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Modeling methods of cylindrical and axisymmetric waterbomb origami based on multi-objective optimization

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Mingyue Zhang
Mingyue Zhang
Mingyue Zhang
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Yan Zhao at Jiangsu University
Yan Zhao
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Abstract and Figures

The basic principle of origami is to use two-dimensional flat materials to obtain various three-dimensional target shapes by folding crease patterns. Among them, the study of waterbomb tessellations inspires the design and functionality implementation of engineering structures. However, there are some optimization spaces when building three-dimensional structures based on waterbomb origami. In this paper, we propose modeling methods to construct cylindrical and axisymmetric three-dimensional waterbomb tessellations with multi-objective optimization. Our methods aim to solve two optimization problems: (I) Constructing waterbomb tessellations to approximate the target surface more accurately. (II) Unifying waterbomb units to construct flat-foldable three-dimensional waterbomb tessellations. In addition, we present waterbomb approximations, the performance of optimization, rigid folding sequences, comparison of crease patterns, and physical origami fabrications to demonstrate the validity of our methods. Our work can expand the exploration of cylindrical and axisymmetric origami-inspired structures, such as foldable roofs, tubular materials, etc. Graphical Abstract
A waterbomb unit, which is intersected by six creases, and a waterbomb crease pattern. a Four types of edges contained in the waterbomb unit, i.e., long, short, horizontal, and bevel edges, whose lengths are denoted by Ll\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\textrm{l}$$\end{document}, Ls\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\textrm{s}$$\end{document}, Lh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\textrm{h}$$\end{document}, and Lb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\textrm{b}$$\end{document}, respectively. b The crease pattern of a waterbomb tessellation consists of four strips (one strip containing three waterbomb units), where Ns=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_\textrm{s}=4$$\end{document}, Nb=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_\textrm{b}=3$$\end{document}. In b, P2P4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_2P_4$$\end{document}, P5P7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_5P_7$$\end{document}, and P8P10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_8P_{10}$$\end{document} are the long edges. P1P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_1P_3$$\end{document}, P3P5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_3P_5$$\end{document}, P4P6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_4P_6$$\end{document}, P6P8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_6P_8$$\end{document}, P7P9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_7P_9$$\end{document}, and P9P11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_9P_{11}$$\end{document} are the short edges. P1P2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_1P_2$$\end{document}, P4P5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_4P_5$$\end{document}, P7P8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_7P_8$$\end{document}, and P10P11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{10}P_{11}$$\end{document} are the horizontal edges. P2P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_2P_3$$\end{document}, P3P4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_3P_4$$\end{document}, P5P6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_5P_6$$\end{document}, P6P7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_6P_7$$\end{document}, P8P9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_8P_9$$\end{document}, and P9P10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_9P_{10}$$\end{document} are the bevel edges. The black dashed frame encloses a waterbomb unit, where αi,k(k=1,2,…,6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{i,k}(k=1,2,\ldots ,6)$$\end{document} is the kth angle of vertex Pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_i$$\end{document}
A waterbomb unit, which is intersected by six creases, and a waterbomb crease pattern. a Four types of edges contained in the waterbomb unit, i.e., long, short, horizontal, and bevel edges, whose lengths are denoted by Ll\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\textrm{l}$$\end{document}, Ls\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\textrm{s}$$\end{document}, Lh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\textrm{h}$$\end{document}, and Lb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_\textrm{b}$$\end{document}, respectively. b The crease pattern of a waterbomb tessellation consists of four strips (one strip containing three waterbomb units), where Ns=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_\textrm{s}=4$$\end{document}, Nb=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_\textrm{b}=3$$\end{document}. In b, P2P4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_2P_4$$\end{document}, P5P7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_5P_7$$\end{document}, and P8P10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_8P_{10}$$\end{document} are the long edges. P1P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_1P_3$$\end{document}, P3P5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_3P_5$$\end{document}, P4P6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_4P_6$$\end{document}, P6P8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_6P_8$$\end{document}, P7P9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_7P_9$$\end{document}, and P9P11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_9P_{11}$$\end{document} are the short