# A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications
## 3 Moment Fitting for Enriched Spectral Elements
### Mass Lumping for Standard Spectral Elements  
In addition to the construction of an effective Ansatz space, GLL nodes can also be used as evaluation points within the Lobatto, or Gauss-Lobatto, integration rule [103]. This is an important prerequisite of the nodal quadrature technique [23; 24; 25], which consists in applying an integration rule defined at the element nodes for the evaluation of Equation (18), thus enforcing the variational formulation whilst producing a diagonal mass matrix. In contrast with other mass lumping techniques, this approach has the great advantage of preserving the order of convergence of the SEM, provided that the quadrature is of sufficient accuracy. The effectiveness of the GLL configuration stems from the fact that, for an SE of degree \(p\), the order of the corresponding rule is \(2p-1\): two degrees less than the Gauss-Legendre (GL) rule (which is a ne plus ultra in one dimension [104]) but far better than what can be achieved with Newton-Cotes formulae [86; 103] relying on equally spaced nodes. Still, this approach leads to a slight under-integration of the stiffness and mass matrices, and thus to some loss of convergence in the computation of distorted elements [105]. For this reason, we employ a GL rule for the evaluation of Equation (19), while nodal quadrature is applied only to the mass matrix (Equation (18)) for the purpose of lumping. It has been shown in several studies [25; 86; 106] that this is effective in preservingthe convergence rates related to SE interpolation. To briefly review the method, let us denote  
\[\int_{\Omega_{\rm e,GLL}}f(\xi)\mathrm{d}\Omega_{\rm e,GLL} \tag{23}\]  
as the numerical evaluation of the integral of a function \(f\) over the GLL-SE domain \(\Omega_{\rm e}\) by means of a quadrature rule that employs its nodes. This technique exploits the fact that the shape functions are rendered orthogonal by the Kronecker delta property (i.e., \(N_{j}(\xi_{k})=\delta_{jk}\)), as long as their support nodes coincide with the quadrature nodes in evaluating the following integral:  
\[\int_{\Omega_{\rm e}}N_{j}(\xi)\ N_{k}(\xi)\ \mathrm{d}\Omega_{\rm e}\approx \int_{\Omega_{\rm e,GLL}}N_{j}(\xi)\ N_{k}(\xi)\ \mathrm{d}\Omega_{\rm e,GLL}=\delta_{jk}\ w_{j}\ \Big{|} \big{|}_{\rm e}(\xi_{j})\Big{|}, \tag{24}\]  
where the approximation sign (\(\approx\)) highlights the aforementioned slight under-integration of the product \(N_{j}^{2}(\xi)\), \(\xi_{j}\) are the reference nodal coordinates in the GLL configuration, \(w_{j}\) are the corresponding integration weights, and \(\big{|}J_{\rm e}\big{|}\) is the determinant of the element Jacobian. By combining the above with Equations (7) and (18), a diagonal mass matrix with the following coefficients is produced:  
\[m_{jk}\approx\sum_{i=1}^{n}w_{i}\,\rho(\xi_{i})\ N_{j}(\xi_{i})\ N_{k}(\xi_{i}) \ \big{|}J_{\rm e}(\xi_{i})\big{|}=\delta_{jk}\ w_{j}\,\rho(\xi_{j})\ \Big{|}J_{\rm e}(\xi_{j})\Big{|}. \tag{25}\]  
It is important to note that the application of this strategy is restricted to elements with a sufficiently smooth mass density and displacement field. When discontinuous integrands appear in Equation (18), customized quadrature rules must be employed (see Section 3.5), which lead to a fully populated mass matrix. When dealing with voids (\(\rho(\xi)=0\)), this can be overcome by additionally performing the diagonal scaling of this matrix [52; 88] or using moment-fitting equations to adapt the weights \(w_{i}\) to the element's cut configuration [56]. In addition, when enrichment functions are present, one is faced with the task of establishing appropriate diagonal mass coefficients for the enrichment parameters (see Equations (15) and (16)). For this, a procedure based on the conservation of kinetic energy has often been used [7; 8; 60]. This approach has also been combined with the Chebyshev version of the SEM [9]. In this contribution, we follow the indications of [PERSON] and [PERSON] [34] in choosing a GLL nodal distribution instead, which makes optimal lumping available for standard SEs.