结合 Origami 和 Kirigami 实现封闭曲面的形状和拓扑形变

[中文正文内容]

封闭曲面通常比开放曲面更耐形变和形状改变。例如，一个空的封闭盒子比打开时更坚固和稳定。开口的存在使其约束更少、更易变形、更容易改变形状，许多关于开放曲面在图案、材料和尺度上形变的研究都证明了这一点。本文介绍了一个利用 Origami 和 Kirigami 原理平衡集成的双稳态封闭曲面形变平台。通过协调面板绕几乎与封闭曲面相切的折痕旋转，以及面板绕几乎与封闭曲面垂直的铰链旋转，我们展示了 Origami-Kirigami 组合体可以在立方体和球体之间进行形状形变，在不同大小的球体之间进行缩放，并在球体和圆环之间改变拓扑结构，同时具有可编程的双稳态。该框架为设计具有封闭构型的双稳态可重构结构和超材料提供了一种有前景的策略。

Abstract

A closed surface is generally more resistant to deformation and shape changes than an open surface. An empty closed box, for example, is stiffer and more stable than when it is open. The presence of an opening makes it less constrained, more deformable, and easier to morph, as demonstrated by several studies on open-surface morphing across patterns, materials, and scales. Here, we present a platform to morph closed surfaces with bistability that harnesses a balanced integration of origami and kirigami principles. By harmonizing panel rotation around creases nearly tangent to the closed surface and panel rotation around hinges nearly perpendicular to the closed surface, we show that origami-kirigami assemblages can shape-morph between a cube and a sphere, scale between spheres of dissimilar size, and change topology between a sphere and a torus, with programmed bistability. The framework offers a promising strategy for designing bistable reconfigurable structures and metamaterials with enclosed configurations.

