The thesis is dedicated to modeling and solving optimization problems of packing
hyperspheres ( 2D , 3D and nD, n 4) into convex containers (HSOA) taking into
account the minimum allowable distances and prohibition areas, the frontiers of which
are formed by cylindrical, spherical surfaces and planes.
Tools of mathematical modeling of the conditions of packing of hyperspheres into
a domain bounded by hyperspherical, hypercylindrical surfaces and hyperplanes making
use of Stoyan phi-function technique are developed.
A mathematical model of the HSOA problem is constructed and its main
characteristics are studied. Variants of the mathematical model are considered according
to the international typology of Cutting&Packing problems depending on the type of
objective function (Open Dimension Problem or Knapsack Problem), dimension and the
peculiarities of the metric characteristics of hyperspheres (congruence, radii distribution,
constraints on the radii values), the spatial shape of the container (hyperrectangle,
hypersphere, hypercylinder, n-polytope), restrictions on the minimum allowable distances
and prohibition zones.
In the methodology, which is created based on the analysis of the source data and
the peculiarities of mathematical models, effective strategies of solving HSOA problems
are proposed. The strategies involve new methods of construction of feasible starting
points, searching for local extrema and approximations to global extrema.
New methods for solving HSOA problems based on nonlinear programming
methods, the greedy algorithm, the branch and bound algorithm, statistical optimization,
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the idea of homothetic transformations of the hyperspheres and the container are
developed.
The effectiveness of the proposed mathematical models and methods is confirmed
by comparing the obtained results with the best world analogues for various
implementations of the HSOA problem published in international journals and available
at http://hydra.nat.uni-magdeburg.de/packing/cst/cst.html, http://www.packomania.com.
Examples of solving practical problems arising in materials science, nuclear power
engineering, powder metallurgy, additive manufacturing, chemical industry, and
medicine are given.
The software developed in the thesis is used at the Department of Applied
Materials Science and Materials Processing of the Lviv Polytechnic National University.
Copyright certificates are registered. The constructed mathematical modeling tools and
methods for solving placement problems are used in the education process at the Kharkiv
National University of Radio Electronics in the "Modeling of geometric objects" and
"Decision making theory" courses.
Keywords: geometric design, packing problem, circle, sphere, hypersphere, phifunction,
mathematical modeling, nonlinear optimization, open dimension problem,
knapsack problem.
