# Integration of Polynomials Times Double Step Function in Quadrilateral Domains for XFEM Analysis
## 5 Discussion  
The integration algorithm presented in this paper, as well as the _DD_EQP_ library implementing it, are helpful tools to evaluate the integral of multiple discontinuous functions by means of any numerical quadrature strategy, removing the need for dividing the domain of integration. The context for the proposed algorithm is mainly XFEM analysis of cracked bodies, in which scenarios involving crossing discontinuities are common [27; 28; 54; 78; 79; 80]. Moreover, as pointed out in Section 2, the existing methods for addressing multiple discontinuities problems make use of integration strategies in which partitioning of the domain is required. The proposed algorithm is a perfect fit in this context, delivering a solution for such an issue and eliminating the need for domain splitting. It can thus potentially be adopted in many XFEM practical applications in which double discontinuities crossing the domain occur, such as hydraulic fractures and multiple cracking in rocks [81; 82; 83; 84], brittle cracking and fracture propagation in brittle and quasi brittle materials [85; 86; 87], and also fracture behaviour in bones and bone-inspired bio-materials [88; 89], where crack branching, junction and kinking are frequent. Nonetheless, the algorithm can be extended to various domains, such as computational geometry, where it is possible to use the library to compute the geometrical characteristics of complex figures made by cutting a simple shape (i.e., a parallelogram) using multiple lines. The proposed formulation could fit problems addressed by various authors in this field, such as in [68; 72; 90; 91; 92; 93; 94]. The use of both the algorithm and the library can also be extended to simulations where the objects' shapes and locations vary dynamically, such as, for instance, an object that breaks into fragments or computer graphics in general [95; 96; 97; 98]. Thus, the effectiveness of the presented algorithm and the simplicity of the use of the library and its ease of implementation into any computational framework make them well-suited means for a broad variety of potential applications, since numerical integration of polynomial functions with jump discontinuities is a frequent problem in various fields [16; 55; 56; 99; 100; 101]. However, it has to be noted that the use of equivalent polynomials leads to a higher computational effort during quadrature than splitting the integration domain since the integrand function doubles its degree [13; 19]. This aspect can be particularly burdensome in analyses containing numerous enriched elements. The optimisation and acceleration of the computation process could be possible improvements.  
Future developments for the proposed formulation include the capability of handling an arbitrary number of discontinuities and the extension of the presented formulation to other domain shapes, both 2D and 3D (such as triangular, tetrahedral and hexahedral elements). Finally, the implementation of the proposed algorithm into computational frameworks for practical use on a larger scale in the context of fracture mechanics and computational geometry may also be addressed.