# A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications
## 4 Numerical Examples
### Example 3  
In this section, we study a glass-fiber-reinforced polymer (GFRP) panel with dimensions of \(500\times 500\) mm and a thickness of about 2 mm. The specimen consisted of twelve layers of VV192T/202 prepregs designed by [PERSON] with orientations of \([0/90/0/90/0/90]_{\mathrm{s}}\). The glass fiber fabric had a twil weave with a density of 202 g/m\({}^{2}\). Delaminations were artificially introduced at the manufacturing stage using Teflon inserts between layers of prepregs. In this particular case, the inserts were located in the middle of the cross-section, i.e., between the sixth and seventh laminae. Four delaminations with different shapes and locations were introduced according to Table 3. Their positions differed slightly with respect to the specifications provided by the manufacturer--all delamination centers were designated to be 125 mm from the edge of the plate and aligned with the center of the plate. The actual delamination locations were identified through signal processing using wavenumber damage imaging (see Figure 11a). A piezoelectric transducer (Noliac, NCE51) of diameter 10 mm and thickness 0.5 mm located at the center of the plate was used for the excitation of guided waves. A Hann windowed signal with a central frequency of 50 kHz and 5 cycles was applied to the piezoelectric transducer (20 Vpp). An SLDV was used to acquire Lamb wave signals on a uniform spatial grid (497 \(\times\) 497 points) on the surface of the plate. The measured area was 496 \(\times\) 496 mm, because a 2 mm margin along the plate edges was omitted. Out-of-plane  
Figure 10: Time history response of the vertical displacement at the sensors (see Figure 9a). For the damaged cases, it is shown that the proposed approach (“XSCM”) was comparable to the SEM, in spite of the substantially different discretization strategies employed (see Figure 9b,c). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Shape** & \(\mathbf{a}\) **(mm)** & \(\mathbf{b}\) **(mm)** & \(\mathbf{x}\) **(mm)** & \(\mathbf{y}\) **(mm)** \\ \hline Ellipse & 30 & 20 & \(-125\) & \(1\) \\ Circle & 20 & 20 & 2.45 & \(-125\) \\ Square & 20 & 20 & 126 & \(-1.5\) \\ Rectangle & 30 & 20 & 1 & 126 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Delamination shapes and locations with origin in the center of the plate. particle vibration velocities were measured using one laser head (Polytec PSV-400) [123] placed perpendicularly to the surface of the plate. The plate was modeled in 3D by a single layer of SEs of order \(p=3\) and element size of \(h=5\) mm. The time integration step measured \(\Delta t=1.8\times 10^{-7}\) s, corresponding to the CFL condition of the standard portion of the mesh. A local time refinement ratio of \(p_{\rm t}=31\) was used in order to ensure the stability of the enriched regions of the model. Establishing appropriate parameters for the GFRP material was not a straight-forward task. After assuming an orthotropic material law, the elastic constants of the GFRP lamina were identified using the inverse method described in Ref. [124], and are reported in Table 4. A layered integration technique based on laminae-conforming element partitions was used to correctly evaluate Equation (19), which was also discontinuous across laminae [125]. For simplicity, the square and rectangular delaminations were modeled by circular and elliptic level sets, respectively. The mass and stiffness of the piezo-electric actuator were also neglected, along with the piezo-electric effect (the mass of the sensor represented roughly 0.03% of the panel mass). Instead, the same loading conditions as in the previous example were applied, with \(p=7.2\times 10^{7}\) [Pa]. This value, as well as the mass-proportional damping coefficient \(\alpha\), was established empirically by matching the amplitudes of the measured and simulated wave velocities. Finally, this material introduced some additional uncertainty with respect to the plate thickness, which could have presented some significant variation from the specified value.