# Cathode Shape Design for Steady-State Electrochemical Machining
## 3 Results and Discussion  
### Cathode Shape Design for Smooth Anode Shapes  
#### 3.1.1 Analytical Solutions  
To illustrate the physical realizability of solutions to the cathode shape design problem, anode profiles in the shape of a smooth indentation are modeled by the graph of a Gaussian:  
\[z=F(x)=-A\exp\biggl{(}-\frac{x^{2}}{2\sigma^{2}}\biggr{)}\quad A>0. \tag{17}\]  
The complex variable method is applied to calculate the required shape of the cathode in the \(x\)-\(z\) plane, resulting in  
\[x_{\rm C}(\xi) = \xi-A\exp\biggl{(}\frac{1-\xi^{2}}{2\sigma^{2}}\biggr{)}\sin \biggl{(}\frac{\xi}{\sigma^{2}}\biggr{)}, \tag{18}\] \[z_{\rm C}(\xi) = 1-A\exp\biggl{(}\frac{1-\xi^{2}}{2\sigma^{2}}\biggr{)}\cos \biggl{(}\frac{\xi}{\sigma^{2}}\biggr{)}. \tag{19}\]  
Figure 3 illustrates surface profiles of anodes and the corresponding cathodes obtained for three combinations of the parameters \(A\) and \(\sigma\). While the shapes of the anodes are smooth in all three panels, the cathodes have qualitatively different shapes, respectively. In Panel (a), the cathode is regular and in line with ECM experience because the cathode is approximately the negative image of the anode. In Panel (c), the cathode surface profile has a \"loop\" at \(x=0\), which is physically not meaningful as the cathode cannot penetrate itself. The cathode shape in Panel (b) with a singularity at \(x=0\) seems to be the limit between physically meaningful (i.e., realizable) and meaningless cathodes.  
The qualitative behavior of the cathode shapes in Figure 3 can be understood by considering the derivative of \(x_{\rm C}(\xi)\) (Equation (18)) with respect to \(\xi\). For \(A<\sigma^{2}\exp\left[-1/\left(2\sigma^{2}\right)\right]\), \(\dot{x}_{\rm C}(\xi)\) is positive for all \(\xi\), i.e., \(x_{\rm C}(\xi)\) increases monotonously. For \(A>\sigma^{2}\exp\left[-1/\left(2\sigma^{2}\right)\right]\), \(\dot{x}_{\rm C}(\xi)\) is negative near \(\xi=0\), resulting in a physically meaningless loop of the cathode surface profile in Figure 3c. For given width \(\sigma\),  
\[A^{*}=\sigma^{2}\exp\biggl{(}-\frac{1}{2\sigma^{2}}\biggr{)} \tag{20}\]  
is the maximum amplitude of a Gaussian-shaped anode described by Equation (17), where an exact and physically realizable cathode shape exists.  
Figure 3: Cathode shapes (solid lines) required to produce Gaussian anode shapes (dashed lines) defined by Equation (17) with \(\sigma=4\) (spatial dimensions are normalized to the width of the standard equilibrium front gap). The solutions in parametric form are given by Equations (18) and (19). The scale on the \(z\)-axis and the legend of Panel (**a**) apply for all three panels. Panel (**a**) shows the example with \(A=8<A^{*}=15.51\), resulting in a regular cathode shape. Panel (**b**) shows the anode with \(A=A^{*}=15.51\) (see Equation (20) in the text), for which, the computed cathode is singular and represents the limit of physically realizable cathode shapes. Panel (**c**) shows a non-physical solution to a cathode shape obtained for a Gaussian anode with \(A=20\).  
In cases where the exact shape of the cathode obtained through the complex variable method has \"loops\" (see, e.g., Figure 3c), it is important to investigate if the numerical approach on the basis of shape optimization yields comparable solutions.