Abstract
Based on the combination of stochastic mathematics and conventional finite difference method, a new numerical computing technique named stochastic finite difference for solving heat conduction problems with random physical parameters, initial and boundary conditions is discussed. Begin with the analysis of steady-state heat conduction problems, difference discrete equations with random parameters are established, and then the computing formulas for the mean value and variance of temperature field are derived by the second-order stochastic parameter perturbation method. Subsequently, the proposed random model and method are extended to the field of transient heat conduction and the new analysis theory of stability applicable to stochastic difference schemes is developed. The layer-by-layer recursive equations for the first two probabilistic moments of the transient temperature field at different time points are quickly obtained and easily solved by programming. Finally, by comparing the results with traditional Monte Carlo simulation, two numerical examples are given to demonstrate the feasibility and effectiveness of the presented method for solving both steady-state and transient heat conduction problems.
Similar content being viewed by others
References
Ganesan R. Vibration analysis for stability of singular non-self-adjoint beam-columns using stochastic FEM. Comput Struct, 1998, 68: 543–554
Rahman S. A dimensional decomposition method for stochastic fracture mechanics. Eng Fract Mech, 2006, 73: 2093–2109
Ghanem R G. Stochastic Finite Element: A Spectral Approach. New York: Springer, 1991
Hien T D, Kleiber M. Stochastic finite element modeling in linear transient heat transfer. Comput Methods Appl Mech Eng, 1997, 144: 111–124
Kaminski M, Hien T D. Stochastic finite element modeling of transient heat transfer in layered composites. Int J Heat Mass Tran, 1999, 26: 801–810
Emery A F. Solving stochastic heat transfer problems. Eng Anal Bound Elem, 2004, 28: 279–291
Xiu D B. A new stochastic approach to transient heat conduction modeling with uncertainty. Int J Heat Mass Tran, 2003, 46: 4681–4693
Tao W Q. Numerical Heat Transfer. 2nd ed. Xi’an: Xi’an Jiaotong University Press, 2009
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, 2006, 45: 1179–1196
Smith G D. Numerical Solutions of Partial Differential Equations (Finite Difference Methods). 3rd ed. Oxford: Clarendon Press, 1985
Zhang Y M, Chen S H. Stochastic perturbation finite elements. Comput Struct, 1996, 59: 425–429
Kocic V L. Global Asymptotic Behavior of Nonlinear Difference Equations of Higher Order with Applications. Dordrect: Kluwer Academic Publishers, 1993
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, C., Qiu, Z. & Wu, D. Numerical analysis of uncertain temperature field by stochastic finite difference method. Sci. China Phys. Mech. Astron. 57, 698–707 (2014). https://doi.org/10.1007/s11433-013-5235-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11433-013-5235-x