Skip to main content
SpringerLink
Log in

Recovering temperature-dependent heat conductivity in 2D and 3D domains with homogenization functions as the bases

  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

The paper solves the parameters identification problem in a nonlinear heat equation with homogenization functions as the bases, which are constructed from the boundary data of the temperature in the 2D and 3D space-time domains. To satisfy the over-specified Neumann boundary condition, a linear equations system is derived and then used to determine the expansion coefficients of the solution. Then, after back substituting the solution and collocating points to satisfy the governing equations, the space-time-dependent and temperature-dependent heat conductivity functions in 2D and 3D nonlinear heat equations are identified by solving other linear systems. The novel methods do not need iteration and solving nonlinear equations, since the unknown heat conductivities are retrieved from the solutions of linear systems. The solutions and the heat conductivity functions recovered are quite accurate in the entire space-time domain. We find that even for the inverse problems of nonlinear heat equations, the homogenization functions method is easily used to recover 2D and 3D space-time-dependent and temperature-dependent heat conductivity functions. It is interesting that the present paper makes a significant contribution to the engineering and science in the field of inverse problems of heat conductivity, merely solving linear equations and without employing iteration and solving nonlinear equations to solve nonlinear inverse problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Cannon JR, Duchateau P (1998) Structural identification of an unknown source term in a heat equation. Inverse Probl 14:535–551

    Article  MathSciNet  MATH  Google Scholar 

  2. Savateev EG, Duchateau P (1995) On problems of determining the source function in a parabolic equation. J Inverse Ill-Posed Prob 3:83–102

    MathSciNet  MATH  Google Scholar 

  3. Borukhov VT, Vabishchevich PN (2000) Numerical solution of the inverse problem of reconstructing a distributed right-hand side of a parabolic equation. Comput Phys Commun 126:32–36

    Article  MathSciNet  MATH  Google Scholar 

  4. Farcas A, Lesnic D (2006) The boundary-element method for the determination of a heat source dependent on one variable. J Eng Math 54:375–388

    Article  MathSciNet  MATH  Google Scholar 

  5. Ling L, Yamamoto M, Hon YC (2006) Identification of source locations in two-dimensional heat equations. Inverse Probl 22:1289–1305

    Article  MathSciNet  MATH  Google Scholar 

  6. Yan L, Fu CL, Yang FL (2008) The method of fundamental solutions for the inverse heat source problem. Eng Anal Bound Elements 32:216–222

    Article  MATH  Google Scholar 

  7. Yang F, Fu CL (2010) The method of simplified Tikhonov regularization for dealing with the inverse time-dependent heat source problem. Comput Math Appl 60:1228–1236

    Article  MathSciNet  MATH  Google Scholar 

  8. Yang L, Dehghan M, Yu JN, Luo GW (2011) Inverse problem of time-dependent heat sources numerical reconstruction. Math Comput Simul 81:1656–1672

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu C-S (2016) Homogenized functions to recover \(H(t)\)/\(H(x)\) by solving a small scale linear system of differencing equations. Int J Heat Mass Transfer 101:1103–1110

    Article  Google Scholar 

  10. Liu C-S (2016) To recover heat source \(G(x)+H(t)\) by using homogenized function and solving rectangular differencing equations. Numer Heat Transfer B 69:351–363

    Article  Google Scholar 

  11. Hasanov A (2012) Identification of spacewise and time dependent source terms in 1D heat conduction equation from temperature measurement at a final time. Int J Heat Mass Transfer 55:2069–2080

    Article  Google Scholar 

  12. Yang L, Yu JN, Luo GW, Deng ZC (2012) Reconstruction of a space and time dependent heat source from finite measurement data. Int J Heat Mass Transfer 55:6573–6581

    Article  Google Scholar 

  13. Liu C-S, Chang CW (2016) A global boundary integral equation method for recovering space-time dependent heat source. Int J Heat Mass Transfer 92:1034–1040

    Article  Google Scholar 

  14. Liu C-S (2016) An integral equation method to recover non-additive and non-separable heat source without initial temperature. Int J Heat Mass Transfer 97:943–953

    Article  Google Scholar 

  15. Liu C-S, Liu D (2017) Recovering a general space-time-dependent heat source by coupled boundary integral equations method. Numer Heat Transfer B 71:283–297

    Article  Google Scholar 

  16. Chang CW, Liu C-S (2012) A new optimal scheme for solving nonlinear heat conduction problems. Comput Model Eng Sci 88:269–291

