## 1 Introduction  
The need for lighter, stronger, and resilient structures across multiple engineering domains, e.g., the aerospace, automotive, and construction industries, necessitates a robust, economical and high-fidelity simulation of failure processes [1; 2; 3]. Failure in structural components subjected to static and/or dynamic loading is commonly associated with complex phenomena, i.e., crack nucleation, propagation, branching, merging and arrest [4; 5]. These phenomena emerge from micro-material discontinuities, which under the action of external stimuli accumulate to cracks and evolve across several length scales eventually leading to structural failure. From a computational standpoint, these physics of damage evolution have proven challenging to resolve. Over the past 20 years, the eXtended Finite Element method (XFEM), the Scaled Boundary Finite Element method (SBFEM) and most recently the Phase Field method (PFM), have emerged as distinct methodologies with the common objective of resolving fracture propagation. In this work, we provide a comparative platform for these methodologies pertinent to both the mathematical treatment of damage evolution and the corresponding algorithmic implications within the framework of Linear Elastic Fracture Mechanics (LEFM). LEFM methods describe damage initiation and propagation within the remit of brittle and quasi-brittle material response. LEFM has been traditionally treated within two distinct methodological frameworks, i.e., computational fracture mechanics (see, e.g., [6]) and continuum damage mechanics (see, e.g., [7]). In the former, damage is explicitly defined as a discrete topological discontinuity. In the latter, damage is effectively homogenised over a representative volume. Diffuse crack approaches effectively lie in the boundary of the two aforementioned methods. The need to predict damage related phenomena precisely, accurately, and economically within the context of LEFM has spurred research into an extensive suite of alternative methodologies. The finite element method (FEM), a representative of the discrete methods class, has reached a mature development status, effectively becoming the industry standard in numerical methods. Yet, challenges remain when characterizing singularities or propagation due to discrete cracks. This is a direct consequence of the following, select FEM shortcomings. The first four challenges primarily originate from the discretization method itself, while the remaining two pertain to difficulties associated with integration of LEFM into the discretization process:  
1. A conforming mesh topology is required to represent the associated crack. 2. The typical polynomial-based interpolation functions cannot reproduce the singular stress field. 3. Tracking crack paths and incorporating branching and merging behaviour is algorithmically challenging. 4. Mesh dependant projection errors arise within the context of nonlinear and dynamic analyses. 5. Nucleation, branching and merging of cracks cannot be treated in a uniform and theoretically sound manner. 6. Calculation of the stress intensity factors (SIFs) requires additional post-processing methods. Several techniques have been developed to tackle the aforementioned issues. First, sophisticated remeshing algorithms [8; 9; 10] and tools [11; 12] have been introduced to model the singular stress field. The utilization of special element types or the introduction of a fine mesh around crack tips contribute to tackling this challenge. Second, specially developed quarter-point elements [13], which are placed around the crack tip, to accurately capture the crack tip singularity. Third, diverse techniques have been proposed to determine the fracture parameters, such as the SIFs. This includes path-independent integrals [14; 15; 16; 17], the virtual crack closure technique [18; 19; 20], the hybrid-element approach [21; 22], and the Irwin's crack closure integral [23].