Geometric-feature-based design of spatially varying multiscale structure with quasi-conformal mapping

For spatially varying multiscale configurations (SVMSCs) that increasingly gain engineering perspective, a design scheme enabling the direct tuning of their microstructural layouts is still desired. Challenges lie behind the fact that an arbitrary setting on the (field) distributions about the microstructural geometric features is likely to violate the underlying conditions of total differential, and the article is aimed to resolve the issue systematically. This is done by representing (two-dimensional) SVMSCs from the viewpoint of quasi-conformal mapping, where the microstructural profile can be summarised by a (complete) family consisting of four geometric feature fields. By modifying the so-called Beltrami equations that automatically satisfy the total differential condition, the relationships among these four geometric feature fields can be explicitly derived, and their degrees of freedom are reduced to two. Different choices for which two field quantities as the free design variables result in various geometric-feature-based modes, which almost cover all scenarios of SVMSC design by means of directly tuning their microstructural profiles. Furthermore, we can now directly manipulate the stretch ratio, one of the four field variables, to avoid ill microstructural distortion, which has been a challenging issue for SVMSCs representation. Fueled by an asymptotic homogenisation scheme (Zhu et al. in J Mech Phys Solids 124:612-633, 2019, https://doi.org/10.1016/j.jmps.2018.11.008), the present method is then employed for geometric-feature-based compliance design of SVMSC.