# Detection and Localization of Multiple Damages through Entropy in Information Theory
## 6 Accounting for Uncertainties  
To conclude this research, a last set of tests were reserved for a more realistic FE model (shown in Figure 14).  
Specifically, having validated the algorithm under ideal conditions, this further test included more realistic assumptions about the material properties. Since the second FE model shares most of the mechanical and geometrical properties of the first case study, only the major differences will be commented on hereafter.  
A random perturbation of the soil parameters over all the springs of the FE model resulted in a random deviation from the nominal value of the first model of about 20% (+10%). This was intended to emulate the inhomogeneity of the surrounding material soil.  
A reduction of 10% of [PERSON]'s modulus of 75 randomly selected finite elements of the SP model was assumed to simulate the presence of manufacturing defects. At the same time, a reduction of stiffness of 20%, 40%, and 60% was assumed to simulate damage on the girth weld, located at 10 m (at the center of the analyzed SP) from 0\({}^{\circ}\) to 90\({}^{\circ}\); while a reduction of stiffness of 40%, 60%, and 80% was assumed to simulate damage on two long seam welds, located at 0\({}^{\circ}\) (from 9.5 to 10 m) and 90\({}^{\circ}\) (from 10 to 9.20 m). This was intended to simulate an S-shaped crack on the pipeline similar to the one shown previously in Figure 1 at the intersection between the girth and two consecutive long seam welds. A  
Figure 13: Automatic detection and localization of the severity of damage (noise = 5%). The highlighted bars (in red for negative values, blue for positives ones) are related to points that fall outside \(\mp\)3 times the standard deviation of data (i.e., 3-sigma method): (**a**) damage scenario 40%/60%/80%; (**b**) damage scenario 0%/20%/80%; (**c**) damage scenario 40%/0%/80%.  
constant width of 0.16 m was applied for the damage in the two long seam welds, while a width of 0.20 m was considered for the damage in the girth weld.  
To summarize, three steps of increasing damage were introduced, considering the following stiffness reduction on the circumferential tract and the two longitudinal ones (in the same order): 20% and 40%; 40% and 60%; 60% and 80%. As for the previous case study, artificial WGN between 5% and 20% of the standard deviation of the output signals was added to the simulated output responses. For brevity, only results for 5% noise are reported in Figure 15, as was already done for the first model.  
Figure 14: The second FE model with damage locations, inhomogeneity of the soil material, and manufacturing defects.  
Figure 15: Increasing stiffness reduction (20–40%, 40–60%, and 60–80%) for damage in the girth weld and the long seam welds, respectively. Graphs are limited between 7 m and 13 m to show the increase of the intensity of damage for increasing stiffness reduction.  
From Figure 15 it is possible to conclude that also in presence of inhomogeneity of the surrounding soil and manufacturing defects the Wiener entropy of axial strains was able to detect and locate quite well the damage. The same considerations can be performed for other percentages of noise, while an instability was observed in the absence of noise in data, for the damage at \(0^{\circ}\). However, as concluded in [10], this is a known behavior of Wiener entropy, which needs the addition of numerical noise to stabilize the health state indicator estimate.