Cybernetics And Systems Analysis logo
Editorial Board Announcements Abstracts Authors Archive
Cybernetics And Systems Analysis
International Theoretical Science Journal
Volume 59 >>> № 1 JANUARY — FEBRUARY 2023
-->

UDC 533.6.013.42
I. Kaliukh1, O. Trofymchuk2, О. Lebid3


1 State Enterprise “State Research Institute of Building Constructions,” Kyiv, Ukraine, and Institute of Telecommunication and Global Information Space of the National Academy of Sciences of Ukraine, Kyiv, Ukraine

 kalyukh2002@gmail.com

2 Institute of Telecommunication and Global Information Space of the National Academy of Sciences of Ukraine, Kyiv, Ukraine

 Trofymchuk@nas.gov.ua

3 Institute of Telecommunication and Global Information Space of the National Academy of Sciences of Ukraine, Kyiv, Ukraine

 o.g.lebid@gmail.com

PECULIARITIES OF APPLYING THE FINITE-DIFFERENCE METHOD FOR CALCULATION
OF NONLINEAR PROBLEMS OF THE DYNAMICS OF DISTRIBUTED SYSTEMS IN A FLOW

Abstract. Peculiarities of application of the finite-difference method (FDM) for calculation of nonlinear dynamic problems of distributed systems (DS) in a flow are considered. The main limitations for the application of the FDM for numerical modeling of wave propagation and reflection in DS are the features of the defining quasilinear equations. They are associated with the need to simultaneously calculate the variables responsible for transient and slow wave processes. The term “singularly perturbed system of equations” is used for such systems of equations. These perturbations are the result of a significant difference in the propagation velocities of longitudinal, configurational, bending, and torsional waves in the DS at the physical level, etc. Therefore, it is necessary to use special step-by-step methods of regularization and filtering of the numerical results. It imposes certain contstraints on the ability to model real processes and the accuracy of the results and forces the use of implicit difference schemes and high-frequency filtering. When solving the system of linear algebraic equations, taking into account the poor conditioning of the matrix of convective terms, the method of regularization was chosen by experimental calculation. Calculations according to the Crank–Nicholson difference scheme, even using coarse grids, can give results with the required degree of accuracy. And the cost of time will be minimal. Another picture is observed when comparing the results on coarse and fine grids for the Euler difference scheme. Irresistible mistakes brought in by errors in approximating the missing boundary conditions lead to greater differences.

Keywords: finite difference method, distributed systems, nonlinearity, singularity, high frequency filtering.


full text

REFERENCES

  1. Troger H., Alperton A.P., Beletsky V.V., Dranovskii V.I., Khoroshilov V.S., Pirozhenko A.V., Zakrzhevskii A.E. Dynamics of tethered space systems. 1st ed. CRC Press, 2017. 245 p.

  2. Amsterdam on stilts. URL: https://www.holandiabeztajemnic.pl/?page_id=32236&lang=ru.

  3. Farenyuk G., Kaliukh I., Farenyuk E., Kaliukh T., Berchun Y., Berchun V. Experimental and theoretical diagnostics of defects in ferroconcrete piles based on reflection of longitudinal and transverse waves. High Tech Concrete: Where Technology and Engineering Meet. Hordijk D., Lukoviс M. (Eds). Cham: Springer, 2018. P. 1307–1317. https://doi.org/10.1007/978 -3-319-59471-2_151.

  4. Kaliukh I., Fareniuk G., Trofymchuk O., Fareniuk I., Berchun Y. Identification of defects in reinforced concrete piles based on multi-wave reflection. Proc. of the fib Symposium 2019: Concrete — Innovations in Materials, Design and Structures, 2019. P. 991–998.

  5. Kaliukh Y., Vusatiuk A. Factorization in problems of control and dynamics of lengthy systems. Cybernetics and Systems Analysis. 2019. Vol. 55, N 2. P. 274–283. https://doi.org/10.1007/s10559-019-00132-9.

