Hypergraph approach in the decomposition of complex technical systems

Bozhko A.N.
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The article considers a mathematical model of decomposition of a complex product into assembly units. This is an important engineering problem, which affects the organization of discrete production and its operational management. A review of modern approaches to mathematical modeling and automated computer-aided of decompositions is given. In them, graphs, networks, matrices, etc. serve as mathematical models of structures of technical systems. These models describe the mechanical structure as a binary relation on a set of system elements. The geometrical coordination and integrity of machines and mechanical devices during the manufacturing process is achieved by means of basing. In general, basing can be performed on several elements simultaneously. Therefore, it represents a variable arity relation, which can not be correctly described in terms of binary mathematical structures. A new hypergraph model of mechanical structure of technical system is described. This model allows to give an adequate formalization of assembly operations and processes. Assembly operations which are carried out by two working bodies and consist in realization of mechanical connections are considered. Such operations are called coherent and sequential. This is the prevailing type of operations in modern industrial practice. It is shown that the mathematical description of such operation is normal contraction of an edge of the hypergraph. A sequence of contractions transforming the hypergraph into a point is a mathematical model of the assembly process. Two important theorems on the properties of contractible hypergraphs and their subgraphs proved by the author are presented. The concept of $s$-hypergraphs is introduced. $S$-hypergraphs are the correct mathematical models of mechanical structures of any assembled technical systems. Decomposition of a product into assembly units is defined as cutting of an $s$-hypergraph into $s$-subgraphs. The cutting problem is described in terms of discrete mathematical programming. Mathematical models of structural, topological and technological constraints are obtained. The objective functions are proposed that formalize the optimal choice of design solutions in various situations. The developed mathematical model of product decomposition is flexible and open. It allows for extensions that take into account the characteristics of the product and its production.

Keywords: assembly, assembly unit, product structure, decomposition into assembly units, computer-aided design system, connection graph, hypergraph, discrete mathematical programming
Citation in English: Bozhko A.N. Hypergraph approach in the decomposition of complex technical systems // Computer Research and Modeling, 2020, vol. 12, no. 5, pp. 1007-1022
Citation in English: Bozhko A.N. Hypergraph approach in the decomposition of complex technical systems // Computer Research and Modeling, 2020, vol. 12, no. 5, pp. 1007-1022
DOI: 10.20537/2076-7633-2020-12-5-1007-1022
URL: http://crm-en.ics.org.ru/journal/article/2980/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

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