# A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications
## 4 Numerical Examples
### Example 2  
The aluminium plate illustrated in Figure 9a presents 34 cylindrical holes of radius \(r_{\rm h}=5\) mm, whereas damage is represented by two planar cracks orthogonal to the xy plane. The bottom crack has a length of 21.2 mm and an inclination of 45\({}^{\circ}\), while the vertical, 10 mm long crack is connected to one of the holes. Piezo-electric excitation was modeled as a circular loading surface of diameter \(2r_{\ell}=10\) mm. The loading followed a radial distribution in space with a maximum of \(p\) at the outside edge of the actuator and a value of zero at its center:  
\[\mathbf{p}(\mathbf{x})=\frac{p}{r_{\ell}}\ \mathbf{r}(\mathbf{x}),\ \ \|\mathbf{r}( \mathbf{x})\|\leq r_{\ell} \tag{41}\]  
Figure 8: Cracked aluminium panel in plane stress (see Figure 7). Topological enrichment scheme. Time history L2 error norm convergence for h-refinement with respect to model size (**a–d**) and simulation time (**e–h**). where \(\mathbf{r}(\mathbf{x})\) points from the center of the circle to \(\mathbf{x}\), \(\|\mathbf{r}(\mathbf{x})\|\) is its Euclidean norm, and \(p=2.5\times 10^{9}\) Pa is the loading magnitude for \(\|\mathbf{r}(\mathbf{x})\|=r_{\ell}\). This assumption is similar to the pin-force model typically used in this context [116], where the loading is, however, applied only along the circumference of the actuator. Modulation in time was achieved via Equation (39) with \(f=200\) kHz and \(n=5\). Given a plate thickness of \(t=1.5\) mm, with these parameters we expected to observe the first symmetric and anti-symmetric Lamb wave modes. It should also be noted that a high number of cycles \(n\) is effective in limiting wave dispersion, albeit at the cost of expanding the excitation window. This example highlights the benefits offered by the XSCM in the modeling of complex components and damage configurations. For the purpose of comparison with the SEM, a setup was chosen that still allowed for discretization with a conforming mesh. In both cases, a second-order mesh was produced with Gmsh [117] and then converted into a third-order GLL SE mesh via the shape functions of the original elements, leading to a sub-parametric representation of the domain. The differences between the meshes are shown in Figure 9b,c. With the SEM, a strucutred nonorthogonal mesh was employed in order to successfully discretize load surfaces, cracks, and holes with hexahedral elements. This required the use of a highly customized Gmsh script and would not be feasible in an automatized fashion should the location of a detail change. With the XSCM, an orthogonal mesh could be used instead, and all domain details, including loading surfaces, were defined at run-time. This enabled the optimal choice of the element size and automatized simulations for, e.g., damage detection applications [118; 119; 120; 121; 122]. In both cases, the plates were one element thick and had an element size of \(h=2.6\) mm in the \(x\) and \(y\) directions, leading to 31 nodes per wavelength of the asymmetric mode. As reported in Table 2, the resulting models had a size of \(1.1\times 10^{6}\) DOF for the SEM and \(9.3\times 10^{5}\) DOF for the XSCM. This difference, in spite of the additional enrichment parameters in the latter model, was due to the fact that, in the former model, the element size had to be frequently reduced to resolve the complex geometry of the modeled structure (see Figure 9b). An undamaged version of the plate, which was added for comparison ('SEM pristine') enabled us to exploit the symmetry of the model along the \(x\) axis, thus halving its size. All solutions employed a time step of \(\Delta t=5\times 10^{-9}\)[s]. This is well below the critical time step of the conforming meshes to promote accurate time integration. The enriched model had a relatively small \(\Delta t_{\mathrm{c}}\), which was expected given the results of Figure 6b. In order to accommodate for this, the leap-frog solver [18; 19] was deployed with a local time refinement ratio of \(p_{\mathrm{t}}=34\)[56].