# A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications
## 3 Moment Fitting for Enriched Spectral Elements
### Mass Lumping for Enriched Spectral Elements
#### 3.6.2 Stability  
By intervening in the mass matrix, mass lumping methods have an impact on element eigenmodes and thus on the stability of explicit schemes. This concern is particularly relevant in the presence of discontinuities, which can lead to a large increase in eigenvalues, the extent of which we wish to quantify in this section. For central difference methods (CDMs), the Courant-Friedrichs-Lewy (CFL) condition is given conservatively by the smallest value of [113]:  
\[\Delta t_{\mathrm{e}}=\frac{2}{\omega_{\mathrm{max}}} \tag{37}\]  
across all elements. In Equation (37), \(\Delta t_{\mathrm{e}}\) is the element critical time step, and \(\omega_{\mathrm{max}}\) is the element spectral radius, i.e., its biggest eigenvalue obtained from [113]:  
\[\mathrm{det}\Big{(}\mathbf{K}_{\mathrm{e}}-\omega_{\mathrm{e}}^{2}\mathbf{M}_{ \mathrm{e}}\Big{)}=0. \tag{38}\]  
We study the stability of the procedure by considering a two-dimensional, unit-size SE traversed by a straight discontinuity parallel to the \(y\) axis and enriched by a modified Heaviside step function, similar to Figure 2. In Figure 6a,b, results are reported for different crack locations \(l_{\mathrm{c}}\), comparing the CMM formulation with the proposed lumping method, where a fixed value of \(\epsilon_{\lambda}=10^{-1}\) is chosen based on Figure 5. All results are scaled by \(\Delta t_{\mathrm{e,0}}\), i.e., the critical time step of a standard SE with \(p=5\). While for the CMM the value of \(\Delta t_{\mathrm{e}}\) remains in the same order of magnitude as \(\Delta t_{\mathrm{e,0}}\), the proposed procedure can lead to values that are from one to two orders of magnitude smaller. If the CDM were used for time integration, this would proportionally multiply the cost of a simulation, risking the negation of the benefits of mass lumping. However, if a leap-frog solver is employed, then this issue will affect only the enriched portion of the domain, which is usually rather small. It was shown in Ref. [56] that, with the CDM, a performance only slightly worse than that of the SEM could be achieved. It should also be noted that Figure 6a does not lead to a performance gain, since this formulation would require the use of an implicit solver. In Figure 6c the effect of \(\epsilon_{\lambda}\) is studied for a fixed configuration (\(l_{\mathrm{c}}=0.3\)). These results indicate that the constraint expressed in Equation (34) is active, as the spectral radius can be controlled by \(\lambda_{\mathrm{min}}\). Finally, even if a leap-frog solver is used, setting \(\epsilon_{\lambda}<10^{-3}\) will lead to prohibitively expensive simulations for negligible accuracy gains (see Figure 5). For these reasons, we recommend using \(\epsilon_{\lambda}\in\left[10^{-2},10^{-1}\right]\), and a value of \(\epsilon_{\lambda}=10^{-2}\) will be used in the following numerical examples.  
Figure 5: Effect of the parameter \(\epsilon_{\lambda}\) on fitting accuracy for selected configurations from Figure 3d–f. See Equations (34) and (35).