Morphing ribbons and their inverse design are usually confined to plane curves, since in most cases only the curvature is considered. Given that curvature and torsion are equally important geometric characteristics of space curves, it is urgent to propose a systematic theoretical framework for the inverse design. Toward this end, we here present a multiscale theoretical framework named biomimetic Turing machine (BTM) to achieve desired target space curves, which is inspired from two microstructural regulation mechanisms behind the hydration-driven morphing of plant tissues: the graded curvature regulated by matrix volume fraction (cm) and the helix-like morphology regulated by fibril orientation angle (FOA). By analogizing to Turing machine encoded by binary mapping, the proposed BTM can inversely encode a morphing ribbon with preset microstructural parameters (FOA and cm) to achieve desired target space curves. The proposed theoretical framework can first bridge the microstructural fiber-matrix swelling and the macroscopic ribbon morphing as a forward problem, in which a twist field is subsequently introduced to create the kinematic map between the target space curve and the ribbon, innovatively posing the inverse design as an initial value problem. To facilitate the experimental implementation of BTM, we further propose an optimization strategy for selecting the twist field and provide design criteria as guidelines for experiments. As a conceptual display, we present a phase diagram in the cm versus FOA plane to illustrate the complex target morphologies (e.g., hemisphere, hyperboloid, and tendril) characterized by various parameters of curvature and torsion designed rationally by the BTM theory, while in previous studies the morphing morphologies (e.g., helices, arcs, and helicoid ribbons) exhibit only constant curvature or torsion. This work presents a novel inverse design strategy for space curves with both curvature and torsion, broadening the potential for the design and fabrication of morphing materials.

中文翻译：

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仿生图灵机：目标空间曲线逆向设计的多尺度理论框架


变形带及其逆向设计通常局限于平面曲线，因为在大多数情况下只考虑曲率。鉴于曲率和扭转是空间曲线同样重要的几何特性，迫切需要为逆向设计提出一个系统的理论框架。为此，我们在这里提出了一个名为仿生图灵机 （BTM） 的多尺度理论框架，以实现所需的目标空间曲线，其灵感来自植物组织水合作用驱动变形背后的两种微观结构调节机制：由基质体积分数 （cm） 调节的渐变曲率和由原纤维取向角 （FOA） 调节的螺旋状形态。通过类比由二进制映射编码的图灵机，所提出的 BTM 可以反向编码具有预设微观结构参数 （FOA 和 cm） 的变形带，以实现所需的目标空间曲线。所提出的理论框架可以首先将微观结构纤维基体膨胀和宏观带状变形作为一个正向问题进行桥接，其中随后引入扭曲场以创建目标空间曲线和条带之间的运动图，创新性地将逆设计作为初始值问题。为了促进 BTM 的实验实现，我们进一步提出了一种选择扭曲场的优化策略，并提供了设计标准作为实验指南。作为概念展示，我们在 cm 与 FOA 平面中提供了一个相图，以说明由 BTM 理论合理设计的曲率和扭转的各种参数所表征的复杂目标形态（例如，半球、双曲面和卷须），而在以前的研究中，变形形态（例如、螺旋、圆弧和螺旋带）仅显示恒定曲率或扭转。这项工作提出了一种新颖的曲率和扭转空间曲线逆向设计策略，拓宽了变形材料设计和制造的潜力。