Stream function approach for determining optimal surface currents

In many areas in industrial engineering, one may be faced with the question how an electromagnetic device has to be designed such that both a rather complex set of requirements such as geometrical constraints has to be fulfilled, and of which the magnetic properties has to be optimal in some sense. Given an electromagnetic design, a variety of methods exist to compute the additional magnetic properties and hence verify the constraints. However, the problem in which the optimal parameters are to be calculated given a set of constraints, is in general harder to solve. In this paper, we focus on quasi-static electromagnetic problems, where the problem is to find a certain conductor shape confined to an arbitrary but given surface, and electromagnetic properties are prescribed. Also conductive surfaces may be present, which affect these electromagnetic properties. With some additional assumptions the shape optimization problem can be formulated as a quadratic optimization problem with linear constraints.