AN INTEGRAL APPROACH TO THE INVERSE ELECTROMAGNETIC SHAPING PROBLEM

High frequency electromagnetic levitation and shaping of molten metals involves a coupling between the molten metal and electromagnetic fields. Modeling a stable molten metal and air interface supported by magnetic pressures requires a self-consistent solution. Considerable work has been done to date on solving for the equilibrium free boundary representing a molten metal and air interface shaped by predefined conductors carrying high frequency alternating currents. From an engineering standpoint, a more practical problem is one in which the desired equilibrium free boundary shape is specified and the alternating currents necessary to achieve this shape are then calculated. In this paper, an integral equation method is presented for solving the inverse shaping problem, where a desired free boundary is specified in addition to the location of a number of source currents. The method results in a matrix equation which is solved for the source current magnitudes necessary to achieve the desired free boundary geometry. The method is tested using a previously developed free boundary solution procedure.