The Miura ori is a very classical origami pattern used in numerous applications in Engineering. A study of the shapes that surfaces using this pattern can assume is still lacking. A constrained nonlinear partial differential equation (PDE) that models the possible shapes that a periodic Miura tessellation can take in the homogenization limit has been established recently and solved only in specific cases. In this paper, the existence and uniqueness of a solution to the unconstrained PDE is proved for general Dirichlet boundary conditions. Then a $H^2$-conforming discretization is introduced to approximate the solution of the PDE coupled to a Newton method to solve the associated discrete problem. A convergence proof for the method is given as well as a convergence rate. Finally, numerical experiments show the robustness of the method and that non trivial shapes can be achieved using periodic Miura tessellations.

翻译：Miura折纸是一种非常经典的折纸图案，在工程领域有诸多应用。然而，目前尚缺乏对这一图案形成的曲面形状的系统研究。近期，研究者建立了一个约束非线性偏微分方程（PDE），用以描述周期性Miura镶嵌在均匀化极限下可能呈现的形状，但该方程仅在特定情况下得到求解。本文首先证明了一般狄利克雷边界条件下无约束PDE解的存在唯一性。随后，引入了一种$H^2$-共形离散化方法来近似该PDE的解，并结合牛顿法求解相应的离散问题。文中给出了该方法的收敛性证明及收敛速率。最后，数值实验表明该方法具有鲁棒性，且周期性Miura镶嵌能够实现非平凡形状。