## 6 Numerical Examples
### Numerical Example 5: Plate with Two Holes and Edge Cracks (PwHC)  
The case of the plate shown in Figure 37 is considered here. This numerical example is studied, since it poses challenges for both diffuse and discrete methods as discussed in [277]. The boundary conditions and material parameters are also shown in Figure 37 according to [9]. A state of plane strain is considered. The Young's modulus, Poisson's ratio, length scale, fracture energy density and crack propagation length are \(E=210\) kN/mm\({}^{2}\), \(v=0.3\), \(l_{0}=0.1\) mm, \(\mathcal{G}_{c}=1.0\) N/mm, \(h_{PFM}\approx 0.06\) mm, and \(\Delta a\leq 1\) mm, where applicable. Furthermore, for the phase-field analysis, a volumetric-deviatoric strain decomposition (similar to Amor et al. [264]) is employed. The bottom edge of the plate is clamped, while on the top edge a prescribed displacement is applied in the vertical direction and displacements in the horizontal direction are prohibited (\(u_{x}=0;u_{y}>0\)). The specimen thickness is \(t=1\) mm. Figure 37: PwHC geometry, material parameters, loading and boundary conditions. Figure 36: LSP comparison of load-displacement curves implementing the anisotropic spectral split vs. hybrid phase field models. In the presence of multiple cracks inside a domain, methods employing discrete crack representations typically implement a stability analysis [163] to ascertain the propagating cracks at each step. However, in this specific case, this involved procedure can be circumvented, due to the symmetric test setup. Nevertheless, the naive approach of simply running the analysis will result in an undesirable outcome, since slight numerical imbalances can result in asymmetric and erroneous results. To counteract these effects, symmetric meshes are employed in the XFEM analysis, while the SBFEM analysis enforces symmetric gSIFs about the diagonal. An average of the gSIFs is calculated to determine the crack propagation angle. Solving this example using the phase-field method produces interesting characteristics with respect to the crack initiation location and crack-paths. It is observed that when there is no restriction imposed on the crack from initiating near the holes, the phase-field initiates simultaneously and symmetrically at the top and bottom hole edges and then propagates almost horizontally as if no notches were present in the structure (Figure 38). However, when the crack evolution is restricted near the hole boundary, e.g., by imposing a very high \(G_{c}\) in the surrounding region, the crack initiates at both notch tips and propagates towards the hole edges simultaneously (Figure 39a). Further loading leads to evolution of multiple cracks initiating at the edges of holes which ultimately merge in the centre of the structure (Figure 39d). This observation is similar to what has been previously reported in [277]. However in the absence of experimental results for this problem, it is currently difficult to deduce which method predicts a realistic crack pattern. Hence, we refrain from reporting the typical load-deflection curves and focus only on the crack paths. Figure 38: PFM crack path without restricting nucleation at the holes. (**a**) Cracks initiating at the holes; (**b**) Growth of cracks originating from the holes; (**c**) Additional cracks nucleate at the holes; (**d**) Nucleated cracks reach the domain boundary. Since the crack paths derived from XFEM/SBFEM have been shown to coincide very well when employing similar discretization levels and crack propagation increments, modified mesh discretizations and crack propagation increments are sampled (Figure 40). The crack paths for all three variants align very well for the initial portion, while separating slightly as they approach the holes due to the crack propagation increment and mesh density variations. Figure 40: Crack path overlay for three variants: XFEM employing a fine mesh with \(\Delta a=0.25\) mm (pink), a coarse mesh with \(\Delta a=0.50\) mm (green) and SBFEM employing an adaptive mesh with \(\Delta a=1.00\) mm (orange).