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INTRODUCTION

Shape morphing, the ability of an object to change shape in response to an external stimulus or control input (1, 2), is commonly observed in natural organisms to achieve various functionalities. Likewise, in the synthetic world, shape morphing is pursued for the design of shape-shifting materials and multifunctional structures, such as hydrogels (2), shape memory alloys (3), living composites (4), liquid crystal elastomers (5), architected dielectric elastomers (6), and origami/kirigami structures (7). One of their common traits stems from their morphable attributes arising from either the base material or/and the material architecture, both contributing to delivering reconfigurability, large dimensional changes, foldability, and other morphing-driven functionalities. Previous research has so far focused on the transformation of shapes either from a flat state to a curved shape (8– 11) or between distinct curved shapes (12, 13). These transformations are all characterized by open surfaces while morphing from an open surface to a closed surface has also been reported (14, 15). Morphing between closed surfaces, however, is now unexplored, yet it bears large scientific implications and practical applications, as described below. In topology, a closed surface is a two-dimensional (2D) manifold that has no infinitely distant points and has no boundary (16). Typical closed surfaces include the sphere and the torus, which are orientable in the 3D space with distinguishable interior and exterior surfaces forming two-sided surfaces, and the non-orientable Klein bottle, which, on the other hand, is one-sided (17). In contrast, while not typically considered a standard mathematical term, open surface in physics often refers to a surface with a defined boundary (18), such as a spherical cap (with one boundary) and cylindrical surface (with two boundaries). Closed surfaces have intrinsic characteristics that can offer advantages over open surfaces in various aspects. On the geometric front, for instance, the existence of a boundary implies that an open surface cannot seal off a volume, while an orientable closed surface can break the 3D space into separate regions, an interior, and an exterior. On the mechanics front, a closed surface provides structural stiffness and stability by distributing internal forces more uniformly over the entire geometry. A typical example is the “shear flow” in the closed cross section of a tube, which can resist torsion much more effectively than its counterpart shear flow in an open cross section (19). On the application front, a closed surface can encapsulate contents within its volume, offering protection against external factors such as contaminants, radiation, or physical impact. The ability to model and predict the morphing of closed surfaces would enable the optimization of both structural and protective properties, benefiting a diverse range of applications involving, for example, controlled release of drugs (20), selective electromagnetic shielding (21), and responsive soft robotics (22). Despite the benefits summarized above, morphing a closed surface appears more challenging than morphing an open surface. The main reason can be attributed to geometric characteristics. Intuitively, a closed surface is typically obtained by sewing the free boundaries of several open patches to create an enclosed volume. As a result, generating and controlling morphing in a closed surface requires the synchronous motion and evolution of the constituent patches so as to preserve the proper adjacency between them across the sewed boundaries, a requirement that imposes stricter constraints than morphing open patches separately. Origami, the art of paper folding, and Kirigami, paper cutting, offer elegant solutions for transforming flat sheets into curved open surfaces (23). They have been successfully applied to develop multifunctional structures and metamaterials at various length scales (24, 25). Both origami and kirigami rely on patch rotations to govern their morphing kinematics, but in distinct ways. In origami, patch rotation occurs around a crease accommodated in the initial plane of a flat sheet, whereas, in kirigami, the rotation axes are out of the initial plane. Compared to origami, kirigami appears to enable shape changes of higher geometric complexity and have more potential to be performed on originally curved surfaces (14, 26), due to the additional degrees of freedom (DOFs) introduced by their slits. On the other hand, origami is favored in cases where the precise control is required over the entire shape transformation (27), whereas kirigami has to seek a trade-off between flexibility and controllability (28). Here, we harness the advantages of both origami and kirigami principles to tackle the challenging task of morphing closed surfaces and imparting bistability. To pursue a morphing platform with high capacity for geometric variation and low complexity for mechanical control, we propose a class of bistable ori-kiri material assemblages that can approximate distinct closed surfaces in both their compact and deployed configurations. In particular, our goal here is to generate bistable morphing between closed surfaces that are dissimilar in size (Fig. 1A), shape (Fig. 1B), and, unprecedentedly, topology (Fig. 1C). The deployed configuration, despite its open slits, covers a closed surface as an inner space is enclosed and bounded by its panels. These slits are comparable in size to the panels, allowing the panels to span the overall closed shape. Based on this consideration, the slits are regarded as an essential component of the closed surface. The ori-kiri assemblages are geometrically constructed by triangular patches with edge connections and vertex connections (Fig. 1, right column). The geometric distribution of creases (i.e., diagonals) and slits (i.e., outer edges) is designed to preserve the congruence between corresponding panels in their compact and deployed states, with the vertices lying on a given closed surface. The panels can fold around their diagonal hinges (origami principle) and rotate at their vertices connected to the neighbor panels (kirigami principle) and, hence, realize the closed-surface morphing. Mechanically, the diagonal connections are realized by origami hinges that are nearly tangent to the closed surfaces, while the vertex connections are realized by kirigami hinges that are nearly perpendicular to the closed surfaces. The bistable reconfiguration of the ori-kiri assemblages is guided by their origami hinges and kirigami hinges, each allowing 1 DOF. Beyond the bistable material assemblages, we further propose a class of unitary-piece ori-kiri metamaterials with functionalities leveraging shape morphing and topology morphing. Fig. 1. Three types of closed-surface morphing (left column) and their conceptual realizations with ori-kiri assemblages (right column). (A) Scaling between spheres of dissimilar radius. (B) Shape morphing between sphere and cube. (C) Topology morphing between sphere (genus-0) and torus (genus-1).

RESULTS

Design strategy

我们实施一个两阶段过程来设计我们的ori-kiri材料组合，双重目标是获得几何兼容性和机械双稳态。在第一阶段，我们将kirigami铰链视为球形关节（每个关节具有3个自由度），并优化ori-kiri材料组合的面板顶点分布，从而在两个封闭曲面上寻求其几何兼容性。第一阶段产生具有多个运动学自由度的ori-kiri机制。在第二步中，我们确定每个kirigami铰链的旋转轴，并冻结所有运动学自由度，从而使每个机制都成为一个双稳态材料组合。

