# The Extreme Function Theory for Damage Detection: An Application to Civil and Aerospace Structures
## 3 Numerical Simulations
### Results for the HAR Wing Spar (2-Dimensional) FE Model  
The results for the damage scenarios depicted in Figure 4b are portrayed in Figure 9. Only the 1 st mode shape is showed for brevity; the 2 nd and 3 rd mode shape returned similar results in terms of accuracy (with similar rates of occurrence for both false positives and false negatives). Artificial white Gaussian was added considering SNR = 60 dB and 40 dB.  
One can see that the method always successfully labels the undamaged and damaged states correctly for SNR = 60 dB, which is already a relatively large amount of noise.  
Note that no damage was inserted in state 00; therefore, no damage was to be identified there, similarly to what expressed for the results of the beam structure (in Figure 6).  
The states 01 and 02, which have too small variations of Young's modulus to be effectively considered as damaged, are rightly identified as false negatives. This proves the reliability of the method in discerning small perturbations, which may happen in real-life  
Figure 8: The similarity between the damaged and undamaged mode shapes in terms of cumulative Euclidean distance (**a**) simply supported beam (**b**) clamped-hinged beam. Points with similarity equal or larger than 80% are highlighted in shades of red, above the dash-dotted line.  
Figure 7: The first mode shape of a cantilevered box beam, without damage and with roving damage. (**a**) Damaged and undamaged mode shapes superimposed; (**b**) similarity between the damaged and undamaged mode shapes in terms of 1-CED (points with similarity equal or larger than 80% are highlighted in shades of red, above the dash-dotted line).  
acquisitions due to the statistical variance of the acquisitions, from the actual damage. On the other hand, with even more white Gaussian noise artificially added to the input data (SNR = 40), the approach still correctly identifies the undamaged states, while it loses accuracy on the damaged mode shapes of the test set. Specifically, it remains able to recognise the most damaged situations (with a 25% or 50% reduction of [PERSON]'s modulus) whenever the damage is inserted at the clamped base. As expected, also by comparison with previous works [10], the scenarios with the damage inserted solely at the mid-length cross-section (states 3 to 5) or on the sides of the base cross-section not fixed to the centre wing box (states 18 to 20).  
Regarding the computational efficiency of the algorithm, the whole simulation of the complete HAR wing dataset was performed in around 31 s. The non-optimised MatLab script elapsed on average 2.2 s to perform a single training fold, 3.9 s for CDF validated estimation and 0.02 s for the damage detection task on a single damage state, with no major differences among the 20 states, on an Intel(r) Core(tm) i7-10750H CPU with 2.6 GHz base frequency and 15.8 GB of available RAM. Similar elapsed times were found for the other numerical and experimental cases.  
Figure 9: Extreme Function Theory (EFT) results for the HAR wing FEM, 1 st mode shape with artificially added White Gaussian Noise (WGN) (**a**): SNR = 60 dB; (**b**): SNR = 40 dB. Effectiveness expressed in percentage.