Разностный метод вычисления потока тепла на недоступной границе в задаче теплопроводности

Разностный метод вычисления потока тепла на недоступной границе в задаче теплопроводности

Сорокин С. Б.
Сибирский журнал индустриальной математики, 26, 3(95), 125-141 (2023)
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УДК 519.632 
DOI: 10.33048/SIBJIM.2023.26.310

Аннотация:

Рассматривается задача продолжения для уравнения теплопроводности. Определение потока тепла на недоступной границе сводится к обратной задаче. Для численного решения обратной задачи применяется неявная разностная схема. На каждом временном шаге для разностного аналога эллиптического уравнения экономичным прямым методом вычисляется поток тепла на недоступной границе. Предложенный алгоритм существенно расширяет круг решаемых задач и может применяться при создании приборов, способных в реальном масштабе времени определять поток тепла на недоступных для измерения частях неоднородных конструкций, например для определение потока тепла на внутреннем радиусе трубы, выполненной из различных материалов.

Литература:
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  5. Alifanov O. M. Inverse Heat Transfer Problems. Springer Sci. & Business Media, 2012.
     
  6. Tikhonov A. N, Arsenin V. Y. Methods of Solution of Ill-Posed Problems. M.: Nauka, 1979.
     
  7. Marin L., Lesnic D. The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtztype equations // Comput. & Structures. 2005. V. 83, N 4–5. P. 267–278.
     
  8. Marin L. A meshless method for the numerical solution of the Cauchy problem associated with threedimensional Helmholtz-type equations // Appl. Math. Comput. 2005. V. 165, N 2. P. 355–374.
     
  9. Jin B., Zheng Y. A meshless method for some inverse problems associated with the Helmholtz equation // Comput. Methods Appl. Mech. Engrg. 2006. V. 195, N 19–22. P. 2270–2288.
     
  10. Wei T., Hon Y. C., Ling L. Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators // Engrg. Analysis with Boundary Elements. 2007. V. 31, N 4. P. 373–385.
     
  11. Marin L., Karageorghis A., Lesni D. The MFS for numerical boundary identification in two-dimensional harmonic problems // Engrg. Analysis with Boundary Elements. 2011. V. 35, N 3. P. 342–354.
     
  12. Wei T., Chen Y. G. A regularization method for a Cauchy problem of Laplace’s equation in an annular domain // Math. Comput. Simulation. 2012. V. 82, N 11. P. 2129–2144.
     
  13. Caille L., Marin L., Delvare F. A meshless fading regularization algorithm for solving the Cauchy problem for the three-dimensional Helmholtz equation // Numer. Algorithms. 2019. V. 82, N 3. P. 869–894.
     
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  15. Qin H. H., Wei T., Shi R. Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation // J. Comput. Appl. Math. 2009. V. 224, N 1. P. 39–53.
     
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  17. Colaço M. J., Alves C. J. S. A fast non-intrusive method for estimating spatial thermal contact conductance by means of the reciprocity functional approach and the method of fundamental solutions // Internat. J. Heat Mass Transfer. 2013. V. 60. P. 653–663.
     
  18. Colaço M. J., Alves C. J. S., Bozzoli F. The reciprocity function approach applied to the non-intrusive estimation of spatially varying internal heat transfer coefficients in ducts: numerical and experimental results // Internat. J. Heat Mass Transfer. 2015. V. 90. P. 1221–1231.
     
  19. Cattani L., Maillet D., Bozzoli F., Rainieri S. Estimation of the local convective heat transfer coefficient in pipe flow using a 2d thermal quadrupole model and truncated singular value decomposition // Internat. J. Heat Mass Transfer. 2015. V. 91. P. 1034–1045.
     
  20. Mocerino A., Colaço M. J., Bozzoli F., Rainieri S. Filtered reciprocity functional approach to estimate internal heat transfer coefficients in 2d cylindrical domains using infrared thermography // Internat. J. Heat Mass Transfer. 2018. V. 125. P. 1181–1195.
     