为了进行演示，我们通过立方体和球体之间的形状变形来示例我们的两阶段平台。关键思想是构建两个四边形面板系统——一个系统的顶点位于立方体上，另一个系统的顶点位于球体上——具有相同数量的面板和相同的顶点和边缘连接规则（图 2A）。为了应用折叠原理，每个四边形面板通过选定的对角线分为两个三角形面板。两个三角形面板的公共边可以看作是四边形面板的折痕。对于立方体，我们自然采用规则的棋盘格作为其六个面的初始几何形状。在每个面内部规定顶点连接，以使其成为具有指定对角线的旋转正方形 (RS) 机制，并在面的交点上规定顶点连接，以使其协同展开（文本 S1 以及图 S1 和 S2）。我们将这个带有其连接的三角形系统表示为 RS 网格。RS 网格的展开版本是通过将展开的 RS 图案映射到球体上来生成的（图 S3）。紧凑和展开的 RS 网格之间的一致性是通过封闭曲面上顶点位置的 2D 参数化（文本 S2 和图 S4）以及参数空间中旋转正方形的展开来实现的（文本 S3 和图 S5）。从平面参数空间到一般弯曲封闭曲面的映射不一定保留大小和形状，但它为在表面内移动顶点铺平了道路。

Fig. 2. Two-stage design of bistable ori-kiri assemblages with features of shape morphing and topology morphing. (A and B) Initial mesh, (C and D) the first design stage to obtain geometric compatibility, and (E and F) the second design stage to obtain mechanical bistability for the shape morphing between a cube and a sphere [(A), (C), and (E)] and the topology morphing between a sphere and a torus (B, D, and F). (G) 3D printed shape-morphing ori-kiri assemblage: compact cube (left) and deployed sphere (right). Inset: Thin metal rods. (H) 3D printed topology-morphing ori-kiri assemblage: compact sphere (left) and deployed torus (right). Inset: Thick screws and nuts.

从两个初始RS网格开始，其中规定了相邻面板的相同对角线和连接规则，我们的首要目标是在立方体上紧凑的RS网格和球体上展开的RS网格中，寻求相应面板之间的一致性（即几何兼容性）（[图2C](https://www.science.org/doi/10.1126/<#F2>））。为此，第一阶段设计采用球形关节用于面板的顶点连接，并通过以下公式优化面板顶点位置 minPc,Pd,acfobj.(Pd,ad)s.t.fcomp.(Pc,Pd,ac,ad,c)=0fsym.(Pc,Pd)=0fpos.(Pc,Pd)≤0fcont.(Pc,Pd)≤0 (1) 其中 P c 和 P d 分别是紧凑和展开的RS网格的顶点位置参数集（图S6）；a c是紧凑构型的大小，等于立方体半边长；a d是展开构型的大小，等于球体的半径；而 c 是对角线分配的数组。对于图2A中的示例，c的分量由下式给出 ci,j,k=+1,∣i−j∣<2−1,∣i−j∣≥2 (2) 其中下标表示RS网格的第_i_列和第_j_行上的四边形，对于_i_ , j = 1, 2, 3, 4和_k_ = 1, 2, …, 6。条目+1和-1分别表示由两个边连接的三角形面板组成的每个四边形的主要和次要对角线（图S7）。

虽然目标函数和约束函数的详细公式在文本S4中给出，但在这里，我们简要描述它们中的每一个。目标函数_f_ obj.控制ori-kiri组合体的展开程度，参考开口角ω（图S8）。兼容性约束函数_f_ comp.控制紧凑和展开构型的等距。为了量化兼容性，我们定义以下度量函数 fmetric(Xc,Xd,c)=1Ne∑k=1Ne(sc,k−sd,k)2 (3) 其中_s_ c,k_和_s d,k_分别是紧凑和展开构型的第_k_个边长；而 X c 和 X d 是紧凑和展开构型的顶点位置。求和遍历所有_N e条边。然而，f metric 不能直接用于优化公式中，因为它不涉及目标变形曲面的形状。兼容性约束函数_f_ comp.应该接收参数 P c 和 P d 和大小变量_a_ c和_a_ d作为输入，而不是顶点位置 X c 和 X d。为此，我们将兼容性的约束函数定义为 fcomp.(Pc,Pd,ac,ad,c)=fmetric[Xc(Pc;ac),Xd(Pd;ad),c]/ℓd2 (4) 特征长度ℓd由展开构型的大小确定。具体而言，半径为_a_ d的展开球体定义ℓd = 2π a d/N。对于紧凑的立方体，映射 X c(P c; a c)表示为 (x,y,z)=gcube−RS(p,q;k,ac);(x,y,z)∈Xc,(p,q)∈Pc,k=1,2,…,6 (5) 对于展开的球体，映射 X d(P d; a d)表示为 (x,y,z)=gsphere−RS(p,q;k,ad);(x,y,z)∈Xd,(p,q)∈Pd,k=1,2,…,6 (6) 值得注意的是，索引_k_表示RS网格上的六个不同的面。函数 g cube−RS 和 g sphere−RS 的显式表达式在文本 S2 中给出。除了 f comp.，我们还有一些其他的约束函数，它们限制了ori-kiri组合体上的顶点分布。对称性约束函数 f sym.将三个正交的镜像对称平面应用于紧凑和展开的RS网格，目的是提供更好的优化RS网格中面板的规则性。位置约束函数 f pos.限制顶点的相对位置，以避免优化RS网格的失真，包括面板的非凸性、边缘的相交以及面板的方向反转（图S9）。连续性约束函数 f cont.对齐相应的顶点并删除六个面彼此相遇的交点上的冗余切口（图S10）。

