Optimal Design of a 90° Superconducting Bending Magnet with active shielding Using Topology Optimization

The development of cyclotrons, MRI or particle therapy systems leads to the design of different kinds of resistive and superconducting bending magnets. Most designs are based on an a priori choice of a geometrically defined solution whose characteristic parameters are then optimized to minimize a cost function while meeting some constraints on the magnetic field distribution. The main drawback of this approach is that the geometry and the topology of the solution cannot evolve during the optimization. The optimality of the final solution is therefore strongly conditioned by the initial choice of the designer. A more open approach consists in seeing the design of these magnets as an inverse problem in the sense that, starting from a predefined field distribution, the goal is to find the winding distribution reproducing it at the lowest cost. This inverse approach, already used for the design of MRI magnets, investigates a much broader space of solutions than the conventional approach. It uses the Biot-Savart law to determine the relation between the currents and the magnetic field at some specific points. The inverse relation is then used to find the optimal current distribution. We use a similar approach except that the relation between the current and the magnetic field is obtained through finite elements analysis. Known as topology optimization, this approach is heavier, but can be applied to non-linear problems like the design of iron-shielded magnets. In this paper, we apply topology optimization to the design of a superconducting bending magnet with active shielding. The cost function to minimize is the superconducting material quantity and the constraints are related to the magnetic field distribution in the magnet aperture and outside the magnet and to manufacturability practical aspects. The optimization tool leads to solutions composed of several coils with non rectangular sections. The study of the current density and of the magnet radius influence shows that these solutions are very stable both in terms of topology and geometry. It also highlights that the stored magnetic energy and the superconducting material quantity tend asymptotically to a minimal value for increasing magnet radii. Taking into account the upper limit of the magnetic field on superconductors, this study finally show that the smallest magnet radius can be reached for intermediate values of current densities.