Free Boundary Problem for the Laplace Equation With Application to ECM Tool Design

The tool design problem of electrochemical machining (ECM) is formulated by the inverted approach in which the spatial coordinates are treated as the dependent variables on the plane of the complex potential. A general solution of this free boundary problem by analytic continuation provides the basis for a series approximation with correct asymptotic behavior using the method of weighted residues. The procedure is used to determine the noninsulated or partially insulated tool shapes which can be used to machine a prescribed workpiece geometry. The method is generally applicable to inverse (design) problems of potential theory which involve a given equipotential (or streamline) boundary along which a Neumann boundary condition is also prescribed as occurs in heat conduction, ideal flow, and electrostatics. The inverted approach not only eliminates the need for trial-and-error design procedures but also provides the advantage of adjustment in geometry by superposition.