Skip to main content
SpringerLink
Log in

Fuzzy finite difference method for heat conduction analysis with uncertain parameters

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

A new numerical technique named as fuzzy finite difference method is proposed to solve the heat conduction problems with fuzzy uncertainties in both the physical parameters and initial/boundary conditions. In virtue of the level-cut method, the difference discrete equations with fuzzy parameters are equivalently transformed into groups of interval equations. New stability analysis theory suited to fuzzy difference schemes is developed. Based on the parameter perturbationmethod, the interval ranges of the uncertain temperature field can be approximately predicted. Subsequently, fuzzy solutions to the original difference equations are obtained by the fuzzy resolution theorem. Two numerical examples are given to demonstrate the feasibility and efficiency of the presented method for solving both steady-state and transient heat conduction 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

Similar content being viewed by others

References

  1. Efe, M.O., Ozbay, H.: Multi input dynamical modeling of heat flow with uncertain diffusivity parameter. Math. Comp. Model Dyn. 9, 437–450 (2003)

    Article  MATH  Google Scholar 

  2. Nicolai, B.M., Egea, J.A., Scheerlinck, N.: Fuzzy finite element analysis of heat conduction problems with uncertain parameters. Int. J. Food Eng. 103, 38–46 (2011)

    Article  Google Scholar 

  3. Lu, Q.: Some results on the controllability of forward stochastic heat equations with control on the drift. J. Funct Anal. 260, 832–851 (2011)

    Article  MathSciNet  Google Scholar 

  4. Moens, D., Vandepitte, D.: Recent advances in nonprobabilistic approaches for non-deterministic dynamic finite element analysis. Arch. Comput. Methods Eng. 13, 389–464 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Stefanou, G.: The stochastic finite element method: Past, present and future. Comput. Methods Appl. Mech. Eng. 198, 1031–1051 (2009)

    Article  MATH  Google Scholar 

  6. Gao, W., Zhang, N.: A new method for random vibration analysis of stochastic truss structures. Finite Elem. Anal. Des. 45, 190–199 (2009)

    Article  MathSciNet  Google Scholar 

  7. Kaminski, M.: A generalized stochastic perturbation technique for plasticity problems. Comput. Mech. 45, 349–361 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Pereira, S.C., Mello, U.T.: Uncertainty in thermal basin modeling: An interval finite element approach. Reliable Comput. 12, 451–470 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Haim, Y.B., Elishakoff, I.: Convex Models of Uncertainty in Applied Mechanics. Elsevier, Amsterdam (1990)

    MATH  Google Scholar 

  10. Wang, Y.G., Wei, G.: Dynamical systems over the space of upper semicontinuous fuzzy sets. Fuzzy Set. Syst. 209, 89–103 (2012)

    Article  MATH  Google Scholar 

  11. Zadeh, L.A.: Fuzzy sets. Inf. Control. 8, 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hanss, M., Turrin, S.: A fuzzy-based approach to comprehensive modeling and analysis of systems with epistemic uncertainties. Struct. Saf. 32, 433–441 (2010)

    Article  Google Scholar 

  13. Balu, A.S., Rao, B.N.: High dimensional model representation based formulations for fuzzy finite element analysis of structures. Finite Elem. Anal. Des. 50, 217–230 (2012)

    Article  Google Scholar 

  14. Farkas, L., Moens, D.: Fuzzy finite element analysis based on reanalysis technique. Struct. Saf. 32, 442–448 (2010)

    Article  Google Scholar 

  15. Hien, T.D., Kleiber, M.: Stochastic finite element modeling in linear transient heat transfer. Comput. Methods Appl. Mech. Eng. 144, 111–124 (1997)

    Article  MATH  Google Scholar 

  16. Kaminski, M., Hien, T.D.: Stochastic finite element modeling of transient heat transfer in layered composites. Int. J. Heat Mass Tran. 26, 801–810 (1999)

    Article  Google Scholar 

  17. Emery, A.F.: Solving stochastic heat transfer problems. Eng. Anal. Bound Elem. 28, 279–291 (2004)

    Article  MATH  Google Scholar 

  18. Xiu, D.B.: A new stochastic approach to transient heat conduction modeling with uncertainty. Int. J. Heat Mass Tran. 46, 4681–4693 (2003)

    Article  MATH  Google Scholar 

  19. Wang, H., Dai, W.: A finite difference method for studying thermal deformation in a double-layered thin film exposed to ultrashort pulsed lasers. Int. J. Therm. Sci. 45, 1179–1196 (2006)

    Article  MathSciNet  Google Scholar 

  20. Smith, G.D.: Numerical Solutions of Partial Differential Equations (Finite Difference Methods), (3rd edn.). Clarendon Press, Oxford (1985)

    Google Scholar 

  21. Tao, W.Q.: Numerical Heat Transfer, (2nd edn.). Xi’an Jiaotong University Press, Xi’an (2009)

    Google Scholar 

  22. Huang, H.Z., Li, H.B.: Perturbation finite element method of structural analysis under fuzzy environments. Eng. Appl. Artif. Intel. 18, 83–91 (2005)

    Article  Google Scholar 

  23. Erdolen, A.: Interval finite element analysis of truss systems. Tek. Dergi. 22, 5305–5318 (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Solid Mechanics, Beihang University, 100191, Beijing, China

    Chong Wang & Zhi-Ping Qiu

Authors
  1. Chong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi-Ping Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chong Wang.

Additional information

The project was supported by the National Special Fund for Major Research Instrument Development (2011YQ140145), 111 Project (B07009), the National Natural Science Foundation of China (11002013), and Defense Industrial Technology Development Program (A2120110001 and B2120110011).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, C., Qiu, ZP. Fuzzy finite difference method for heat conduction analysis with uncertain parameters. Acta Mech Sin 30, 383–390 (2014). https://doi.org/10.1007/s10409-014-0036-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-014-0036-7

Keywords

Search

Navigation