This paper evaluates the effect that different imposed crystal symmetries have on the topology design of two-phase isotropic elastic composites ruled by the target of attaining extreme theoretical properties. Extreme properties are defined by the Cherkaev–Gibiansky bounds, for 2D cases, or the Hashin–Shtrikman bounds, for 3D cases.

The topology design methodology used in this study is an inverse homogenization technique which is mathematically formulated as a topology optimization problem. The crystal symmetry is imposed on the material configuration within a predefined design domain, which is taken as the primitive cell of the underlying Bravais lattice of the crystal system studied in each case.

The influence of imposing crystal symmetries to the microstructure topologies is evaluated by testing five plane groups of the hexagonal crystal system for 2D problems and four space groups of the cubic crystal systems for 3D problems.

A discussion about the adequacy of the tested plane or space groups to attain elastic properties close to the theoretical bounds is presented. The extracted conclusions could be meaningful for more general classes of topology design problems in the thermal, phononic or photonic fields.

本文评估了不同的施加晶体对称性对以达到极限理论特性为目标的两相各向同性弹性复合材料的拓扑设计的影响。对于2D情况，极限属性由Cherkaev–Gibiansky边界定义；对于3D情况，极限属性由Hashin–Shtrikman边界定义。

本研究中使用的拓扑设计方法是一种逆均化技术，其在数学上被公式化为拓扑优化问题。晶体对称性强加于预定义设计域内的材料配置上，该设计域被视为在每种情况下研究的晶体系统的基础Bravais晶格的原始单元。

通过测试六边形晶体系统的五个平面组用于2D问题，以及测试立方晶体系统的四个空间组用于3D问题，来评估施加晶体对称性对微观结构拓扑的影响。

提出了关于被测平面或空间群是否足以获得接近理论界限的弹性的讨论。所得出的结论对于热，声子或光子领域中更一般的拓扑设计问题类别可能是有意义的。