## 4 Inverse Analysis of Structural Damage Using the Developed Approach
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Damage Scenario**} & \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{**Mean**} & \multicolumn{2}{c}{**Standard**} & \multirow{2}{*}{**Min**} & \multirow{2}{*}{**Max**} & \multicolumn{2}{c}{**Wilcoxon Sign**} \\  & & & & **Deviation** & & **Rank (_p_-Value)** \\ \hline \multirow{4}{*}{Scenario 1} & UPSO & \(5.7\times 10^{-4}\) & \(5.7\times 10^{-4}\) & \(6.5\times 10^{-6}\) & \(1.2\times 10^{-3}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & GBO & \(6.3\times 10^{-3}\) & \(2.3\times 10^{-4}\) & \(6.1\times 10^{-3}\) & \(6.5\times 10^{-3}\) & \(9.0\times 10^{-3}\) \\ \cline{2-6}  & OL-UPSGBO & \(1.1\times 10^{-6}\) & \(2.2\times 10^{-6}\) & \(6.2\times 10^{-7}\) & \(6.5\times 10^{-6}\) & \\ \hline \multirow{4}{*}{Scenario 2} & UPSO & \(1.3\times 10^{-2}\) & \(1.4\times 10^{-5}\) & \(1.3\times 10^{-2}\) & \(1.3\times 10^{-2}\) & \(9.0\times 10^{-3}\) \\ \cline{2-6}  & GBO & \(1.6\times 10^{-2}\) & \(4.8\times 10^{-3}\) & \(1.1\times 10^{-2}\) & \(2.1\times 10^{-2}\) & \(9.0\times 10^{-3}\) \\ \cline{2-6}  & OL-UPSGBO & \(9.3\times 10^{-8}\) & \(1.1\times 10^{-7}\) & \(4.9\times 10^{-7}\) & \(2.8\times 10^{-7}\) & \\ \hline \multirow{4}{*}{Scenario 3} & UPSO & \(5.3\times 10^{-3}\) & \(1.1\times 10^{-4}\) & \(5.2\times 10^{-3}\) & \(5.4\times 10^{-3}\) & \(9.0\times 10^{-3}\) \\ \cline{2-6}  & GBO & \(6.8\times 10^{-2}\) & \(5.4\times 10^{-4}\) & \(6.8\times 10^{-2}\) & \(6.9\times 10^{-2}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & OL-UPSGBO & \(5.5\times 10^{-7}\) & \(8.2\times 10^{-7}\) & \(3.2\times 10^{-8}\) & \(2.0\times 10^{-6}\) & \\ \hline \multirow{4}{*}{Scenario 4} & UPSO & \(1.3\times 10^{-2}\) & \(3.0\times 10^{-5}\) & \(1.3\times 10^{-2}\) & \(1.3\times 10^{-2}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & GBO & \(5.3\times 10^{-2}\) & \(1.7\times 10^{-2}\) & \(4.1\times 10^{-2}\) & \(7.6\times 10^{-2}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & OL-UPSGBO & \(3.9\times 10^{-6}\) & \(4.6\times 10^{-6}\) & \(8.8\times 10^{-8}\) & \(1.2\times 10^{-5}\) & \\ \hline \multirow{4}{*}{Scenario 5} & UPSO & \(3.5\times 10^{-2}\) & \(3.7\times 10^{-4}\) & \(3.5\times 10^{-2}\) & \(3.5\times 10^{-2}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & GBO & \(2.8\times 10^{-2}\) & \(1.9\times 10^{-3}\) & \(2.7\times 10^{-2}\) & \(3.3\times 10^{-2}\) & \(9.1\times 10^{-3}\) \\ \cline{2-6}  & OL-UPSGBO & \(2.2\times 10^{-5}\) & \(3.1\times 10^{-6}\) & \(1.9\times 10^{-5}\) & \(2.6\times 10^{-5}\) & \\ \hline \multirow{4}{*}{Scenario 6} & UPSO & \(1.3\times 10^{-2}\) & \(1.9\times 10^{-5}\) & \(1.3\times 10^{-2}\) & \(1.3\times 10^{-2}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & GBO & \(2.3\times 10^{-2}\) & \(7.3\times 10^{-4}\) & \(2.3\times 10^{-2}\) & \(2.4\times 10^{-2}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & OL-UPSGBO & \(1.6\times 10^{-6}\) & \(1.9\times 10^{-6}\) & \(1.9\times 10^{-8}\) & \(5.2\times 10^{-6}\) & \\ \hline \multirow{4}{*}{Scenario 6 (3\% noise)} & UPSO & \(1.3\times 10^{-2}\) & \(2.5\times 10^{-5}\) & \(1.3\times 10^{-2}\) & \(1.3\times 10^{-2}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & GBO & \(2.5\times 10^{-2}\) & \(1.1\times 10^{-5}\) & \(2.5\times 10^{-2}\) & \(2.5\times 10^{-2}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & OL-UPSGBO & \(4.0\times 10^{-4}\) & \(7.5\times 10^{-7}\) & \(4.0\times 10^{-4}\) & \(4.0\times 10^{-4}\) & \\ \hline \multirow{4}{*}{Scenario 6 (5\% noise)} & UPSO & \(1.3\times 10^{-2}\) & \(2.7\times 10^{-5}\) & \(1.3\times 10^{-2}\) & \(1.3\times 10^{-2}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & GBO & \(2.5\times 10^{-2}\) & \(5.3\times 10^{-4}\) & \(2.5\times 10^{2}\) & \(2.6\times 10^{-4}\) & \(3.9\times 10^{-3}\) \\ \cline{2-6}  & OL-UPSGBO & \(5.8\times 10^{-4}\) & \(4.0\times 10^{-6}\) & \(5.8\times 10^{-4}\) & \(5.9\times 10^{-4}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Inverse analysis of structural damage using the proposed method.