## Initial model
Therefore MAC-based Objective Function I presents itself as a good candidate for damage detection purposes due to less sparseness in the results. Due to wrong detections with different objective functions especially in noisy conditions, a suitable technique needs to be applied to address the damage detection issue in non-damaged elements. A sensitivity analysis of all the elements was performed. It has been revealed that elements closer to the supports have very low sensitivities. Such elements can cause ill-conditioning of the problem, and a small change in the value results in a large change in the modal data. Furthermore, elements at equal distance from the center of the beam have equal sensitivities, e.g., Element Nos. 5 and 6 influence the modal data in a similar manner. Such elements can also lead to similar modal values. Regularization techniques in the context of model updating have been applied by various researchers using SM. SM has the drawback of being stuck in local minima and involves finding the inverse of the sensitivity matrix. The choice of suitable regularization parameter was also based on the sensitivity matrix, and corresponding filter factors have been calculated using techniques such as GCV, L-curve or quasi-optimality functions [37]. However, in the context of this paper, a regularization technique was applied using GA to harness the benefits of both techniques. The method adopted in this study is a forward analysis method, which does not involve solving the inverse of matrices. Furthermore, EAs have a limitation that they work on a number of points in parallel contrary to SM-based methods, which work on a single point at a time. Therefore, in this paper, based on the basic concept of the L-curve, Pareto fronts using multiobjective optimization were employed, as these can be applied graphically in the context of EAs. The tradeoff value can be selected where the curvatures of the three corresponding points change with maximum bend angle [25]. Therefore, the Pareto optimal front has been used in this study to find out the tradeoff between the objective function and regularization part as explained in Equation (15). Multiobjective GA implemented in MATLAB using function _gamultiobj_ has been used. A population size of 100, maximum generation limit of 1000 and function tolerance of 1 \(\times\) 10\({}^{-10}\) havebeen used for the multiobjective function. A typical Pareto front was obtained for Damaged Beam 1 and is shown in Figure 5 for 5% noise in frequencies and mode shapes. It has been found that a value of 6 \(\times\) 10\({}^{-6}\) gives a suitable tradeoff between the two axes. Likewise, other tradeoff points have been selected for other noise levels based on their corresponding Pareto fronts and are reported in Table 1. It can be seen that the value of these parameter decreases with low noise and increases with high noise. Likewise for noise in mode shapes only, the values of these parameters are generally less than noise both in frequencies and mode shapes. This is logical as the parameter essentially is a penalty function, which weights the need for inclusion of the function \(\|x-x_{o}\|\) in the overall objective function value. The more the noise, the more will be the tradeoff value. GA was applied using Equation (15) with Objective Function I, II and III. The effect of regularization on all the objective functions is shown in Figures 6-8. It can be seen that Objective Function I has shown promising results as compared to the other two objective functions. This advocates that noise errors cannot be eliminated through regularization without a proper objective function. Figure 6 shows the damage index of the updated damaged model for all noise levels for regularized Objective Function I. For the damaged beam, the damage index for Element No.