  6. Turk stream. URL: http://turkstream.info/.

  7. Kaliukh I., Lebid O. Application of asymptotic and numerical methods to determine stability boundaries of distributed systems in a flow. Cybernetics and Systems Analysis. 2022. Vol. 58, N 2. P. 233–241. https://doi.org/10.1007/s10559-022-00455-0.

  8. Kaliukh I., Lebid O. Constructing the adaptive algorithms for solving multi-wave problems. Cybernetics and Systems Analysis. 2021. Vol. 57, N 6. P. 938–949. https://doi.org/10.1007/s10559-021-00419-w.

  9. Kaliukh I., Trofymchuk O., Lebid O. Numerical solution of two-point static problems for distributed extended systems by means of the Nelder–Mead method. Cybernetics and Systems Analysis. 2019. Vol. 55, N 4. P. 616–624. https://doi.org/10.1007/s10559-019-00170-3 .

  10. AN/ALE-50 towed decoy system. URL: https://en.wikipedia.org/wiki/AN/ALE-50_towed_decoy_system.

  11. Zhu Z.H. Dynamic modeling of cable system using a new nodal position finite element method. International Journal for Numerical Methods in Biomedical Engineering. 2010. Vol. 26, Iss. 6. P. 692–704. https://doi.org/10.1002/cnm.1161.

  12. Deep-sea mining. URL: https://dic.academic.ru/dic.nsf/enc_geolog/1445/.

  13. Kaliukh I., Senatorov V., Khavkin O., Kaliukh T., Khavkin K. Experimental and analytic researches on technical state, design and operation of reinforced concrete anti-landslide structures for seismic dangerous regions of Ukraine. Proc. of the Fib Symposium: Engineering a Concrete Future: Technology, Modeling and Construction (Apr. 22–24, 2013, Tel-Aviv). 2013. P. 625–628.

  14. Gladkiy A.V., Sergienko I.V., Skopetsky V.V. Numerical-analytical methods for studying wave processes. Kyiv: Nauk. Dumka, 2001. 452 p.

  15. Gladky A.V., Skopecky V.V. On numerical modeling and optimization of unidirectional wave processes in inhomogeneous media. Kibernetika i sistemnyj analiz. 2010. N 5. P. 177–186.

  16. Gubarev V.F. Rational approximation of systems with distributed parameters. Kibernetika i sistemnyj analiz. 2008. N 2. P. 99–115.

  17. Selezov I.T., Kryvonos Yu.G., Gandzha I.S. Wave propagation and diffraction. Mathematical methods and applications. Ser. Foundations of Engineering Mechanics. Singapore: Springer, 2018. 237 p. https://doi.org/10.1007/978-981-10-4923-1.

  18. Trofimchuk A.N. Dynamic interaction of a rigid slab with a water-saturated poroelastic foundation. International Applied Mechanics. 1996. Vol. 32, N 1. P. 60–64. https://doi.org/10.1007/BF02084849.

  19. Saltanov N.V. Flexible threads in streams [in Russian]. Kyiv: Nauk. thought, 1974. 140 p.

  20. Timoshenko S.P. Fluctuations in engineering [in Russian]. Moscow: Fizmatgiz, 1959. 450 p.

  21. Swan O.H. A five-mode quasi-linear model of the nonlinear dynamics of an extended system. Ekolohichna bezpeka ta pryrodokorystuvannya. 2021. N 38(2). P. 104–120. https://doi.org/10.32347/ 2411-4049.2021.2.104-120.

  22. Selezov I.T. Development and application of the Cauchy–Poisson method to layer elastodynamics and the Timoshenko equation. Kibernetika i sistemnyj analíz. 2018. N 3. P. 106–115.

  23. Kurant R. Equations with partial derivatives [Russian translation]. Moscow: Mir, 1964. 830 p.

  24. Berto G.O. Oceanographic buoys [in Russian]. Leningrad: Sudostroyeniye, 1979. 216 p.

  25. Bukharin S.N., Kalikov V.N., Nekrasov I.V., Ordanovich A.E. On the accuracy of approximation of a flexible thread by a finite-dimensional model. Proceedings of the III Republican Conference on Applied Hydromechanics. Kyiv: In-t gidromekhaniki AN USSR, 1984. P. 31.

