# Integration of Polynomials Times Double Step Function in Quadrilateral Domains for XFEM Analysis
## 6 Conclusions  
The integration of polynomials times double step functions within quadrilateral domains has been explored in this paper. The study focused on investigating the behaviour and properties of these integrals and their applications in various fields, especially XFEM analysis. The integration of polynomials times double step functions within quadrilateral domains represents a challenging task, mainly due to the piecewise nature of the double step functions. However, an integration algorithm to effectively perform these integrations by means of equivalent polynomials has been developed and presented in this paper, along with a software library for its implementation. Both the library and the presented algorithm are ready-to-use handy tools for numerically computing integrals of polynomial functions across generic subdomains generated by two discontinuities partitioning a regular integration domain. In this paper, two practical numerical examples of the library usage have been carried out, and the results demonstrated the ease of calculation, the removal of complicated subdomain processing and the absolute accuracy of the proposed algorithm. Moreover, a comparison with other integration methods for the same numerical examples has been carried out and validated the proposed formulation. In addition, the utter generality and extensibility of the mathematical framework that underlies the library and the proposed formulation are among their key points. Furthermore, in addition to XFEM analysis in the context of fracture mechanics, the integration of polynomials times double step functions in quadrilateral domains exhibited promising applications in various fields, such as numerical simulations, computational geometry and engineering problems that involve modelling phenomena with sharp transitions or abrupt changes. The capability to accurately integrate such functions within quadrilateral domains without subdividing the integration domain could define pathways for more precise calculations in these contexts. Future research can further explore advanced numerical techniques and optimisation approaches to improve the efficiency and accuracy and enable the possibility of taking into account an arbitrary number of discontinuities. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/a16060290/s1](https://www.mdpi.com/article/10.3390/a16060290/s1)]([https://www.mdpi.com/article/10.3390/a16060290/s1](https://www.mdpi.com/article/10.3390/a16060290/s1)), The source code of the _DD_EQP_ Fortran library embedding the proposed integration algorithm, as well as the numerical examples used in this paper and software documentation, can be found at: equivalent-polynomials.net ([[https://sites.google.com/view/eqpols](https://sites.google.com/view/eqpols)]([https://sites.google.com/view/eqpols](https://sites.google.com/view/eqpols)) accessed on 1 March 2023). A maintained GitHub repository ([[https://github.com/sebfichera/DD_EQP_Library](https://github.com/sebfichera/DD_EQP_Library)]([https://github.com/sebfichera/DD_EQP_Library](https://github.com/sebfichera/DD_EQP_Library)) accessed on 1 March 2023) for the _DD_EQP_ library is also available. Conceptualisation, G.V. and M.C. ; methodology, M.C., G.M. and S.F. ; software, S.F. and G.M. ; validation, S.F., G.M. and M.C. ; formal analysis, S.F. and G.M. ; investigation, S.F. and G.M. ; resources, G.V. and M.C. ; data curation, S.F. and G.M. ; writing--original draft preparation, S.F., G.M., M.C. and G.V. ; writing--review and editing, S.F., G.M., M.C. and G.V. ; visualisation, S.F. and G.M. ; supervision, G.V. and M.C. All authors have read and agreed to the published version of the manuscript. Ministry of University and Research (MUR), research project XFAST-SIMS: Extra fast and accurate simulation of complex structural systems (Grant agreement n. 20173C478N) partially funded this research. **Data Availability Statement:** No new data were created or analysed in this study. Data sharing is not applicable to this article.