Skip to main content
Log in

An iterative method to solve the heat transfer problem under the non-linear boundary conditions

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

The aim of the paper is to determine the approximation of the tangential matrix for solving the non-linear heat transfer problem. Numerical model of the strongly non-linear heat transfer problem based on the theory of the finite element method is presented. The tangential matrix of the Newton method is formulated. A method to solve the heat transfer with the non-linear boundary conditions, based on the secant slope of a reference function, is developed. The contraction mapping principle is introduced to verify the convergence of this method. The application of the method is shown by two examples. Numerical results of these examples are comparable to the ones solved with the Newton method and the commercial software COMSOL for the heat transfer problem under the radiative boundary conditions.

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

Access this article

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

Similar content being viewed by others

References

  1. Arslanturk C (2009) Correlation equations for optimum design of annular fins with temperature dependent thermal conductivity. Heat Mass Transf 45:519–525

    Article  Google Scholar 

  2. Blobner J, Bialecki RA, Kuhn G (1999) Transient non-linear heat conduction-radiation problems: a boundary element formulation. Int J Numer Methods Eng 46:1865–1882

    Article  MathSciNet  MATH  Google Scholar 

  3. Bouaziz MN, Hanini S (2007) Efficiency and optimisation of fin with temperature-dependent thermal conductivity: a simplified solution. Heat Mass Transf 44:1–9

    Article  Google Scholar 

  4. Budacova J (1977) Method of finite elements for solving some heat-conduction problems. J Eng Phys Thermophys 33:728–733

    Google Scholar 

  5. Campo A (1977) Unsteady heat transfer from a circular fin with nonlinear dissipation. Heat Mass Transf 10:203–210

    Google Scholar 

  6. Campo A (1982) Estimate of the transient conduction of heat in materials with linear thermal properties based on the solution for constant properties. Heat Mass Transf 17:1–9

    Google Scholar 

  7. Comini G, Guidice SD, Lewis RW, Zienkiewicz OC (1974) Finite element solution of non-linear heat conduction problems with special reference to phase change. Int J Numer Methods Eng 8:613–624

    Article  MATH  Google Scholar 

  8. Gururaja Rao C, Nagabhushana Rao V, Krishna Das C (2008) Simulation studies on multi-mode heat transfer from an open cavity with a flush-mounted discrete heat source. Heat Mass Transf 44:727–737

    Article  Google Scholar 

  9. Hosseini SM, Akhlaghi M, Shakeri M (2007) Transient heat conduction in functionally graded thick hollow cylinders by analytical method. Heat Mass Transf 43:669–675

    Article  Google Scholar 

  10. Khattabi A, Steinhagen P (1993) Analysis of transient nonlinear heat conduction in wood using finite-difference solutions. Holz als Roh-und Werkstoff 51:272–278

    Article  Google Scholar 

  11. Kim S, Huang CH (2006) The approximate solutions to the non-linear heat conduction problems in a semi-infinite medium. Heat Mass Transf 42:727–738

    Article  Google Scholar 

  12. Koizumi M, Utamura M, Kotani K (1985) Three-dimensional transient heat conduction analysis with non-linear boundary conditions by boundary element method. J Nucl Sci Technol 22:972–982

    Article  Google Scholar 

  13. Kwon YW, Bang H (1997) The finite element method using matlab. CRC Press, Boca Raton

    Google Scholar 

  14. Lees M (1966) A linear three-level difference scheme for quasilinear parabolic equations. Math Comput 20:516–522

    Article  MathSciNet  MATH  Google Scholar 

  15. Lewis RW, Nithiarasu P, Seetharamu K (2004) Fundamentals of the finite element method for heat and fluid flow. Wiley, Chichester

    Book  Google Scholar 

  16. Liu FL, Du RY (2005) Nonlinear fem for calculating temperature field. J Hebei Norm Univ (Nat Sci Edition) 29:21–24

    Google Scholar 

  17. Mehta RC, Jayachandran T (1988) Finite element analysis of conductive and radiative heating of a thin skin calorimeter. Heat Mass Transf 22:227–230

    Google Scholar 

  18. Onur N, Sivrioglu M (1993) Transient heat conduction with uniform heat generation in a slab subjected to convection and radiation cooling. Heat Mass Transf 28:345–349

    Google Scholar 

  19. Peaceman DW, Rachford HH Jr (1955) The numerical solution of parabolic and elliptic differential equations. J Soc Ind Appl Math 3:28–41

    Article  MathSciNet  MATH  Google Scholar 

  20. Qin QH (1994) Unconditionally stable fem for transient linear heat conduction analysis. Commun Numer Methods Eng 10:427–435

    Article  MATH  Google Scholar 

  21. Rheinboldt WC (1998) Methods for solving systems of nonlinear equations. 2nd edn. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  22. Simo JC, Taylor RL (1986) A return mapping algorithm for plane stress elastioplasticity. Int J Numer Methods Eng 22:649–670

    Article  MathSciNet  MATH  Google Scholar 

  23. Singh IV, Tanaka M, Endo M (2007) Meshless method for nonlinear heat conduction analysis of nano-composites. Heat Mass Transf 43:1097–1106

    Article  Google Scholar 

  24. Vujičić MR (2006) Finite element solution of transient heat conduction using iterative solvers. Eng Comput 23:408–431

    Article  MATH  Google Scholar 

  25. Wang XC (2003) Finite element method (in Chinese). Tsinghua University Press, Beijing

    Google Scholar 

  26. Yang H (1999) A new approach of time stepping for solving transfer problems. Commun Numer Methods Eng 15:325–334

    Article  MATH  Google Scholar 

  27. Yang H, Gao Q (2003) A precise time stepping scheme to solve hyperbolic and parabolic heat transfer problems with radiative boundary condition. Heat Mass Transf 39:571–577

    Article  Google Scholar 

  28. Yu J, Yang Y, Campo A (2010) Approximate solution of the nonlinear heat conduction equation in a semi-infinite domain. Math Probl Eng 2010:1–24

    MathSciNet  Google Scholar 

  29. Zienkiewicz O, Taylor R (2000) The finite element method—volume 2—solid mechanics. 5th edn. Butterworth-Heinemann, Oxford

    Google Scholar 

Download references

Acknowledgments

The authors acknowledge gratefully the financial support of the German Academic Exchange Service (DAAD).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhenggang Zhu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhu, Z., Kaliske, M. An iterative method to solve the heat transfer problem under the non-linear boundary conditions. Heat Mass Transfer 48, 283–290 (2012). https://doi.org/10.1007/s00231-011-0881-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-011-0881-x

Keywords

Navigation