# A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications
## 3 Moment Fitting for Enriched Spectral Elements  
### The Spectral Element Method  
While low-order standard Lagrangian FE schemes have seen widespread adoption due to their robustness and simplicity, Gibbs (or Runge) phenomena (see, e.g., [82] Chapter 4.2 and [83] Chapter 3) must be overcome as the polynomial order is increased in pursuit of more efficient discretizations. In this context, the SEM might be considered as a version of the FEM with more sophisticated choices of interpolating polynomials, which are produced by embedding additional properties in the determination of their supporting nodes. For instance, a Chebyshev nodal distribution [32; 84; 85] can be derived by minimizing the interpolation error. With two nodes fixed at the boundaries of the domain, internal Gauss-Lobatto-Legendre (GLL) nodes can be derived as optimal quadrature points in one dimension [17; 23]. It can also be shown that the node location corresponds with the maximum of the respective shape function, with a value of one [83; 86]. We refer the interested reader to [33; 83] for these derivations and their applications in the SEM. For reasons that are highlighted in Section 3.4, the GLL version of the SEM has been extensively applied in structural dynamics [25; 34; 87], including previous versions of the SCM [54; 56; 88], and is also adopted herein.  
Consider a generic one-dimensional GLL SE of order \(p>1\) with \(n=p+1\) nodes. While the two vertices delimit the element boundaries in the reference system \(\xi\in[-1,1]\), the remaining nodes are chosen as the roots of the Lobatto polynomial \(L_{p-1}\) of order \(p-1\). Thus, the element nodes can be defined as the solution of  
\[\Big{(}1-\xi^{2}\Big{)}L_{p-1}(\xi)=0, \tag{4}\]  
Figure 1: (a) Generic physical domain \(\Omega\) with geometrical details (e.g., river holes), damage (crack), and time-dependent surface excitations. (b) SE model consisting of a structured mesh and implicit descriptions of the void (\(\Gamma_{\text{c}}\)), crack (\(\Gamma_{\text{cr}}\)), and loading surfaces (\(\Gamma_{\text{cs}}\)).  
which is known as the completed Lobatto polynomial [89]. Then, shape functions can be constructed as Lagrangian interpolations supported at these nodes:  
\[N_{i}(\xi)=\prod_{j=1,j\
eq i}^{p+1}\frac{\xi-\xi_{j}^{p}}{\xi_{i}^{p}-\xi_{j}^{ p}}, \tag{5}\]  
where \(i\) is the node number. The local coordinates and shape functions for quadrilateral and hexahedral elements can be generated by applying Equations (4) and (5) in each dimension of the reference system and taking the sparse products of the respective results. For a generic element in \(d\) dimensions, the unknown displacement field \(\mathbf{u}(\xi,t)\) at a given time \(t\) is interpolated from its nodal values \(\mathbf{u}_{i}(t)\), which are collected in the vector \(\mathbf{u}_{\mathbf{e}}(t)\):  
\[\mathbf{u}(\xi,t)=\sum_{i=1}^{n}N_{i}(\xi)\ \mathbf{u}_{i}(t)=\mathbf{N}(\xi) \ \mathbf{u}_{\mathbf{e}}(t), \tag{6}\]  
with  
\[\mathbf{N}(\xi)=\begin{bmatrix}N_{1}\mathbb{I}_{d}&N_{2}\mathbb{I}_{d}&\dots&N _{n}\mathbb{I}_{d}\end{bmatrix} \tag{7}\]  
and \(\mathbb{I}_{d}\) being the identity matrix.