# Integration of Polynomials Times Double Step Function in Quadrilateral Domains for XFEM Analysis
## 2 Literature Review
### Numerical Methods in Fracture Mechanics
In XFEM, a discontinuous displacement field along the crack surface is introduced by simply adding more basis functions to the approximation. Additionally, when XFEM and level sets are used together, the geometry and displacement field of a crack can be built in terms of the original mesh nodal values. These benefits are especially significant when the geometry changes, as in the case of a developing crack [12]. Still, in the context of Galerkin-based methods, although computationally more expensive than standard FEM, mesh-free methods represent a solid formulation to deal with fracture mechanics and large deformation problems and provide an alternative approach to traditional mesh-based methods [44]. These methods, such as smoothed particle hydrodynamics (SPH) [45] and the reproducing kernel particle method (RKPM) [46], offer advantages in handling crack propagation and complex fracture behaviour without the need for a predefined mesh. Mesh-free methods utilise a set of scattered nodes to discretise the problem domain, allowing for adaptive refinement and efficient computation [40]. The efficient use of mesh-free methods in the context of crack propagation, branching and multiple cracks has been extensively discussed in [11; 22; 47; 48; 12]. Mesh-free methods have gained major attention lately; however, FEM is still a convenient method, and XFEM offers numerous advantages for problems involving discontinuities. Some of the aforementioned benefits of FEM/XFEM are:  
* FEM employs a structured mesh to discretise the domain into elements, resulting in efficient data representation, storage and manipulation and enabling the use of optimised algorithms and data structures to improve computational performance [49];
* FEM demonstrates excellent convergence properties. The accuracy of the solution improves as the mesh is refined. Convergence analysis plays a crucial role in assessing the reliability of numerical simulations [50];
* FEM generally provides higher accuracy for problems with smooth solutions. This advantage arises from the use of polynomial interpolation functions within each element, resulting in accurate approximations [51];
* XFEM is specifically designed to handle problems with discontinuities, and it can accurately represent the crack geometry without explicitly meshing the crack surfaces [8]. This eliminates the need for crack-tracking procedures that may be required in mesh-free methods [24; 52];
* XFEM provides a more stable solution and higher accuracy compared to many mesh-free methods in capturing stress and displacement fields near the crack tip by means of enrichment approximation functions [44];
* Mesh-free shape functions do not possess Kroneker delta properties; this results in a bigger computational effort compared to XFEM [44; 53];
* In XFEM, the crack geometry is implicitly represented within the finite elements, which reduces the dependency on the mesh density. This leads to a more efficient computational process and reduces the computational cost [12]. Mesh-free methods may require a large number of nodes or particles to accurately capture localised phenomena, such as cracks [44]. It is important to note that the advantages mentioned above are specific to certain aspects of FEM/XFEM over mesh-free methods and that the choice between them depends on the specific requirements and characteristics of the problem at hand.