  26. Leonard J.M., Nath J.H. Comparison of finite element and lumped parameter methods for oceanic cables. J. Eng. Struct. 1981. Vol. 3, N 6. P. 153–167.

  27. Paul B., Soler A.I. Cable dynamics and optimum owing strategies for submersibles. Mar. Technol. Soc. J. 1972. Vol. 2. P. 34–42.

  28. Peyrot A.H., Asce M. Marine cable structures. J. Struct. Div.: Proc. Amer. Soc. Civ. Eng. 1980. Vol. 106, N 12. P. 2391–2404.

  29. Ilyin R.F., Zhigula T.I. On the dynamics of elastic-viscous threads (ropes) with multi-rope lifting of a load. Prikladnaya mekhanika. 1976. Vol. 12, N 8. P. 58–63.

  30. Lin J.-M., Sansalone М. Impact-echo response of hollow cylindrical concrete structure surrounded by soil and rock: Part I — Numerical studies. Geotechnical Testing. 1994. Vol. 17, N 2. P. 207–219.

  31. Lin J.-M., Sansalone М. Impact-echo response of hollow cylindrical concrete structure surrounded by soil and rock: Part II — Experimental studies. Geotechnical Testing. 1994. Vol. 17, N 2. P. 220–226.

  32. Liao S.T., Roesset J.M. Identification of defects in piles through dynamic testing. Int. J. for Numer. and Analyt. Meth. in Geomech. 1997. Vol. 21, N 4. P. 277–291.

  33. Liao S.-T., Tong J.-H., Chen C.-H., Wu T.-T. Numerical simulation and experimental study of Parallel Seismic test for piles. International Journal of Solids and Structures. 2006. Vol. 43. P. 2279–2298.

  34. Ambrosini D., Ezeberry J. Long piles integrity trough impact echo technique. Mecanica Computacional. Vol. XXIV. Buenos Aires, Argentina, 2005. P. 651–669.

  35. Kim D.S., Seo W.S., Lee K.M. IE-SASW method for nondestructive evaluation of concrete structure. NDT&E International. 2006. Vol. 39, N 2. P. 143–154.

  36. Ablow C.M., Schechter S. Numerical simulation of undersea cable dynamics. Ocean Eng. 1983. Vol. 10, N 6. P. 443–457.

  37. Valeev V.I., Kozhevnikova G.L. Rope systems in liquid flow Moscow: TsAGI, 1976. N 489. 57 p.

  38. Lebid А.G. Control and dynamics of a distributed system with variable length. Journal of Automation and Information Sciences. 2020. Vol. 52, Iss. 9. P. 39–50. https://doi.org/10.1615/JAutomatInfScien.v52.i9.40.

  39. Kovaleva M.A., Rutkovsky A.L. Singularly perturbed optimal control systems [in Russian]. Moscow: Mir nauki, 2020. 95 p. URL: https://www.twirpx.com/file/3444935/.

  40. Samarsky A.A. Theory of difference schemes [in Russian]. Moscow: Nauka, 1983. 616 p.

  41. Khrushch V.K. Numerical methods of gas dynamics [in Russian]. Dnepropetrovsk: Dnepropetrovsk University, 1981. 102 p.

  42. Kaliukh Y.I. Specific features of using the linearization method for the analysis of low-frequency oscillations of a towed system. Int. Appl. Mech. 2021. Vol. 57, N 1. P. 103–110. https://doi.org/10.1007/s10778-021-01055-9.

  43. Yanenko N.N. Method of fractional steps for solving multidimensional problems of mathematical physics [in Russian]. Novosibirsk: Nauka, 1967. 196 p.




Editorial Board Announcements Abstracts Authors Archive
about scope office editorial board instructions for authors subscriptions contacts
© 2023 Kibernetika.org. All rights reserved.