edges. P1P2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_1P_2$$\end{document}, P4P5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_4P_5$$\end{document}, P7P8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_7P_8$$\end{document}, and P10P11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{10}P_{11}$$\end{document} are the horizontal edges. P2P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_2P_3$$\end{document}, P3P4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_3P_4$$\end{document}, P5P6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_5P_6$$\end{document}, P6P7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_6P_7$$\end{document}, P8P9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_8P_9$$\end{document}, and P9P10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_9P_{10}$$\end{document} are the bevel edges. The black dashed frame encloses a waterbomb unit, where αi,k(k=1,2,…,6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{i,k}(k=1,2,\ldots ,6)$$\end{document} is the kth angle of vertex Pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_i$$\end{document}
… 
A strip and profile curve for cylindrical and axisymmetric waterbomb tessellations. a A strip of the cylindrical waterbomb tessellation and the corresponding profile curve, where D is the distance of positive translation along the y-axis applied to the profile curve Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document} to obtain the curve Γc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _\textrm{c}$$\end{document}. b A strip of the axisymmetric waterbomb tessellation and the corresponding profile curve, where Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta$$\end{document} is the rotation angle of the profile curve Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document} about the z-axis, Θ=π/Ns\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta =\pi /N_\textrm{s}$$\end{document}
A strip and profile curve for cylindrical and axisymmetric waterbomb tessellations. a A strip of the cylindrical waterbomb tessellation and the corresponding profile curve, where D is the distance of positive translation along the y-axis applied to the profile curve Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document} to obtain the curve Γc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _\textrm{c}$$\end{document}. b A strip of the axisymmetric waterbomb tessellation and the corresponding profile curve, where Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta$$\end{document} is the rotation angle of the profile curve Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document} about the z-axis, Θ=π/Ns\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta =\pi /N_\textrm{s}$$\end{document}
… 
Resulting waterbomb tessellations for approximating cylindrical and axisymmetric target surfaces based on Class I optimization. a U-shaped structure. b Roof. c S-shaped structure. d Ridge. e Gourd. f Funnel. g Hourglass. h Magic ball
Resulting waterbomb tessellations for approximating cylindrical and axisymmetric target surfaces based on Class I optimization. a U-shaped structure. b Roof. c S-shaped structure. d Ridge. e Gourd. f Funnel. g Hourglass. h Magic ball
… 
Resulting cylindrical and axisymmetric waterbomb tessellations with uniform waterbomb units based on Class II optimization. a Tunnel. b Circle. c Plane. d Pottery. e Cylinder. f Ellipse. g Stadium. h Tire. Among them, a, d are composed of uniform waterbomb units, and those of b, c, e–h consist of uniform rectangular waterbomb units
Resulting cylindrical and axisymmetric waterbomb tessellations with uniform waterbomb units based on Class II optimization. a Tunnel. b Circle. c Plane. d Pottery. e Cylinder. f Ellipse. g Stadium. h Tire. Among them, a, d are composed of uniform waterbomb units, and those of b, c, e–h consist of uniform rectangular waterbomb units
… 
Cylindrical and axisymmetric waterbomb tessellations and their crease patterns with uniform square waterbomb units based on Class II optimization. a Bridge and b Magic ball and their crease patterns. These models are made of uniform square waterbomb units
+4
Cylindrical and axisymmetric waterbomb tessellations and their crease patterns with uniform square waterbomb units based on Class II optimization. a Bridge and b Magic ball and their crease patterns. These models are made of uniform square waterbomb units
… 
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Vol.:(0123456789)
Multimedia Systems (2024) 30:135
https://doi.org/10.1007/s00530-024-01326-8
REGULAR PAPER
Modeling methods ofcylindrical andaxisymmetric waterbomb
origami based onmulti‑objective optimization
MingyueZhang1,2· YanZhao2
Received: 25 December 2023 / Accepted: 25 March 2024 / Published online: 25 April 2024
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024, corrected publication 2024
Abstract
The basic principle of origami is to use two-dimensional flat materials to obtain various three-dimensional target shapes by
folding crease patterns. Among them, the study of waterbomb tessellations inspires the design and functionality implemen-
tation of engineering structures. However, there are some optimization spaces when building three-dimensional structures
based on waterbomb origami. In this paper, we propose modeling methods to construct cylindrical and axisymmetric three-
dimensional waterbomb tessellations with multi-objective optimization. Our methods aim to solve two optimization problems:
(I) Constructing waterbomb tessellations to approximate the target surface more accurately. (II) Unifying waterbomb units
to construct flat-foldable three-dimensional waterbomb tessellations. In addition, we present waterbomb approximations,
the performance of optimization, rigid folding sequences, comparison of crease patterns, and physical origami fabrications
to demonstrate the validity of our methods. Our work can expand the exploration of cylindrical and axisymmetric origami-
inspired structures, such as foldable roofs, tubular materials, etc.
