The code for the first part of the calculation from Multi-functional topology optimization of Victoria cruziana veins

The growth and development of biological tissues and organs strongly depend on the requirements of their multiple functions. Plant veins yield efficient nutrient transport and withstand various external loads. Victoria cruziana, a tropical species of the Nymphaeaceae family of water lilies, has evolved a network of three-dimensional and rugged veins, which yields a superior load-bearing capacity. However, it remains elusive how biological and mechanical factors affect their unique vein layout. In this paper, we propose a multi-functional and large-scale topology optimization method to investigate the morphomechanics of Victoria cruziana veins, which optimize both the structural stiffness and nutrient transport efficiency. Our results suggest that increasing the branching order of radial veins improves the efficiency of nutrient delivery, and the gradient variation of circumferential vein sizes significantly contributes to the stiffness of the leaf. In the present method, we also consider the optimization of the wall thickness and the maximum layout distance of circumferential veins. Furthermore, biomimetic leaves are fabricated by using the three-dimensional printing technique to verify our theoretical findings. This work not only gains insights into the morphomechanics of Victoria cruziana veins, but also helps the design of, e.g. rib-reinforced shells, slabs and dome skeletons.生物组织与器官的生长发育高度依赖于其各项功能的实际需求。植物叶脉兼具高效养分运输与抵御多种外部载荷的双重性能。作为睡莲科（Nymphaeaceae）睡莲属的热带物种，克鲁兹王莲（Victoria cruziana）演化出三维立体且粗壮坚韧的叶脉网络，从而具备卓越的载荷承载能力。然而，目前学界仍未明确生物与力学因素如何塑造其独特的叶脉布局结构。本研究提出一种兼顾多功能性的大规模拓扑优化（topology optimization）方法，用以探究克鲁兹王莲叶脉的形态力学特性，该方法可同时优化结构刚度与养分运输效率。研究结果表明，提升径向叶脉的分支级数可改善养分输送效率，而环向叶脉尺寸的梯度变化则对叶片刚度具有显著贡献。本研究所提出的优化方法同时兼顾了壁厚与环向叶脉最大布局间距的优化设计。此外，本研究通过三维打印（three-dimensional printing）技术制备仿生叶片，以验证理论分析结果的有效性。本研究不仅深入揭示了克鲁兹王莲叶脉的形态力学机制，同时也可为肋加固壳体、平板结构以及穹顶骨架等工程结构的设计提供参考。