DUAL PROBABILISTIC ANALYSIS OF THE TRANSIENT HEAT TRANSFER BY THE STOCHASTIC FINITE ELEMENT METHOD WITH OPTIMIZED POLYNOMIAL BASIS

The main aim of this work is to contrast three various probabilistic computational techniques, namely analytical, simulation and perturbation-based, in a solution of the transient heat transfer problem in specific axisymmetric problem with Gaussian uncertainty in physical parameters. It is done thanks to a common application of the Finite Element Method program ABAQUS (for the deterministic part) and symbolic algebra system MAPLE, where all probabilistic procedures have been programmed. We determine up to the fourth order probabilistic characteristics of the resulting temperatures, i.e. expectations, coefficients of variation, skewness and kurtosis together with the histograms – all as the functions of the input coefficient of variation of random heat conductivity coefficient. Stochastic perturbation technique is implemented here using the tenth order Taylor series expansion and traditional Least Squares Method released with polynomial basis whose final order is a subject of the separate statistical optimization. Probabilistic results computed show almost perfect agreement of all the probabilistic characteristics under consideration, which means that the traditional simulation method may be replaced due to the time and computer scale savings with the stochastic perturbation method.

Full text (pdf)

[1]  Chorin A.: Gaussian fields and random flow. Journal of Fluid Mechanics 63, 1974, pp. 21-32.

[2]  Emery A.F.: Solving stochastic heat transfer problems. Engineering Analysis with Boundary Elements 28(3), 2004, pp. 279-291.

[3]  Binder K., Heermann D.W.: Monte-Carlo Simulation in Statistical Physics. An Introduction. 3rd edition, Springer-Verlag, Berlin-Heidelberg, 1997.

[4]  Xiu D., Karniadakis G.E.: A new stochastic approach to transient heat conduction modelling with uncertainty. International Journal of Heat & Mass Transfer 46(24), 2003, pp. 4681-4693.

[5]  Kamiński M.: The Stochastic Perturbation Method for Computational Mechanics. Chichester, Wiley, 2013.

[6]  Wang C., Qiu Z.: Interval analysis of steady-state heat convection-diffusion problem with uncertain-but-bounded parameters. International Journal of Heat & Mass Transfer 91, 2015, pp. 355-362.

[7]  Sakji S., Soize C., Heck J.V.: Computational stochastic heat transfer with model uncertainties in a plasterboard submitted to fire load and experimental validation. Fire Materials 33(3), 2009, pp. 109-127.

[8]  Carslaw H.S., Jaeger J.C.: Conduction of Heat in Solids. Oxford University Press, London, 1959.

[9]  Wan X., Karniadakis G.: Stochastic heat transfer enhancement in a grooved channel. Journal of Fluid Mechanics 565, 2006, pp. 255-278.

[10] Kamiński M., Hien T.D.: Stochastic finite element modelling of transient heat transfer in layered composites. International Communications in Heat & Mass Transfer 26(6), 1999, pp. 791-800.

[11] Blackwell B., Beck J.V.: A technique for uncertainty analysis for inverse heat conduction problems. International Journal of Heat & Mass Transfer 53(4), 2010, pp. 753-759.