Optimal random packing of spheres and extremal effective conductivity

Mityushev V. Zhunussova Z.
June 2021 MDPI AG

Symmetry
2021 #13 Issue 6

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

Effective conductivity , Optimal packing of spheres , Periodic graph , Structural optimization , Voronoi tessellation

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Department of Computer Science and Computational Methods, Institute of Computer Sciences, Pedagogical University of Krakow, Podchorażych˛ 2, Krakow, 30-084, Poland
Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan

Department of Computer Science and Computational Methods
Institute of Mathematics and Mathematical Modeling
Faculty of Mechanics and Mathematics

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