# Cathode Shape Design for Steady-State Electrochemical Machining
## 3 Results and Discussion
### Cathode Shape Design for Smooth Anode Shapes
#### 3.1.2 Numerical Computations  
The numerical simulation tool for the cathode shape design was applied to the anode shape defined by Equation (17) with \(\sigma=4\) and several values of \(A\). Figure 4 exemplarily illustrates the results for an anode with \(A=8\) (black dashed line in Panel (a)), for which, a regular shape of the cathode was obtained in the analytical calculation (cf. Figure 3a). For the iterative computation based on continuous adjoint shape optimization, the offset curve of the anode with a normal distance of 1 (dotted black line in Figure 4a) is assumed to be the surface profile of the initial cathode. The computational domain was discretized by a \(200\times 20\) mesh; the computation was carried out for 200 iteration steps with constant step length \(\tau=0.4\) (see Equation (12)). The surface profile of the cathode obtained after 200 iterations is shown by the solid line in Figure 4a, which overlaps with the analytical solution given by Equations (18) and (19) (dash dotted line) on the scale of the figure. For clearer visualization, the absolute deviation \(\Delta\) between these solutions, measured in surface normal direction, was computed and is shown in Figure 4b. The maximum of \(\Delta\), denoted by \(\Delta_{\max}\), is less than 0.02, i.e., 2% of the width of the standard steady-state front gap. The results of the simulation were also assessed in terms of the mesh dependence of its accuracy. The surface profiles of the anode and the initial cathode remain the same as those shown in Figure 4. The maximum normal deviation (\(\Delta_{\max}\)) of the numerically obtained cathode shapes from the analytical solution (Equations (18) and (19) was computed in each case. The computational domain was discretized by \(N_{x}\times N_{z}\) meshes. For \(N_{z}=20\) and \(N_{x}\) varying between 200 and 800, no significant changes in \(\Delta_{\max}\) are observed (not displayed). Figure 5 shows the quantity \(\Delta_{\max}\) as a function of \(N_{z}\) that ranges from 5 to 160 while \(N_{x}=200\) is kept fixed. As evident from the figure, \(\Delta_{\max}\) is inversely proportional to \(N_{z}\), indicating an improvement in accuracy due to mesh refinement in the case where the shape of the cathode is regular. Figure 4: Two-dimensional cathode shape design example. In Panel (**a**), the black dashed line shows the anode shape defined by Equation (17) with \(A=8\) and \(\sigma=4\); the geometry of the cathode at the beginning of the tool design iteration is represented by the dotted black line obtained from equidistantly offsetting the anode geometry by 1; the cathode shape obtained from iterative optimization (solid line) overlaps with the analytical solution (dash-dotted line) obtained from Equations (18) and (19). Panel (**b**) shows the deviation of the numerical result from the analytical solution in surface normal direction. As an example of a case in which a non-physical shape of the cathode was obtained in the analytical calculation (cf. Figure 3c), the numerical computation procedure was applied to an anode shape with \(A=20\) and \(\sigma=4\). During the iterative optimization procedure, the cathode boundary forms a loop before the computation diverges due to errors resulting from the degenerated mesh. The solid curve in Figure 6a illustrates the cathode surface profile in the vicinity of the vertex in the last iteration before the computation diverges. Figure 6b shows the evolution of the convergence indicator \(\delta_{\max}\) (Equation (14)). The loop of the numerically computed cathode surface profile agrees qualitatively with the analytical solution. Quantitatively, the numerically and analytically computed cathode shapes cannot be compared with each other since the numerical computation is stopped before convergence.