我们使用序列二次规划 (SQP) 算法来解决方程1中的优化问题。为了获得图2C中优化的ori-kiri组合体，我们将展开球体的半径规定为_a_ d = 1；紧凑立方体的初始边长指定为 2ac=2；在目标函数中，参考开口角指定为常数 ω = 0.5π（文本S4）。通过使用桁架网络建模组合体，其中杆表示面板边缘（图S11），我们可以计算优化ori-kiri组合体的运动学不确定度（DOKI）（文本S5）。我们注意到，DOKI是在无限小节点位移下，杆和铰链系统的自由度。在其余文本中，我们简单地使用DOF来指代DOKI。分析表明，紧凑立方体具有180个自由度，展开球体具有138个自由度，结果表明获得的ori-kiri组合体在运动学上是不确定的（即，具有多自由度的运动）；因此，从物理角度来看，它在其紧凑和展开的构型之间具有零能量跃迁路径（[图2C](https://www.science.org/doi/10.1126/<#F2>））。过多的自由度可能导致两个不良问题：首先，控制两个零能量状态之间的跃迁；其次，在零能量状态下保持所需的形状。为了解决这些问题，我们在第二设计阶段通过用旋转关节替换球形关节来抑制所有自由度，从而使ori-kiri组合体具有运动学确定性和双稳态。

第二设计阶段旨在确定指导紧凑和展开构型之间转换的旋转关节的方向（[图2E](https://www.science.org/doi/10.1126/<#F2>））。为此，我们计算每对顶点连接的三角形面板的相对旋转矩阵，然后计算其旋转轴的方向；我们还从旋转矩阵的分量中确定旋转角度（文本S5）。在转换之前和之后（无论是展开还是不展开），每个旋转轴的最终方向相对于面板（共享该轴）作为参考对象是不变的（图S12）。结果，旋转关节可以引导面板绕其轴旋转，并引导面板在紧凑和展开的构型之间转换。与具有3个自由度的球形关节相比，每个旋转关节仅允许面板绕旋转轴相对旋转，这可以大大降低ori-kiri组合体的自由度。再次采用桁架模型，其中杆代替面板边缘，我们执行运动学不确定度分析，发现图2E中的铰接ori-kiri组合体在紧凑和展开状态下都具有0个自由度（文本S5）。该结果证明该组合体已转换为具有两个零能量稳定状态的结构。

Deployment simulation

运动学不确定度分析是在杆和铰链（即旋转关节）的桁架模型上进行的，表明ori-kiri组合体在紧凑立方体和展开球体的状态下是稳定的。为了进一步说明组合体在其两个兼容状态之间的全局双稳态性，我们使用桁架模型的简化能量公式模拟展开路径。基本上，我们假设能量由两个来源贡献（图S13）：杆的拉伸能量（用_E_ S表示）和铰链处的离轴旋转能量（用_E_ R表示）。我们将拉伸能量_E_ S写成 ES(X)=12∑n=1Nbar[kS,n(s′n−sn)2] (7) 其中_s_ _n_和s′n分别是第_n_个杆的原始长度和变形长度；k S,n_是第_n_个杆的刚度；X是桁架的节点位置；而_N bar是杆的总数（N bar = N e）。

