## 7 Discussion and Conclusions
### Contrasting Discrete and PFM Crack Representation Approaches
The PFM, as previously described in Section 7.3, comprises either monolithic or staggered approaches within an iterative solution scheme. In the quasi-static regime, displacement or generalised control solution procedures are typically employed. This however necessitates that either Equations (71) and (72) must be solved with very small time-increments (typically \(~{}10^{-5}\)-\(10^{-6}\)), or stagger iterations must be performed between both equations to ensure energy convergence. Often both of these options lead to high computational cost. Therefore, the corresponding load-deflection curve follows from the solution at every time step. In such quasi-static analyses, displacement controlled analysis automatically yields the load-deflection curve along with the softening branch. The discrete crack methods derive the load deflection curve in back-calculation. To this end, an arbitrary loading, e.g., force- or displacement-based, is applied. The resulting equivalent SIF is compared to the critical stress intensity factor. Hence, a scaling factor is derived for the loads and displacements at which crack propagation is initiated. This implies that recovery of the linear branch is a one-step process. Recovering an explicit linear elastic branch with the PFM requires either a linear phase field approximation as in Section 5.2.3 or cubic degradation functions [80]. Absence of these approaches will yield deviations from the linear elastic behaviour contingent on the evolution of material degradation in the process zone. Since the overall system stiffness is underestimated, the associated displacements are overestimated accordingly. In the PFM a crack is never explicitly propagated, but associated with the evolution of the phase field that emerges from the solution of the phase field governing equation. This is driven by the definition of the crack driving force as discussed in Section 5.2. Depending on the PFM formulation employed, the crack driving force can be established on the basis of either energy or limit-stress criteria. The discrete crack methods, within the LEFM framework, require the calculation of the crack propagation angle and some crack propagation increment. Examples of the later are either user specified or provided by Paris' equation. The crack is assumed to propagate in a straight line, originating from the crack tip determined in the previous analysis step. Hence, the history variables required are none other than the polyline for SBFEM, while XFEM propagates the associated level sets as well. PFM require updating the scalar phase field and specific realization of the PFM require further history variables to impose the crack-irreversibility condition, preventing the crack from healing during cyclic loading. The fact that the solution of the phase field governing equations emerge from an energy minimization problem, opposite to discrete fracture approaches, enables the resolution of crack initiation without the requirement for a crack path to be defined a priori. Furthermore, crack nucleation, growth and coalescence happen automatically; this results in a robust method with enormous flexibility to model complex cracking patterns including the simulation of curvilinear cracks, crack merging, and crack branching without the need for ad-hoc crack tracking methods. Finally, the method is naturally extended to 3D [229], considering also the case of fracture under multi-physics scenaria, e.g., temperature induced fracture [256; 286], hydraulic fracturing [81; 82; 84; 287; 288], and diffusion [289]. These advantages, render the phase-field approach a robust crack prediction method. Compared to discrete fracture approaches, the variational structure upon which the phase field theory emerges, equips it with significant capabilities for modelling diverse and complex fracture problems in a unified and consistent manner. The major advantage of the diffuse crack methods and the PFM specifically lies in their generality.