Optimal Random Packing of Spheres and Extremal Effective Conductivity

A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) composites with ideally conducting spherical inclusions is established. The location optimal-design problem yields inclusions. geometrical-packing physical-conductivity problems are stated a periodic toroidal d-dimensional space an arbitrarily fixed number n nonoverlapping per periodicity cell. Energy depends on Voronoi tessellation (Delaunay graph) associated centers ak (k=1,2,…,n). All Delaunay graphs divided into classes isomorphic graphs. For any n, such finite. estimated framework structural approximations reduced to study elementary function variables. minimum over locations attained at within class optimal-packing unique up translations can be found from linear algebraic equations. Such approach useful for random where initial balls randomly chosen; hence, dynamically change following prescribed rules. finite algorithm constructed determine Rd.