The code for the first part of the calculation from Multi-functional topology optimization of <i>Victoria cruziana</i> 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. <i>Victoria cruziana</i>, 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 <i>Victoria cruziana</i> 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 <i>Victoria cruziana</i> 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，高度依赖于其自身多项功能的需求适配。植物叶脉兼具高效的养分运输能力与抵御多种外部载荷的性能。克鲁兹王莲（Victoria cruziana）作为睡莲科（Nymphaeaceae）的热带睡莲物种，演化出了三维立体且粗壮的叶脉网络，具备优异的承载能力。然而，目前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。本研究提出一种多功能大规模拓扑优化方法，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，而周向叶脉尺寸的梯度变化则对叶片刚度具有显著贡献。本优化方法同时考虑了壁厚参数与周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技术制备仿生叶片，以验证理论研究结果的正确性。本工作不仅深化了对克鲁兹王莲叶脉形态力学特性的认知，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。