Methodology to Solve Optimal Placement Problems for 3D Objects

Journal Journal of Mechanical Engineering
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 23, no. 2, 2020 (June)
Pages 60-71
Cited by J. of Mech. Eng., 2020, vol. 23, no. 2, pp. 60-71



Yurii H. Stoian, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi St., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0002-8053-0276

Andrii M. Chuhai, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi St., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0002-4079-5632



This paper is devoted to solving optimization problems of packing 3D objects both by constructing exact mathematical models and by developing approaches based on the application of non-linear optimization methods and modern solvers. Developed are constructive tools for both mathematical and computer modeling of relations between oriented and non-oriented 3D objects, whose boundaries are formed by cylindrical, conical, and spherical surfaces and planes in the form of new classes of both Stoyan’s Φ-functions (further referred to as phi-functions) and quasi-phi-functions. Based on the developed mathematical modeling tools, constructed and investigated is the basic mathematical model of the problem of optimally packing 3D objects, whose boundaries are formed by cylindrical, conical, and spherical surfaces and planes, as well as the model’s various implementations, which cover a wide class of scientific and applied problems of packing 3D objects. Developed is the methodology for solving the problems of packing 3D objects that allow both continuous rotations and translations at the same time. Proposed are strategies, methods and algorithms for solving the optimization problems of packing 3D objects with taking into account technological constraints (minimum admissible distances, prohibited zones, the possibility of continuous translations and rotations). On the basis of the proposed mathematical modeling tools, mathematical models, methods, and algorithms, developed is the software that uses parallel computing technology to automatically solve the optimization problems of packing 3D objects. The results obtained can be used for solving problems of optimizing layout solutions; for computer modeling in materials science, powder metallurgy, and nanotechnologies; in optimizing the 3D printing process for the SLS technology of additive production; in information and logistics systems that optimize transportation and storage of goods.


Keywords: packing, 3D objects, geometric design, phi-functions, mathematical modeling, continuous rotations, nonlinear optimization.


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Received 11 February 2020