# A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications
## 3 Moment Fitting for Enriched Spectral Elements
### The Extended Finite Element Method  
In the extended or generalized FEM (XFEM/GFEM) [41; 90], the polynomial solution space of the underlying mesh is supplemented by additional _enrichment_ functions, thus enabling one to model the singular and/or discontinuous character ofdamage [41; 42] and material interfaces [45; 91]. Herein, only damage in the form of cracks and delaminations is considered; thus, enrichment is limited to discontinuous and singular functions. Within the XFEM framework, cracks are typically represented implicitly by means of two level-set functions:  
* The normal level set \(\phi\), representing the signed distance from the crack surface: \[\phi(\mathbf{x})=\min_{\mathbf{x}\in\Gamma_{c}}\|\mathbf{x}-\bar{\mathbf{x}} \|\text{sign}\big{(}\mathbf{n}^{+}\cdot(\mathbf{x}-\bar{\mathbf{x}})\big{)},\] (8) where \(\mathbf{n}^{+}\) is the outward normal to the crack surface and \(\text{sign}(\square)\) denotes the sign function. * The tangential level set \(\psi\), representing a signed distance function satisfying the conditions \[\
abla\phi\cdot\
abla\psi=0,\] (9) \[\phi(\mathbf{x})=0\] (10) where \(\Gamma_{f}\) represents the crack front/tips. Using these functions, a polar coordinate system, with its origin at the crack tip/front, can be defined: \[r=\sqrt{\phi^{2}+\psi^{2}},\qquad\theta=\arctan\biggl{(}\frac{\phi}{\psi} \biggr{)}.\] (11)  
It is common practice for level-set functions to only be evaluated at nodal points and FE shape functions to be used to interpolate these values in element interiors. Since this provides a relationship between the level sets/polar coordinates and the local element coordinates \(\xi\), in what follows, these terms will be used interchangeably. Then, enrichment functions can be defined using the level sets of polar coordinates. For cracks and delaminations, two types of enrichment are necessary: discontinuous functions to represent displacement jumps along crack/delamination faces; and asymptotic functions to represent the singularities at the crack tips/fronts. For the former, the Heaviside function is typically used:  
\[H(\phi)= \begin{cases}-1&\text{if }\phi<0\\ 1&\text{if }\phi\geq 0.\end{cases}  \tag{12}\]  
For the latter, a set of four [41] or twelve asymptotic functions [92] are typically used for strong discontinuities in homogeneous materials or bimaterial interfaces, respectively. Although there is a vast body of research that applies these principles in the study of composite delamination (e.g., [93; 94; 95; 96; 97]), the present contribution is concerned with modeling small areas of damage at minimal cost, rather than studying the phenomena of fracture mechanics in detail. Therefore, the asymptotic behavior in the immediate vicinity of the crack/delamination is not of interest. It has been shown [98] that acceptable accuracy can also be obtained using only the first singular enrichment function or even with enrichment functions that do not include singularities [99]. For these reasons, a single enrichment function is used herein, which also allows us to simplify the mass lumping process:  
\[F_{asy}(r,\theta)=\sqrt{r}\sin\left(\frac{\theta}{2}\right). \tag{13}\]  
To facilitate the assignment of different enrichment functions to nodes, the following nodal sets are defined:  
* is the set of nodes enriched with the discontinuous function (Equation (12)). This set includes all nodes belonging to elements that are split in two by cracks but not those that contain crack tips/fronts. * is the set of nodes enriched with the asymptotic function (Equation (13)). This set includes all nodes belonging to elements that contain crack tips/fronts. The above definition implies a topological enrichment scheme, as opposed to geometrical enrichment, where all nodes within a certain distance from the crack tips/fronts would be enriched [100].