**Abstract** : Electromagnetic Casting (EMC) and Magnetic Suspension Melt Processing (MSMP) are very important technologies in the metallurgical industry. They make use of an electromagnetic field for contactless heating, shaping and control of solidification of hot melts. Advantages of these techniques are high surface quality, high cleanness, low contamination and near-net shape manufacturing. The ECM has primarily been employed for containerless continuous casting but is mainly used to prepare ingots of aluminum alloy. Another important application, extensively used in aeronautics, astronautics, energy and chemical engineering, is in the manufacturing of components of engines made of superalloy materials (Ni,Ti,\ldots). These technologies are based on the repulsive forces that an alternating electromagnetic field produces on the surface of this kind of materials. This electromagnetic field is induced by an externally imposed alternating current. Under suitable assumptions, the mathematical model is described by a set of equations expressing an equilibrium relation on the boundary between electromagnetic pressures and surface tensions, as well as gravity forces in the three-di\-men\-sion\-al problem. The boundary shape of the liquid metal such that the equilibrium is attained can be found as the solution of a nonlinear free surface problem. In this work we study the electromagnetic shaping of a vertically falling molten metal column with a magnetic field induced by a set of vertical electric wires. In the direct problem the position and shape of the inductors are given and the magnetic field created by them produces a surface pressure on the vertical column of liquid metal to change its shape until an equilibrium relation on the boundary between the electromagnetic pressures and surface tensions is satisfied. The numerical method to solve the direct problem is based on an energetic variational formulation for the equilibrium. The inverse problem consists of locating the inductors in order to have an horizontal cross-section of the molten metal as close as possible to a prescribed shape. In this work we consider a realistic case where each inductor is a set of bundled insulated strands. In this case we represent inductors by a set of domains in the plane. The electric current density is assumed uniform on the entire cross-section of the inductor. This is a very reasonable approximation for the case where the inductors are made up of multiple individually insulated strands twisted or woven together. In order to design suitable inductors using numerical methods two different approaches are proposed, the first one looks for a set of inductors such that the distance between the computed shape and the given target one is minimized. In the second approach the error of the equilibrium equation for the target shape is minimized. In this last formulation only shape variables concerning the inductors are considered. This fact allows us to solve the inverse problem with a minor computational effort. In order to develop a numerical method we consider a Simultaneous Analysis and Design optimization technics, SAND, which is a formulation that includes the state variables as unknowns of the mathematical program and the state equations as equality constraints. In this work we employ a SAND formulation to solve the inverse problem for both approaches. The finite dimensional optimization problem obtained after discretization was solved employing the \emph{Feasible Arc Interior Point Algorithm}, FAIPA, a line search interior-point algorithm for nonlinear optimization. Numerical computations for several examples of the direct and inverse problems are presented to show the efficacy of the proposed formulations in the design of suitable inductors.