# Cathode Shape Design for Steady-State Electrochemical Machining
## 3 Results and Discussion
### Anode Shapes Produced by Regularized Cathodes  
Perimeter regularization is shown to be an admissible regularization technique and enables the computation of stable approximate solutions to the cathode shape design problem in the case where the desired shape of the anode is not given by an analytic function. However, every regularization technique modifies the original problem to a certain extent. Hence, the cathode computed using regularization, which will be abbreviated by \"regularized cathode\" in the following, usually produces an anode shape (in an ECM process) that deviates from the shape of the desired anode. In practical applications, this deviation should be quantified to assess whether the regularized cathode can be used in ECM to produce the desired anode shape within predefined tolerance levels. The following example illustrates the influence of the regularization parameter chosen for the cathode shape design procedure on the anode shape produced by the regularized cathode. The focus is on the rounding of originally sharp corners on the anode surface. The shapes of the desired anodes are similar to the study presented in Figure 7. Figure 13 exemplarily illustrates the results of the cathode shape design problem with \(\gamma=135^{\circ}\) for \(\alpha=0.01\) and \(0.2\). In the region \(|x|>2\), which is far from the tip of the corner, the cathode shapes obtained for both values of \(\alpha\) agree with each other. Large differences are observed in the vicinity of the tip of the corner: the widths of the front gaps at \(x=0\) differ from each other by \(0.5\), which is half of the width of the standard equilibrium front gap. Furthermore, the cathode shape obtained for \(\alpha=0.01\) has two \"peaks\" at \(\pm 1.5\), which are not observed in the case of \(\alpha=0.2\). The steady-state shapes of anodes obtained in an ECM process using the regularized cathodes were computed to verify the quality of the designed cathodes. As illustrated in Figure 13a, the anode shapes produced by regularized cathodes have a rounded corner, where the radius of the rounding is larger for \(\alpha=0.2\) than for \(\alpha=0.01\). The largest deviation of the resulting obtained shapes from the desired shapes of the anode is found at \(x=0\), which is quantified by \(\Delta A\), as indicated in the inset of Figure 13a. Regularized cathode shapes were computed for a series of anodes with values of the opening angle \(\gamma\) between \(90^{\circ}\) and \(150^{\circ}\) and values of the regularization parameter ranging from \(0.005\) to \(0.2\) (in the cases with \(\gamma\leq 120^{\circ}\), the simulation is unstable for \(\alpha<0.02\)). The quantity \(\Delta A\) was determined for all anodes produced by the computed regularized cathode shapes. As illustrated in Figure 13b, \(\Delta A\) increases with increasing values of \(\alpha\), indicating that the least regularized cathode (with sharp \"peaks\" on its surface) that can be computed  
Figure 13: Anode shapes produced by regularized cathodes (spatial dimensions are normalized to the width of the standard equilibrium front gap). (**a**) The prescribed anode (dash-dotted line) has an opening angle of \(\gamma=135^{\circ}\). The panel shows details of the anode around \(x=0\) and \(z=A\) with \(\Delta A\) exemplarily denoting the difference in the amplitude of the computed anode from the nominal value \(A\). (**b**) The values of \(\Delta A\) obtained from the numerical computation at different angles \(\gamma\) are illustrated by the symbols. stably would produce the anode with the most precise shape. This phenomenon is also observed in Figure 7, where the precision of the shape of the produced anode increases whereas the shape of the cathode \"deteriorates\" due to the formation of singularities and loops (cf. Figure 7). Furthermore, the value of \(\Delta A\) obtained for a given regularization parameter decreases with an increasing value of the angle \(\gamma\).