Three-dimensional tool design for steady-state electrochemical machining by continuous adjoint-based shape optimization

• We propose a method to solve the tool design problem in electrochemical machining in two and three dimensions.
• We use continuous adjoint-based shape optimization combined with elements of shape calculus.
• A comparison with exact analytic solution demonstrates the accuracy of the method.
• We investigate the influence of electrode curvature on the front gap width.
• The inaccuracies of the cos θ-method for curved electrodes can be estimated.

In electrochemical machining (ECM), it is important to design the shape of an appropriate tool capable of producing a workpiece of desired shape. This work presents a numerical approach to solving the two- and three-dimensional tool design problem in steady-state ECM. The tool design problem is transformed into a shape optimization problem and then solved using the continuous adjoint method combined with elements of shape calculus, ensuring high efficiency and the maximum possible degrees of freedom. A numerical experiment on a two-dimensional Gaussian-shaped workpiece shows a good agreement of the calculated tool shape with the exact analytical solution. Tool design is carried out for a series of two- and three-dimensional workpiece shapes to investigate the influence of the curvature of the desired workpiece on the front gap width in steady-state ECM.