# 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
Secondly, we should mention that, in this work, \(\rho\) is considered constant within the physical portion of the element, and the effect of voids is accounted for by evaluating the vector of moments (\(\mathbf{b}_{jk}\)) in a consistent sense, according to one of the procedures described in Section 3.5. If no voids are present, then \(\mathbf{w}_{11}\) corresponds to the standard GLL weights. A third important remark is that a positive definite mass matrix is required for explicit solvers to converge; therefore, negative weights must be avoided when \(j=k\) in Equation (29). Similarly, certain values of the off-diagonal weights \(\mathbf{w}_{12}\) might lead to complex eigenvalues for some of the blocks \(\mathbf{M}_{i}\). These issues call for the application of box constraints on the weights, which cannot be applied directly to Equation (29). To overcome this, a residual is allowed in its evaluation:  
\[\mathbf{r}_{jk}=\mathbf{A}_{jk}\mathbf{w}_{jk}-\mathbf{b}_{jk}. \tag{30}\]Then, the computation of the weights can be expressed as an optimization problem, seeking to minimize the \(L_{2}\) norm of this residual, leading to the following quadratic programming problem:  
\[\underset{\mathbf{w}}{\text{minimize}}\ \ \frac{1}{2}\mathbf{w}^{\mathsf{T}} \bar{\mathbf{A}}\mathbf{w}-\mathbf{w}^{\mathsf{T}}\bar{\mathbf{b}} \tag{31}\]  
where \(\bar{\mathbf{A}}=\mathbf{A}^{\mathsf{T}}\mathbf{A}\) and \(\bar{\mathbf{b}}=\mathbf{A}^{\mathsf{T}}\mathbf{b}\). In Ref. [56], we discussed appropriate constraints for the set \(\mathbf{w}_{11}\) that are necessary in the presence of voids. Constraints for the fitting problem in enriched elements are formulated herein. From the evaluation of Equation (26), we have  
\[\mathbf{M}_{i}\approx\rho\bigg{[} \begin{matrix}w_{i,11}&w_{i,12}F_{i}\\ w_{i,12}F_{i}&w_{i,22}F_{i}^{2}\end{matrix} \bigg{]}, \tag{32}\]  
where \(F_{i}=F(\xi_{i})\). To ensure the positive definiteness of this block, its eigenvalues \(\lambda_{1,2}\) must be positive [110]. They can be computed as  
\[\lambda_{1,2}=\frac{\rho}{2}\Big{[}w_{i,11}+w_{i,22}F_{i}^{2}\mp\sqrt{(w_{i,2 2}F_{i}^{2}-w_{i,11})^{2}+4w_{i,12}^{2}F_{i}^{2}}\Big{]}. \tag{33}\]  
Due to their impact on stability and the properties of the mass matrix, we wish to formulate the constraints for Equation (31) in terms of a smallest admissible eigenvalue \(\lambda_{\text{min}}\in\mathbb{R}^{+}\). From Equation (33), the following condition for \(\lambda_{1,2}>\lambda_{\text{min}}\) is derived in Appendix A:  
\[w_{i,22}>\frac{\lambda_{\text{min}}^{2}-\lambda_{\text{min}}\rho w_{i,11}- \rho^{2}w_{i,12}^{2}F_{i}^{2}}{\rho F_{i}^{2}(\lambda_{\text{min}}-\rho w_{i, 11})}, \tag{34}\]  
where \(w_{i,11}\) is readily available, and \(w_{i,12}\) can be computed with Equation (29). The natural upper bound for \(\lambda_{\text{min}}\) is \(\lambda_{\text{min}}<\rho w_{i,11}\), thus, appropriate values can be computed as follows:  
\[\lambda_{\text{min}}=\epsilon_{\lambda}\ \rho\ \min_{0<i\leq n}\{w_{i,11}\}, \tag{35}\]  
where a tuning parameter \(\epsilon_{\lambda}\in[0,1)\), which will be established in Sections 3.6.1 and 3.6.2, is multiplied by the mass density and the smallest weight of the set \(\mathbf{w}_{11}\). In essence, Equations (30), (34) and (35) aim at a compromise between an advantageous mass matrix structure and an accurate evaluation of Equation (18) within the aforementioned restrictions. As long as the ordering of standard and enriched DOF related to a node is contiguous (see Appendix B), the mass matrix has a \(2\times 2\) block structure in enriched regions of the domain and is diagonal elsewhere. In linear elastodynamics, Equations (18)-(20) can be assembled once and then reused; thus, the computational cost overhead introduced by the moment-fitting procedure is negligible. These advantages, however, come with several caveats. This method is restricted to the use of one enrichment function per node, as the emergence of additional interaction coefficients in Equation (26) quickly complicates the formulation of constraints.