Stochastic finite element method homogenization of heat conduction problem in fiber composites

The main idea behind the paper is to present two alternative methods of homogenization of<br />the heat conduction problem in composite materials, where the heat conductivity coefficients are assumed<br />to be random variables. These two methods are the Monte-Carlo simulation (MCS) technique and the second<br />order perturbation second probabilistic moment method, with its computational implementation known as<br />the Stochastic Finite Element Method (SFEM). From the mathematical point of view, the deterministic<br />homogenization method, being extended to probabilistic spaces, is based on the effective modules approach.<br />Numerical results obtained in the paper allow to compare MCS against the SFEM and, on the other hand,<br />to verify the sensitivity of effective heat conductivity probabilistic moments to the reinforcement ratio. These<br />computational studies are provided in the range of up to fourth order probabilistic moments of effective<br />conductivity coefficient and compared with probabilistic characteristics of the Voigt-Reuss bounds.