  21. Bazán F. S. V., Bedin L., Bozzoli F. New methods for numerical estimation of convective heat transfer coefficient in circular ducts // Internat. J. Thermal Sci. 2019. V. 139. P. 387–402.
     
  22. Sorokin S. B. An efficient direct method for numerically solving the Cauchy problem for Laplace’s equation // Numer. Anal. Appl. 2019. V. 12. P. 87–103.
     
  23. Sorokin S. B. An implicit iterative method for numerical solution of the Cauchy problem for elliptic equations // J. Appl. Indust. Math. 2019. V. 13. P. 759–770.
     
  24. Kozlov V., Maz’ya V., Fomin A. An iterative method for solving the Cauchy problem for elliptic equations // Comput. Math. Math. Phys. 1991. V. 31, N 1. P. 45–52.
     
  25. Marin L., Elliott L., Heggs P. J., Ingham D. B., Lesnic D., Wen X. An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation // Comput. Methods Appl. Mech. Engrg. 2003. V. 192, N 5–6. P. 709–722.
     
  26. Marin L. Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems // Engrg. Analysis with Boundary Elements. 2011. V. 35, N 3. P. 415– 429.
     
  27. Kabanikhin S. I., Karchevsky A. L. Optimizational method for solving the Cauchy problem for an elliptic equation // J. Inverse Ill-Posed Probl. 1995. V. 3, N 1. P. 21–46.
     
  28. Marin L., Elliott L., Heggs P., Ingham D., Lesnic D., Wen X. Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations // Comput. Mech. 2003. V. 31, N 3–4. P. 367–377.
     
  29. Marin L., Elliott L., Heggs P., Ingham D., Lesnic D., Wen X. Bem solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method // Engrg. Analysis with Boundary Elements. 2004. V. 28, N 9. P. 1025–1034.
     
  30. Kabanikhin S. I., Shishlenin M. A., Nurseitov D. B., Nurseitova A. T., Kasenov S. E. Comparative analysis of methods for regularizing an initial boundary value problem for the Helmholtz equation // J. Appl. Math. 2014. V. 2014. Article 786326; https://doi.org/10.1155/2014/786326
     
  31. Johansson B. T., Lesnic D. A method of fundamental solutions for transient heat conduction // Engrg. Analysis with Boundary Elements. 2008. V. 32, N 9. P. 697–703.
     
  32. Johansson B. T., Lesnic D., Reeve T. A method of fundamental solutions for two-dimensional heat conduction // Internat. J. Comput. Math. 2011. V. 88, N 8. P. 1697–1713.
     
  33. Cao R. Numerical computation based on the method of fundamental solutions for a Cauchy problem of heat equation // Turkish J. Anal. Number Theory. 2014. V. 2, N 3. P. 70–74.
     
  34. Reeve T. H. The method of fundamental solutions for some direct and inverse problems. Thesis Ph. D. Univ. Birmingham, 2013.
     
  35. Yaparova N. M. Numerical methods for solving a boundary-value inverse heat conduction problem // Inverse Probl. Sci. Engrg. 2014. V. 22, N 5. P. 832–847.
     
  36. Solodusha S. V., Yaparova N. M. Numerical solving an inverse boundary value problem of heat conduction using Volterra equations of the first kind // Numer. Anal. Appl. 2015. V. 8, N 3. P. 267–274.
     
  37. Belonosov A. S., Shishlenin M. A. Continuation problem for the parabolic equation with the data on the part of the boundary // Sib. Electron. Math. Reports. 2014. V. 11. P. 22–34.
     
  38. Belonosov A. S., Shishlenin M. A. Regularization methods of the continuation problem for the parabolic equation // Internat. Conf. Numerical Analysis and Its Applications. Cham: Springer-Verl., 2017. P. 220– 226.
     
  39. Belonosov A., Shishlenin M., Klyuchinskiy D. A comparative analysis of numerical methods of solving the continuation problem for 1D parabolic equation with the data given on the part of the boundary // Adv. Comput. Math. 2019. V. 45, N 2. P. 735–755.
     
