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Title: A stochastic analysis of steady and transient heat conduction in random media using a homogenization approach

Abstract

We present a new stochastic analysis for steady and transient one-dimensional heat conduction problem based on the homogenization approach. Thermal conductivity is assumed to be a random field K consisting of random variables of a total number N. Both steady and transient solutions T are expressed in terms of the homogenized solution (symbol) and its spatial derivatives (equation), where homogenized solution (symbol) is obtained by solving the homogenized equation with effective thermal conductivity. Both mean and variance of stochastic solutions can be obtained analytically for K field consisting of independent identically distributed (i.i.d) random variables. The mean and variance of T are shown to be dependent only on the mean and variance of these i.i.d variables, not the particular form of probability distribution function of i.i.d variables. Variance of temperature field T can be separated into two contributions: the ensemble contribution (through the homogenized temperature (symbol)); and the configurational contribution (through the random variable Ln(x)Ln(x)). The configurational contribution is shown to be proportional to the local gradient of (symbol). Large uncertainty of T field was found at locations with large gradient of (symbol) due to the significant configurational contributions at these locations. Numerical simulations were implemented based on a directmore » Monte Carlo method and good agreement is obtained between numerical Monte Carlo results and the proposed stochastic analysis.« less

Authors:
Publication Date:
Research Org.:
Idaho National Lab. (INL), Idaho Falls, ID (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1133856
Report Number(s):
INL/JOU-14-31365
DOE Contract Number:  
DE-AC07-05ID14517
Resource Type:
Journal Article
Journal Name:
Applied Mathematical Modelling
Additional Journal Information:
Journal Volume: 38; Journal Issue: 13
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS; heat conduction; homogenization; random field; stochastic; uncertainty

Citation Formats

Xu, Zhijie. A stochastic analysis of steady and transient heat conduction in random media using a homogenization approach. United States: N. p., 2014. Web.
Xu, Zhijie. A stochastic analysis of steady and transient heat conduction in random media using a homogenization approach. United States.
Xu, Zhijie. 2014. "A stochastic analysis of steady and transient heat conduction in random media using a homogenization approach". United States.
@article{osti_1133856,
title = {A stochastic analysis of steady and transient heat conduction in random media using a homogenization approach},
author = {Xu, Zhijie},
abstractNote = {We present a new stochastic analysis for steady and transient one-dimensional heat conduction problem based on the homogenization approach. Thermal conductivity is assumed to be a random field K consisting of random variables of a total number N. Both steady and transient solutions T are expressed in terms of the homogenized solution (symbol) and its spatial derivatives (equation), where homogenized solution (symbol) is obtained by solving the homogenized equation with effective thermal conductivity. Both mean and variance of stochastic solutions can be obtained analytically for K field consisting of independent identically distributed (i.i.d) random variables. The mean and variance of T are shown to be dependent only on the mean and variance of these i.i.d variables, not the particular form of probability distribution function of i.i.d variables. Variance of temperature field T can be separated into two contributions: the ensemble contribution (through the homogenized temperature (symbol)); and the configurational contribution (through the random variable Ln(x)Ln(x)). The configurational contribution is shown to be proportional to the local gradient of (symbol). Large uncertainty of T field was found at locations with large gradient of (symbol) due to the significant configurational contributions at these locations. Numerical simulations were implemented based on a direct Monte Carlo method and good agreement is obtained between numerical Monte Carlo results and the proposed stochastic analysis.},
doi = {},
url = {https://www.osti.gov/biblio/1133856}, journal = {Applied Mathematical Modelling},
number = 13,
volume = 38,
place = {United States},
year = {Tue Jul 01 00:00:00 EDT 2014},
month = {Tue Jul 01 00:00:00 EDT 2014}
}