Inverse and direct magnetic shaping problems

In this work, we study the direct and inverse problems arising from electromagnetic shaping in applications such as continuous casting processes. The magnetic field produces a surface pressure which forces a molten metal to change its shape until it reaches an equilibrium state between the magnetic pressure and the surface tension. The arising direct problem is a free boundary problem which is to determine the shape of molten metals for a given magnetic field. On the other hand, the inverse problem is to seek a configuration of electromagnetic field generators (i.e. inductors) in order that molten metals have prescribed shapes. A level set method will be presented to determine an equilibrium shape formed by a given configuration of magnetic fields. Also a computational method as well as uniqueness results for the inverse problem will be introduced and illustrated with examples.