Effective-medium approximation for composite media: Realizable single-scale dispersions

It is known that the popular effective medium approximation (EMA) for the effective conductivity ${\sigma }_e$ of a composite is exactly realizable by certain multiscale hierarchical microstructures. We have found a class of periodic, single-scale dispersions that achieve the EMA function at a given phase conductivity ratio for a two-phase, two-dimensional composite over all volume fractions. Moreover, to an excellent approximation (but not exactly), the same structures realize the EMA for almost the entire range of phase conductivities and volume fractions. The inclusion shapes are given analytically by the generalized hypocycloid, which in general has a nonsmooth interface. To find these structures, we utilized target optimization techniques and a theorem concerning the spectral function.