## 4 Inverse Analysis of Structural Damage Using the Developed Approach  
Based on the theoretical background outlined in Section 2, this work utilizes the hybrid objective function in Equation (40). Physically, the objective function utilizes the \(MSEn\) and \(MKEn\) in order to derive the maximum dynamic characteristics related to the damage situation of the structure. The \(MSEn\) subobjective serves to identify the damage changes related to stiffness alterations of the structure and helps to identify linear damages in general, as well as to detect the occurrence of structural cracks based on the coupled effect of mode shapes and stiffness. In addition, the \(MKEn\) can contribute by coupling the effects of both natural modes, as well as mode shapes, together with any change in the mass matrix of the structure, which provides a prominent damage identifier and achieves a better generalized framework for damage diagnosis. In addition, with the help of modal assurance criterion, the mode shape-based subobjective in Equation (40) is also implemented in order to check the effect of damage on the mode shapes, regardless of the system stiffness and mass matrices, which helps in case of noisy conditions. To verify the performance of the proposed approach, the damage is considered to occur due to the reduction of stiffness corresponding to some selected damage scenarios of structural elements along the structure. The combination of the presented OL-UPSGBO algorithm and the new hybrid objective function, as well as their integration into the inverse analysis of structural damage identification problems, can be observed in Figure 7, as well as in the pseudo-code, as follows. 1. Initialization stage 1. Develop the FE model of the structure using a commercial software or a self-coded model. 2. Set a population of search agents, which are the damage indicators related to the overall structural or substructural elements, where each search agent represents one configuration of the FE model of the structure. 3. Extract the modal features related to the intact and damaged structure, and calculate the MSEn-, MKEn-, and mode shape-based sub-objectives using Equations (37)-(39). Thereafter, calculate the objective function using Equation (40) for each corresponding FE model configuration related to each candidate search agent. 4. Evaluate all the search agents using the developed objective function (as in Equation (40)). 5. Initialize all the stochastic parameters of the OL-UPSOGBO algorithm, as in Table 1. 6. Define the initial best global and local solutions, worst global solution, and the best performance of each search agent. 7. Apply the OL paradigm as in Equation (42). B. Iterative stage 1. Start the UPSO framework by calculating the global and local velocities as in Equation (45). 2. Update the population using Equation (46). 3. Update the best global and local solutions, worst global solution, and the best performance of each search agent. 4. Start the GBO stage by employing Equation (61). 5. Apply the local escaping operator, as in Equation (63). 6. Apply the OL paradigm, as in Equation (42). 7. Update the best global and local solutions, worst global solution, and the best performance of each search agent. 8. Break if termination criteria are satisfied. C. Damage identification stage 1. After termination of the iterative process, the best performed search agent is selected and registered. 2. The best solution contains the damage parameters \(\delta_{j}\) corresponding to each element \(j\). 3. Elicit the damage locations and calculate the damage severities using Equation (41), and analyze the results. To test the performance of the proposed approach for the inverse analysis of structural damage, the well-known American Society of Civil Engineering (ASCE) benchmark model is studied [67; 68]. The frame model is widely utilized by the SHM society to study the robustness of damage identification approaches, as can be seen in [67; 68].