## 1 Introduction
The computational toll for such analyses is significant, with the majority of the effort stemming from the remeshing algorithm and the need for a fine mesh in the vicinity of the crack tip. Due to these limitations several novel numerical methods treating discrete cracks, such as meshless methods (MM), material point methods (MPM), boundary element methods (BEM), the extended/generalized finite element method (XFEM/GFEM), and the scaled boundary finite element method (SBFEM) have been applied, all distancing themselves from FEM in the way they define their support. MM [24; 25; 26] were conceived with the aim of eliminating difficulties associated with the reliance on a mesh. Hence, the interpolation in MMs is solely based on a set of distributed nodes, thus eliminating FEM issues commonly associated with mesh distortion and remeshing. Crack path extensions are effortlessly accounted for by introducing additional nodes. However, certain drawbacks remain. The MM shape functions require higher order integration and the treatment of essential boundary conditions is intricate, since the shape functions do not necessarily satisfy the Kronecker delta property. Generally, the computational toll of MMs results higher to that of the FEM [27]. MPM [28] is an extension to Particle-In-Cell methods [29], which efficiently treat history-dependent variables. In MPM, the continuum is represented by a set of material points that are moving within a non-deformable (Eulerian) computational grid where contrary to MM, solution of the governing equations is performed. Treatment of discrete cracks is accounted for by the introduction of multiple velocity fields [30] or more recently phase fields [31; 32; 33]. MPM has been found to offer significant computational advantages when compared to purely meshless methods since it does not require time-consuming neighbour searching. The BEM [34] solves initial value problems described as boundary integral equations hence reducing dimensionality by one. This significantly reduces the complexity of mesh generation, since only the boundary and the crack front need be discretised. Furthermore, compared to the FEM, BEM can often achieve greater accuracy, due to the nature of integrals used in the problem description. However, this is simultaneously the source of disadvantages. This formulation results in fully populated, dense matrices necessitating tailored numerical methods [35; 36] to efficiently solve the resulting discrete equations. The introduction of isogeometric analysis (IGA) [37; 38] suggests profound implications on practical engineering design. The key concept entails employing the Non-Uniform Rational B-spline (NURBS) not only for the geometric representation, but also for the discretization employed in the subsequent analysis. NURBS substitute standard FEM shape functions with the solution obtained on their support. A hybrid isogeometric boundary element method has been proposed [39; 40; 41] coupling many of the advantages of its parent methods. The direct adoption of the geometry representation as given by CAD software, greatly facilitates the integration of design and analysis, since no volume parametrization is required for crack propagation. Additionally, when applied to fracture [42; 43], the delivered higher-continuity can increase the accuracy of the stress field around the crack tip. An effective means of tackling the issues of mesh dependency and treatment of singularities, is provided by the extended/generalized [44; 45] finite element method (XFEM/GFEM), whose use is wide spread both in academia and industry. The most characteristic trait of this method is the use of partition of unity (PU) enrichment [46; 47; 48], to incorporate known features of the solution in the finite element approximation space through appropriate enrichment functions. For fracture mechanics problems, discontinuous and singular enrichment functions are employed locally, i.e., in the vicinity of the crack, to allow the representation of discrete cracks independently of the underlying mesh.