Keywords Waterbomb tessellation· Multi-objective optimization· Distance constraint· Uniform waterbomb units
1 Introduction
Origami is an ancient art of paper folding. The principle of
origami is to use flat paper as the material and fold it along
creases to obtain origami structures with various shapes.
The rationality of origami structures and the variability of
their shapes make it possible for scientists and engineers to
study the design and function of origami. Origami struc-
tures are widely used in scientific and engineering fields,
such as DNA origami [1], a self-folding microscale con-
tainer that could be used for controlled drug delivery [2],
a self-deployable origami stent graft based on waterbomb
origami [3], a centimeter-scale crawling robot that is self-
folded from shape-memory composites [4], and simulation
of an origami-based deployable solar array for space satel-
lites [5]. Common types of origami include Miura-ori [6],
Resch origami [7], Yoshimura origami [8], and waterbomb
origami [9].
One of the traditional types of origami is the waterbomb
origami also known as the waterbomb tessellation. There are
two main types of waterbomb units. The first type of water-
bomb unit consists of eight alternating mountain and valley
folds around a central vertex. The second type of waterbomb
origami consists of creases of two mountain folds and four
valley folds around the central vertex. As shown in Fig.1a, the
red line represents mountain folds and the blue line represents
valley creases, which indicate the folding direction of each
crease in the plane. In this paper, we focus on the six-crease
waterbomb tessellation. Owing to its large deployable ratio
between expanded and packaged states, the six-crease water-
bomb tessellation can be potentially used to fold large flat
roofs and space solar panels [10]. We exploit the symmetrical
properties of the waterbomb pattern to inverse the design of
the waterbomb tessellation based on the profile curve of the
Communicated by B. Bao.
* Yan Zhao
yanzhao_cs@ujs.edu.cn
Mingyue Zhang
zmy8833@icloud.com
1 School ofFilm andTelevision, Wuxi City College
ofVocational Technology, 12 Qianou Road, Wuxi214153,
Jiangsu, China
2 School ofComputer Science andCommunication
Engineering, Jiangsu University, 301 Xuefu Road,
Zhenjiang212013, Jiangsu, China
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
ResearchGate has not been able to resolve any citations for this publication.
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Computational Design Methods for Cylindrical and Axisymmetric Waterbomb Tessellations
Article
  • Sep 2021
  • COMPUT AIDED GEOM D
Origami has provided a potential way to construct 3D curved structures by folding flat sheet materials without cutting or stretching. As a traditional origami, waterbomb tessellation is widely studied from aspects of science and engineering. However, users cannot easily utilize this kind of origami to fit curved target surfaces because the underlying geometric constraints limit the design space. In this study, we propose computational design methods for approximating cylindrical and axisymmetric curved surfaces based on waterbomb tessellations. With consideration of symmetry and periodic repetition, a single strip of the waterbomb tessellation is first modelled and then longitudinally and circumferentially replicated to construct cylindrical and axisymmetric waterbomb tessellations, respectively. To fulfill flat-foldability, an optimization process is introduced for minimizing flat-foldable residuals iteratively and then a regulation process of the crease pattern is applied for further reducing such residuals. In addition, we demonstrate waterbomb-derivative tessellations with quad-paddings to expand the design variations. Furthermore, rigid-folding sequences and several physically engineered origami pieces are presented. The proposed methods can be utilized to facilitate the design of origami-inspired structures for various engineering design purposes, such as foldable shelters, tubular structures, metamaterials, and so on.
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Inverse design of deployable origami structures that approximate a general surface
Article
  • Sep 2021
  • INT J SOLIDS STRUCT
Shape-morphing finds widespread utility, from the deployment of small stents and large solar sails to actuation and propulsion in soft robotics. Origami structures provide a template for shape-morphing, but rules for designing and folding the structures are challenging to integrate into broad and versatile design tools. Here, we develop a sequential two-stage optimization framework to approximate a general surface by a deployable origami structure. The optimization is performed over the space of all possible rigidly and flat-foldable quadrilateral mesh origami. So, the origami structures produced by our framework come with desirable engineering properties: they can be easily manufactured on a flat reference sheet, deployed to their target state by a controlled folding motion, then to a compact folded state in applications involving storage and portability. The attainable surfaces demonstrated include those with modest but diverse curvatures and unprecedented ones with sharp ridges. The framework provides not only a tool to design various deployable and retractable surfaces in engineering and architecture, but also a route to optimizing other properties and functionality.
