三浦折叠曲面的$H^2$-共形逼近
$H^2$-conformal approximation of Miura surfaces
AI总结：
本文针对周期性三浦镶嵌形状建模的无约束偏微分方程，证明一般狄利克雷边界下解的存在唯一性，提出结合牛顿法的$H^2$-协调离散化方法并证明收敛性，数值实验验证方法鲁棒性且可生成非平凡形状。
摘要：
三浦折（Miura ori）是一种极为经典的折纸图案，在工程领域有着众多应用。目前，针对采用该图案的曲面所能呈现的形状，相关研究仍较为匮乏。近期，学界已建立起一个带约束的非线性偏微分方程（PDE），用于刻画周期性三浦镶嵌在均匀化极限下可呈现的形状，但该方程仅在特定情形下得到求解。本文中，针对一般狄利克雷边界条件，证明了无约束偏微分方程解的存在性与唯一性。随后，本文引入一种$H^2$-协调离散化方法来逼近该偏微分方程的解，并结合牛顿法求解对应的离散问题。文中给出了该方法的收敛性证明以及收敛速率。最后，数值实验验证了该方法的鲁棒性，同时表明周期性三浦镶嵌可生成非平凡的形状。
英文摘要：
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.
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