On the problem of the parameters determination for a stationary source in a semi-bounded layered medium

Authors

UDC

517.9: 539.3

DOI:

https://doi.org/10.31429/vestnik-19-4-76-82

Abstract

In this paper, when solving the inverse problem of the parameters determination for a stationary radiation source (coordinates and intensity), formulated as an optimization problem for the residual functional of functions depending on the source parameters, we use conditional optimization methods based on the exact solution of the direct problem of the concentration calculation for a substance in a multilayer medium emitted by a stationary source using the matrix method based on the integral approach. The applied method of numerical inversion of Fourier integrals is based on the method of direct contour integration. The source is identified by the response of the pollutant distribution medium to the action of a stationary emission – a change in the concentration of an impurity at a given height based on additional "measurements", as which we chose the numerical values of the solution for the direct problem. To solve the optimization problem, local and global search methods, as well as genetic algorithms for global search, were used. Calculations carried out for various characteristics of the layered medium and the position of the source showed that the pattern search method allows us to find a more accurate solution than genetic algorithms, the convergence rate of which decreases near the minimum point. At the same time, the accuracy of restoring the initial intensity value of the pollutant source depends on the structure of the wind characteristics in the layered medium; an increase in wind speeds worsens the solution of the inverse problem. The number of measurement points has a significant impact on the accuracy of the source power value recovery when the measurement data has noise.

Keywords:

diffusion-convection, multilayer medium, matrix method, direct problem, inverse problem, conditional optimization methods

Author info

  • Olga N. Lapina

    канд. физ.-мат. наук, доцент кафедры вычислительных технологий Кубанского государственного университета

  • Aleksandr G. Nesterenko

    канд. физ.-мат. наук, доцент кафедры физики и информационных систем Кубанского государственного университета

  • Yuri G. Nikitin

    канд. физ.-мат. наук, доцент кафедры теоретической физики и компьютерных технологий Кубанского государственного университета

