2D Analytical Magnetic Optimizations for Accelerator Dipole Block Designs

A new magnetic design method, which optimizes the position of the conductors in the cross-section of a race-track-shaped magnet, has been set up. This method is based on the computation of analytical formulas of field, harmonics and forces. The conductors are modeled with rectangles, which is particularly well adapted to a block design. A first equation is used to calculate the contribution of the rectangles, as a function of their four spatial coordinates. All the block contributions are then added up to give the entire equation as a function of all the rectangles coordinates. Finally a non-linear optimization, using the analytical formulas, computes these coordinates to fulfill the magnet specifications. The objective can be either to maximize the central field, or to minimize a criterion, for instance the conductor volume, the harmonics, the electromagnetic forces. Various kinds of constraints can also be imposed: zeroed harmonics, blocks position and number, central field. Two magnets designs are proposed as practical examples: a 13 T Nb
₃
Sn magnet and a 20 T graded magnet.