基于生物分枝网对称与对称性破缺作用机理的仿生拓扑优化方法研究

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 networks in various biological systems. It is universal that the bio branching networks represent geometrically symmetric configurations, while its symmetry breaking reflects the mechanics self-adaptability to different loading conditions. This project starts from the basic contradictive relationship between symmetry and symmetry breaking, explores the mechanism of symmetry and symmetry breaking in the perspective of morphological mechanics principles and reveals the physical and mechanical behaviors of load path bifurcation in branching structures. Both the distribution and transfer characterization methods of load path under the unified dimension are studied to clarify the adaptive reorganization mechanism of branching networks in global and local load-bearing force systems and to condense the optimal cutting mechanism of load path. On the basis of fully exploiting the superiority of Lagrangian and Eulerian descriptions, the framework which combines both the bionic mechanism (i.e., branching, growth and degeneration) and explicit topological description is established. Based on the symmetry metrics, a continuous and smooth diversity constraint function is constructed and the adaptive growth of the structures is utilized to simulate the topology optimization process. This project aims to break through the massive computational burden caused by the huge design variables of traditional topology optimization methods and to break through the complicated post-processing burden caused by the implicit topological description, which promised to provide a new solution principle and technical support for the optimization design of mechanical load-bearing structures.

自然界中各种性能卓越的网状拓扑图式为人类在机械工程领域的创新设计提供了无穷尽的灵感源泉。生物分枝网的对称性是一种几何普遍性，而其对称性破缺则体现了一种力学自适应性。本项目正是从‘对称与对称性破缺’这一基本矛盾关系入手，以形态力学的视角，探究分枝网对称与对称性破缺的作用机理，揭示分枝网力流分歧的真实物理力学行为。研究统一量纲下力流的分布与传递特性表征方法，阐明分枝网全局与局部承载力系的自适应重组机理，并从中凝练出载荷路径的最优裁剪机制。在充分挖掘拉格朗日描述与欧拉描述各自优势的基础上，将分歧、生长、退化等仿生机制与显式拓扑描述框架相融合。基于对称性度量标准，构造连续光滑的多样性约束函数，利用结构的自适应生长来模拟拓扑优化过程。本项目旨在突破传统拓扑优化由于设计变量庞大所导致的巨量计算瓶颈以及由于隐式拓扑描述所导致的复杂后处理瓶颈，可望为机械承载结构的优化设计提供全新的解决原理与技术支撑。

本项目从‘对称与对称性破缺’这一基本矛盾关系入手，以形态力学的视角，探究分枝网对称与对称性破缺的作用机理，揭示其高效承载形态背后的真实物理成因，为力/热/流承载结构优化提供量化准则与设计工具。本项目由生物分枝网络拓扑图式的数字化提取着手，提炼分枝网络几何形态规律，以结构维持能耗（力问题）与热/质输运能耗（热流耦合问题）最小化为基础，构建生物分枝系统形态参数与结构承载效能函数模型，获得分枝网络尺寸、分歧角度及分歧点位的解析列式与最优解，解释了生物分枝网络全局对称及局部对称破缺的几何形态成因，从而破解了多元载荷流（力/热/流载荷）最优剪裁机制。以此为基础，本项目由结构拓扑优化基本数学性态着手，将生物分枝网络（传质传热结构）抽象为经典面点问题/体点问题，采用稳态传热模型探讨了生物分枝结构最优解存在性与最优拓扑特征。研究发现宏观载荷传递系统近似全局最优设计呈现裸子植物所特有的针状对称形态，而在最小制造尺寸约束下则逐步趋向被子植物常见的树状对称形态。针对最优拓扑特征的讨论揭示了生物分枝网络在不同边界条件下的演变规律，为结构优化设计提供数学建模与参数定义支撑。通过对自然分歧网络“对称及对称破缺”基本分歧规律研究、载荷承载机制量化分析及其形态背后数学最优性的讨论，本项目提出了一种基于自主生长范式的拓扑优化方法，并配套构建了基于显式几何描述的形态表征算法与结构计算算法，形成了完整的生长式拓扑优化理论与数值实现框架。为进一步拓展生长式拓扑优化理论在多尺度结构设计中的应用，本项目采用构型理论解析生物分枝特征在宏/微观设计域中的共性与特性，为多尺度工程设计提供了最优拓扑辨识依据。本项目将以上研究应用于力/热/流等多种载荷的结构设计中，搭建了面向多元复杂承载结构的优化设计框架。