# Entropy Wavelet-Based Method to Increase Efficiency in Highway Bridge Damage Identification
## 4 Numerical Results and Analysis  
In this section, the EWM is numerically validated through simulations with the finite element model previously described. For this purpose, the EWM is applied in the FEM model of the RPB in its healthy condition, as well as with different damage scenarios. The results indicate that the detection and localization of all the damage cases is possible with the EWM in a quick, accurate, and efficient manner. Since, in this article, the EWM is presented for the first time, in this section, several diagrams will be shown and analyzed in detail, which will allow for the sequential observations of how the idea of making the WEAM more efficient arose, by implementing an algorithm based on the Shannon entropy in order to give way to the EWM. Moreover, the EWM was applied to identify diverse damage conditions, and the corresponding results/diagrams will be also presented. The most important challenges that led to the development of the EWM were the reduction of the steps required to apply the WEAM and the reduction of the computing time necessary to obtain the results, which allow for the detection and localization of damage without losing any precision. Therefore, since these are processes that involve a significant computational burden which are related to other stages, it was identified that the definition of the CWT scale range, useful for identifying damage with the WEAM via the use of colored 3D CWT diagrams, is the most critical stage of that method (stage 5 in [1]), along with the generation of new colored 3D CWT diagrams to identify damage in the area of interest (stage 8 in [1]). Thus, considering the stages of the WEAM, which were described in detail in [1], seven of those steps were eliminated via the EWM (steps 4, 5, 7, 8, 9, 10, and 11); that is, 64% of the WEAM stages were eliminated and replaced with a few steps requiring low computing time consumption. It should be noted that around 95% of the computing time needed to obtain the results via the application of the WEAM and determine the health condition of the analyzed structure relates to the stages 5, 7, 8, 9, and 10, and all of them were eliminated with the EWM. Consequently, the EWM began to be designed, processing the corresponding data from the numerical simulations with the scenario of the damaged bridge, with an intermediate severity damage on the deck at 25% of the length (simulated by reducing 30% the area of the cross-section at 25% of the 203 m length of the bridge deck between towers), and the CWT coefficients were obtained for each scale value, from 1 to 1000, measuring at 25% of L. In the corresponding diagram that was obtained (see Figure 9), it can be observed that it is difficult to detect damage due to the large number of curves. However, it can be noted that there are certain curves with magnitude increments of CTWcoefficients around 25% of L. Therefore, it is necessary to define what scales these curves correspond to, discarding those that are not useful. To clarify the diagram shown in Figure 9, the CWT coefficients were obtained again for the scale range from 1 to 1000, but now with increments of 50, and the corresponding results are presented in Figure 10. Thus, in Figure 10, it is possible to observe that most of the curves that are not useful for detecting damage no longer appear, while a significant number of the useful curves that suggest the existence of damage at 25% of L are preserved. Therefore, for this analyzed scenario, there are more useful curves than there are curves that do not reveal the presence of damage. Therefore, in order to have a reliable parameter that allows for the systematic defining of the useful scale range to identify damage without the need of obtaining CTW coefficient curves for different ranges and increments of scale, an algorithm based on the Shannon entropy was implemented (see Section 3.1, Equation (1)).