基于自然分叉结构载荷传递路径构筑机制的生长式拓扑优化方法研究

Nature has evolved many ingenious topologies that provide inexhaustible inspiration for designers to solve mechanical engineering problems. One conspicuous category of such topologies is the branching structures in various biological systems, such as the leaf venation, root and axis system of plants. The motivation of this project is to explore the potential of natural branching structures as concept generators for creating the optimal load-bearing topology for mechanical structures. The configuration laws of natural branching structures are analyzed regarding different geometrical scales and symmetrical features. The mechanics characterization method of load paths is developed to explore the generating, coupling and transferring principles of load flow inside natural branching structures. The possible relationship between the distribution patterns of load paths and the load capacity of mechanical structures is investigated so as to clarify the construction mechanism of optimal load paths. The growth principles of natural branching structures including the mechanics robustness and adaptability are explored, which provide informative guidelines to reveal the internal connection between the macro load-bearing topology and the micro growth competitive behaviors. The modeling of growth competition is transformed into a conditional extremum problem expressed by two dual forms. The growth-based topology optimization is manipulated by mimicking the competition among individual growth elements, which always try to develop into a configuration with optimal load paths. Based on this, a well-founded methodology and a complete numerical package is proposed to implement the optimal growth of structural elements, which can serve engineering purpose for the innovative design of mechanical load-bearing structures.

自然界中各种性能卓越的拓扑构型为人类在机械工程领域的结构创新设计提供了无穷尽的灵感源泉。本项目力求从不同几何尺度与不同对称程度两个层面剖析自然分叉结构的特殊构型规律。研究载荷传递路径的力学表征方法，分析自然分叉结构中载荷流的形成、耦合与传递规律，探索载荷传递路径分布形态与机械结构承载效能的内在联系，阐明最优载荷传递路径的力学构筑机理。研究自然分叉结构典型拓扑图式的自优化生长机制（鲁棒性与自适应性），揭示微/细观水平的生长竞争特性对于宏观拓扑图式的影响规律。以承载结构中载荷传递路径的分布形态作为拓扑优化的目标控制函数，利用数学上互为对偶的条件极值模型来定量表征结构生长竞争过程，将生长式拓扑优化问题转化为一个由自优化生长机制驱动的载荷传递路径寻优问题，提出一套指导结构鲁棒自适应生长的仿生拓扑优化方法与数值实现技术，为机械装备承载结构的创新设计提供基础理论与使能技术支撑。

本项目力求从不同几何尺度与不同对称程度两个层面剖析自然分叉结构的特殊构型规律，包括中轴图式与回环图式的构型规律。分别从应力与应变能两种不同的视角研究了载荷传递路径的力学表征方法，首次利用摩尔圆图成功解释了与主流观点（主应力迹线理论）的力学机理差异。研究发现，在以拉/压效应为主导的承载状态下，载荷传递路径（植物叶脉）分布走向与主流观点结果一致，但在剪切效应为主导的承载状态下，载荷传递路径（植物叶脉）分布走向与主流观点结果相差90o。在此基础上，分析了自然分叉结构中载荷流的形成、耦合与传递规律，探索载荷传递路径分布形态与机械结构承载效能的内在联系，严格验证了弯曲度与疏密度分布准则、均匀性与连续性分布准则，并利用上述准则反演了著名的Constructal 构型的进化过程，并首次从载荷流的视角展示了自然分叉构型的形态发生过程，阐明了最优载荷传递路径的力学构筑机理。研究自然分叉结构典型拓扑图式的自优化生长机制（鲁棒性与自适应性），揭示微/细观水平的生长竞争特性对于宏观拓扑图式的影响规律。在植物中轴主脉分歧特性描述的基础上，进一步分析植物维管束主干与支干间的截面尺寸关系，拟合出多源载荷流在分歧节点切口处的耦合规律（分歧方向）与传递规律（分歧速率）。依据中心性准则、同伦性准则、单位性准则构建维管束的拓扑图式，研究维管束拓扑图式的对称性度量方法，及破缺度评价模型。提出并验证了一种基于混合几何描述框架的典型生物拓扑图式自适应生长算法，以载荷传递路径的分布形态作为拓扑优化的目标控制函数，利用数学上互为对偶的条件极值模型来定量表征结构生长竞争过程，将生长式拓扑优化问题转化为一个由自优化生长机制驱动的载荷传递路径寻优问题，提出一套指导结构自适应生长的仿生拓扑优化方法与数值实现技术，为机械装备承载结构的创新设计提供基础理论与使能技术支撑。