Computational design of multi-phase microstructural materials for extremal conductivity

This paper presents a systematic investigation into the computational design of multi-phase microstructural composites with tailored isotropic and anisotropic thermal conductivities. The composites are assumed to be periodically ranked by base cells (representative volume elements) whose best possible geometric configurations make the composite’s bulk or effective thermal conductivity attaining to the target Milton–Kohn bounds. To avoid checkerboard patterns and generate edge-preserving results in topology optimization, a nonlinear diffusion technique is exploited by introducing the generalized interface energy into the objective function. The adjoint variable method is used to formulate the sensitivity of the objective functions with respect to multi-phase design variables (“relative density”), which guides the method of moving asymptotes to converge along the steepest direction. Unlike the typical density-based method (e.g. SIMP), the penalty factor is no longer needed in this present method after the local conductivity is interpolated by the Hashin–Shtrikman bound other than commonly-used arithmetic bound. In addition to the conventional Vigdergauz-like structures, three new classes of single-length-scale microstructures are generated to closely approach the isotropic Hashin–Strikman bounds in three-phase and two-dimensional cases. This paper also generated sandwich-like microstructures attaining to the anisotropic Milton–Kohn bounds.