基于图论方法与优化算法的六折痕折纸结构构型研究

近年来，折纸结构快速发展并成功应用于诸多领域，但其仍存在折纸新构型难以构建、构型与折展运动难以协调等难题。为了研发多折痕折纸结构，考虑折纸构型的规整性及可折叠性，基于图论方法描述折纸折痕构型，建立折纸顶点的几何约束方程组，综合采用图解法及优化方法，求解单元内部可平折点，得到一系列基于不同对称单元的六折痕折纸结构构型。研究表明，该折痕设计方法精确可行，且具有并行性，能求得给定单元形状的所有内部可平折点。该折痕设计方法适用于任意边界条件的六折痕折纸结构，可为折纸结构创新与应用发挥积极作用。

Research on origami patterns for six-fold origami structures using graph theory and optimization algorithms

Recent years, origami structures develope rapidly and have been successfully applied in various engineering fields. However, there are still some fundamental scientific problems to be solved such as difficult construction of new origami patterns and difficult coordination between configurations and folding motions. In order to explore multi-fold origami structure, the origami crease configuration is described based on the graph theory method considering the regularity and foldability of the origami configuration, and the geometric constraint equations of origami vertices are established. Through flat foldability of each vertex, nodal coordinates of the inner points of basic units are evaluated using graphic and optimization methods. A series of novel six-fold origami structures based on different symmetrical units are proposed. The numerical results show that the proposed origami design method is not only accurate and feasible, but also parallel. All interior points of a given boundary shape can be effectively obtained to satisfy the flat-foldability of six-fold origami. The presented method is suitable for six-fold origami structures with arbitrary boundary conditions. It can play a positive role in design and engineering applications of origami structures.