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 mixed formulation. It is also proved that the constraints propagate from the boundary to the interior of the domain for well-chosen boundary conditions. Then, a numerical method based on a least-squares formulation, Taylor--Hood finite elements and a Newton method is introduced to approximate Miura surfaces. The numerical method is proved to converge at order one in space and numerical tests are performed to demonstrate its robustness.

翻译：Miura曲面是约束非线性椭圆方程组的解。该方程组通过对Miura折痕进行均匀化推导得到，Miura折痕是一种在工程领域具有多种应用的折纸结构。先前研究给出了解存在的次优条件，并提出了基于$H^2$协调有限元法的近似方法。本文采用混合公式研究Miura曲面的存在性，同时证明在精心选取边界条件时，约束条件会从边界传播到区域内部。随后引入一种基于最小二乘公式、Taylor-Hood有限元以及牛顿法的数值方法对Miura曲面进行近似。该方法被证明在空间上具有一阶收敛性，并通过数值实验验证了其鲁棒性。