## 5 Numerical Experiments  
In this section we investigate the performance of the proposed algorithms. A simple example consisting of a two-dimensional bar fixed at one end and subjected to an applied load at the other is used to gain confidence that the proposed algorithms are functioning as desired. A more complicated example, consisting of a two-dimensional idealization of a compact test specimen is used to evaluate how well the algorithms address the use of different meshes, by visually comparing the resulting damage fields on each triangulation. The geometry of a two-dimensional bar fixed at one end and subjected to a Neumann (traction) boundary condition at the other is given in Figure 3.  
**Figure 3**. Geometry and boundary conditions of a two-dimensional bar fixed at \(X_{1}=0\) and subject to an applied traction at \(X_{1}=L\). The essential boundary conditions are \(\mathbf{u}|_{X_{1}=0}=\mathbf{0}\) and \(\mathbf{v}|_{X_{1}=0}=\mathbf{0}\), and the applied traction force is defined as the smooth function  
\[\mathbf{s}_{\kappa}(X,t)|_{X_{1}=L}= \begin{cases}\frac{\mathbf{s}_{0}}{2}\left( 1-\cos\frac{2\pi t}{\tau}\right)&t<\frac{\tau}{2}\\ \mathbf{s}_{0}&t\geq\frac{\tau}{2}\end{cases} , \tag{40}\]  
where \(\mathbf{s}_{0}=[s_{x},0]^{T}\). The initial conditions are \(\mathbf{u}(X,t)=\mathbf{0}\), \(\mathbf{v}(X,t)=\mathbf{0}\), and \(\phi(X,t)=0.001\). This simple initial boundary value problem corresponds to a two-dimensional bar which is clamped at one end, and initially at rest with uniform damage, being smoothly loaded up to a particular load at which time the load is held constant. Thus, a wave will propagate along the bar, eventually striking the fixed end. The particular material properties and parameters for the following simulations are given in Table 1, and have been chosen to be representative of a brittle material. We see in Figure 4 that the mesh refinement is localized to the fixed (left) end of the bar, indicating that this is where the damage is increasing. This is expected, because as the wave strikes the fixed end \begin{table}
\begin{tabular}{c c c c} \hline \hline Elastic Modulus & 9 \(\times\) 10\({}^{9}\,\mathrm{Pa}\) & Density & 1.7 \(\times\) 10\({}^{3}\,\mathrm{kg/m^{3}}\) \\ Poisson Ratio & 0.3 & \(G_{cr}\) & 3 \(\times\) 10\({}^{7}\,\mathrm{J/m^{3}}\) \\ \(\eta_{c}\) & 0.1\(\,\mathrm{m^{3}/J\cdot s}\) & \(\beta\) & 2 \\ \(\tau\) & 20\(\,\mathrm{\mu s}\) & \(s_{x}\) & 5 \(\times\) 10\({}^{7}\,\mathrm{Pa}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties representative of a brittle material, and the loading parameters. Figure 3: Geometry and boundary conditions of a two-dimensional bar fixed at \(X_{1}=0\) and subject to an applied traction at \(X_{1}=L\). **Figure 4**. Damage-based refinement scheme: from top to bottom, triangulations of the two-dimensional bar at times \(t=0\), \(t=\)88.5 \(\mu\)s, \(t=\)106.1 \(\mu\)s, and \(t=\) 115.9 \(\mu\)s. The refinement is concentrated at the fixed end of the bar, which is where we expect the damage to increase. of the bar, we expect from the theory of linear elasticity to have very high strains at the corners of the domain, and thus an energy release rate that is higher than other locations in the domain. We plot the values of the damage variable in Figure 5, and see that the damage nucleates at the corners of the domain, and subsequently, as the corner cells become fully damaged we see the adjacent cell become damaged, and so on. The damage-based refinement scheme only allows for damage evolution at the smallest mesh size. We can see this in Figure 5, where the piecewise constant damage is clearly localized to an individual cell at the smallest refinement level. We see that there is no damage evolution in larger cells, as desired. We now want to compare the performance of the damage based and energy release rate based error estimation schemes. **Figure 5**.