# A Moment-Fitted Extended Spectral Cell Method for Structural Health Monitoring Applications
## 5 Conclusions  
When the SEM is complemented with PU enrichment functions in pursuit of fast simulations, high-oder accuracy, and relief from meshing issues, preserving and combining all these qualities hinges on the effectiveness of the mass lumping of enriched SEs. In this contribution, a novel method based on the moment-fitting framework and the nodal quadrature method was presented: while a block-diagonal mass matrix was produced by evaluating its integral at the element nodes, integration weights were delivered by a nonlinear moment-fitting procedure. This was intended to minimize the mass lumping error under the constraints of a positive definite mass matrix. Due to these constraints, the mass matrix was under-integrated, and the variational formulation was violated by a degree that was quantified in Figures 3-5. This approach offered some room to balance accuracy and stability thanks to Equation (34); however, some further research could be devoted to improving its stability (Figure 6b,c).  
The method yielded an accuracy that was comparable to that of the CMM formulation for several models, with convergence loss emerging only for relatively large models (\(5\times 10^{5}\) DOF in two dimensions) (Figure 8a-d). By alleviating stability issues via a leap-frog solver [18; 19; 56] and leveraging the fact that only a portion of the domain was cut or enriched, this approach resulted in efficient explicit dynamics simulations (Figure 8e-h). The XSCM applied in Section 4.2 stemmed from enhancing the SCM [51; 52; 53; 54; 55] with PU enrichment functions and enabled the automatic generation of complex damage configurations (Figure 9c) while delivering comparable results to the conforming SEM, thus saving meshing effort. These damage modeling capabilities were finally applied to composite delamination, where good agreement between the model and experimental data could be achieved.  
Figure 12: Time histories of velocities \(\dot{u}_{1-6}\) at the respective sensors \(\mathfrak{s}_{1-6}\), positioned according to Figure 11c.  
**Author Contributions:** Conceptualization, S.N. and K.A.; Methodology, S.N. and K.A.; Software, S.N.; Validation, S.N.; Resources, P.K. and E.C.; Data curation, P.K.; Writing--original draft, S.N.; Writing--review & editing, K.A., P.K. and E.C.; Supervision, K.A. and E.C.; Project administration, P.K. and E.C.; Funding acquisition, P.K. and E.C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was conducted as part of the project \"Fusion of Models and Data for Enriched Evaluation of Structural Health\" (2019/01/Y/STS/00060) financed by the National Science Center (Poland) and the Swiss National Science Foundation.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data will be made available upon request.  
**Acknowledgments:** The scanning laser Doppler vibrometer data were kindly provided by [PERSON]. The data were acquired during his research on the project entitled \"Phenomenon of conversion of the acoustic waves to elastic waves due to interactions with solids\" funded by the National Science Center, Poland (agreement no. 2018/31/B/ST8/01298).  
**Conflicts of Interest:** The authors declare no conflict of interest.