Advertisement

Inverse Heat Conduction Problem in Two-Dimensional Anisotropic Medium

  • Surbhi AroraEmail author
  • Jaydev Dabas
Original Paper
  • 20 Downloads

Abstract

The study proposes a novel mesh-free scheme for a two-dimensional space and tests its efficiency for the inverse heat conduction problem in an anisotropic medium. It aims at determining unknown data at the inaccessible boundary with the help of an approximate solution constructed as a linear combination of fundamental solutions and heat polynomials. The advantage of using this combination lies in the fact that it yields ‘shape functions’ which are in turn solution of the considered equation. The crucial point to note here is that the proposed method yields highly accurate results and the number of collocation points required by the altered scheme are lesser than the standard method of fundamental solutions (MFS). As a direct consequence of the ill-posed problem, a highly ill-conditioned system is obtained, which is solved using regularization. Further, numerical results are provided for two-dimensional convex and non-convex domains to prove that the efficiency of the new scheme is slightly better than the traditional MFS, especially for noisy data.

Keywords

Anisotropic medium Fundamental solution Heat polynomial Regularization 

Mathematics Subject Classification

35A08 35K05 35R25 35R30 

Notes

References

  1. 1.
    Al-Khalidy, N.: A general space marching algorithm for the solution of two-dimensional boundary inverse heat conduction problems. Numer. Heat Transf. Part B 34(3), 339–360 (1998)Google Scholar
  2. 2.
    Beck, J.V., Blackwell, B., Clair Jr., C.R.S.: Inverse Heat Conduction. Wiley, Hoboken (1985)zbMATHGoogle Scholar
  3. 3.
    Bogomolny, A.: Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22(4), 644–669 (1985)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Springer, Berlin (2006)zbMATHGoogle Scholar
  5. 5.
    Chantasiriwan, S.: An algorithm for solving multidimensional inverse heat conduction problem. Int. J. Heat Mass Transf. 44(20), 3823–3832 (2001)zbMATHGoogle Scholar
  6. 6.
    Chen, B., Chen, W., Cheng, A.H., Sun, L.L., Wei, X., Peng, H.: Identification of the thermal conductivity coefficients of 3D anisotropic media by the singular boundary method. Int. J. Heat Mass Transf. 100, 24–33 (2016).  https://doi.org/10.1016/j.ijheatmasstransfer.2016.04.024 CrossRefGoogle Scholar
  7. 7.
    Chen, T.C., Tuan, P.C.: Input estimation method including finite-element scheme for solving inverse heat conduction problems. Numer. Heat Transf. Part B 47(3), 277–290 (2005)Google Scholar
  8. 8.
    Cheng, W., Zhang, Y.Q., Fu, C.: A wavelet regularization method for an inverse heat conduction problem with convection term. Electron. J. Differ. Equ. 2013(122), 1–9 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Demir, A., Hasanov, A.: Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach. J. Math. Anal. Appl. 340(1), 5–15 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dong, C.F., Sun, F.Y., Meng, B.Q.: A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium. Eng. Anal. Bound Elem. 31(1), 75–82 (2007)zbMATHGoogle Scholar
  11. 11.
    Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9(1–2), 69–95 (1998)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Fu, C., Zhu, Y., Qiu, C.Y.: Wavelet regularization for an inverse heat conduction problem. J. Math. Anal. Appl. 288(1), 212–222 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Futakiewicz, S.: Heat functions method for solving direct and inverse heat conduction problems. Ph.D. thesis (1999)Google Scholar
  14. 14.
    Gu, Y., Hua, Q., Zhang, C., He, X.: The generalized finite difference method for long-time transient heat conduction in 3D anisotropic composite materials. Appl. Math. Model. 71, 316–330 (2019)MathSciNetGoogle Scholar
  15. 15.
    Gu, Y., Fan, C.M., Xu, R.P.: Localized method of fundamental solutions for large-scale modelling of two-dimensional elasticity problems. Appl. Math. Lett. 93, 8–14 (2019)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Guo, L., Murio, D.: A mollified space-marching finite-different algorithm for the two-dimensional inverse heat conduction problem with slab symmetry. Inverse Probl. 7(2), 247 (1991)zbMATHGoogle Scholar
  17. 17.
    Hansen, P.C.: The truncatedsvd as a method for regularization. BIT Numer. Math. 27(4), 534–553 (1987)MathSciNetGoogle Scholar
  18. 18.
    Hansen, P.C.: Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank. SIAM J. Sci. Stat. Comput. 11(3), 503–518 (1990)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34(4), 561–580 (1992)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hansen, P.C.: Regularization tools: a matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6(1), 1–35 (1994)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hansen, P.C.: The L-curve and its use in the numerical of inverse problems In: Johnston, P. (ed.) Computational Inverse Problems in Electrocardiology, Advances in Computational Bioengineering, pp. 119–142. WIT Press (2000)Google Scholar
  22. 22.
    Hansen, P.C., O Leary, D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14(6), 1487–1503 (1993)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hon, Y.C., Wei, T.: A fundamental solution method for inverse heat conduction problem. Eng. Anal. Bound. Elem. 28(5), 489–495 (2004)zbMATHGoogle Scholar
  24. 24.
    Hozejowski, L.: Heat polynomials and their applications to direct and inverse heat conduction problems. Ph.D. thesis, Kielce (in Polish) (1999)Google Scholar
  25. 25.
