## 4 Conclusions  
In this research, three different objective functions based on frequencies, MAC, modal strain energy and flexibility have been analyzed for damage detection. Two simulated beams and an  
Figure 15: Damage distribution for Case 3 using (**a**) Objective Functions I, II and III without regularization and (**b**) Objective Functions I, II and III with regularization.  
experimental beam were investigated to check the performance of these objective functions using GA. The following conclusions were drawn from the study:  
1. The simulated beam (Case 1) was investigated with a single damage scenario. First, three mode shapes have been considered for damage assessment for the different objective functions. Two different noise cases, i.e., noise in both frequencies and mode shapes and noise in mode shapes only at 1%, 3%, 5% and 10% have been investigated in the frequency domain. It has been found that with no noise, all the objective functions performed well and detected the damage correctly. However, results indicate that with an increase in the noise, the damage detection capabilities of all the objective functions decreased. Objective Function I based on frequencies and MAC has worked better in damage detection than Objective Functions II and III in noisy conditions. However, less degradation has been seen in the noise in mode shapes-only case as compared to noise both in frequencies and mode shapes. Probable reasons for the better performance of Objective Function I as compared to the other two were also discussed. It was found that the function value of Objective Function I is lesser than Objective Functions II and III.
2. A regularization function was added in the objective functions based on the a priori modeling of the structure. The multi-objective GA was used to find the optimal tradeoff between the objective function and the regularization part. The results show that the regularization function performed well even in noisy conditions for Objective Function I.
3. The multiple damage scenario has been simulated in Case 2 where three elements were damaged. At 0% noise, all the functions detected the damage correctly. However, the performance degraded when noise levels were increased. The regularization function has performed well for Objective Function I only, which proves its adequacy in multiple damage scenarios.
4. The simulations were later verified on an experimentally tested beam. Convergence analysis of the SAP 2000 model was performed for the first three natural frequencies, and the 20-node beam was selected for the FE model. It was found that detections in non-damaged elements were of greater magnitude for Objective Functions II and III as compared to Objective Function I. Regularization was attempted for the experimental beam. The damage index was much improved for the damaged, as well as for the undamaged elements for Objective Function I with regularization, which further proves the performance of the proposed approach in actual experimental conditions.  
The paper has proposed the use of EA for beam structures. Further studies on 3D complex structures where noise has been added directly to time series data are recommended.  
[PERSON] conceived of the idea of the paper and designed the experiments. [PERSON] contributed towards simulation tools. [PERSON] helped in the simulations. [PERSON] and [PERSON] analyzed the data. [PERSON] and [PERSON] helped in the revisions of the paper.  
The authors declare no conflict of interest.