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 T∼ and its spatial derivatives T(x,t)=T∼+∑n=1∞Ln(x)∂nT∼/∂xn, where homogenized solution T∼ 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 T∼); and the configurational contribution (through the random variable Ln(x)). The configurational contribution is shown to be proportional to the local gradient of T∼. Large uncertainty of T field was found at locations with large gradient of T∼ 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.