    Jin, B.: A meshless method for the Laplace and biharmonic equations subjected to noisy boundary data. Comput. Model Eng. Sci. 6, 253–262 (2004)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Jin, B., Zheng, Y.: A meshless method for some inverse problems associated with the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 195(19), 2270–2288 (2006)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Johansson, B.T.: Properties of a method of fundamental solutions for the parabolic heat equation. Appl. Math. Lett. 65, 83–89 (2017).  https://doi.org/10.1016/j.aml.2016.08.021 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Karageorghis, A.: A practical algorithm for determining the optimal pseudo-boundary in the method of fundamental solutions. Adv. Appl. Math. Mech. 1(4), 510–528 (2009)MathSciNetGoogle Scholar
  29. 29.
    Karageorghis, A., Fairweather, G.: The method of fundamental solutions for the numerical solution of the biharmonic equation. J. Comput. Phys. 69(2), 434–459 (1987)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems, vol. 120. Springer, Berlin (2011)zbMATHGoogle Scholar
  31. 31.
    Kreyszig, E.: Introductory Functional Analysis with Applications, vol. 1. Wiley, New York (1978)zbMATHGoogle Scholar
  32. 32.
    Kupradze, V.D., Aleksidze, M.A.: The method of functional equations for the approximate solution of certain boundary value problems. USSR Comput. Math. Math. Phys. 4(4), 82–126 (1964)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kurpisz, K., Nowak, A.J.: BEM approach to inverse heat conduction problems. Eng. Anal. Bound. Elem. 10(4), 291–297 (1992)Google Scholar
  34. 34.
    Lagier, G.L., Lemonnier, H., Coutris, N.: A numerical solution of the linear multidimensional unsteady inverse heat conduction problem with the boundary element method and the singular value decomposition. Int. J. Therm. Sci. 43(2), 145–155 (2004)Google Scholar
  35. 35.
    Lesnic, D., Elliott, L., Ingham, D.B.: Application of the boundary element method to inverse heat conduction problems. Int. J. Heat Mass Transf. 39(7), 1503–1517 (1996)zbMATHGoogle Scholar
  36. 36.
    Lin, J., Chen, W., Wang, F.: A new investigation into regularization techniques for the method of fundamental solutions. Math. Comput. Simulat. 81(6), 1144–1152 (2011)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Lin, J., Chen, C.S., Liu, C.S., Lu, J.: Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions. Comput. Math. Appl. 72(3), 555–67 (2016)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Lin, J., Reutskiy, S.Y., Lu, J.: A novel meshless method for fully nonlinear advection-diffusion-reaction problems to model transfer in anisotropic media. Appl. Math. Comput. 339, 459–476 (2018)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Lin, J., Reutskiy, S.Y.: An accurate meshless formulation for the simulation of linear and fully nonlinear advection diffusion reaction problems. Adv. Eng. Softw. 126, 127–146 (2018)Google Scholar
  40. 40.
    Lu, S., Liu, J., Lin, G., Zhang, P.: Modified scaled boundary finite element analysis of 3D steady-state heat conduction in anisotropic layered media. Int. J. Heat Mass Transf. 108, 2462–2471 (2017)Google Scholar
  41. 41.
    Marin, L.: A meshless method for solving the Cauchy problem in three-dimensional elastostatics. Comput. Math. Appl. 50(1–2), 73–92 (2005a)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Marin, L.: A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations. Appl. Math. Comput. 165(2), 355–374 (2005b)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Marin, L., Lesnic, D.: The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity. Int. J. Solids Struct. 41(13), 3425–3438 (2004)zbMATHGoogle Scholar
  44. 44.
    Marin, L., Lesnic, D.: The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput. Struct. 83(4), 267–278 (2005)MathSciNetGoogle Scholar
  45. 45.
    Mathon, R., Johnston, R.L.: The approximate solution of elliptic boundary-value problems by fundamental solutions. SIAM J. Numer. Anal. 14(4), 638–650 (1977)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Murio, D.A.: The Mollification Method and the Numerical Solution of Ill-posed Problems. Wiley (2011)Google Scholar
  47. 47.
    Raynaud, M., Bransier, J.: A new finite-difference method for the nonlinear inverse heat conduction problem. Numer. Heat Transf. Part A Appl. 9(1), 27–42 (1986)Google Scholar
  48. 48.
    Reinhardt, H.J.: A numerical method for the solution of two-dimensional inverse heat conduction problems. Int. J. Numer. Methods Eng. 32(2), 363–383 (1991)zbMATHGoogle Scholar
  49. 49.
    Rosenbloom, P.C., Widder, D.V.: Expansions in terms of heat polynomials and associated functions. Trans. Am. Math. Soc. 92(2), 220–266 (1959)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Rostamian, M., Shahrezaee, A.: A meshless method for solving 1D time-dependent heat source problem. Inverse Probl. Sci. Eng. 26(1), 51–82 (2018)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Ruzhansky, M., Cho, Y.J., Agarwal, P., Area, I., et al.: Advances in Real and Complex Analysis with Applications. Springer, Berlin (2017)zbMATHGoogle Scholar
  52. 52.
    Shigeta, T., Young, D.L.: Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points. J. Comput. Phys. 228(6), 1903–1915 (2009)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Sun, Y., He, S.: A meshless method based on the method of fundamental solution for three-dimensional inverse heat conduction problems. Int. J. Heat Mass Transf. 108, 945–960 (2017)Google Scholar
  54. 54.
    Ushijima, T., Chiba, F.: A fundamental solution method for the reduced wave problem in a domain exterior to a disc. J. Comput. Appl. Math. 152(1), 545–557 (2003)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Yano, H., Fukutani, S., Kieda, A.: A boundary residual method with heat polynomials for solving unsteady heat conduction problems. J. Frankl. Inst. 316(4), 291–298 (1983)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of Applied Sciences and EngineeringIndian Institute of TechnologyRoorkeeIndia

Personalised recommendations