# A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications
## 1 Introduction
Despite the effectiveness of high-order methods, when modeling damage, holes, or material interfaces, one is faced with the limitations of meshing software and the unsuitability of polynomial shape functions in approximating phenomena of a singular and/or discontinuous nature. Extended or generalized FE methods (XFEMs/GFEMs) [41; 42; 43; 44] have enabled to overcome these issues by deploying mesh-independent geometrical descriptions [45] of these features, which are then used to locally supplement the solution space with partition-of-unity (PU) enrichment functions [46; 47; 48] and to generate appropriate quadrature rules for the evaluation of the weak form (e.g., [49; 50]). In the presence of voids, the spectral cell method (SCM) [51; 52; 53; 54; 55; 56] combines this kind of integration with a mass lumping procedure and a GLL spectral element (SE) mesh in order to perform dynamic analysis explicitly. On the other hand, the adoption of an XFEM in this context has been mainly driven by the study of dynamic crack propagation, which at first involved the use of IMEX [57] or fully implicit [58] solvers. This was due to the absence of mass lumping strategies for enriched elements and the near-zero mass coefficients [59] of nodes in the proximity of discontinuities. To overcome this, [PERSON] et al. developed customized mass lumping routines for Heaviside enriched elements in 2D [7] and 3D [8] and demonstrated their excellent stability properties. This formulation, which assumed the conservation of discrete kinetic energy for rigid-body modes, was then generalized to arbitrary enrichment functions [60] and combined with the Chebyshev version of the SEM [9]. [PERSON] et al. [22] explored the possibility of enforcing energy conservation using only the standard mass coefficients; however, this necessitated the adoption of a node-based IMEX solver [61; 62] to handle the resulting zero-valued enriched masses. It should be noted that these methods generally lead to a loss of convergence, as they introduce a customized expression of the mass matrix. In order to preserve optimal convergence rates, [PERSON] and [PERSON] [63] and [PERSON] et al. [64] applied [PERSON]'s variationally consistent lumping method [65], which could deliver block-diagonal mass matrices for non-negative shape functions. In this contribution, we aimed to provide an efficient high-order method for explicit dynamic analysis in the presence of damage. Our main strategy consisted of extending the SCM with local PU enrichment functions in an attempt to combine the qualities of both approaches. Our discretization was similar to that of [9], with the difference that a GLL grid was employed in order to achieve the optimal lumping of standard SEs. Variationally consistent lumping via [65] of enriched SEs cannot be performed, since SE shape functions can assume negative values. Therefore, we developed a novel mass lumping method for enriched elements, which can be summarized as a nodal quadrature with customized weights. While integration at the element nodes delivered a (block-)diagonal mass matrix by construction, the weights were tailored to the enrichment functions and cut configuration of an element via moment-fitting equations. This setup is an established and versatile framework to generate quadrature rules. It can be used to integrate general polygons [66] and handle singular and discontinuous integrands [67; 68; 69; 70]. To decrease the cost of the procedure, node locations could be chosen a priori [67; 69; 71; 72]; however, one can also optimize them for accuracy [66; 68; 73]. In fictitious domain methods, moment fitting enables a drastic reduction in the number of integration points, with important performance implications [74; 75], particularly in nonlinear applications [72; 76]. In these methods, the positiveness of the resulting weights is a prerequisite for the stability of the models. For this reason, recent contributions have focused on non-negative moment fitting [77; 78].