  40. Prikhodko A., Shishlenin M. Comparative analysis of the numerical methods for 3d continuation problem for parabolic equation with data on the part of the boundary // J. Physics. Conf. Ser. 2021. V. 2092, N 1. Article 012010.
     
  41. Karchevsky A. L. Development of the heated thin foil technique for investigating nonstationary transfer processes // Interfac. Phenom. Heat Transfer. 2018. V. 6, N 3. P. 179–185.
     
  42. Marchuk G. I., Brown A. A. Methods of Numerical Mathematics. V. 2. N. Y.: Springer-Verl., 1982.
     
  43. Samarskii A. A. The Theory of Difference Schemes. CRC Press, 2001.
     
  44. Samarskii A. A., Andreev V. B. Difference Methods for Elliptic Equations. M.: Nauka, 1976.
     
  45. Samarskii A. A., Nikolaev E. S. Methods of Solution of Grid Equations. M.: Nauka, 1978.
     
  46. Kabanikhin S. I. Inverse and Ill-Posed Problems: Theory and Applications. V. 55. Walter De Gruyter, 2011.

Работа выполнена в рамках государственного задания ИВМиМГ СО РАН (проект 0251-2021-0001).

С. Б. Сорокин
  1. Институт вычислительной математики и математической геофизики СО РАН, 
    просп. Акад. Лаврентьева, 6, г. Новосибирск 630090, Россия
  2. Новосибирский государственный университет, 
    ул. Пирогова, 1, г. Новосибирск 630090, Россия

E-mail: sorokin@sscc.ru

Статья поступила 28.02.2023 г. 
После доработки — 03.03.2023 г.
Принята к публикации 27.04.2023 г.

Abstract:

The continuation problem for the heat equation is considered. Determining the heat flux at an inaccessible boundary reduces to an inverse problem. For the numerical solution of the inverse problem, an implicit difference scheme is used. At each time step, for the difference analogue of the elliptic equation, the heat flux at the inaccessible boundary is calculated by an economical direct method. The proposed algorithm significantly expands the range of tasks to be solved and can be used for creation of devices capable of real-time determination of the heat flux on inaccessible measurements of parts of inhomogeneous structures. For example, to determine the heat flux on the internal radius of a pipe made of various materials.

References:
  1. Zachár A. Analysis of coiled-tube heat exchangers to improve heat transfer rate with spirally corrugated wall. Internat. J. Heat Mass Transfer, 2010, Vol. 53, No. 19–20, pp. 3928–3939.
     
  2. Kundan A., Plawsky J. L., Wayner Jr. P. C. Thermophysical characteristics of a wickless heat pipe in microgravity–constrained vapor bubble experiment. Internat. J. Heat Mass Transfer, 2014, Vol. 78, pp. 1105–1113.
     
  3. Karchevsky A. L., Marchuk I. V., Kabov O. A. Calculation of the heat flux near the liquid–gas–solid contact line. Appl. Math. Model., 2016, Vol. 40, No. 2, pp. 1029–1037.
     
  4. Cheverda V. V., Karchevsky A. L., Marchuk I. V., Kabov O. A. Heat flux density in the region of droplet contact line on a horizontal surface of a thin heated foil. Thermoph. Aeromech., 2017, Vol. 24, No. 5, pp. 803–806.
     
  5. Alifanov O. M. Inverse Heat Transfer Problems. Springer Sci. & Business Media, 2012.
     
  6. Tikhonov A. N., Arsenin V. Y. Methods of Solution of Ill-Posed Problems. Moscow: Nauka, 1979.
     
  7. Marin L., Lesnic D. The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtztype equations. Comput. & Structures, 2005, Vol. 83, No. 4–5, pp. 267–278.
     
  8. Marin L. A meshless method for the numerical solution of the Cauchy problem associated with threedimensional Helmholtz-type equations. Appl. Math. Comput., 2005, Vol. 165, No. 2, pp. 355–374.
     
