# Adaptive Probabilistic Optimization Approach for Vibration-Based Structural Health Monitoring
## 4 Results and Discussion
#### Case 1
The probability of damage existence was never reported below 53%. Similarly, the actual position of damage was within the predicted range of crack location for most trials. The crack size, however, was consistently underestimated. Figure 4: Frequency diagrams (probability density) of (**a**) crack location and (**b**) crack size for Case 1. **Case 2**  
In this case, the noise level was increased to 9% and the crack size was kept unchanged at 30% of the beam thickness. The crack was introduced closer to the left boundary at 725 mm from the origin. The sensor location was moved to the middle of the beam, i.e., data collected from a node located at 500 mm with the crack size being 30% of the beam thickness. The results of the adaptive algorithm are summarized in Table 4.  
The results in Table 4 strongly suggest that the simply supported beam is damaged with a probability of approximately 66% at a location between 713 mm and 831 mm. There was a noticeable reduction in the probability of damage existence in comparison to Case 1, which could be attributed to the increase in the level of noise and the locations of damage. The sensor location also was expected to have a significant impact on the analysis of damage identification, which is a research topic that has been pursued by many researchers [32, 33, 34]. The crack size range in stage 2 was underestimated and slightly below the actual value, and the probability saw a moderate increase of about 3%. On the contrary, the ranges of crack location predicted by the algorithm in both stages included the actual crack position, but the probability sustained a drop of approximately 5%, resulting in a decrease in the probability of damage existence in stage 2. The predicted range of crack size was about 5% of the beam thickness. However, the crack location in stage 2 was further shifted to the right and the actual crack location moved closer to the lower bound of the predicted range. It can be concluded that the algorithm produced less favorable results in stage 2, especially for crack location with a drop in the probability accompanied by an increase in the predicted range. Given that the algorithm was designed to yield the most probable scenario rather than optimization of the probability of damage, this behavior may be considered to be realistic for a structural health monitoring system. Despite a decline in the probability of damage existence in stage 2, all parameters have been reported with a probability greater than 50% indicating the structure could potentially be damaged. In Figure 5, the frequency diagrams of crack size and crack location from both stages are superimposed. The 9% noise added to the signal has introduced short side bars spread out around the main bars in the frequency diagrams of crack location. Despite a small drop in the probability of damage location as reported in Table 4, it is worth noting from Figure 5a that the algorithm yields better results for damage location by virtually limiting 100% of occurrences within 50% of the beam length. Likewise, the frequency diagrams of crack size in Figure 5b show noticeable improvements in stage 2 with a higher peak around 2.5 mm and a significant reduction of occurrences at both extremes. Finding a false negative in damage detection is quite common from deterministic methods because damage detection is an ill-posed problem with no unique solution, making a probabilistic approach more appropriate to provide meaningful results in such problems. Despite several damage scenarios presented in Figure 5, the algorithm successfully determined the optimal solution representing the status of the structure.