## 1 Introduction
This in turn significantly decreases or even removes the remeshing burden, while also increasing the accuracy with which asymptotic fields are represented. Alternatively, the scaled boundary finite element method (SBFEM) [49] naturally incorporates the singular stress field, providing an elegant extraction of the generalized stress intensity factors (gSIFs) in post-processing at negligible additional computational cost [50]. This is a consequence of SBFEM retaining an analytical solution in the radial direction, while only requiring discretization along the tangential boundary in the standard FEM sense. However, double nodes are introduced to accommodate strong discontinuities. This is partially mitigated due to the polytope nature of SBFEM, which only imposes the condition of star-convexity on elements. Exploiting balanced quadtrees as hierarchical meshes in conjunction with polygon clipping the majority of meshing effort is circumvented [51]. XFEM and SBFEM receive in-depth treatment in Sections 3 and 4. Alternative discrete fracture methods based on cohesive theories have been utilized to overcome stress singularities in LEFM also accounting for the nonlinear separation phenomena [6]. Barenblatt [52] originally introduced the cohesive zone method (CZM) to model fracture in brittle materials. Later, Dugdale [53] extended the CZM to simulate the plastic fracture process zone around the crack tips. In cohesive fracture theory, the material is not considered perfectly brittle as in Griffith's theory. Rather, there is a small zone in front of the crack that can exhibit some ductility. The fracture energy is gradually released at the crack tip based on the crack opening and equals the critical fracture energy at full crack opening. If the cohesive zone is sufficiently small, the ductility zone becomes unimportant and the theory of LEFM can be applied. The CZM has been employed within a FEM, see, e.g., [54; 55] and a BEM, see, e.g., [56] setting, also in conjunction with a partition of unity approach [47]. Furthermore, CZM has been introduced within a particle based approach as in the case of SPH [57], reproducing kernel particles [58], and the Element-Free Galerkin method [59]. A popular partition of unity approach and a reasonable extension of the CZM is the Cohesive Segments Method (CSM) [60]. The CSM introduces arbitrary cohesive segments within the finite elements that act as discontinuities in the displacement field hence alleviating the CZM requirement for the definition of cohesive elements at the finite element interface. In CSM, the cracks are modelled as a set of overlapping cohesive segments with their support nodes being enriched with jump and tip enrichment functions similar to the XFEM. A combination of overlapping crack cohesive segments results in a continuous crack. Remmers et al. [60] originally applied the CSM in quasi-static brittle fracture problems mainly focused on mode I separation problems and further extended the method for the simulation of dynamic crack propagation problems [61]. Using the CSM as point of departure,various PUM with cohesive theories have been successfully introduced with a meshless discritization approach, see, e.g., [62; 63]. Rather than attempting to model the actual, discrete crack topology, either as a strong discontinuity in the displacement field (e.g., XFEM) or an explicitly defined boundary (e.g., SBFEM), diffuse approximations of cracks incorporate the effects associated with the crack formation, e.g., the stress release or the stiffness degradation into the constitutive model [64]. Such approaches initiated with the pioneering work of Rashid [65], who defined a cracking criterion for pre-stressed concrete pressure vessels on the basis of loss of material stiffness in the direction normal to a crack as this evolves. In the past 10 years, several methodologies pertinent to diffuse crack models emerged, such as gradient enhanced damage methods [66; 67], Thick Level Set methods [68], and Phase field methods [69].