  9. Jin B., Zheng Y. A meshless method for some inverse problems associated with the Helmholtz equation. Comput. Methods Appl. Mech. Engrg., 2006, Vol. 195, No. 19–22, pp. 2270–2288.
     
  10. Wei T., Hon Y. C., Ling L. Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Engrg. Analysis with Boundary Elements, 2007, Vol. 31, No. 4, pp. 373– 385.
     
  11. Marin L., Karageorghis A., Lesni D. The MFS for numerical boundary identification in two-dimensional harmonic problems. Engrg. Analysis with Boundary Elements, 2011, Vol. 35, No. 3, pp. 342–354.
     
  12. Wei T., Chen Y. G. A regularization method for a Cauchy problem of Laplace’s equation in an annular domain. Math. Comput. Simulation, 2012, Vol. 82, No. 11, pp. 2129–2144.
     
  13. Caille L., Marin L., Delvare F. A meshless fading regularization algorithm for solving the Cauchy problem for the three-dimensional Helmholtz equation. Numer. Algorithms, 2019, Vol. 82, No 3, pp. 869–894.
     
  14. Cheng J., Hon Y. C., Wei T., Yamamoto M. Numerical computation of a Cauchy problem for Laplace’s equation. J. Appl. Math. Mech., 2001, Vol. 81, No. 10., pp. 665–674.
     
  15. Qin H. H., Wei T., Shi R. Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation J. Comput. Appl. Math. 2009. V. 224, No. 1, pp. 39–53.
     
  16. Karchevsky A. L. Reformulation of an inverse problem statement that reduces computational costs. Euras. J. Math. Comput. Appl., 2013, Vol. 1, No. 2, pp. 4–20.
     
  17. Colaço M. J., Alves C. J. S. A fast non-intrusive method for estimating spatial thermal contact conductance by means of the reciprocity functional approach and the method of fundamental solutions. Internat. J. Heat Mass Transfer, 2013, Vol. 60, pp. 653–663.
     
  18. Colaço M. J., Alves C. J. S., Bozzoli F. The reciprocity function approach applied to the non-intrusive estimation of spatially varying internal heat transfer coefficients in ducts: numerical and experimental results. Internat. J. Heat Mass Transfer, 2015, Vol. 90, pp. 1221–1231.
     
  19. Cattani L., Maillet D., Bozzoli F., Rainieri S. Estimation of the local convective heat transfer coefficient in pipe flow using a 2d thermal quadrupole model and truncated singular value decomposition. Internat. J. Heat Mass Transfer, 2015, Vol. 91, pp. 1034–1045.
     
  20. Mocerino A., Colaço M. J., Bozzoli F., Rainieri S. Filtered reciprocity functional approach to estimate internal heat transfer coefficients in 2d cylindrical domains using infrared thermography. Internat. J. Heat Mass Transfer, 2018, Vol. 125, pp. 1181–1195.
     
  21. Bazán F. S. V., Bedin L., Bozzoli F. New methods for numerical estimation of convective heat transfer coefficient in circular ducts. Internat. J. Thermal Sci., 2019, Vol. 139, pp. 387–402.
     
  22. Sorokin S. B. An efficient direct method for numerically solving the Cauchy problem for Laplace’s equation. Numer. Anal. Appl., 2019, Vol. 12, pp. 87–103.
     
  23. Sorokin S. B. An implicit iterative method for numerical solution of the Cauchy problem for elliptic equations J. Appl. Indust. Math. 2019. V. 13, pp. 759–770.
     
  24. Kozlov V., Maz’ya V., Fomin A. An iterative method for solving the Cauchy problem for elliptic equations. Comput. Math. Math. Phys., 1991, Vol. 31, No. 1, pp. 45–52.
     
  25. Marin L., Elliott L., Heggs P. J., Ingham D. B., Lesnic D., Wen X. An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation. Comput. Methods Appl. Mech. Engrg., 2003, Vol. 192, No. 5–6, pp. 709–722.
     
  26. Marin L. Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems. Engrg. Analysis with Boundary Elements, 2011, Vol. 35, No. 3, pp. 415–429.
     
