## 2 Research Methodology
### Modeling Methods Improvements
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Method** & **Advantages** & **Disadvantages** & **Domain * \\ \hline Time-varying harmonic analysis [87] & Improved adaptive frequency-shift filtering allows for more efficient harmonic analysis. & Requires complex algorithms for adaptive signal processing. & \\ \cline{2-4}  & Reduces the loss of spectral information within the standard Fourier transform. & Sensitive to the choice of filtering parameters, which affects the accuracy. & TFD \\ \hline Stochastic finite element method (SFEM) [91] & Provides full statistical information—means, variances, and confidence bands. & Requires reliable probability-density functions for every uncertain parameter; scarce or poor statistics can distort the results. & TDD/FD \\ \cline{2-4}  & Non-intrusive: the existing FE model is reused, with randomness introduced via Karhunen–Loeve expansion of material fields. & Assumes Gaussian, independent inputs in the paper; real-world correlations or non-Gaussian tails may lead to over- or under-estimating risk. & TD/FD \\ \hline Fuzzy finite element method (FFEM) [91] & No probability-density functions are required; triangular fuzzy numbers can be elicited from expert judgment or sparse test data. & Produces conservative (potentially over-wide) response bounds, which may mask useful sensitivity information. & TDD/FD \\ \cline{2-4}  & Guarantees that the true response lies within the \(\alpha\)-level envelopes—valuable when only epistemic uncertainty is present. & The choice of membership-function shape and spread is subjective; inconsistent expert inputs can skew the results. & TDD/FD \\ \hline Polynomial chaos expansion (PCE) [92] & Provides a full probabilistic output. & Requires a reliable PDF for every uncertain parameter. & \\ \cline{2-4}  & Non-intrusive and differentiable, enabling variance-based sensitivity analysis and robust optimization. & “Curse of dimensionality”: accuracy and grid size grows rapidly beyond & FD \\ \hline Monte Carlo simulation (MCS) & Scales naturally with added random variables and nonlinearities [92; 93]. & Requires 1000+ FEM runs even for a moderate problem [92; 93]. & \\ \cline{2-4}  & It provides a statistical picture of the response distribution, which helps validate interval surrogates [93]. & Impractical for iterative design loops or real-time monitoring due to computational burden [92]. & TD/FD \\ \hline Chebyshev interval method (CIM) [93] & Requires no probability- density functions. & Empirical tuning. & \\ \cline{2-4}  & A relatively small number of FEM evaluations. & It ignores correlations among parameters and has not been tested with defect-induced nonlinearities. & TD/FD \\ \hline Markov-chain Monte Carlo (MCMC) on sub-interval similarity (SIS) metric [94] & Integrates SIS directly into the likelihood, enabling consistent treatment of sparse, irregular experimental data without assuming Gaussian errors. & Needs prior PDFs; posterior sampling can still be expensive for high-dimensional parameter spaces. & \\ \cline{2-4}  & Universally applicable: the same metric works when data are intervals, single samples, or complete distributions. & Performance and convergence depend on the empirical choice of the sub-interval number. & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 9: _Cont._Along with the key issues, they challenge researchers to apply their ingenuity to solve them. Ways of such problems are also important, so it is also reasonable to pay attention to them. The issues, such as measurement noise, computational complexity, iteration divergence, and nonlinear relationship of parameters, will be observed below.