based on a stabilized least-square formulation, conforming finite elements and a Newton method, is introduced to approximate Miura surfaces. The numerical method is proved to converge and numerical tests are performed to demonstrate its robustness.