# A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications
## 4 Numerical Examples
### Example 1  
Figure 7 displays an aluminum panel in plane stress with parameters \(E=70\) GPa, \(\
u=0.33\), and \(\rho=2700\) kg/m\({}^{3}\). Fixed boundary conditions are applied on its left end, while on the right-hand side the transient load  
\[\mathbf{p}_{\mathrm{s}}(\mathbf{x},t)=\mathbf{p}(\mathbf{x})\sin\left(\omega t \right)\sin^{2}\!\left(\frac{\omega t}{2n}\right),\ \ \ t\in\left[0;\frac{n}{f}\right] \tag{39}\]  
is applied. In Equation (39), a spatially uniform load is given by \(\mathbf{p}(\mathbf{x})\equiv p\mathbf{e}_{\mathrm{x}}\), with \(\mathbf{e}_{\mathrm{x}}=[1,0]^{T}\) and \(p=10^{6}\) N/m, while time modulation is produced by a Hann window of circular frequency \(\omega=2\pi f\). The excitation window \(n/f\) is determined by the carrier frequency \(f=200\) kHz and the number of cycles \(n=5\). This use of a smoothed signal is effective in producing narrow frequency band waves and thus limiting the dispersive effects [114]. The damage consists of an oblique, planar crack of length \(\sqrt{2}a\), (\(a=5\) cm), which is discretized according to Section 3.2. In this benchmark, the proposed method is compared to a CMM formulation paired with the implicit version of the Newmark algorithm [115]. For this, the displacement time history is recorded at sensors  
Figure 6: Critical time step ratios \(\frac{\Delta t_{c}}{\Delta t_{c0}}\) for Heaviside-step enriched SEs of different orders. Different locations \(l_{c}\) of the discontinuity (see Figure 2) are studied for (**a**) the CMM and (**b**) the fitted lumping procedure with \(\epsilon_{\lambda}=10^{-1}\). In (**c**), stability can be improved by increasing \(\epsilon_{\lambda}\), which constrains the fitting problem via Equations (34) and (35).  
\(s_{1}\) and \(s_{2}\) positioned according to Figure 7, which are used to measure the accuracy of a simulation as follows:  
\[\epsilon_{\text{h,t}}=\sqrt{\frac{\sum\limits_{i=0}^{n_{\text{t}}}[u_{\text{h}}(i \Delta t)-u_{\text{ref}}(i\Delta t)]^{2}}{\sum\limits_{i=0}^{n_{\text{t}}}u_{ \text{ref}}^{2}(i\Delta t)}}, \tag{40}\]  
where \(n_{\text{t}}\) is the number of time steps; \(u_{\text{h}}\) is the numerical solution at a sensor; and \(u_{\text{ref}}\) is a reference solution computed with the CMM formulation, discretized with \(3.2\times 10^{6}\) DOF, \(p=6\), and \(\Delta t=10^{-10}\). The simulation time measures 0.1 ms and, for all models to be tested, a fixed time step of \(\Delta t=10^{-9}\) s is used, which roughly represents the CFL condition of the finest meshes.  
The convergence of \(\epsilon_{\text{h,t}}\) at both sensors is reported for models of third to sixth orders with respect to the number of DOF in Figure 8a-d and the amount of time necessary for the solution of Equation (17) in Figure 8e-h. In the former set, the proposed method performed similarly to the CMM formulation; however, some loss of convergence emerged for the largest models. This likely coincided with the stage at which the integration errors discussed in Section 3.6 exceeded those of the SE interpolation and, thus, created an accuracy bottleneck. In the latter set (Figure 8e-h), the proposed approach increased the speed by roughly one order of magnitude for \(p=3,4\) and more modestly for \(p=5,6\). Although the CMM formulation paired with an implicit solver prevailed in terms of maximum accuracy, one should keep in mind that this required substantial additional memory storage for the factorization of the effective stiffness matrix. Finally, the same formulation was used for the reference solution, and thus the accuracy of the CMM could have been overstated for the largest models presented here.  
Figure 8: _Cont._  
Figure 7: Cracked aluminium panel in plane stress: 2D SE mesh of generic element size \(h\), crack with \(a=0.05\), and sensors \(s_{1,2}\). The unit is m.