Numerical Methods for Uncertainty Estimation in Mechanical Systems: A Literature Review

Uncertainty and reliability analyses are essential for evaluating the performance and safety of engineering systems subject to variability in parameters, operating conditions, and modeling assumptions. This paper presents a comprehensive review of classical and modern approaches for uncertainty quantification and reliability assessment. Three main methodological frameworks are addressed: probabilistic, fuzzy, and interval-based methods. Probabilistic approaches, such as the Monte Carlo Simulation (MCS), First-Order Reliability Method (FORM), and Second-Order Reliability Method (SORM), are discussed as rigorous tools for modeling randomness when sufficient statistical data are available. Fuzzy and interval methods are examined as alternatives to represent epistemic uncertainty when probabilistic information is limited or unavailable. The comparative analysis highlights the advantages, limitations, and applicability of each method in engineering contexts. The results emphasize that the choice of approach depends on the nature of uncertainty, data availability, and computational constraints. The study concludes that integrating probabilistic and non-probabilistic frameworks offers a promising pathway toward more robust and interpretable uncertainty analysis in complex systems.