Abstract Adaptive growth method is a structural topology optimization approach based on the growth mechanism of natural biological branching systems, and it has been successfully applied for the static reinforcement of three-dimensional box structures. However, due to the effect of structural characteristics towards natural frequencies, it is difficult for adaptive growth method to effectively grow stiffeners inside three-dimensional box structures considering dynamic performance. And this paper proposes an improved version of adaptive growth method to solve this problem. Firstly, apart from the growth mechanism of natural branching systems in the initial adaptive growth method, the morphology of root tips is introduced into the growth process of internal stiffeners. The thickness of stiffeners during the optimization can be divided into four different zones: initial thickness zone, immediate thickness zone, allowable thickness zone and maturation zone. Then, a material interpolation scheme with different thickness zones is introduced to simultaneously penalize the elastic modulus and density of material of internal stiffeners according to their thicknesses. By applying this bio-inspired method, obstacles of the ground structure in dynamic design of stiffeners in three-dimensional box structures can be effectively eliminated. And notably, the internal stiffeners can successfully grow from “seed lines” and gradually bifurcate or degenerate along the optimal direction to improve the natural frequencies of structures with a clear layout pattern. Finally, the effectiveness and advantages of the proposed method are illustrated by several numerical examples.

中文翻译：

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三维箱形结构加劲肋相对于固有频率的改进自适应增长方法

摘要 自适应生长法是一种基于自然生物分枝系统生长机制的结构拓扑优化方法，已成功应用于三维箱形结构的静态加固。然而，由于结构特性对固有频率的影响，自适应增长方法难以在考虑动态性能的三维箱形结构内有效地增长加强筋。并且本文提出了一种改进版的自适应增长方法来解决这个问题。首先，除了初始适应性生长方法中自然分枝系统的生长机制外，将根尖形态引入内部加强筋的生长过程。优化过程中加强筋的厚度可分为四个不同的区域：初始厚度区、中间厚度区、允许厚度区和成熟区。然后，引入了具有不同厚度区域的材料插值方案，以根据其厚度同时惩罚内部加强筋材料的弹性模量和密度。通过应用这种仿生方法，可以有效消除三维箱形结构加强筋动力设计中地面结构的障碍。值得注意的是，内部加强筋可以成功地从“种子线”生长，并沿着最佳方向逐渐分叉或退化，以提高结构的固有频率，并具有清晰的布局模式。最后，