Two-Dimensional Dispersed Composites on a Square Torus

The article is devoted to the recently established connection between the problem of packing disks on a torus and the effective conductivity of composites with circular inclusions. The packing problem is usually investigated by geometric considerations, the conductivity problem by means of elliptic functions. An algorithm has been developed for determining the optimal arrangement of three disks on a torus formed by a hexagonal lattice. The corresponding minimization function is constructed in terms of expressions consisting of elliptic functions with unknown arguments. This makes it possible to optimize packaging for a hexagonal screen. The numerically found roots coincide with the previously established optimal points, as well as with a purely geometric study.