A data-driven integrated optimization method of macroscopic topology and microscopic configuration for the graded functional cellular structures

In the topology optimization of the multiscale structure, how to ensure the connectivity between adjacent microstructures, how to control the design space of microstructures, and how to reduce the amount of calculation and improve calculation efficiency are three basic challenging issues currently faced. To this end, this paper proposes a data-driven approach to achieve the integrated optimization of macroscopic topology and microscopic configuration of the graded functional cellular structures. At the macro level, a topological description function is introduced to realize the topological control of the macrostructure. At the micro level, several cutting functions are used to realize the control of the configuration and size of the microstructure. The integrated optimization design of macro and micro cellular structures can be realized. Based on the computational homogenization method and numerical integration technology, an optimization problem independent offline microstructure database is established at the microscopic scale, where the relationship between the equivalent elastic parameters, relative pseudo-density, and design variables of the microstructure is stored. Based on this offline database, the entire topology optimization process is completed only on a macro scale, which greatly reduces the amount of calculation and improves calculation efficiency. In addition, implicit geometric modeling of full-scale cellular structures can be achieved using the reconstruction technique introduced in this work, which ensures smooth connection between adjacent microstructures. Finally, numerical examples are used to verify the effectiveness of the algorithm and the superiority of gradient cellular structures compared with single-scale structures.

在多尺度结构（multiscale structure）的拓扑优化（topology optimization）中，如何保障相邻微结构（microstructure）间的连通性、管控微结构的设计空间（design space），以及缩减计算量、提升计算效率，是当前面临的三大基础性挑战性问题。为此，本文提出一种数据驱动方法，实现梯度功能胞状结构（graded functional cellular structures）的宏观拓扑与微观构型一体化优化。宏观层面，引入拓扑描述函数（topological description function）以实现宏观结构的拓扑管控；微观层面，采用若干切割函数（cutting functions）实现微结构的构型与尺寸调控，从而完成宏微观胞状结构的一体化优化设计。基于计算均匀化方法（computational homogenization method）与数值积分技术（numerical integration technology），本文在微观尺度建立离线微结构数据库（offline microstructure database），存储微结构的等效弹性参数（equivalent elastic parameters）、相对伪密度（relative pseudo-density）与设计变量（design variables）之间的映射关系。依托该离线数据库，整个拓扑优化流程仅需在宏观尺度完成，大幅缩减计算量并提升计算效率。此外，通过本文提出的重构技术（reconstruction technique）可实现全尺寸胞状结构的隐式几何建模（implicit geometric modeling），保障相邻微结构间的平滑衔接。最后，通过数值算例验证了所提算法的有效性，以及梯度胞状结构相较于单尺度结构的性能优势。