# Cathode Shape Design for Steady-State Electrochemical Machining
## 3 Results and Discussion
### Cathode Shape Design for Anode Shapes with Limited Smoothness  
#### 3.2.1 Analytical Solutions  
For producing an anode whose surface profile is represented by a graph of a continuous but non-analytic function \(F\) on an interval \([0,L]\), the required shape of the cathode can be calculated using the method proposed by [PERSON] et al. [22]. In this method, \(F\) is thought to be continued periodically and approximated by Fourier polynomials  
\[F_{N}(x)=\frac{a_{0}}{2}+\sum_{n=1}^{N}\left(a_{n}\cos\frac{2\pi x}{L}+b_{n} \sin\frac{2\pi x}{L}\right) \tag{22}\]  
with Fourier coefficients \(a_{n}\) and \(b_{n}\). Then, the complex variable method is applied to \(F_{N}\), resulting in  
\[x_{\rm C}(\xi) = \xi+\sum_{n=1}^{N}\sinh\left(\frac{2n\pi}{L}\right)\times\] \[\times\left[a_{n}\sin\left(\frac{2n\pi}{L}\xi\right)-b_{n}\cos \left(\frac{2n\pi}{L}\xi\right)\right]\] \[z_{\rm C}(\xi) = 1+\frac{a_{0}}{2}+\sum_{n=1}^{N}\cosh\left(\frac{2n\pi}{L} \right)\times\] \[\left[a_{n}\cos\left(\frac{2n\pi}{L}\xi\right)+b_{n}\sin\left( \frac{2n\pi}{L}\xi\right)\right]\]  
as the parametric representation of the cathode. In the following example, the surface profile of the anode is defined by the continuous function  
\[F(x)=A\left(1-\frac{4|x|}{L}\right)\text{, for }|x|\leq\frac{L}{4} \tag{25}\]  
which has a sharp bend at \(x=0\) (see Figure 7) intended to model sharp edges or corners in reality. The approximating Fourier polynomial is given by  
\[F_{N}(x) = \frac{A}{4}+\sum_{n=1}^{N}a_{n}\cos\left(\frac{2n\pi}{L}x\right), \tag{26}\] \[a_{n} = \frac{4A}{(n\pi)^{2}}\left[1-\cos\left(\frac{n\pi}{2}\right) \right]. \tag{27}\]With Equations (23) and (24), the surface profile of the cathode required to produce the approximated anode with surface profile \(F_{N}\) is given by the parametric representation  
\[x_{\rm C}(\xi) = \xi+\sum_{n=1}^{N}c_{n}\sin\biggl{(}\frac{2n\pi}{L}\xi\biggr{)}, \tag{28}\] \[c_{n} = \frac{4A}{(n\pi)^{2}}\Bigl{[}1-\cos\Bigl{(}\frac{n\pi}{2}\Bigr{)} \Bigr{]}\sinh\biggl{(}\frac{2n\pi}{L}\biggr{)};\] \[x_{\rm C}(\xi) = \xi+\sum_{n=1}^{N}\sinh\left(\frac{2n\pi}{L}\right)\times\]0 (29) \[x_{\rm C}(\xi) = \xi+\sum_{n=1}^{N}\sinh\left(\frac{2n\pi}{L}\right)\times\]1  
Figure 7 shows examples in which the surface profile of the anode described by Equation (25) with \(L=40\) and \(A=5\) is approximated by Fourier polynomials (26) of degree \(N=10\), \(20\), and \(30\), respectively. The precision of the approximation increases with an increasing \(N\) since, for a continuous function \(F\), the function sequence \(\{F_{N}\}_{N\in\mathbb{N}}\) converges uniformly. The maximum deviation of the anode surface profiles \(F_{10}\) and \(F_{30}\) from each other, which is found at \(x=0\), is below \(0.15\); however, the maximum deviation of the corresponding cathode surface profiles from each other is larger than \(1.5\). Furthermore, with an increasing \(N\), the cathode shape becomes increasingly \"distorted\": The cathode surface profile changes from regular (\(N=10\), solid line) over singular (\(N=20\), dashed line) to physically meaningless (\(N=30\), dotted line). The size of the \"loops\" in the physically unrealizable case increases with a decreasing distance to the sharp bends of the anode surface at \(x=0\) and \(x=\pm 10\). The behavior of the cathode shapes shown in Figure 7 can be understood by analyzing the magnitude of the coefficients \(a_{n}\), \(c_{n}\), and \(d_{n}\) of the harmonic terms in Equations (26)-(29). For the approximate anode, the coefficients \(a_{n}\) decrease like \(1/n^{2}\). For the resulting cathodes, however, the coefficients \(c_{n}\) and \(d_{n}\) increase like \(\sinh(2n\pi/L)/n^{2}\) and \(\cosh(2n\pi/L)/n^{2}\). When the degree \(N\) of the Fourier polynomial increases, the approximation of the shape of the anode improves. However, the shape of the approximated cathode becomes physically meaningless for \(N\) larger than a certain limit, \(N_{\rm max}\).