Mathematical Modeling in the Textile Industry

TL;DR

This paper introduces a mathematical model for weaving structures, defines new concepts like self-mirror and rotation-stable structures, and develops algorithms and software for analyzing and classifying these structures efficiently.

Contribution

It presents novel definitions, algorithms, and a software implementation for analyzing weaving structures using binary matrices and object-oriented programming.

Findings

Algorithms improve analysis speed and memory efficiency.

Binary matrix representation aids in classifying weaving structures.

Results describe the topology of various weaving structures.

Abstract

A mathematical model, describing some different weaving structures, is made in this article. The terms self-mirror and rotation-stable weaving structure are initiated here. There are used the properties and operations in the set of the binary matrices and an equivalence relation in this set. Some combinatorial problems about finding the cardinal number and the elements of the factor set according to this relation is discussed. We propose an algorithm, which solves these problems. The presentation of an arbitrary binary matrix using sequence of nonnegative integers is discussed. It is shown that the presentation of binary matrices using ordered n-tuples of natural numbers makes the algorithms faster and saves a lot of memory. Implementing these ideas a computer program, which receives all of the examined objects, is created. In the paper we use object-oriented programming using the syntax and the semantic of C++ programming language. Some advantages in the use of bitwise operations are shown. The results we have received are used to describe the topology of the different weaving structures.