# The Extreme Function Theory for Damage Detection: An Application to Civil and Aerospace Structures
## 3 Numerical Simulations  
The procedure is firstly validated on a numerically-simulated beam with different boundary conditions. Then, for more complex structures, the proposed approach is firstly validated on two other numerical datasets. The first case study comes from an aerospace application and involves the High Aspect Ratio (HAR) aluminium spar of the XB-1 wing prototype [33]. The spar--portrayed in Figure 4a--can be seen as a thin, plate-like structural element with a peculiar shape. Due to its large flexibility, the spar may undergo large flap-wise oscillations [34; 35] and it is therefore highly subject to the insurgence of fatigue damage, especially close to its clamped end and in its mid-length at the section where the width taper angle change. The simple Finite Elements (FE) model utilised here was realised in Ansys(r) Mechanical APDL; 400 8-noded, 6-degrees-of-freedom-per-node quadratic shell elements were utilised. The main mechanical and geometrical properties are reported in Table 2 (the values of the parameters derive from preovius studies described in [36]). The model parameters were updated with vibrational data coming from laser Doppler vibrometer (LDV) and video acquisitions [36], extracted by means of the Fast Relaxed Vector Fitting (FRVF) approach [37; 38]. Five output channels (located at the positions of the 4 Inertial Measurement Units and of the LDV target used during the experimental acquisitions, as represented in Figure 4a) were utilised to define the mode shapes. Several levels of reduced Young's modulus were applied to the portions highlighted in Figure 4b to simulate 20 scenarios [10]. In particular, the first two states (01 and 02) are intended to represent small perturbations from the undamaged baseline, where the variations are not sufficiently marked to be defined certainly as actual damage. This simulates the possible statistical variability of the identified mechanical properties; therefore, these states should preferably be labelled as false positives by a reliable (i.e., not hypersensitive) damage detection procedure. Indeed, it was found that, when trained on noiseless mode shapes only, the procedure does not discern these small perturbations from the normal (undamaged) model. The second numerical case study comes from a civil engineering application and is intended to model a simple multi-storey frame structure, which will serve as the experimental benchmark in the next Section. The FE (Figure 5) model, developed utilising the StaBil 2.0 MatLab Toolbox [39], was set as follows. 8 beam elements per column and per beam were used, totalling 72. The [PERSON] model was applied for all elements. To mimic the experimental setup which will be described in the next section, the Young's modulus of the beams \(E_{beam}\) was set as two orders of magnitude larger than its counterpart for the columns (\(E_{column}\)). This allows to approximate the structural system as a shear-type frame structure; therefore, a single channel per floor is enough to capture its main vibrational modes. Three output channels were considered as virtual sensors, located at nodes #4, #12, and #20 (indicated by the purple dots in Figure 5). The main technical details are reported in Table 3. The column tracts highlighted in green are the ones where the damage was inserted (as a stiffness reduction equal to 25% or 50%). Figure 4: The High Aspect Ratio (HAR) wing spar. (**a**) The experimental prototype. (**b**) The damage scenarios considered in the FE analysis.