对于每个铰链（由_m_索引），如果可以将两个铰接面板之间的旋转分解为绕铰链轴的旋转（用γ _m_表示）和绕垂直于铰链轴的轴的旋转（用δ _m_表示），则便于公式化能量。可以证明，当离轴旋转δ m_和杆应变εn=(s′n−sn)/sn较小时，此分解存在且是唯一的（文本S6）。我们假设旋转γ m_不消耗能量；因此，旋转能量可以写成 ER(X)=12∑m=1Nhinge(kRδm2) (8) 其中_k R是离轴旋转刚度，而_N hinge是铰链的总数。

我们假设所有杆都具有恒定的轴向刚度_EA_；因此，k S,n = EA /s n。我们引入一个特征长度ℓc来对变形能量进行无量纲化。具体而言，对于从立方体展开到球体的组合体，我们定义ℓc = a d，即紧凑立方体的半边长。然后，缩放的拉伸能量和旋转能量表示为 E¯S(X)=ES(X)/(EAℓc)andE¯R(X)=ER(X)/(EAℓc) (9) 分别。最后，我们获得总的缩放能量 E¯T(X)=E¯S(X)+E¯R(X) (10) 我们假设kR=cREAℓc，其中_c_ R是一个调整E¯S和E¯R在E¯T中的比例的系数。该系数可以用刚度表示为cR=kR/(kS,nsnℓc)。

我们可以最小化E¯T(X)来模拟ori-kiri组合体的展开，从而获得中间构型以及展开路径的每个步骤的能量景观。为了应用加载条件，我们选择一组控制节点x ctrl.，规定它们在展开时的位置，并优化其他节点的位置。具体而言，对于从立方体展开到球体的组合体，我们选择位于RS网格的六个面的中心的六对节点（参见电影S1）。选择这些节点的主要原因是，与自由点相比，这些节点行进的距离相对较长，从而可以有效地监控展开的每个步骤。有关变形能量、加载条件和优化设置的详细公式，请参阅文本S6。

作为第一个初始猜测，c R = 0.1，以便获得小的离轴旋转角度和低的杆应变（图S14），这与我们在_E_ R公式化中的假设一致。模拟的能量景观具有一个能量势垒，两端都有两个零点（电影S1），证实了铰接ori-kiri组合体的双稳态性。为了验证分析和模拟，我们使用挤压空心圆柱体3D打印面板，这些圆柱体的轴是旋转关节的轴，并用细金属杆将它们固定在一起（图2G）。展开后，此原型显示了封闭立方体状态和展开球形状态之间的双稳态转换（电影S2）。最后，关于每个旋转关节的一个值得注意的观察是，它的方向是相对于其所属的面板的局部参考系规定的。结果，从立方体到球体的变形伴随着折纸折叠角度和切纸开口角度的变化，而不是旋转关节的方向的变化。

Extension to other morphing categories

所提出的两阶段设计策略是通用的，因为它可应用于不同封闭曲面上的不同类型的网格。这种优势使我们能够探索具有不同拓扑的封闭曲面之间的变形。例如，我们可以使用标准的UV网格来离散球体（图2B，左图；文本S1；以及图S1）。该网格在后文中被称为“UV映射”，这是一种纹理映射技术，可以将图像从3D模型的表面展开到2D表面（[29](https://www.science.org/doi/10.1126/<#R29>））。标准UV网格由其主要部分的梯形和极地区域中的两个三角形簇组成。与RS网格类似，我们分配对角线以激活折纸折叠变形，目的是实现曲率变化。我们定义一个对角线分配数组c，其分量定义为 ci,j=+1,eveni+j−1,oddi+j (11) 其中下标表示UV网格的第_i_列和第_j_行上的梯形，对于_i_ = 1, 2, …, 16 和 j = 2, 3, …, 7。数组 c 描述了图2B中紧凑球形UV网格的对角线分配。

