## 4 Inverse Analysis of Structural Damage Using the Developed Approach
The model is composed of four stories and 116 structural elements, as in Figure 8. To avoid the lengthy theoretical background, a complete explanation about the structural properties and physical parameters of the structure can be studied [67; 68]. Hence, by utilizing the aforementioned structure, six damage scenarios are considered in this paper, which can provide the big picture regarding the efficiency of the developed approach. The six scenarios were made by reducing the structural stiffness of some selected brace elements by 25%. Such a small damage severity can be challenging to detect and localize using traditional approaches, as it has a small effect on structural responses. Additionally, the modal responses of one damage scenario, which is scenario 6, are subjected to two levels of Gaussian noise, specifically 3% and 5%. Furthermore, in this work, incomplete modal data which contains maximum dynamic information, are considered to better simulate practical engineering. The six damage scenarios are written as in Table 5. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Damage Scenario** & **Damage Locations** & **Damage Severity** \\ \hline Scenario 1 & Brace elements 83, and 114 & 25\% \\ Scenario 2 & Brace elements 26, and 55 & 25\% \\ Scenario 3 & Brace elements 52, and 109 & 25\% \\ Scenario 4 & Brace elements 24, 82, and 112 & 25\% \\ Scenario 5 & Brace elements 23, 51, and 111 & 25\% \\ Scenario 6 & Brace elements 22, 51, 80, and 109 & 25\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Damage scenarios. Figure 7: The pseudocode of the inverse analysis of structural damage using the proposed method. The proposed method has the remarkable merits of utilizing several search mechanisms that are able to overcome the multimodal and complex optimization features of the problem of inverse analysis of structural damage with very few stochastic parameter settings. Using the stochastic parameters in Table 2, several population sizes were first tested and a population size of 100 search agents was selected. The performance of the proposed OL-UPSGBO algorithm is compared with the component algorithms, namely the UPSO and the GBO. Each algorithm is run 10 times after considering the high computational time required to solve the inverse analysis of the structural damage problem. Two types of statistical tests are conducted, which are the parametric tests, using the mean, minimum, maximum, and standard deviation of solutions, and the non-parametric test, using the well-known Wilcoxon Sign Rank Test [62]. The results of the inverse analysis of the structural damage problem are shown in Table 6. Moreover, the detailed damage identification results and the evolution of damage parameters of the considered elements related to damage scenarios 1, 2, 3, 4, 5, and 6 are shown in Figures 9-14. In addition, the noise immunity of the proposed approach can be seen in Figures 15 and 16, where the two noisy conditions of 3% and 5% Gaussian noise of scenario 6 are illustrated. By studying Table 6, it is well-observed that the proposed OL-UPSGBO algorithm has shown great performance when employed to solve the new hybrid objective function, with remarkable damage identification abilities even under noisy conditions. The \(p\)-value taken from the Wilcoxon test is under the confidence level of 0.05, which proves the statistical significance of the OL-UPSGBO algorithm compared to the UPSO and GBO algorithms. The reason why the proposed OL-UPSGBO has shown an outstanding performance is because the main global search paradigm of GBO is improved by using two search mechanisms, namely by oppositional-based learning (OL), which is able to evaluate the oppositional of a considered solution and compares the two solutions in order to choose a better one. This efficient tool helps to accelerate the convergence of the algorithm and explore more search areas, which suits problems concerning multimodality and multiple local optima of the inverse problem of structural damage identification.