具有几何弹性的一般曲面设计用编织纸条
Weaving paper strips for designing of general curved surface with geometrical elasticity
摘要：
本研究提出了一种新的方法来创建一个一般的曲面“amigami”。基于黎曼流形上的非线性弹性理论，对编织纸带进行了形状优化。通过沿着其坐标切割介质，将目标表面分割成小的弯曲条带，并且每个条带嵌入到平坦的纸张中，以最小化由于平面内变形而引起的应变能泛函。弱形式的平衡方程是从一个李导数与虚拟位移矢量场，并使用Galerkin方法与非均匀B样条流形数值求解方程。作为演示，我们制作了悬链面和螺旋面，这些曲面是由54张纸条波浪而成的。纸艺提醒我们从悬链面到螺旋面的等距变换，反之亦然。我们还提供了严格的数学证明纸带的应变估计。这个估计过程是经典的欧拉-伯努利梁理论到现代几何弹性理论的推广。
英文摘要：
This study proposes 'amigami' as a new method of creating a general curved surface. It conducts the shape optimization of weaving paper strips based on the theory of nonlinear elasticity on Riemannian manifolds. The target surface is split into small curved strips by cutting the medium along with its coordinates, and each strip is embedded into a flat paper sheet to minimize a strain energy functional due to the in-plane deformation. The weak form equilibrium equation is derived from a Lie derivative with the virtual displacement vector field, and the equation is solved numerically using the Galerkin method with a non-uniform B-spline manifold. As a demonstration, we made catenoid and helicoid surfaces which are made by waving 54 paper strips. The papercraft reminds us of the isometric transformation from the catenoid to the helicoid and vice versa. We also provide strain estimates for paper strips with rigorous mathematical proof. This estimating process is a generalization of the classical beam theory of Euler-Bernoulli to a modern geometrical elasticity.
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