We describe a Newton method applied to the evaluation of a critical point of a total energy associated to a shape optimization problem. The key point of this methods is the Hessian of the shape functional. We give an expression of the Hessian as well as the relation with the second-order Eulerian semi-derivative. An application to the electromagnetic shaping of liquid metals process is studied. The unknown surface i represented by piecewise linear closed Jordan curves. Each step of the algorithm requires solving two exterior elliptic boundary values problems. This is done by using an integral representation of solutions on this surfaces. A comparison with a Quasi-Newton algorithm is worked out
