Uncertainty Quantification in Structural Systems Using Universal Grey Theory

Abstract

In recent years, increasing attention has been directed toward addressing uncertainties in engineering systems, which may arise from both aleatory and epistemic sources. Aleatory uncertainty originates from the inherent randomness of physical systems, whereas epistemic uncertainty stems from incomplete or limited knowledge. The present study focuses exclusively on the quantification of epistemic uncertainty in engineering systems. When system information is imprecise or partially available in the form of intervals or ranges, methods such as the combinatorial approach, interval methods (IM), and Universal Grey Theory (UGT) are commonly employed. However, as system dimensionality increases, significant challenges arise. Combinatorial optimization becomes computationally expensive for large-scale systems, while interval methods often lead to overestimation due to dependency issues and violations of physical laws. To overcome these limitations, UGT offers a promising alternative by satisfying distributive laws, thereby effectively addressing dependency problems and improving computational efficiency. Nevertheless, traditional UGT faces limitations in situations where one or both interval bounds are negative and the upper bound has a smaller absolute value. To address this issue, the present study proposes a necessary modification to the arithmetic operations within the UGT framework. Another critical aspect of efficiently quantifying epistemic uncertainty in large-scale systems is computational cost. Finite element–based uncertainty analysis is often computationally demanding for high-dimensional systems. Although metamodeling techniques such as the Response Surface Method (RSM) and Kriging can approximate original models, their computational expense increases significantly with system dimensionality. High Dimensional Model Representation (HDMR), a variant of RSM, emerges as a computationally efficient alternative for handling large-scale systems. To effectively quantify epistemic uncertainty in high-dimensional systems, this study proposes an integrated HDMR–UGT–based formulation. In this approach, HDMR is employed to construct response surfaces using finite element simulations, while UGT is utilized for explicit uncertainty propagation and prediction of response bounds. The proposed methodology is validated through numerical examples, demonstrating high accuracy and significant computational efficiency in exclusively quantifying epistemic uncertainty. Comparative studies with conventional techniques further confirm the effectiveness of the proposed HDMR–UGT formulation in reducing computational effort while maintaining high accuracy in the analysis of engineering systems subjected to epistemic uncertainty.