    MathSciNet  MATH  Google Scholar 

  17. Domairry G, Nadim N (2008) Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation. Int Commun Heat Mass Transf 35:93–102

    Article  Google Scholar 

  18. Ganji DD, Sadighi A (2007) Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. J Comput Appl Math 207:24–34

    Article  MathSciNet  MATH  Google Scholar 

  19. Hashemi MS (2015) Constructing a new geometric numerical integration method to the nonlinear heat transfer equations. Commun Nonlinear Sci Numer Simul 22:990–1001

    Article  MathSciNet  MATH  Google Scholar 

  20. Moore TJ, Jones MR (2015) Solving nonlinear heat transfer problems using variation of parameters. Int J Thermal Sci 93:29–35

    Article  Google Scholar 

  21. Yaghoobi H, Torabi M (2011) The application of differential transformation method to nonlinear equations arising in heat transfer. Int Commun Heat Mass Transf 38:815–820

    Article  Google Scholar 

  22. Zogheib B, Tohidi E (2017) An accurate space-time pseudospectral method for solving nonlinear multi-dimensional heat transfer problems. Mediterr J Math 14:30

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu C-S (2014) On-line detecting heat source of a nonlinear heat conduction equation by a differential algebraic equation method. Int J Heat Mass Transfer 76:153–161

    Article  Google Scholar 

  24. Liu C-S (2014) An iterative method to recover heat conductivity function of a nonlinear heat conduction equation. Numer Heat Transfer B 65:80–101

    Article  Google Scholar 

  25. Liu C-S, Qiu L, Wang F (2019) Nonlinear wave inverse source problem solved by a method of \(m\)-order homogenization functions. Appl Math Lett 91:90–96

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu C-S, Chen YW, Chang JR (2019) Solving a nonlinear convection-diffusion equation with source and moving boundary both unknown by a family of homogenization functions. Int J Heat Mass Transfer 138:25–31

    Article  Google Scholar 

  27. Liu C-S, Chang CW (2019) Solving the inverse conductivity problems of nonlinear elliptic equations by the superposition of homogenization functions method. Appl Math Lett 94:272–278

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu C-S, Qiu L, Lin J (2019) Solving the higher-dimensional nonlinear inverse heat source problems by the superposition of homogenization functions method. Int J Heat Mass Transfer 141:651–657

    Article  Google Scholar 

  29. Qiu L, Lin J, Wang F, Qin QH, Liu C-S (2021) A homogenization function method for inverse heat source problems in 3D functionally graded materials. Appl Math Model 91:923–933

    Article  MathSciNet  MATH  Google Scholar 

  30. Qiu L, Hu C, Qin QH (2020) A novel homogenization function method for inverse source problem of nonlinear time-fractional wave equation. Appl Math Lett 109:106554

    Article  MathSciNet  MATH  Google Scholar 

  31. Chen HT, Lin JY (1998) Simultaneous estimations of temperature-dependent thermal conductivity and heat capacity. Int J Heat Mass Transfer 41:2237–2244

    Article  MATH  Google Scholar 

  32. Alifanov OM, Mikhailov VV (1978) Solution of the non-linear inverse thermal conductivity problem by the iteration method. J Eng Phys 35:1501–1506

    Article  Google Scholar 

  33. Huang CH, Yan JY (1995) An inverse problem in simultaneously measuring temperature-dependent thermal conductivity and heat capacity. Int J Heat Mass Transfer 38:3433–3441

    Article  Google Scholar 

  34. Huang CH, Yan JY, Chen HT (1995) Function estimation in predicting temperature-dependent thermal conductivity without internal measurements. AIAA J Thermophys Heat Transf 9:667–673

    Article  Google Scholar 

  35. Huang CH, Ozisik MN (1991) A direct integration approach for simultaneously estimating temperature dependent thermal conductivity and heat capacity. Numer Heat Transfer A 20:95–110

    Article  Google Scholar 

  36. Yang CY (1998) A linear inverse model for the temperature-dependent thermal conductivity determination in one-dimensional problems. Appl Math Model 22:1–9

    Article  MathSciNet  MATH  Google Scholar 

  37. Yang CY (1999) Estimation of the temperature-dependent thermal conductivity in inverse heat conduction problems. Appl Math Model 23:469–478

    Article  MATH  Google Scholar 

  38. Yang CY (2000) Determination of the temperature dependent thermophysical properties from temperature responses measured at medium’s boundaries. Int J Heat Mass Transfer 43:1261–1270

