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 and numerical tests are performed to demonstrate its robustness.

翻译：Miura曲面是约束非线性椭圆系统的解，该方程组通过对Miura折痕（一种在工程中具有多种应用的折纸结构）进行均匀化推导得出。前期研究给出了解存在的次优条件，并提出了用于逼近解的$H^2$协调有限元方法。本文采用混合格式研究Miura曲面的存在性，同时证明在适当选择的边界条件下，约束条件从边界传播至区域内部。随后引入基于最小二乘格式、Taylor-Hood有限元及牛顿法的数值方法逼近Miura曲面，并证明该数值方法收敛。数值实验验证了方法的鲁棒性。