# 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
#### 3.6.1 Accuracy  
Figure 2 displays the reference system of a jump-enriched 1D element of unit density and order \(p=5\), along with the standard (\(N_{3}(\xi)\)) and enriched (\(N_{3}(\xi)H(\xi)\)) interpolation functions related to its third node. In this section, we study the accuracy of the procedure in its intended application, namely the integration of the terms \(m_{i,12}\) and \(m_{i,22}\) (see Equation (27)). To this end, the integration error \(\epsilon_{\rm I}\) is computed with respect to a reference value \(I_{\rm ref}\), which is evaluated with a boundary-conforming quadrature rule, according to Section 3.5:  
\[\epsilon_{\rm I}=\left|\frac{f_{i,jk}(\xi_{i})w_{i,jk}-I_{\rm ref}}{I_{\rm ref }}\right|. \tag{36}\]  
In Figure 3, \(\epsilon_{\rm I}\) is reported for different element orders \(p\), integrands \(f_{i,jk}=(N_{p,j}H)^{j}\), crack locations \(l_{c}\), and a fixed value of \(\epsilon_{\lambda}=1-10^{-4}\). We should note that, by choosing \(H\) as an enrichment function, the integrands (as well as the moments in Equation (29)) are discontinuous only for the case with \(j=1\). In general, it can be observed that the procedure is less successful whenever the discontinuity lies near an element node: an effect that is most pronounced in the interaction terms for \(p=3\) (Figure 3a). With this exception, the accuracy achieved in the off-diagonal terms is satisfactory, as shown in Figure 3b.c. For diagonal terms, however, (Figure 3d-f) the errors are extremely elevated. In this context, we should underline that this strategy is first and foremost a mass lumping method, and secondarily an integration method. In other words, the quality of the weight set \(\mathbf{w}_{22}\) is heavily restricted by the constraints expressed by Equation (34), which are necessary in order to maintain a positive definite mass matrix. We should also note that, with few exceptions [25], several mass lumping strategies provide little to no a priori guarantees in terms of accuracy, and the validity of the resulting model will be studied in Sections 4.1-4.3. In Figure 4, the convergence of the integration error is studied with integrands of the form \(H(\xi)^{j}(1+\xi^{\prime}H)\) and a fixed discontinuity at \(l_{c}=1/3\). Although errors in the order of \(10^{-10}\) are not as low as machine precision, Figure 4a suggests that an integration order of \(p\) can be achieved for the weight set \(\mathbf{w}_{12}\). This is consistent with the performance of other moment-fitting procedures that do not optimize the Gauss point location [69,75] and thus perform similarly to Newton-Cotes quadrature rules [86,103]. In comparison, the error convergence for the weight set \(\mathbf{w}_{22}\) (Figure 4b) is modest, and, given the simplicity of the integrand, can be again attributed to the presence of constraints, as the incompleteness of the fitting basis \(G_{22}\) (see Table 1) should not play a role in this case. Figure 2: 1D enriched element for the benchmarks presented in Figure 3. Exemplary plot of a standard shape function (\(N_{3}(\xi)\)) and enriched interpolation function (\(N_{3}(\xi)H(\Phi)\)) for \(p=5\). To gain further insight into this issue, we study the impact of the minimum mass eigenvalue \(\lambda_{\text{min}}\) on the accuracy of the weight set \(\mathbf{w}_{22}\). Figure 5 displays the integration error \(\epsilon_{1}\) [-] by varying \(\epsilon_{\lambda}\) for a selection of cases from Figure 3. For crack locations unfavorably near a node (cases (a) and (b)), this parameter has no significant impact, while in more favorable configurations (cases (c) and (d)), smaller values of \(\epsilon_{\lambda}\) allow for a better solution of Equation (31). Some local minima of \(\epsilon_{1}\) [-] are sporadically detected for \(\epsilon_{\lambda}\in[0.1,1)\) (e.g., case (e)), which are likely related to the nonlinear character of Equation (34). Figure 4: Integration of discontinuous polynomials with the procedure presented in Section 3.6. Cracked element with \(l_{c}=1/3\), \(\epsilon_{\lambda}=1-10^{-4}\).