# 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  
In this section, a mass lumping method for enriched elements is derived by combining nodal quadrature with a moment-fitting procedure that tailors the integration weights to the character of the integrand. The starting point is to combine Equations (16), (18) and (24). In one dimension, the mass coefficients of an enriched node \(i\) are then given by:  
\[\mathbf{M}_{i}= \begin{bmatrix}m_{i,11}&m_{i,12}\\ m_{i,12}&m_{i,22}\end{bmatrix} \approx\int_{\Omega_{\mathrm{e,GLL}}}\rho(\bar{ \boldsymbol{\xi}}) \begin{bmatrix}N_{i}^{2}(\bar{\boldsymbol{\xi}})&N_{i}(\bar{ \boldsymbol{\xi}})F_{i}(\bar{\boldsymbol{\xi}})\\ N_{i}(\bar{\boldsymbol{\xi}})F_{i}(\bar{\boldsymbol{\xi}})&N_{i}{}^{2}(\bar{ \boldsymbol{\xi}})F_{i}^{2}(\bar{\boldsymbol{\xi}})\end{bmatrix} \mathrm{d} \Omega_{\mathrm{e,GLL}}, \tag{26}\]  
i.e., although not fully diagonal, the mass matrix of enriched elements assumes an advantageous block-diagonal structure. In the spirit of nodal quadrature, our goal is now to evaluate Equation (26) with the best possible accuracy. For this, the three mass coefficients \(m_{i,jk}\), with \(j,k\in\{1,2\}\), can be considered as distinct integrals of functions \(f_{i,jk}\):  
\[m_{i,jk}\approx\int_{\Omega_{\mathrm{e,GLL}}}f_{i,jk}(\bar{\boldsymbol{\xi}}) \;\mathrm{d}\Omega_{\mathrm{e,GLL}}=\sum_{l=1}^{n}f_{i,jk}(\bar{\boldsymbol{ \xi}}_{l})\;w_{l,jk}=f_{i,jk}(\bar{\boldsymbol{\xi}}_{i})\;w_{i,jk}, \tag{27}\]  
where \(n\) is the number of coinciding element _and_ quadrature nodes, and thus the Kronecker delta property can be applied to achieve the last expression on the right side. As these nodes \((\bar{\boldsymbol{\xi}}_{i})\) are necessarily fixed, it only remains to establish three independent weight sets \(\mathbf{w}_{jk}\) that are best-suited to integrate the respective function sets \(\mathcal{F}_{jk}=\left\{f_{1,jk},f_{2,jk},\ldots,f_{n,jk}\right\}\). For this task, we first decompose these integrands into \(m\) basis functions \(\mathcal{G}_{jk}=\left\{g_{1,jk},g_{2,jk},\ldots,g_{n,jk}\right\}\), such that, in the ideal case,  
\[f_{i,jk}\in span\Big{\{}g_{1,jk},g_{2,jk},\ldots,g_{m,jk}\Big{\}}\forall i. \tag{28}\]  
Then, by applying the above to Equation (27), the relation between a set of weights \(\mathbf{w}_{jk}\) and basis functions \(\mathcal{G}_{jk}\) can be expressed in terms of the moment-fitting equations  
\[ \begin{bmatrix}g_{1,jk}(\bar{\boldsymbol{\xi}}_{1})&g_{1,jk}(\bar{\boldsymbol {\xi}}_{2})&\ldots&g_{1,jk}(\bar{\boldsymbol{\xi}}_{n})\\ g_{2,jk}(\bar{\boldsymbol{\xi}}_{1})&g_{2,jk}(\bar{\boldsymbol{\xi}}_{2})&& \vdots\\ \vdots&&\ddots&\vdots\\ g_{m,jk}(\bar{\boldsymbol{\xi}}_{1})&\ldots&\ldots&g_{m,jk}(\bar{\boldsymbol{ \xi}}_{n})\end{bmatrix}\begin{bmatrix}w_{1,jk}\\ w_{2,jk}\\ \vdots\\ w_{n,jk}\end{bmatrix} = \begin{bmatrix}\int_{\Omega_{\mathrm{e}}}g_{1,jk}(\bar{ \boldsymbol{\xi}})\;\mathrm{d}\Omega_{\mathrm{e}}\\ \int_{\Omega_{\mathrm{e}}}g_{2,jk}(\bar{\boldsymbol{\xi}})\;\mathrm{d} \Omega_{\mathrm{e}}\\ \vdots\\ \int_{\Omega_{\mathrm{e}}}g_{m,jk}(\bar{\boldsymbol{\xi}})\;\mathrm{d} \Omega_{\mathrm{e}}\end{bmatrix} \iff\mathbf{A}_{jk}\mathbf{w}_{jk}=\mathbf{b }_{jk}, \tag{29}\]  
where the repetition of indices does not imply summation, and the choice of basis functions \(\mathcal{G}_{jk}\) is reported in Table 1. Three considerations are due at this point. Firstly, we should note that the number of basis functions \(m\) is limited by the number of nodes \(n\), as the system would be overdetermined for \(m>n\). This is in contrast to most other instances of this method, which exploit an under-determined system [71; 77]. For this reason, only the basis monomials up to order \(m\) are considered, and the higher-order terms appearing in the diagonal integrands \(\mathcal{F}_{11}\), \(\mathcal{F}_{22}\) will be under-integrated, as they are not represented in Equation (29). For standard SEs, however, this choice still leads to weights that match the GLL rule, and thus to only minor under-integration of the mass matrix [56].