References

  1. Пененко, В.В., Рапута, В.Ф., Быков, А.В., Планирование эксперимента в задаче оценивания мощности источников примеси. Физика атмосферы и океана, 1985, т. 21, № 9. с. 913–920. [Penenko, V.V., Raputa, V.F., Bykov, A.V., Designing an experiment in the problem of estimating the power of impurity sources. Fizika atmosfery i okeana = Physics of the Atmosphere and Ocean, 1985, vol. 21, no. 9, pp. 913–920. (in Russian)]
  2. Besk, J.V., Blackwell, В., Clair, C.St., Inverse conduction ill-posed problems. Wiley, New York, 1985.
  3. Самарский А.А., Вабищевич П.Н., Численные методы решения обратных задач математической физики. Эдиториал УРСС, Москва, 2004. [Samarskii A.A., Vabishchevich P.N., Chislennye metody resheniya obratnykh zadach matematicheskoy fiziki = Numerical methods for solving inverse problems of mathematical physics. Editorial URSS, Moscow, 2004. (in Russian)]
  4. Бакушинский, А.Б., Гончарский А.В., Итеративные методы решения некорректных задач. Наука, Москва, 1988. [Bakushinsky, A.B., Goncharsky A.V., Iterativnye metody resheniya nekorrektnykh zadach = Iterative methods for solving ill-posed problems. Nauka, Moscow, 1988. (in Russian)]
  5. Тихонов, A.H., Арсенин, В.Я., Методы решения некорректных задач. Наука, Москва, 1979. [Tikhonov, A.H., Arsenin, V.Ya., Methods for solving ill-posed problems. Nauka, Moscow, 1979. (in Russian)]
  6. Марчук, Г.И., Математическое моделирование в проблеме окружающей среды. Наука, Москва, 1982. [Marchuk, G.I., Matematicheskoe modelirovanie v probleme okruzhayushchey sredy = Mathematical Modeling in the problem of the environment. Nauka, Moscow, 1982. (in Russian)]
  7. Алексеев, Г.В., Разрешимость обратных экстремальных задач для стационарных уравнений тепломассопереноса. Сиб. мат. журн., 2001, т. 42, № 5, с. 971–991. [Alekseev, G.V., Solvability of inverse extremal problems for stationary equations of heat and mass transfer. Sibirskiy matematicheskiy zhurnal = Siberian Mathematical Journal, 2001, vol. 42, no. 5, pp. 971–991. (in Russian)]
  8. Алексеев, Г.В., Адомавичюс, Э.А., О разрешимости неоднородных краевых задач для стационарных уравнений массопереноса. Дальневост. мат. журн., 2001, т. 2, № 2, с. 138–153. [Alekseev G.V., Adomavichyus E.A., On the solvability of inhomogeneous boundary value problems for stationary equations of mass transfer. Dal'nevostochnyy matematicheskiy zhurnal = Far Eastern Mathematical Journal, 2001, vol. 2, no. 2, pp. 138–153. (in Russian)]
  9. Алексеев, Г.В., Обратные экстремальные задачи для стационарных уравнений теории массопереноса. Ж. вычисл. матем. и матем. физ., 2002, т. 42, № 3, с. 380-–394. [Alekseev, G.V., Inverse extremal problems for stationary equations of mass transfer theory. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki = Journal of Computational Mathematics and Mathematical Physics, 2002, vol. 42, no. 3, pp. 380–394. (in Russian)]
  10. Алексеев, Г.В., Соболева, О.В., Об устойчивости решений экстремальных задач для стационарных уравнений массопереноса. Дальневосточный математический журнал, 2009, вып. 1–2, с. 5–14. [Alekseev, G.V., Soboleva, O.V., On the stability of solutions to extremal problems for stationary equations of mass transfer. Dal'nevostochnyy matematicheskiy zhurnal = Far Eastern Mathematical Journal, 2009, iss. 1–2, pp. 5–14.(in Russian)]
  11. Сыромятников, П.В., Матричный метод построения символа функции Грина для стационарных задач турбулентной диффузии в многослойных средах. Экологический вестник научных центров Черноморского экономического сотрудничества, 2018, т. 15, № 3, с. 62–71. [Syromyatnikov, P.V., Matrix method for constructing the Green's function symbol for stationary problems of turbulent diffusion in multilayer media. Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva = Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2018, vol. 15, no. 3, pp. 62–71. (in Russian)] DOI 10.31429/vestnik-15-3-62-71
  12. Кривошеева, М.А., Лапина, О.Н., Нестеренко, А.Г., Никитин, Ю.Г., Сыромятников, П.В., Аналитическое и численное моделирование стационарной краевой задачи диффузии–конвекции–распада для однородного слоя на основе уравнений турбулентной диффузии. Экологический вестник научных центров Черноморского экономического сотрудничества, 2020, т. 17, № 3, с. 37–47. [Krivosheeva, M.A., Lapina, O.N., Nesterenko, A.G., Nikitin, Yu.G., Syromyatnikov, P.V., Analytical and numerical modeling of the stationary boundary value problem of diffusion-convection-decay for a homogeneous layer based on the equations of turbulent diffusion. Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva = Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2020, vol. 17, no. 3, pp. 37–47. (in Russian)] DOI 10.31429/vestnik-17-3-37-47
  13. Сыромятников, П.В., Матричный метод решения нестационарных задач конвекции–диффузии в полуограниченных многослойных и градиентных средах. Наука Юга России, 2018, т. 14, № 4, с. 3–13. [Syromyatnikov, P.V., Matrix method for solving non-stationary problems of convection – diffusion in semi-bounded multilayer and gradient media. Nauka Yuga Rossii = Science of the South of Russia, 2018, vol. 14, no. 4, pp. 3–13. (in Russian)]
  14. Лапина, О.Н., Нестеренко, А.Г., Никитин, Ю.Г., Павлова, А.В., Моделирование процесса диффузии–конвекции загрязняющей примеси от периодического источника. Экологический вестник научных центров Черноморского экономического сотрудничества, 2022, т. 19, № 1, с. 26–35. [Lapina, O.N., Nesterenko, A.G., Nikitin, Yu.G., Pavlova, A.V., Modeling of the process of diffusion– convection of a pollutant from a periodic source. Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva = Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2022, vol. 19, no. 1, pp. 26–35. (in Russian)] DOI 10.31429/vestnik-19-1-25-34
  15. Liu, G.R., Han, X., Computational inverse techniques in nondestructive evaluation. CRS Press, 2003.
  16. Gablonsky, J.N., DIRECT. Version 2.0. User Guide. Technical Report CRSC-TR01-08. Center for Research and Scientific Computation, North Carolina State University, 2001.
  17. Karmazin, A., Kirillova, E., Seemsnn, W., Syromyatnikov, P., On the solution of crack identification problem in composite materials. In Advanced Problems of Mechanics: Proceedings of the XXXIX International Summer School. July 1–5 2011. St-Peterburg, Russia, pp. 227–234.
  18. Рапута, В.Ф., Крылова, А.И., Оптимизационные модели управления и контроля источников аэрозолей в приземном слое атмосферы. Оптика атмосферы и океана, 1994, т. 7, № 8, с. 1120–1125. [Raputa, V.F., Krylova, A.I., Optimization models for the control and monitoring of aerosol sources in the surface layer of the atmosphere. Optika atmosfery i okeana = Atmospheric and oceanic optics, 1994, vol. 7, no. 8, pp. 1120–1125. (in Russian)]

Downloads

Download data is not yet available.

Downloads

Issue

Vol. 19 No. 4 (2022)

Pages

76-82

Section

Mechanics

Dates

Submitted

November 20, 2022

Accepted

November 21, 2022

Published

November 30, 2022

How to Cite

[1]
Lapina, O.N., Nesterenko, A.G., Nikitin, Y.G., On the problem of the parameters determination for a stationary source in a semi-bounded layered medium. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2022, т. 19, № 4, pp. 76–82. DOI: 10.31429/vestnik-19-4-76-82

License

Copyright (c) 2022 Лапина О.Н., Нестеренко А.Г., Никитин Ю.Г.

Creative Commons License

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

Similar Articles

1-10 of 537

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

Most read articles by the same author(s)