# Research on Multi-Physical Field Coupling Simulation of Local Electrochemical Machining
## 2 Materials and Methods  
### Principle Introduction  
Figure 2 is the schematic diagram of local electrochemical machining. During the machining process, the workpiece is stationary and only the cathode tool carries out vertical feed motion, and the initial convex structure of the workpiece is the pre-shaped convex structure by CRECM (Figure 2a). The side wall of the convex is an inverted cone, which does not meet the accuracy requirements of the target side wall. After local electrolytic machining, the final convex obtains a vertical convex side wall. From the sectional view (Figure 2b), it can be seen that the high-speed electrolyte passes through the narrow gap between the inner wall of the cathode tool and the insulating cavity, and then flows through the processing area. The external insulating layer of the cathode can prevent the stray corrosion of the cathode tool on the workpiece surface, and the internal insulating cavity can prevent the stray corrosion of the convex to ensure the final taper of the convex. Under the continuous feed of the cathode tool, the excess material on the side wall of the convex is dissolved by electrochemical corrosion and fed to the final position to obtain a convex that meets the target.  
In order to ensure that the contour near the root of the machining convex coincides with the contour of the target circle, the shape of the cathode end is designed according to the traditional cathode design method \"cos\(\theta\) method\". The principle is shown in Figure 3. The machining gap at angle \(\theta\) is:  
\[\Delta_{\theta}=\frac{\Delta_{b}}{\cos\theta} \tag{1}\]  
Figure 2: Principle of local electrochemical machining of pre-shaped convex structure: (**a**) overall view, (**b**) sectional view.  
where \(\Delta_{b}\) is the equilibrium gap along the feed direction.  
The width \(L\) of the cathode end is the key to affect the final machining accuracy. It can be seen from the local magnification of Figure 2 that the machining allowance at the root of the pre-shaped convex structure by CRECM is uneven, and it is difficult to determine the optimal width \(L\). Cathode ends that are too narrow or too wide will produce joint marks. Therefore, it is necessary to obtain the optimal cathode end width \(L\) through accurate simulation.  
The simulation is based on COMSOL Multiphysics software. The processing object is the pre-shaped rectangular convex structure by CRECM, which has the same shape in different cross sections, so it can be simplified into a 2D section model, mainly because the calculation amount of 3D model simulation is much larger than that of a 2D model. Figure 4 shows the machining area of the local electrochemical machining, and the width \(L\) of the cathode end is initially set at 3.3 mm, \(\Gamma_{1}\) is the initial anode workpiece surface and is also the pre-shaped convex contour by CRECM, \(\Gamma_{5}\Gamma_{9}\) is the surface of the cathode end, \(\Gamma_{4}\Gamma_{7}\Gamma_{10}\) is the insulating boundary through which no current passes, \(\Gamma_{6}\Gamma_{8}\) is the inlet of the electrolyte, and \(\Gamma_{2}\Gamma_{3}\Gamma_{11}\Gamma_{12}\) is the outlet of the electrolyte.  
Figure 4: Local electrochemical machining area.  
Figure 3: cos\(\theta\) method.