Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 22, no. 1, 2019 (March)
Pages 67-75
Cited by J. of Mech. Eng., 2019, vol. 22, no. 1, pp. 67-75



Georgiy N. Yaskov, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi Str., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0002-1476-1818



This paper discusses the problem of optimally packing spheres of various dimensions into containers of arbitrary geometrical shapes. According to the international classification, this problem belongs to Sphere Packing Problems (SPPs). The problem is to pack a set of spheres (circles, hyperspheres) with given radii into a container with given metric characteristics. The aim of this work is to create an integrated methodology for solving SPPs. The basic formulations of the problem are presented: in the form of the knapsack problem (KP), open dimension problem (ODP), and their corresponding mathematical models. The solution strategy selection is influenced by the form of problem statement, dimension of the space where the spheres are to be packed, metric peculiarities of the spheres (equal or unequal), number of the spheres to be packed, geometric shape of the container, presence of technological restraints, and count time limit. The structural elements of the methodology are mathematical models, methods for constructing initial packings, and methods of local and global optimization. In developing the solution method, we construct the initial feasible packings by using both the random and lattice methods, using a greedy algorithm and solving an auxiliary nonlinear programming problem. As local optimization methods, we consider the modifications of the feasible direction method, interior point method, Lagrange multiplier method, and method of optimization in groups of variables. For global optimization, we use the method of enumerating the subsets of spheres of a given set and method of enumerating the extreme points of the feasible region, which are implemented by using the branch and bound algorithm, the modifications of the decremental neighborhood search method, method of smooth transition from one local minimum to another by increasing problem dimensionality and introducing additional variable metric characteristics, solution method implemented as a sequence of non-linear programming problems of increasing dimensionality, and a multi-start method. Strategies for solving different SPP statements are proposed.


Keywords: sphere, hypersphere, sphere packing, knapsack problem, open dimension problem, nonlinear optimization.


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Received 14 January 2019

Published 30 March 2019