# Cathode Shape Design for Steady-State Electrochemical Machining
## 3 Results and Discussion
### Perimeter Regularization for Cathode Shape Design
The example shown in Figure 9a was recalculated using the perimeter regularization technique with \(\alpha\) ranging from 0.0002 to 0.6. To illustrate the effect of regularization, the region around \(x=0\) of two numerically computed cathode shapes is shown in Figure 11a and compared with the exact shape of the cathode. Whereas, for \(\alpha=0.0005\), the cathode surface exhibits \"peaks\" similar to the unregularized solution (cf. Figure 9a), the vertex at \(x=0\) of the cathode obtained for \(\alpha=0.5\) is truncated compared with the analytical solution. To quantify the overall difference between the analytical and numerical solutions, the mean normal deviation  
\[\Delta_{\mathrm{rms,cathode}}=\left(\frac{1}{|\Gamma_{C}|}\int_{\Gamma_{C}} \Delta^{2}\mathrm{d}S\right)^{1/2} \tag{35}\]  
of the numerically computed cathode (required to produce the spline-interpolated anode) from the exact cathode (required to produce the anode defined by the analytic function) is plotted in Figure 11b as a function of the regularization parameter, \(\alpha\). With an increasing \(\alpha\), \(\Delta_{\mathrm{rms,cathode}}\) first decreases, then reaches a minimum at an optimum regularization parameter of \(\alpha_{\mathrm{opt}}=0.12\) and increases for \(\alpha>0.12\). Figure 10: Smoothing effect of the regularization term on the boundary motion in a convex (Panel **(a)**) or concave (Panel **(b)**) region of the cathode. The function \(\Delta_{\text{rms,cathode}}(\alpha)\) behaves in a way that is commonly observed in the regularization of ill-posed problems: for \(\alpha\to 0\), the \"error\" (\(\Delta_{\text{rms,cathode}}\)) increases since the instability of the ill-posed inverse problem dominates; for large \(\alpha\), the approximation of the original objective function (6) by the regularized objective function (32) becomes less accurate and \(\Delta_{\text{rms,cathode}}\) is dominated by the approximation error. In a further study, the number of the control points for the spline interpolation of the anode shape is varied. A higher number of equidistantly distributed control points corresponds to a decrease in the mean normal deviation  
\[\Delta_{\text{rms,anode}}=\left(\frac{1}{|\Gamma_{\text{A}}|}\int_{\Gamma_{ \text{A}}}\Delta^{2}\,\text{d}S\right)^{1/2}\]  
of the anode, or a reduction in the \"perturbation\" of the given anode data. As shown in Figure 12a, the value of \(\alpha_{\text{opt}}\) approaches zero when \(\Delta_{\text{rms,anode}}\to 0\). Likewise, the quantity \(\Delta_{\text{rms,cathode}}(\alpha_{\text{opt}})\) approaches zero when \(\Delta_{\text{rms,anode}}\to 0\) (Figure 12b). Both results indicate the _admissibility_ of the perimeter regularization method (cf. [55] Definition 2.3). Figure 11: Influence of the regularization parameter \(\alpha\) on the computed cathode shape (spatial dimensions are normalized to the width of the standard equilibrium front gap). The prescribed anode is the cubic spline connecting 13 control points shown in Figure 9. Panel (**a**) shows the region around the vertex of the cathode (cf. Figure 9a). Panel (**b**) shows the mean normal deviation of the numerically computed cathode from the analytical solution (see Equation (35)) as a function of \(\alpha\). Figure 12: Optimum regularization parameter (Panel (**a**)) and error in cathode shape (Panel (**b**)) as a function of the error in anode shape. The deviation of the prescribed anode from the ideal anode is varied by varying the number of control points in spline interpolation (cf. Figure 9a).