Formalization of Technological Processes Based on Vector and Tensor Analysis

Authors

DOI:

https://doi.org/10.20535/2521-1943.2025.9.4(107).341433

Keywords:

vector, tensor, basis, Kronecker symbol, scalar product, cosine similarity, ADS, CTS, TP, PCM

Abstract

The work further develops the results of studies [1]–[5] based on a systematic analysis using an interpretative-formal approach in a new scientific and technical direction – the technological interpretation of provisions of vector and tensor analysis that are similar in meaning. This allows expanding the formal field of representation of technological processes within the concept of their “technological meaning” and increasing the formal capacity of TP description. The results of research are revealed in the practical and production aspects related to TP, providing for the application of vector and tensor analysis in the coordinate approach, technological space, scalar product, matrix tensor, as well as examples of tensor and vector analysis in composite AKZ technology. It is determined that vectors (tensors) can be specified in different ways, depending on the technological context (polymer composite materials technology – PCM), and the set of components is only its representation in a certain (in terms of detail) basis. A coordinate approach is used, as well as the possibility of other methods of specifying and working with vectors (tensors) using the example of ordinary vectors and simple second-rank tensors, characterized by the powerful idea of orthogonality. Since the second vector and tensor represent real technological objects, including: autonomous dynamic systems (ADS), structural and technological solutions (STS), technological processes (TP), in the form of contravariant and covariant vectors, etc.
The interpretative correspondence of the technological interpretation of contravariant and covariant coordinates of a vector is shown, and the nature of the relationships between technological contravariant and technological covariant coordinates is established.
The example demonstrates the invariance of the enlarged stages of a complex technological process in different coordinate systems, which confirms the invariance of the technological vector under the condition of transformation of its coordinates.

