A Stochastic Analysis of Steady and Transient Heat Conduction in Random Media Using a Homogenization Approach

TL;DR

This paper develops a stochastic homogenization approach to analyze steady and transient heat conduction in random media, providing analytical expressions for mean and variance of temperature fields, and validating results with Monte Carlo simulations.

Contribution

It introduces a novel stochastic homogenization method for heat conduction in random media, deriving analytical mean and variance of temperature fields based on i.i.d. random conductivity variables.

Findings

Variance of temperature depends on local gradients and random conductivity statistics.

Large temperature uncertainty occurs at locations with high temperature gradients.

Analytical results agree well with Monte Carlo simulations.

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 large number N. Both steady and transient solutions T are expressed in terms of the homogenized solution and its spatial derivatives, where homogenized solution 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 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 ); and the configurational contribution (through different configrations) . The configurational contribution is shown to be proportional to the local gradient of homogenized solution. Large uncertainty of T field was found at locations with large gradient of homogenized solution 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.