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A novel scheme of folding discretized surfaces of revolution inspired by waterbomb origami
Article
  • Nov 2021
  • MECH MACH THEORY
Origami research is always devoted to exploring the transformation from a paper-like material into a three-dimensional (3D) object, which is widely used in the storage of planar structures. However, origami only has a two-dimensional crease map, so it performs limited effect on the storage of 3D surface. Inspired by waterbomb origami, this paper presents a scheme of folding discretized surfaces of revolution to create foldable 3D geometries. The geometries, which are the approximation of target surfaces of revolution, consist of discretizing surfaces of revolution and have 3D crease maps to guide folding. The discretized surfaces constructed by the approaching scheme can satisfy the error requirements and retain the features of original surfaces. The paper verifies the feasibility of the folding scheme by showing the formula of four typical surfaces, including convex shape, concave shape, concave-convex shape and capsule shape. This scheme of folding discretized surfaces of revolution shows the potential of origami to 3D deployable structures in the field of aerospace.
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DNA origami
Article
  • Dec 2021
Biological materials are self-assembled with near-atomic precision in living cells, whereas synthetic 3D structures generally lack such precision and controllability. Recently, DNA nanotechnology, especially DNA origami technology, has been useful in the bottom-up fabrication of well-defined nanostructures ranging from tens of nanometres to sub-micrometres. In this Primer, we summarize the methodologies of DNA origami technology, including origami design, synthesis, functionalization and characterization. We highlight applications of origami structures in nanofabrication, nanophotonics and nanoelectronics, catalysis, computation, molecular machines, bioimaging, drug delivery and biophysics. We identify challenges for the field, including size limits, stability issues and the scale of production, and discuss their possible solutions. We further provide an outlook on next-generation DNA origami techniques that will allow in vivo synthesis and multiscale manufacturing.
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Bayesian statistics and modelling
Article
  • Dec 2021
Bayesian statistics is an approach to data analysis based on Bayes’ theorem, where available knowledge about parameters in a statistical model is updated with the information in observed data. The background knowledge is expressed as a prior distribution and combined with observational data in the form of a likelihood function to determine the posterior distribution. The posterior can also be used for making predictions about future events. This Primer describes the stages involved in Bayesian analysis, from specifying the prior and data models to deriving inference, model checking and refinement. We discuss the importance of prior and posterior predictive checking, selecting a proper technique for sampling from a posterior distribution, variational inference and variable selection. Examples of successful applications of Bayesian analysis across various research fields are provided, including in social sciences, ecology, genetics, medicine and more. We propose strategies for reproducibility and reporting standards, outlining an updated WAMBS (when to Worry and how to Avoid the Misuse of Bayesian Statistics) checklist. Finally, we outline the impact of Bayesian analysis on artificial intelligence, a major goal in the next decade.
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Constructing Rigid-Foldable Generalized Miura-Ori Tessellations for Curved Surfaces
Article
  • Feb 2021
Origami has shown the potential to approximate three-dimensional curved surfaces by folding through designed crease patterns on flat materials. The Miura-ori tessellation is a widely used pattern in engineering and tiles the plane when partially folded. Based on constrained optimization, this article presents the construction of generalized Miura-ori patterns that can approximate three-dimensional parametric surfaces of varying curvatures while preserving the inherent properties of the standard Miura-ori, including developability, flat foldability, and rigid foldability. An initial configuration is constructed by tiling the target surface with triangulated Miura-like unit cells and used as the initial guess for the optimization. For approximation of a single target surface, a portion of the vertexes on the one side is attached to the target surface; for fitting of two target surfaces, a portion of vertexes on the other side is also attached to the second target surface. The parametric coordinates are adopted as the unknown variables for the vertexes on the target surfaces, while the Cartesian coordinates are the unknowns for the other vertexes. The constructed generalized Miura-ori tessellations can be rigidly folded from the flat state to the target state with a single degree-of-freedom.
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