This paper presents a general formulation and solution method for the problem of two-scale concurrent topology optimization of cellular structure and its anisotropic materials. This formulation seeks simultaneous determination of the macro and micro structural topologies and the orientations of microstructures, such as a unit cell with anisotropic and inhomogeneous material properties. In the present formulation, the microstructure of cellular material is uniform in the entire macrostructure but with spatially-varying orientations. The solution method consists of two new features: (a) a mutual strain energy density-based approach proposed for analytically determining the orientations of fully anisotropic materials in compliant mechanism problems, which can also be modified to address minimum compliance designs and (b) an extended and fully-coupled moving iso-surface threshold (MIST) method and algorithm developed for solving the formulated concurrent optimization problem of cellular structures using two-scale physical response functions. The present formulation and algorithm are validated via studying numerical examples of concurrent optimum design of: (i) macro and micro topologies; (ii) macro topology and orthotropic material orientations; and (iii) macro and micro topologies and material orientations.

本文提出了一种用于蜂窝结构及其各向异性材料的两尺度并发拓扑优化问题的通用公式和求解方法。该公式寻求同时确定宏观和微观结构拓扑以及微观结构的取向，例如具有各向异性和非均匀材料特性的晶胞。在本配方中，蜂窝材料的微观结构在整个宏观结构中是均匀的，但具有空间变化的取向。该求解方法包括两个新特征：(a) 提出了一种基于互应变能密度的方法，用于分析确定柔顺机构问题中完全各向异性材料的方向，等表面阈值 (MIST) 方法和算法开发用于使用两尺度物理响应函数解决蜂窝结构的公式化并发优化问题。本公式和算法通过研究以下并行优化设计的数值实例得到验证：(i) 宏观和微观拓扑；(ii) 宏观拓扑和正交各向异性材料取向；(iii) 宏观和微观拓扑结构和材料取向。