  27. Kabanikhin S. I., Karchevsky A. L. Optimizational method for solving the Cauchy problem for an elliptic equation. J. Inverse Ill-Posed Probl., 1995, Vol. 3, No. 1, pp. 21–46.
     
  28. Marin L., Elliott L., Heggs P., Ingham D., Lesnic D., Wen X. Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations. Comput. Mech., 2003, Vol. 31, No. 3–4, pp. 367–377.
     
  29. Marin L., Elliott L., Heggs P., Ingham D., Lesnic D., Wen X. Bem solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method. Engrg. Analysis with Boundary Elements, 2004, Vol. 28, No. 9, pp. 1025–1034.
     
  30. Kabanikhin S. I., Shishlenin M. A., Nurseitov D. B., Nurseitova A. T., Kasenov S. E. Comparative analysis of methods for regularizing an initial boundary value problem for the Helmholtz equation. J. Appl. Math., 2014, Vol. 2014, article 786326; https://doi.org/10.1155/2014/786326
     
  31. Johansson B. T., Lesnic D. A method of fundamental solutions for transient heat conduction. Engrg. Analysis with Boundary Elements, 2008, Vol. 32, No. 9, pp. 697–703.
     
  32. Johansson B. T., Lesnic D., Reeve T. A method of fundamental solutions for two-dimensional heat conduction. Internat. J. Comput. Math., 2011, Vol. 88, No. 8, pp. 1697–1713.
     
  33. Cao R. Numerical computation based on the method of fundamental solutions for a Cauchy problem of heat equation. Turkish J. Anal. Number Theory, 2014, Vol. 2, No. 3, pp. 70–74.
     
  34. Reeve T. H. The method of fundamental solutions for some direct and inverse problems. Thesis Ph. D. Univ. Birmingham, 2013.
     
  35. Yaparova N. M. Numerical methods for solving a boundary-value inverse heat conduction problem. Inverse Probl. Sci. Engrg. 2014. V. 22, No. 5, pp. 832–847.
     
  36. Solodusha S. V., Yaparova N. M. Numerical solving an inverse boundary value problem of heat conduction using Volterra equations of the first kind. Numer. Anal. Appl., 2015, Vol. 8, No. 3, pp. 267–274.
     
  37. Belonosov A. S., Shishlenin M. A. Continuation problem for the parabolic equation with the data on the part of the boundary. Sib. Electron. Math. Reports, 2014, Vol. 11, pp. 22–34.
     
  38. Belonosov A. S., Shishlenin M. A. Regularization methods of the continuation problem for the parabolic equation. Numer. Anal. Appl., 2017, pp. 220–226.
     
  39. Belonosov A., Shishlenin M., Klyuchinskiy D. A comparative analysis of numerical methods of solving the continuation problem for 1D parabolic equation with the data given on the part of the boundary. Adv. Comput. Math., 2019, Vol. 45, No. 2, pp. 735–755.
     
  40. Prikhodko A., Shishlenin M. Comparative analysis of the numerical methods for 3d continuation problem for parabolic equation with data on the part of the boundary. J. Physics. Conf. Ser., 2021, Vol. 2092, No. 1, article 012010.
     
  41. Karchevsky A. L. Development of the heated thin foil technique for investigating nonstationary transfer processes. Interfac. Phenom. Heat Transfer, 2018, Vol. 6, No. 3, pp. 179–185.
     
  42. Marchuk G. I., Brown A. A. Methods of Numerical Mathematics. V. 2. N. Y.: Springer-Verl., 1982.
     
  43. Samarskii A. A. The Theory of Difference Schemes. CRC Press, 2001.
     
  44. Samarskii A. A., Andreev V. B. Difference Methods for Elliptic Equations. Moscow: Nauka, 1976.
     
  45. Samarskii A. A., Nikolaev E. S. Methods of Solution of Grid Equations. Moscow: Nauka, 1978.
     
  46. Kabanikhin S. I. Inverse and Ill-Posed Problems: Theory and Applications. V. 55. Walter De Gruyter, 2011.