  'Coil design' is an inverse problem in which arrangements of wire are
designed to generate a prescribed magnetic field when energized with electric
current. The design of gradient and shim coils for magnetic resonance imaging
(MRI) are important examples of coil design. The magnetic fields that these
coils generate are usually required to be both strong and accurate. Other
electromagnetic properties of the coils, such as inductance, may be considered
in the design process, which becomes an optimization problem. The maximum
current density is additionally optimized in this work and the resultant coils
are investigated for performance and practicality. Coils with minimax current
density were found to exhibit maximally spread wires and may help disperse
localized regions of Joule heating. They also produce the highest possible
magnetic field strength per unit current for any given surface and wire size.
Three different flavours of boundary element method that employ different basis
functions (triangular elements with uniform current, cylindrical elements with
sinusoidal current and conic section elements with sinusoidal-uniform current)
were used with this approach to illustrate its generality.
