Computation of Miura surfaces

Miura surfaces are the solutions of a system of nonlinear elliptic 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, further insight into Miura surfaces is presented along with a proof of existence and uniqueness when using appropriate boundary conditions. A numerical method based on a least-squares formulation, $\mathbb{P}^1$-Lagrange finite elements and a Newton method is introduced to approximate them. The numerical method presents an improved convergence rate with respect to previous work and it is more efficient. Finally, numerical tests are performed to demonstrate the robustness of the method.