    Article  MATH  Google Scholar 

  39. Tervola P (1989) A method to determine the thermal conductivity from measured temperature profiles. Int J Heat Mass Transfer 32:1425–1430

    Article  MATH  Google Scholar 

  40. Kim S, Kim MC, Kim KY (2003) Non-iterative estimation of temperature-dependent thermal conductivity without internal measurements. Int J Heat Mass Transfer 46:1801–1810

    Article  MATH  Google Scholar 

  41. Lesnic D (2002) The determination of the thermal properties of a heat conductor in a nonlinear heat conduction problem. Z Ang Math Phys (ZAMP) 53:175–196

    Article  MathSciNet  MATH  Google Scholar 

  42. Lesnic D, Elliott L, Ingham DB (1995) A note of the determination of the thermal properties of a material in a transient nonlinear heat conduction problem. Int Commun Heat Mass Transf 22:475–482

    Article  Google Scholar 

  43. Lesnic D, Elliott L, Ingham DB (1996) Identification of the thermal conductivity and heat capacity in unsteady nonlinear heat conduction problems using the boundary element method. J Comput Phys 126:410–420

    Article  MATH  Google Scholar 

  44. Lin JH, Chen CK, Yang YT (2001) Inverse method for estimating thermal conductivity in one-dimensional heat conduction problems. AIAA J Thermophys Heat Transf 15:34–41

    Article  Google Scholar 

  45. Yeung WK, Lam TT (1996) Second-order finite difference approximation for inverse determination of thermal conductivity. Int J Heat Mass Transfer 39:3685–3693

    Article  Google Scholar 

  46. Liu C-S (2007) Identification of temperature-dependent thermophysical properties in a partial differential equation subject to extra final measurement data. Numer Methods Partial Differ Eq 23:1083–1109

    Article  MathSciNet  MATH  Google Scholar 

  47. Liu C-S (2008) An LGSM to identify nonhomogeneous heat conductivity functions by an extra measurement of temperature. Int J Heat Mass Transfer 51:2603–2613

    Article  MATH  Google Scholar 

  48. Kaipio JP, Fox C (2011) The Bayesian framework for inverse problems in heat transfer. Heat Transfer Eng 32:718–753

    Article  Google Scholar 

  49. Chang CW, Kuo CC (2013) A Lie-group algorithm to estimate the source coefficients in the advection-dispersion equations. Numer Heat Transfer B 64:147–173

    Article  Google Scholar 

  50. Xu WX, Zhang YF, Jiang JY, Liu ZY, Jiao Y (2021) Thermal conductivity and elastic modulus of 3D porous/fractured media considering percolation. Int J Eng Sci 161:103456

    Article  MATH  Google Scholar 

  51. Liu C-S (2011) A Lie-group adaptive method to identify spatial-dependence heat conductivity coefficients. Numer.Heat Transfer B 60:305–323

    Article  Google Scholar 

  52. Liu C-S (2020) An energetic boundary functional method for solving the inverse heat conductivity problems in arbitrary plane domains. Int J Heat Mass Transfer 151:119418

    Article  Google Scholar 

  53. Liu C-S (2006) One-step GPS for the estimation of temperature-dependent thermal conductivity. Int J Heat Mass Transfer 49:3084–3093

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The work was supported by the National Natural Science Foundation of China (No. 12072103), the Fundamental Research Funds for the Central Universities (No. B200202126), the Natural Science Foundation of Jiangsu Province (No. BK20190073), the State Key Laboratory of Acoustics, Chinese Academy of Sciences (No. SKLA202001), and the China Postdoctoral Science Foundation (Nos. 2017M611669, 2018T110430).

Author information

Authors and Affiliations

  1. Key Laboratory of Coastal Disaster and Defence (Ministry of Education), College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu, 210098, China

    Ji Lin & Chein-Shan Liu

  2. Center of Excellence for Ocean Engineering, Center of Excellence for the Oceans, National Taiwan Ocean University, Keelung, 202-24, Taiwan

    Chein-Shan Liu

Authors
  1. Ji Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chein-Shan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Ji Lin or Chein-Shan Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, J., Liu, CS. Recovering temperature-dependent heat conductivity in 2D and 3D domains with homogenization functions as the bases. Engineering with Computers 38 (Suppl 3), 2349–2363 (2022). https://doi.org/10.1007/s00366-021-01384-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-021-01384-w

Keywords

Search

Navigation