通过球体的2D参数化，我们可以在参数空间中展开具有指定顶点连接规则的相应UV网格，并将展开的图案映射到圆环上（文本S2和S3以及图S4、S15和S16）。结果，我们在圆环上获得了UV网格的展开版本。圆环有一个孔，即亏格1，该孔没有嵌入面板中，因此它在拓扑方面与球体不同（图2B，右图）。与形状变形组合体的初始几何形状相比（图2A），我们已经改变了封闭曲面的形状和拓扑，以及用于设置拓扑变形问题的网格类型。然而，我们仍然可以采用两阶段设计框架，如下所述。

在第一阶段，我们求解方程1，优化球体和圆环上的顶点位置参数（图S17），以实现紧凑和展开的UV网格之间的几何兼容性。为此，我们需要修改封闭曲面的参数化。对于紧凑球体，映射 X c(P c; a c)变为 (x,y,z)=gsphere−UV(p,q;ac);(x,y,z)∈Xc,(p,q)∈Pc (12) 对于展开圆环，映射 X d(P d; a d)变为 (x,y,z)=gtorus−UV(p,q;R,r);(x,y,z)∈Xd,(p,q)∈Pd,ad=(R,r) (13) 函数 g sphere−UV 和 g torus−UV 的显式表达式在文本 S2 中给出。参考方程1，我们注意到，展开构型的大小已成为一个数组 a d = (R , r)，因为圆环由主半径 R 和短半径 r 共同确定。为了获得图2D中优化的UV网格，我们将主半径和短半径分别规定为 R = 1 和 r = 0.5。紧凑球体_a_ c的半径分配初始值1。在目标函数中，参考开口角指定为常数 ω = 0.4π。

通过分配球形关节，我们获得了具有紧凑球形构型和展开环形构型的ori-kiri组合体（图2D）。两个构型都是运动学上不确定的，具有138个自由度，因此，通过不确定的展开路径以零能量连接，如运动学不确定度分析所研究的。在第二阶段，我们通过计算顶点连接的面板的相对旋转矩阵来获得旋转关节的方向（[图2F](https://www.science.org/doi/10.1126/<#F2>））。通过分配旋转关节，我们将ori-kiri组合体的自由度降低到零，并使其具有双稳态，这是通过对其能量景观的分析进行验证的结果（电影S1）。为了验证设计，我们组装了带有挤压空心圆柱体的3D打印面板，这些圆柱体的轴是旋转关节的轴（图2H）。我们观察到，球体的拓扑变形在赤道和极点处承受更大的力，导致在两个兼容构型之间转换时，这些区域中的面板断裂。为了解决这个问题，我们调整了连接的紧密度，用螺钉和螺母实现（图2H，插图）。当螺栓拧得更紧时，铰链中的接触会增加，产生少量摩擦，从而降低模型的移动性。在这方面，我们强调螺栓永远不会完全拧紧；目的不是完全限制铰链运动，而是调整铰链刚度，以便更好地捕获变形并欣赏稳定的构型。使用两个级别的紧密度。第一个级别更紧，允许紧凑球体和展开圆环在外部载荷下保持其形状（电影S3）。第二个级别比第一个级别稍松，降低了转换的能量势垒，同时避免了面板的断裂（电影S3）。这种方法能够在组合体内协调稳定性和可重构性，而不会使铰链失效。

除了形状变形和拓扑变形之外，我们还可以利用我们的框架来实现封闭曲面的缩放。这可以通过为紧凑和展开的构型选择形状相同但大小不同的两个曲面来实现。例如，图3A显示了通过带有RS网格的ori-kiri组合体实现的两个球体之间的缩放。此外，通过将展开的RS网格的球体更改为立方体，我们可以设计另一个双稳态形状变形组合体，该组合体在紧凑状态下近似于球体，而在展开状态下近似于立方体（图3B）。 Fig. 3. Energy landscapes and geometric metrics evolving along the deployment path of the bistable ori-kiri assemblages. (A) Scaling between two spheres. (B) Shape morphing between a compact sphere and a deployed cube. (C) Topology morphing between a compact sphere and a deployed torus. (**