## 5 Numerical Experiments
### Idealized Compact Test Specimen  
To further test and demonstrate the capabilities of the proposed algorithm, we choose to model the well known experiment of a compact test specimen in tension, in order to simulate conditions under which Mode I failure would normally occur. However, unlike many experiments of this nature, the loading on the specimen is monotonic and highly dynamic (as opposed to standard fatigue tests with slow, cyclic loading). The geometry and initial meshes for the simulations are given in Figure 9, and the boundary conditions are  
\[\mathbf{s}_{\kappa}(X,t) =0 X\in\partial\mathcal{B}_{\kappa}\backslash\Gamma_{N} \tag{41}\] \[\mathbf{s}_{\kappa}(X,t) = \begin{cases}\frac{\mathbf{s}_{0}}{2}\left(1-\cos\frac{2\pi t}{ \tau}\right)&t<\frac{\tau}{2}\\ \mathbf{s}_{0}&t\geq\frac{\tau}{2}\end{cases}  X\in\Gamma_{N}, \tag{42}\]  
where \(\mathbf{s}_{0}=[0,s_{y}]^{T}\). The initial conditions are \(\mathbf{u}(X,t)=\mathbf{0}\), \(\mathbf{v}(X,t)=\mathbf{0}\), and \(\phi(X,t)=0.001\), and the material properties are listed in Table 1, with the exception that now we take \(\eta_{c}=0.05\,\mathrm{m}^{3}/\mathrm{J}\cdot\mathrm{s}\) and \(G_{cr}=1\times 10^{7}\,\mathrm{J}/\mathrm{m}^{3}\). For this specific loading condition and set of material properties (including \(h_{\text{min}}\)), we see in Figure 10 that both the damage and energy release rate based refinement schemes show similar results for the damage solution on both initial meshes, thus demonstrating that the solution is somewhat independent of the mesh. By this we mean that, while the two damage distributions are clearly not identical, they do concentrate on the horizontal symmetry line of the compact specimen, as expected. As it turns out, the differences between the solutions obtained using different initial grids can be traced to the  
Figure 9: Geometry and initial meshes for the simulations of an idealized 2D compact test specimen. Vertical traction loading is applied to the faces denoted as \(\Gamma_{N}\), creating a situation likely to result in Mode I failure. The meshes were created using the _pave_ (left) and _submap_ (right) methods in Cubit [17]. specific values used for the parameter \(h_{\text{min}}\) in relation to that of the parameter \(G_{cr}\). Before addressing this issue, we remark that for the results shown in Figure 10, the parameter in the energy release rate based refinement algorithm is \(c\!=\!0.5\), and as such, we notice that the corresponding solutions in Figures 10(c) and 10(d) have a slightly greater extent of refinement. However, comparing the damage solutions in Figures 10(a) and 10(b) with those in Figures 10(c) and 10(d), we see that the preemptive refinement of the energy release rate based algorithm yielded essentially the same damage solution as that produced by the refinement indicator based on the time rate of change of the damage. Therefore, we conclude again that in this instance, either scheme is adequate. Going back to discussing the mesh dependence issue, the most noticeable difference between the solutions on the grid in Figures 10(a) and 10(c) and the grid in Figures 10(b) and 10(d) is the degree of localization of the damage solution. In the first set of aforementioned figures, the damage is localized to a single cell, while in the second set the damage is clearly spread over numerous cells across the horizontal mid-line of the solution domain. Figures 11(a) and 11(b) show the region of interest in Figures 10(a) and 10(b), respectively. It is clear in these figures that, with the choice of piecewise constant elements, the damage solution seems to evolve in cells that are adjacent to the face of an already damaged cell, as opposed to one which is connected only at a corner. We discovered that this effect becomes more pronounced as we increase the value of \(G_{cr}\) while keeping the value of \(h_{\text{min}}\) fixed.