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.

翻译：Miura曲面是一类约束非线性椭圆方程组的解。该方程组通过Miura折叠的均匀化方法导出，Miura折叠是一种在工程中具有多种应用的折纸结构。先前的研究给出了解存在的次优条件，并提出了一种$H^2$-协调有限元方法来逼近这类解。本文采用梯度形式研究Miura曲面的存在性，并证明了在一定假设下约束条件会从边界传播到区域内部。随后，提出了一种基于稳定化最小二乘格式、协调有限元及牛顿法的数值方法来逼近Miura曲面。该数值方法被证明具有收敛性，并通过数值实验验证了其鲁棒性。