## 7 Discussion and Conclusions
### Contrasting Discrete and PFM Crack Representation Approaches  
The merits of each method within the LEFM setting are discussed by contrasting key features and analysis steps. For the discrete methods, the representation of the crack is typically available in explicit form. Crack propagation analysis yields a polyline description of the crack topology. Since SBFEM employs polygon clipping, it does not require further information. XFEM, if chosen to employ an implicit enrichment representation, models the crack additionally by associated level sets. Crack path extraction is not necessary, since it is already given as a polyline. A crack consisting of a one-segment polyline is usually provided as input. For the PFM, the crack is represented by a scalar phase-field, with the phase-field variable directly embedded into the constitutive equations. The crack is represented as the region of fully degraded material with \(c=0\). Hence, no explicit crack representation is required during the analysis, albeit readily available in post-processing, if required. Initial defects are introduced in the system by specifying sets of points with corresponding phase field values. Meshing requirements for analysis by XFEM are largely decoupled due to the level set representation, yet substituted by more involved numerical integration procedures. This permits the use of a constant mesh during crack propagation analysis. This is contrary to analysis by SBFEM, where the initial quadtree decomposition, i.e., the mesh, is updated during each step incrementally. Discontinuities introduced by polygon clipping result in double nodes, such that the nDOF of the system increase gradually as the analysis proceeds. Furthermore, in select cases, clipping can result in non star-convex elements, which the method cannot treat. Delaunay triangulation of the element is required in such instances. Furthermore, due to clipping, elements with poor aspect ratios, in the conventional FEM sense, may arise. Empirically, this does not seem to be as severe an issue manifesting itself in erroneous numerical integration results, when employing SBFEM. In order to adequately represent the fracture process zone, the PFM requires a highly refined mesh in the regions of expected  
Figure 42: Steps comprising SBFEM analysis. crack propagation as well as at the crack tip, rendering the phase-field method computationally expensive for solving large-scale problems, especially when compared to discrete fracture approaches. It is now accepted that an element size \(h\leq l_{0}\) is required to accurately resolve the crack path. However, this computational burden is effectively addressed using parallel solvers, adaptive mesh refinement [229; 283], multiscale computation techniques [284] also within a local/global solution context [285]. The methods further differ in the hyper-parameters that ought to be specified by the analyst. XFEM requires the specification of crack tip enrichment type and radius, as well as the region where the interaction integral is to be calculated. Special care must be taken to exclude blending elements from the calculation of the SIFs, which may affect final results. SBFEM similarly requires the analyst to specify the homogenization region about the crack tip. With the exception of the cohesive phase field model, in the PFM implementations discussed in Section 5.2 the specification of the length scale regulates the response, imposing guidelines on mesh discretization and scaling of the critical fracture energy. As further discussed in Section 5.2, early efforts to treat this regulatory effect by introducing  
Figure 43: Steps comprising Phase field analysis. stress-based crack-driving forces have been reported in [256] whereas, and most notably, Wu and Nguyen [251] provided length-scale insensitive formulations that also preserve \(\Gamma\)-convergence. The solution process for both XFEM and SBFEM involves a single elastic solution step.