# Cathode Shape Design for Steady-State Electrochemical Machining
## 3 Results and Discussion
### Cathode Shape Design for Anode Shapes with Limited Smoothness
#### 3.2.2 Numerical Computations  
To illustrate the behavior of numerical computations of cathode shapes for anode shapes with limited smoothness, curves described by  
\[F(x)=A\exp\left(-\frac{2\sigma|x|+x^{2}}{2\sigma^{2}}\right)\quad A\
eq 0,\, \sigma>0, \tag{30}\]  
with a discontinuous first derivative at \(x=0\) were chosen as surface profiles of 2D anodes that have a sharp \"corner\". The shapes of cathodes corresponding to anodes with \(A=\pm 0.5\) and \(\sigma=10\) were computed by applying the continuous adjoint-based shape optimization method; the mesh cell size in \(x\) direction, \(\Delta x\), was varied from 0.1 to 0.5.  
Figure 8 shows numerically computed cathode shapes in the vicinity of a sharp corner of a convex (Panel (a)) or concave (Panel (b)) anode.  
To evaluate the plausibility of the numerical results, the limit of physically meaningful cathode shapes, which will be referred to as the \"analytical limit solution\", was computed by the Fourier approximation method of the previous section.  
Details of the interelectrode gap region near \(x=0\) are shown in Figure 8. All in all, the numerically computed cathode shape approaches the \"analytical limit solution\" when the mesh becomes finer. The \"analytical limit solution\" has two singularities at \(x=\pm 0.7\) when the anode is convex (Panel (a)) and one singularity at \(x=0\) when the anode is concave (Panel (b)). At the same positions, the numerically computed cathode surface profiles also have \"peaks\", which become increasingly sharp when \(\Delta x\) decreases. For \(\Delta x<0.05\) (not shown), the computation diverges since the cathode boundary starts forming loops at the locations of the \"peaks\", similar to the situation shown in Figure 6.  
The behavior of the results shown in Figure 8 bears some similarity to the behavior of the results obtained through the Fourier approximation method: with an increasing degree of the Fourier polynomial, the accuracy of the approximation of the anode improves, but the shape of the obtained cathode becomes less smooth and eventually physically meaningless (cf. Figure 7). In fact, both refining the mesh and increasing the degree of the Fourier polynomial involves an increase in the degree of freedom.  
The formation of \"peaks\" when using a fine mesh was also mentioned by [PERSON] and [PERSON] [49] in a related inverse problem associated with steady-state heat conduction. How  
Figure 8: Cathode shapes in the vicinity of a sharp corner of a convex (Panel (a), \(A=5\) in Equation (30)) or concave (Panel (b), \(A=-5\) in Equation (30)) anode. Spatial dimensions are normalized to the width of the standard equilibrium front gap. The quantity \(\Delta x\) denotes the mesh cell size in \(x\) direction used in the computation. The circles on the lines mark the positions of the mesh vertices. The upper solid line illustrates the limit of the physically realizable shape of the cathode.  
ever, the authors consider this phenomenon to be \"unrealistic\" and see its cause as due to the motion of the mesh in the numerical iteration process. As revealed by the comparison with the results obtained through the Fourier approximation method (see Figure 8), the formation of the \"peaks\" is not a numerical artifact but a consequence of the nature of the cathode shape design problem.