Block element method in solving vector boundary problems using scalar

Authors

  • Babeshko V.A. Kuban State University, Krasnodar, Russian Federation
  • Evdokimova O.V. Southern Scientific Center, Russian Academy of Science, Rostov-on-Don, Russian Federation
  • Babeshko O.M. Kuban State University, Krasnodar, Russian Federation

UDC

539.3

DOI:

https://doi.org/10.31429/vestnik-17-4-6-13

Abstract

This paper presents for the first time a solution of a vector boundary value problem decomposed over packed block elements that are solutions of scalar boundary value problems in a non-classical domain. Solutions of a number of vector partial differential equations in continuum mechanics, electromagnetic phenomena, and field theory allow representations in the form of decompositions based on solutions of scalar equations. This approach is convenient for solving problems in the entire space. When solving boundary value problems, the difficulty of applying this approach is the difficulty of satisfying boundary conditions. In a number of classical fields, this can be done and exact solutions to boundary value problems can be obtained. These classic areas include the half-space, the ball, the cylinder, and some areas obtained from views of transformation groups spaces. However, for a number of important areas other than classical ones, such as wedge-shaped ones, this approach has not yet been able to build accurate solutions. In this paper, probably for the first time, this approach is used to construct an exact solution in the first quadrant of a plane boundary value problem of the second kind for dynamic Lamé equations. The solution is compared with the obtained direct application of the block element method to the vector boundary value problem. It is known that the unbounded domain makes it not effective to use numerical methods in this boundary value problem. The solution is constructed using the block element method under arbitrary boundary conditions. This makes it possible to study different properties of solutions by changing the effects on the boundary.

Keywords:

boundary value problems, block element method, packed block elements, Lame and Helmholtz equations

Funding information

This work was supported by the Russian Foundation for Basic Research (projects 19-41-230003, 19-41-230004, 19-48-230014, 18-08-00465, 18-01-00384, 18-05-80008), the GZ UNC RAS reg. 01201354241 (project 00-20-13), Ministry of the science Russian Federation (project FZEN-2020-0022) charged on 2020 year.

Author info

  • Vladimir A. Babeshko

    академик РАН, д-р физ.-мат. наук, заведующий кафедрой математического моделирования Кубанского государственного университета, директор Научно-исследовательского центра прогнозирования и предупреждения геоэкологических и техногенных катастроф Кубанского государственного университета, заведующий лабораторией Южного федерального университета

  • Olga V. Evdokimova

    д-р физ.-мат. наук, главный научный сотрудник Южного научного центра РАН

  • Olga M. Babeshko

    д-р физ.-мат. наук, главный научный сотрудник научно-исследовательского центра прогнозирования и предупреждения геоэкологических и техногенных катастроф Кубанского государственного университета

