Stochastic heat transfer problems are often solved using a perturbation approach that yields estimates of mean values and standard deviations for properties and boundary conditions that are random variables. Methods based on polynomial chaos and Wick products can be used when the randomness is a random field or white noise to describe specific realizations and to determine the statistics of the response. Polynomial chaos is best suited for problems in which the properties are strongly correlated, while the Wick product approach is most effective for variables containing white noise components. A transient lumped capacitance cooling problem and a one-dimensional fin are analyzed by both methods to demonstrate their usefulness.
Issue Section:
Conduction
1.
Emery
, A. F.
, 2002, “Transient and Steady State Free Convection from a Horizontal Cylinder
,” Proc. Inverse Problems in Engineering
, Agra dos Reis, Brazil.2.
Bouleau
, N.
, and Lepingle
, D.
, 1993, Numerical Methods for Stochastic Processes
, J. Wiley and Sons
, New York.3.
Ghanem
, R.
, and Spanos
, P.
, 1991, Stochastic Finite Elements: A Spectral Approach
, Springer-Verlag Publ.
, New York.4.
Ghanem
, R.
, 1998, “Probabilistic Characterization of Transport in Heterogeneous Media
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 158
, pp. 199
–220
.5.
Le Maitre
, O. P.
, Knio
, O. M.
, Najm
, H. N.
, and Ghanem
, R. G.
, 2001, “A Stochastic Projection Method for Fluid Flow
,” J. Comput. Phys.
0021-9991, 173
, pp. 481
–511
.6.
Holden
, H.
, Oksendal
, B.
, Uboe
, J.
, and Zhang
, T. S.
, 1995, Stochastic Partial Differential Equations—A Modeling, White Noise Functional Approach
, Birkauser Publ.
, Boston, MA.7.
Soong
, T. T.
, 1973, Random Differential Equations in Engineering and Science
, Academic Press
, New York.8.
Gard
, T. C.
, 1988, Introduction to Stochastic Differential Equations
, Marcel Dekker
, New York.9.
Oksendal
, B.
, 1998, Stochastic Differential Equations: An Introduction with Applications
, Springer
, New York.10.
Xiu
, D.
, and Karniadakis
, G. E.
, 2002, “The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations
,” SIAM J. Sci. Comput. (USA)
1064-8275, 24
(2
), pp. 619
–644
.11.
Sakamoto
, S.
, and Ghanem
, R.
, 2002, “Polynomial Chaos Decomposition for the Simulation of Non-Gaussian Non-stationary Stochastic Processes
,” J. Eng. Mech.
0733-9399, 128
(2
), pp. 190
–201
.12.
Keese
, A.
, and Matthies
, H. G.
, 2002, “Efficient Solvers for Nonlinear Stochastic Problem
,” Fifth World Congress on Computational Mechanics
, Vienna, Austria.13.
Guttman
, I.
, Wilks
, S. S.
, and Hunter
, J. S.
, 1982, Introductory Engineering Statistics
, J. Wiley and Sons
, New York.14.
Ditlevsen
, O.
, and Tarp-Johansen
, N. J.
, 1999, “Choice of Input Fields in Stochastic Finite Elements
,” Probab. Eng. Mech.
0266-8920, 14
, pp. 63
–72
.15.
Keese
, A.
, 2003, “A Review of Recent Developments in the Numerical Solution of Stochastic Partial Differential Equations (Stochastic Finite Elements)
,” http://opus.tu-bs.de/opus/volltexte/2003/504/pdf/review_sfern.pdfhttp://opus.tu-bs.de/opus/volltexte/2003/504/pdf/review_sfern.pdf16.
Benth
, F. E.
, and Theting
, T. G.
, 2000, Some Regularity Results for the Stochastic Pressure Equation of Wick-Type www.maphysto.dk/oldpages/publications/publications2000_static.htmlwww.maphysto.dk/oldpages/publications/publications2000_static.htmlpp.17.
Theting
, T. G.
, 2000, “Solving Parabolic Wick-Stochastic Boundary Value Problems Using a Finite Element Method
,” Stochastics and Stochastics Reports
, 75
(1–2
), pp. 57
–92
.Copyright © 2007
by American Society of Mechanical Engineers
You do not currently have access to this content.
