The relation of constant mean curvature surfaces to multiphase composites with extremal thermal conductivity

The behaviour of a periodic composite material depends not only on the properties of the constituent phases but also strongly on the microstructural texture of those phases such as spheres, lamellae and needles. This paper shows how to design the microstructure for a specific extremal bulk (effective) thermal conductivity in a three-phase composite medium. An inverse homogenization technique that is driven by the computational topology optimization algorithm is presented. Apart from benchmarking examples such as the Vigdergauz-type and sandwich-like architectures, a series of new single length-scale designs of microstructures are generated from this procedure. The topological design results are validated by comparing their conductivities against the empirical formulae in the two-phase composites. This study interestingly finds that the phase interfaces yielded from the topology optimization highly resemble the constant mean curvature surfaces. A comparison of their respective attainability with the Milton–Kohn physical bounds is made and the equivalence of these two sets of topologies is consequently justified.

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