References

  1. Бабешко В.А., Евдокимова О.В., Бабешко О.М., Евдокимов В.С. Метод блочного элемента в разложении решений сложных задач механики // ДАН. 2020. Т. 495. №6. С. 34–38. DOI: 10.31857/S2686740020060048 [Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M., Evdokimov, V.S. Metod blochnogo elementa v razlozhenii resheniy slozhnykh zadach mekhaniki [Block element method in the decomposition of solutions to complex problems of mechanics]. Doklady Akademii nauk [Reports of the Academy of Sciences], 2020, vol. 495, no. 6, pp. 34–38. DOI: 10.31857/S2686740020060048 (In Russian)]
  2. Babeshko V.A., Evdokimova O.V., Babeshko O.M. On the possibility of predicting some types of earthquake by a mechanical approach // Acta Mechanica. 2018. Vol. 229. Iss. 5. P. 2163–2175. DOI: 10.1007/s00707-017-2092-0
  3. Babeshko V.A., Evdokimova O.V., Babeshko O.M. On a mechanical approach to the prediction of earthquakes during horizontal motion of litospheric plates // Acta Mechanica. 2018. Vol. 229. P. 4727–4739. DOI: 10.1007/s00707-018-2255-7
  4. Babeshko V.A., Evdokimova O.V., Babeshko O.M. A new type of cracks adding to Griffith–Irwin cracks // Doklady Physics. 2019. Vol. 64. No. 2. P. 102–105. DOI: 10.1134/S10283358191030042
  5. Гельфанд И.М., Минлос Р.А., Шапиро З.Я. Представления группы вращений и группы Лоренца, их применения. М: Физматгиз, 1958. 367 с. [Gelfand, I.M., Minlos, Z.A., Shapiro, Z.Ya. Representations of the rotation group and the Lorentz group, their applications. Fizmatgiz, Moscow, 1958. (In Russian)]
  6. Улитко А.Ф. Метод собственных векторных функций в пространственных задачах теории упругости. Киев: Наукова Думка, 1979. 262 с. [Ulitko, A.F. Method of eigenvector functions in spatial problems of elasticity theory. Kiev, Naukova Dumka, 1979. (In Russian)]
  7. Гринченко В.Т., Мелешко В.В. Гармонические колебания и волны в упругих телах. Киев: Наукова Думка, 1981. 284 с. [Grinchenko, V.T., Meleshko, V.V. Harmonic vibrations and waves in elastic bodies. Kiev, Naukova Dumka, 1981. (In Russian)]
  8. Nowacki W. Teoria sprezystosci. Panstwowe Wydawnictwo Naukowe, Warsaw, 1970.
  9. Nowacki W. Dynamiczne zagadnienia termosprezystosci. Panstwowe Wydawnictwo Naukowe, Warsaw, 1966. [Nowacki, W. Dynamiczne zagadnienia termosprezystosci. Panstwowe Wydawnictwo Naukowe, Warsaw, 1966.]
  10. Nowacki W. Efekty elektromagnetyczne w stalych cialach odksztalcalnych. Panstwowe Wydawnictwo Naukowe, Warsaw, 1981. [Nowacki W. Efekty elektromagnetyczne w stalych cialach odksztalcalnych. Panstwowe Wydawnictwo Naukowe, Warsaw, 1981.]
  11. Бабич В.М. О коротковолновой асимптотике функции Грина для уравнения Гельмгольца // Математический сборник. 1964. Т. 65. С. 577–630. [Babich, V.M. On the Short-Wave Asymptotic Behaviour of the Green's Function for the Helmholtz Equation. Mat. Sb. (N.S.), 1964, vol. 65, iss. 4, pp. 576–630.]
  12. Babich V.M., Buldyrev V.S. Asymptotic methods in short-wavelength distraction theory. Alpha Science International Ltd, 2009.
  13. Мухина И.В. Приближенное сведение к уравнениям Гельмгольца уравнений теории упругости и электродинамики для неоднородных сред // ПММ. 1972. Т. 36. С. 667–671. [Mukhina I.V. Approximate reduction of the equations of the theory of elasticity and electrodynamics for inhomogeneous media to the Helmholtz equations. Prikladnaya Matematika i Mekhanika, 1972, vol. 36, iss. 4, pp. 667–671.]
  14. Молотков Л.А. Исследование распространения волн в пористых и трещиноватых средах на основе эффективных моделей Био и слоистых сред. С.-Пб.: Наука, 2001. 348 с. [Molotkov, L.A. Wave propagation in porous and cracked media, studied on the basis of effective biot models and layered media. Nauka, Saint-Petersburg, 2001 (In Russian).]
  15. Tkacheva L.A. Vibrations of a floating elastic plate due to periodic displacements of a bottom segment // J. Appl. Mech. Tech. Phys. Vol. 46. Iss. 5. P. 754–765.
  16. Tkacheva L.A. Plane problem of vibrations of an elastic floating plate under periodic external loading // J. Appl. Mech. Tech. Phys. 2004. Vol. 45. Iss. 3. P. 420–427.
  17. Tkacheva L.A. Behavior of a floating elastic plate during vibrations of a bottom segment // J. Appl. Mech. Tech. Phys. 2005. Vol. 46. Iss. 2. P. 230–238.
  18. Tkacheva L.A. Interaction of surface and flexural-gravity waves in ice cover with a vertical wall // J. Appl. Mech. Tech. Phys. 2013. Vol. 54. Iss. 4. P. 651–661.
  19. Brekhovskikh L.M. Waves in layered media. Academic Press, 1960.
  20. Babeshko V.A., Evdokimova O.V., Babeshko O.M. On the problem of acoustic and hydrodynamic properties of a medium occupying the area of a three-dimensional rectangular wedge // J. Appl. Mech. Tech. Phys. 2019. Vol. 60. Iss. 6. P. 90–96. DOI: 10.15372/PMTF20190610

Downloads

Downloads

Issue

Vol. 17 (2020) No. 4

Pages

6-13

Section

Mechanics

Dates

Submitted

November 24, 2020

Accepted

December 4, 2020

Published

December 27, 2020

How to Cite

[1]
Babeshko, V.A., Evdokimova, O.V., Babeshko, O.M., Block element method in solving vector boundary problems using scalar. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2020, т. 17, № 4, pp. 6–13. DOI: 10.31429/vestnik-17-4-6-13

License

Copyright (c) 2020 Бабешко В.А., Евдокимова О.В., Бабешко О.М.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Similar Articles

1-10 of 1069

You may also start an advanced similarity search for this article.

Most read articles by the same author(s)

1 2 3 4 5 6 7 8 9 10 > >>