Abstract
A new finite element method is presented for the analysis of thermal and structural problems using universal grey number theory. The universal grey 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 universal grey number-based fuzzy analysis for the analysis of finite element problems. This method would yield significantly improved and more accurate results compared to the conventional interval-based fuzzy analysis. In order to verify the results obtained with universal grey number method, a probabilistic approach, namely, the polynomial chaos expansion method is also applied to find the uncertain responses. The universal grey number method is used to investigate the finite element results of heat transfer problems, static structures, and structural dynamics problems. These problems have different dimensionalities with distinctive solution approaches. It is shown that, in each case, the interval values of the response parameters such as nodal temperatures, displacements, natural frequencies, and stresses given by the universal grey number theory are consistent with the ranges of the input parameters, while the interval method has significantly overestimated the computed results. It is also revealed that the universal grey number method is more efficient compared to both the interval analysis and polynomial chaos expansion methods. The interval method is found to produce not only higher levels of uncertainty, but also, in some cases, the results would be physically invalid and unacceptable. Finally, a different approach to find the uncertain bounds using the particle swarm optimization method is presented. The results computed with the optimization methods can enhance the performance of interval analysis approach.