Optimization of 3D Objects Layout into a Multiply Connected Domain with Account for Shortest Distances
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OPTIMIZATION OF 3D OBJECTS LAYOUT INTO A MULTIPLY CONNECTED DOMAIN WITH ACCOUNT FOR SHORTEST DISTANCES Yu. G. Stoyan,a† V. V. Semkin,a‡ and A. M. Chugay a††
UDC 519.859
Abstract. The layout of 3D objects into a multiply connected domain formed by a circular cylinder and right rectangular prisms is optimized. Constraints are imposed on the shortest distances between objects. In order to construct a mathematical model, the Ô-function technique is employed. A special approach is proposed to construct different starting points. The approach is based on solving an auxiliary problem with an increased dimension of the solution space. The numerical examples are given. Keywords: mathematical modeling, layout problem, optimization, 3D objects. INTRODUCTION The problems that have been intensively solved in many branches of science and engineering in the last decades include problems of searching for the most favorable mutual arrangement of various objects, namely, automated layout of various equipment [1], layout of instruments in aircraft compartment [2], automated design of master plans at the stage of layout and space-planning design for workshops of industrial enterprises [3], engineering design of the heat power complexes, etc. In most cases, such problems are notable for a large number of possible alternatives of admissible solutions, unknown characteristics of the best alternative solution, great amount of original information, etc. An analysis of the studies [1–6], where packing of three-dimensional geometrical objects is considered, allows concluding that the complexity of the solution of the problem under study is due to the absence of adequate mathematical models and efficient solution methods. Because of high computational complexity of layout problems, their solution employs heuristic approaches in most cases, i.e., instead of exhaustive search, which takes significant time (or is impossible from the technical standpoint), a much faster yet insufficiently theoretically justified algorithm is applied. Since heuristic methods are not completely mathematically justified, they do not guarantee finding the optimal solution. Therefore, it is of importance to construct adequate mathematical models, to enhance existing ones, and develop new methods for the solution of problems of layout of three-dimensional geometrical objects. PROBLEM FORMULATION For the mathematical statement of the problem under study, it is necessary to perform at least three stages: to describe the layout objects, to propose an optimization criterion, and to construct a mathematical model of the problem. The time of problem solution and its quality depends in many respects on the correct description of layout objects. Let a set of 3D objects Oi Î R 3 , i Î I =
6
U I k , be specified. Object Oi , depending on the value of index i, is one of the
k =1
following sets: · a ball S i = { X = ( x, y, z ) Î R 3 : x 2 + y 2 + z 2 - ( ri0 ) 2 £ 0}, i Î I 1 = {1, 2, K , n1}; a
A. N. Podgorny Institute for Mechanical Engineering Problems, National Academy
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