Universal Gray Finite Elements for Heat Transfer Analysis in the Presence of Uncertainties

A new finite element method is presented for the analysis of uncertain heat transfer problems using universal gray number theory. The universal gray number representation involves normalization of the uncertain parameters based on their lower and upper bound values with its own distinctive rules of arithmetic operations which makes this method distinctive from conventional interval analysis approaches. This work introduces the concept of fuzzy finite element-based heat transfer analysis using universal gray number theory, that compared to the interval-based fuzzy analysis, would yield significantly improved and more accurate results. Heat transfer problems, including a one-dimensional tapered fin, a two-dimensional hollow rectangle representing a thin slice of a chimney of a thermal power plant, and a three-dimensional (axisymmetric) solid body with different boundary conditions, were considered for the uncertainty analysis. It is shown that, in each case, the interval values of the response parameters given by the universal gray number theory are consistent with the ranges of the input parameters, compared to those given by the interval analysis. It is also revealed that universal gray number theory is more inclusive and less computationally intensive compared to the interval analysis.