# Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers
## 4 Inverse Problem
### Pseudo-Experimental Data
#### 4.1.1 Damage Located at a Sensor Point  
In this test case, we discuss only the second damage scenario D2.A, in which case the notch is located at \(s^{D}=0.25\) below sensor #2, and its depth is \(0.25H^{U}\) (the residual height is \(h^{D}=0.75\)). This enables capturing the large peak of curvature variation at the damage location. Similar results are obtained for other residual heights and are not reported here for the sake of brevity.  
Note that the minimization of the objective function in Equation (9) refers to minimization of the distance; therefore, the identified damage parameters correspond to its minimum. Conversely, the beamformers are a measure of the _match_ between measured and replica vectors; hence, the identified damage parameters are the coordinates of its maximum. In the contour plots, light colors indicate small values, while dark colors indicate large values. Therefore, the estimate of the objective function appears as the lightest color, while the estimate of the beamformers appears as the darkest.  
Figure 6 presents the contour plots of the objective function, including one modal curvature only (a \(i=1\), b \(i=2\), c \(i=3\)) and their superposition (\(i=1\), \(2\), \(3\)) for damagescenario D2.A. As the input data derive from the forward solution, the estimate matches the exact damage parameters \(s^{D}\) and \(h^{D}\), which are indicated by a circle in all the following figures. There are, however, several other local minima, which, in the presence of errors, could lead to incorrect results.  
The contour plots of the Bartlett processor for the different modes and their superposition are plotted in Figure 7. The processor has unsatisfactory resolution both in terms of damage depth and location. Even though the position estimate is correct in this case, it may be strongly influenced by measurement and modeling errors. In fact, several local maxima, called sidelobes, are apparent, even though their magnitude is smaller than the global maximum. This is consistent with the results reported in the literature [29].  
The contourplots of the MVDR processor are provided in Figure 8. This beamformer suppresses the sidelobes and therefore the local maxima, clearly pointing to the damage parameters. The use of the MVDR processor in this damage scenario provides a very clear maximum for the first three modes individually, as well as for their superposition.  
In conclusion, all three processors provide a unique solution, even when using a single mode, although the sensitivity of the estimators to the two damage parameters is different. In fact, the values of the estimators vary considerably with \(s^{D}\), which hints at the high sensitivity of the estimator to this parameter. On the contrary, both the objective function and Bartlett beamformer have limited sensitivity to variations in \(h^{D}\), while the sensitivity of MVDR is high with regard to this parameter. This is particularly relevant for the early-stage identification of damage.  
Figure 6: Contour plots of the objective function including one ((**a**) \(i=1\), (**b**) \(i=2\), (**c**) \(i=3\)) and three numerical modal curvatures ((**d**) \(i=1\), \(2\), \(3\)) (coordinates of the circle are \(s^{D}\) and \(h^{D}\) for D2.A).  
Figure 7: Contour plots of the Bartlett beamformer including one ((**a**) \(i=1\), (**b**) \(i=2\), (**c**) \(i=3\)) and three numerical modal curvatures ((**d**) \(i=1\), \(2\), \(3\)) (coordinates of the circle are \(s^{D}\) and \(h^{D}\) for D2.A).