# Cathode Shape Design for Steady-State Electrochemical Machining
## 1 Introduction
For each point on the workpiece boundary \(\hat{\Gamma}_{\rm A}\), \(\theta\) is defined as the angle between the outer boundary unit normal vector \(\underline{\mu}\) and the feed direction unit vector \(\underline{\epsilon}_{\rm H}\). “Front gap” refers to an inter-electrode gap region where the workpiece surface is perpendicular to the feed direction (i.e. \(\theta=0\)). “Standard front gap” (with steady-state width \(\tilde{h}_{\rm e}^{\rm std}\)) is a front gap region where the electrodes are plane (figure adapted from Ref. [8]). to the number of degrees of freedom in the complex plane, this method is inherently limited to 2D applications. * _Numerically solving an \"initial value problem\"_: Some authors solve the cathode shape design problem either via computing a solution to a system of ordinary differential equations [23] or by imposing both anode boundary conditions on the finite element discretized equation [24; 25; 26]. These methods, especially the latter, require a non-standard numerical scheme different from those used when solving conventional boundary value problems of partial differential equations. Furthermore, the method of [PERSON] et al. [24] and [PERSON] et al. [25] is assumed to only be applicable to electrode shapes with gentle curvatures [9]. * _Correction methods_: [PERSON] et al. [27] proposed a correction factor method that makes local corrections of the cathode shape based on the shape of the anode obtained as the numerical solution to the direct problem. The underlying ECM model extends the potential model by taking into account a variable electrolyte conductivity that depends on the temperature and void fraction distribution within the electrolyte [19; 28; 29; 30]. The correction factor method was included in an integrated tool design approach described by [PERSON] and [PERSON] [31]. [PERSON] et al. [32] and [PERSON] and [PERSON] [33] corrected the shape of the cathode based on the deviation of the current density on the anode. In several studies, correction methods for cathode shape design are proposed, taking multiphysics ECM models into account [34; 35; 36; 37]. Correction methods have a wide field of use, including cathode shape design for non-steady-state ECM, such as electrochemical drilling [27] or electrochemical trepanning [37]. For these methods, much numerical expertise and effort are required to develop a computational environment. * _Embedding method:_ [PERSON] [38] approximated the shape of the cathode, computed anode, and required anode by sums of basis functions and computed the shape of the cathode by solving a nonlinear equation system for the corresponding coefficients (i.e., the design variables of the cathode). The method was applied by [PERSON] and [PERSON] [39] to solve the cathode shape design problem in two dimensions, taking multiphase flow into account. Since the solution of the nonlinear equation system involves the computation of the Jacobian for the Newton method, the anode shape prediction problem has to be solved \(M+1\) times per iteration (\(M\) is the number of coefficients), making the method computationally expensive. * _Optimization methods:_ Several approaches approximate the shape of the cathode, computed anode, and required anode by sums of basis functions and use gradient-based minimization of an objective function to compute the coefficients of electrode shape basis functions [39; 40; 41]. However, the approaches are also computationally expensive and require the anode shape prediction problem to be solved \(M+1\) times per iteration. This work applied the continuous adjoint shape optimization method that is capable of solving the steady-state cathode shape design problem in three dimensions [8]. The method is efficient since only two Laplace equations per optimization iteration have to be solved, although the number of design variables \(M\) corresponds to the degree of freedom of the mesh nodes at the cathode surface.