## 5 Conclusions  
This study has discussed an efficient approach for solving the inverse analysis of structural damage problem using a new framework based on a new modal feature-based objective function, as well as a new evolutionary computation algorithm. The new hybrid objective function was formulated based on the modal strain and modal kinetic energy terms, which helps to contrast the maximum modal information from the structure under consideration. Moreover, in order to improve noise immunity and separate the modal responses from system matrices, which helps to cover more damage types, the modal assurance criterion of mode shapes was utilized and added as a third term to the objective function. In addition, to overcome the complex nature of the considered inverse problem, the OL framework for population initialization and convergence acceleration was first adopted. Thereafter, the UPSO elitism mechanism was merged with the efficient search mechanisms of the GBO. The newly developed algorithm OL-UPSGBO, with the characteristics of the convergence acceleration feature of OL, enhanced balanced exploration-exploitation of UPSO, and the local escaping operator of GBO was designed and tested. The OL-UPSGBO was tested using the CEC 2017, the most complex and well-known benchmark objective func  
Figure 19: Damage identification for Scenario 6 using \(F_{3}\): (**a**) Damage locations and severity, and (**b**) convergence of damage parameters related to damaged elements.  
Figure 18: Damage identification for Scenario 6 using \(F_{2}\): (**a**) Damage locations and severity, and (**b**) convergence of damage parameters related to damaged elements.  
tion set containing unimodal, multimodal, hybrid, and composite objective functions, and was compared to available state-of-the-art algorithms in the literature. The benchmarking results showed a remarkable performance of the proposed OL-UPSGBO algorithm compared to other approaches. Furthermore, as the main aim of this study, the proposed algorithm was implemented to solve the new hybrid objective function related to the inverse problem of structural damage detection. The ASCE frame structure was studied and several damage scenarios were tested. Compared to the component algorithms, the OL-UPSGBO algorithm showed a robust and stable computational performance, and remarkable damage deductibility for all damage scenarios even under noisy conditions. This can be well-observed by the parametric and non-parametric statistical tests after executing the damage identification framework. Hence, the developed approach can be highly recommended to solve the inverse analysis of structural damage problem.  
Conceptualization, N.F.A. and L.S.; methodology, N.F.A.; software, N.F.A.; validation, N.F.A., L.S., T.A.-h., X.Q. and M.C.; formal analysis, N.F.A., L.S. and T.A.-h.; investigation, N.F.A., L.S. and T.A.-h.; resources, N.F.A.; data curation, N.F.A. and T.A.-h.; writing\(-\)original draft preparation, N.F.A.; writing\(-\)review and editing, N.F.A.; visualization, N.F.A.; supervision, X.Q. and M.C.; project administration, X.Q., and M.C.; funding acquisition, X.Q. and M.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Research Fund for International Young Scientists of the National Natural Science Foundation of China (No. 52250410359), the 2022 National Young Foreign Talents Program of China (No. QN2022143002L), the Fundamental Research Funds for the Central Universities (No. B220204002), and the Anhui Provincial International Joint Research Center of Data Diagnosis and Smart Maintenance on Bridge Structures (No. 2021 AHGHZD01).  
Not applicable.  
Data Availability StatementData sharing does not apply to this article as no datasets were generated or analyzed during the current study.  
The authors declare no conflict of interest.