References

  1. D. S. Kiva and V. F. Zabashta, “About the fundamental properties of polymeric composite materials in the context of creation and production of efficient designs”, Technological Systems, no. 72/3, pp. 45–56, 2015. Available: https://www.technological-systems.com/index.php/Home/article/view/170.
  2. G. A. Krivov, Yu. M. Tarasov, A. G. Gromashev and V. F. Zabashta, “Technologies for non-autoclave molding of aircraft airframe structures from polymer composite materials”, Technological Systems, no. 49/5, pp. 47–70, 2009. Available: https://www.technological-systems.com/index.php/Home/article/view/539.
  3. D. S. Kiva and V. F. Zabashta, “Composite wing box of transport aircraft (constructive and technological aspects)”, Technological Systems, no. 84/3, pp. 7–31, 2018. Available: https://www.technological-systems.com/index.php/Home/article/view/144.
  4. A. V. Kovalenko, "Study of the properties of a binder for molding products by the method of impregnation under pressure", Proceedings of VIAM, no. 1, pp. 28-35, 2015.
  5. V. F. Zabashta, "Technological preparation for the production of polymer composite materials", in Mechanical Engineering. Encyclopedia, vol. III-6, 2006, pp. 104-143.
  6. S. A. Bychkov, A. V. Gaydachuk, A. V. Andreev and W. Bo, “Development of new structural and technological solutions for aircraft wings and repair of composite structures”, Open Information and Computer Integrated Technologies, no. 86, pp. 76–90, 2019. DOI: https://doi.org/10.32620/oikit.2019.86.05.
  7. GOST R 58932-2020. Technological support for the development and production of aircraft. Procedure for the development and content of directive technological materials.
  8. D. Kiva and V. Zabashta, “Alternative technologies of composite high-loaded aircraft constructions: a qualitative method of making multicriterial decisions. Part I. Initial stages in the problem of decision-making”, Mechanics and Advanced Technologies, vol. 5, no. 2, pp. 203–211, 2021. DOI: https://doi.org/10.20535/2521-1943.2021.5.2.245000.
  9. V. Zabashta, “Alternative technologies of composite highly loaded of aircraft structures: a qualitative method of making multi-criteria decisions. Part II. Modeling in multi-criteria evaluation of alternatives”, Mechanics and Advanced Technologies, vol. 6, no. 2, pp. 203–220, 2022. DOI: https://doi.org/10.20535/2521-1943.2022.6.2.265371.
  10. V. Zabashta, “Alternative technologies of composite of highly loaded aircraft structures: a qualitative method for making multicriteria decisions: Part III. Research of the methodological basis in decision-making: technological constructions in the assessment toolkit”, Mechanics and Advanced Technologies, vol. 7, no. 3 (99), pp. 374–388, 2023. DOI: https://doi.org/10.20535/2521-1943.2023.7.3.293231.
  11. V. Zabashta, “Alternative technologies of composite highly loaded aircraft structures: a qualitative method of making multi-criteria decisions. Comprehensive presentation: Part IV. Equilibrium and stability of the dynamic model of the ADS”, Mechanics and Advanced Technologies, vol. 8, no. 3(102), pp. 316–331, 2024. DOI: https://doi.org/10.20535/2521-1943.2024.8.3(102).302969.
  12. V. Zabashta, “Formalization of technological processes based on non-Euclidean geometries: spherical and Riemann geometry”, Mechanics and Advanced Technologies, vol. 9, no. 2(105), pp. 238–253, 2025. DOI: https://doi.org/10.20535/2521-1943.2025.9.2(105).325989.
  13. L. S. Lisovska and N. T. Hryniv, "Ensuring synergy of innovation processes at the enterprise", Bulletin of the National University "Lviv Polytechnic". Logistics, no. 846, pp. 103-109, 2016.
  14. D. Pritykin, The Magic of Tensor Algebra: Part 5 – Tensor Operations and Other Theoretical Topics. Habr.com, 2015. Available: https://habr.com/ru/articles/261991/.
  15. Cosine Similarity. Sciencedirect.com. Available: https://www.sciencedirect.com/topics/computer-science/cosine-similarity.
  16. M. T. Senkiv, Vector and Tensor Analysis. Lviv: Lviv University Publishing House, 1990, 148 p.
  17. S. M. Grebenyuk, Yu. M. Strelyaev and M. I. Klymenko, Tensor Analysis. Zaporizhzhia: ZNU, 2015, 90 p.
  18. Zh. V. Khuda, Lecture notes on the discipline “Fundamentals of Vector and Tensor Analysis”. Kamianske: DSTU, 2019, 65 p.
  19. O. V. Nikulin and T. V. Nakonechna, Fundamentals of vector and tensor calculus: theoretical information and tests. Dnipropetrovsk: Bila K. O., 2011, 72 p. Available: https://www.dstu.dp.ua/Portal/Data/1/5/3-21-b.pdf.
  20. G. M. Zrazhevsky and V. F. Zrazhevska, Applied vector analysis. Kyiv: Taras Shevchenko National University of Kyiv, 2023, 143 p. Available: https://mechmat.knu.ua/wp-content/uploads/2023/01/applied_vector_analysis.pdf.
  21. A. J. McConnell, Introduction to tensor analysis with applications to geometry, mechanics and physics. Moscow: Fizmatgiz, 1963, 411 p.
  22. M. A. Razumova and V. M. Khotyaintsev, Fundamentals of vector and tensor analysis. Kyiv: VPTs "Kyyivskyy universytet", 2011, 216 p. Available: https://iht.knu.ua/sites/default/files/ovta-posibnyk_0.pdf.

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Published

2025-12-29

How to Cite

[1]
V. Zabashta, “Formalization of Technological Processes Based on Vector and Tensor Analysis”, Mech. Adv. Technol., vol. 9, no. 4(107), pp. 456–468, Dec. 2025.

Issue

Vol. 9 No. 4(107) (2025)

Section

Aviation Systems and Technologies

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Copyright (c) 2025 Володимир Забашта

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