# Cathode Shape Design for Steady-State Electrochemical Machining
## 1 Introduction  
_Electrochemical machining_ (ECM) is a non-conventional machining technique. Its principle is shown in Figure 1. Since it is contactless, ECM can be applied in the manufacturing of complex-shaped parts made of high-strength materials without introducing residual stresses into the workpiece and without tool wear. Today, applications of ECM include the production of turbine disks and blades in aerospace industry [1, 2], manufacturing of fuel injection devices in automotive industry [3], production of gun barrels in weapon industry [4], fabrication of medical implants [5], or shaping three-dimensional (3D) structures with submicrometer precision [6]. An overview of the recent developments of ECM is presented in review articles [2, 7]. One of the main disadvantages of ECM is the long production preparation cycle, especially for the tool cathode that is required for producing workpieces of complex shapes. In this context, two problems exist: the _direct-_ or _anode shape prediction problem_ of ECM refers to anticipating the evolution of the anode shape in a defined standard ECM process with a given set of process parameters and a known shape of the cathode. In industrial applications, however, the known information is often the desired shape of the workpiece, resulting in a problem that is referred to as the _inverse-_ or _cathode shape design problem_ of ECM. Regarding the solution of the direct problem, the main focus is--apart from the prediction of the anode shape--on modeling different interdependent physical and chemical processes within the interelectrode gap during machining. The results of numerous studies are summarized in a comprehensive review [9]. In recent years, the focus has been turned to verifying multiphysics models with experiments [10], or taking further processes such as short pulses [11], electrochemical reaction kinetics [12], or the mechanical oscillation of the cathode [13] into account. Recently, machine learning techniques have been applied to predict workpiece profiles after ECM [14] or the microstructure and mechanical properties of materials [15; 16; 17]. Due to the relevance for industrial applications, numerous researchers have developed algorithms for solving the inverse or cathode shape design problem. These methods can be classified into the following groups:  
* _Approximative analytical methods_: [PERSON] [18] proposed the so-called _cos \(\theta\)-method_, assuming the width of the interelectrode gap to be inversely proportional to \(\cos\theta\), where \(\theta\) is the angle between the normal of the local anode surface and the cathode feed direction (see Figure 1b). However, this method is only applicable in gap regions where \(\theta\) is less than \(45^{\circ}\)[19]. * _Exact analytical methods_: The complex variable method applied by [PERSON] [20], [PERSON] and [PERSON] [21], and extended by [PERSON] et al. [22] makes use of the harmonic property of holomorphic functions and the principle of analytic continuation. However, due  
Figure 1: Principle of electrochemical machining (ECM). (**a**) Initial configuration consisting of a cathode tool of a defined shape and a plane anode workpiece to be machined. The tool is advanced towards the workpiece at a defined feed rate. Electrical current flows when a voltage is applied across the gap; the thin red lines between the electrodes illustrate the current density. (**b**) Steady-state configuration of tool and workpiece. The final shape of the workpiece is approximately the negative image of the tool electrode. The inter-electrode gap is the calculation domain \(\hat{\Omega}\) of the model used in this work. The boundaries \(\hat{\Gamma}_{\rm C}\) and \(\hat{\Gamma}_{\rm A}\) are defined by tool (cathode) and workpiece (anode) surfaces, respectively, and \(\hat{\Gamma}_{\rm N}\) is the non-metallic side boundary through which the electrical current flux is zero.