# Integration of Polynomials Times Double Step Function in Quadrilateral Domains for XFEM Analysis
## 2 Literature Review  
### Numerical Methods in Fracture Mechanics  
The development of numerical methods to address fracture mechanics problems has been tackled by various authors. The issue is not trivial since the FEM, being a piecewise differentiable polynomial approximation, is ill-suited for problems involving discontinuities (such as cracks) [29]. The use of FEM in such problems depends on an accurate and time-consuming discretisation process [29; 30], where the finite element mesh has to be carefully defined following the discontinuity interface in order to obtain exact results. Moreover, as the discontinuity evolves (i.e., crack growth problems), the mesh has to be regenerated at each step of the analysis to track the discontinuity growth path [30], leading to high computational costs. Methods such as the cohesive zone model (CZM) proposed by [PERSON] [31] and [PERSON] [32] and the boundary element method (BEM) [33] were a first step towards the development of effective numerical methods for fracture mechanics problems. The recent evolution of CZM, such as in [34; 35; 36], and advancements in BEM, such as the scaled boundary finite element method (SBFEM) [37; 38], proved their effectiveness in fracture mechanics, delivering accurate results. Along with these methods, partition of unity methods (PUM) and element-free Galerkin-based formulations were developed by various authors [39; 40]. PUM allows for the definition of solution spaces with local properties established by the user [5; 41]. The method has been developed by [PERSON] and [PERSON] [41] specifically to solve problems in which standard FEM fails or the results are excessively expensive. PUM relies mainly on two features:  
* Allowing the introduction of _a priori_ knowledge about the solution in the approximation space [41];
* Permitting the building of approximation spaces of any desired regularity [41]. These properties allow us to overcome FEM limits, such as the restriction of local approximation to polynomials and the poor approximation properties of polynomial functions for certain problems [41]. The PUM enrichment approximation field can be generally defined as  
\[u(x)=\sum_{I\in n}N_{I}(x)\mathbf{u}_{I}+\text{enriched terms} \tag{1}\]  
where \(n\) is the nodal point set and \(N_{I}(x)\) are the standard shape functions [29]. The PUM is the base of methods especially designed to tackle problems involving discontinuities, singularities, localised deformations and complex geometries such as GFEM and XFEM [12]. GFEM has been introduced by [PERSON], [PERSON] and [PERSON] [42], and it is an extension of the traditional FEM used in numerical analysis. It allows for the incorporation of additional enrichment functions or enrichment techniques to better capture localised phenomena, such as singularities, discontinuities or highly varying solutions. GFEM enhances the capabilities of FEM by enriching the basis functions used in the approximation, thereby improving the accuracy of the solution in regions of interest [43]. The elements incorporating enrichment functions have additional degrees of freedom associated with the enrichment, allowing for a more accurate representation of the solution. The enrichment functions are often defined based on the local behaviour of the solution. The main features of this method are the use of meshes that are partially or totally independent of the domain geometry and the approximation enrichment by special functions of interest [43]. XFEM, on the other hand, is a specific type of GFEM developed by [PERSON] et al. [7; 14] for engineering problems involving discontinuous and singular functions. XFEM introduces additional degrees of freedom and enrichment functions, such as the Heaviside step function, that can model the displacement field around discontinuities without the need for explicit meshing of the crack surfaces [12]. This makes XFEM particularly useful for fracture mechanics problems involving crack propagation or complex geometries.