Efficient Calculation of and Coil Shape Optimization for Lorentz Forces*

The Lorentz forces on electromagnetic coils play an important limiting role in the design of practical magnetic confinement fusion reactors as designs must demonstrate that the forces are within prescribed limits. Unfortunately, the Lorentz force is time-consuming to numerically evaluate due to a source point singularity in the calculation of the self-force and, as a result, is typically reserved for engineering design. A method to rapidly calculate the self-force is desirable as Lorentz forces could be more easily incorporated into the physics optimization itself, though naive attempts to treat coils as infinitesimally thin in order to reduce computational time fail as the magnetic field nonphysically diverges. Instead, we present a novel method for calculating the Lorentz self-force (in addition to similar models for the self-inductance and self-field) using non-singular integral formulae of reduced dimensions that were derived rigorously by dividing the domain of integration of the magnetic vector potential into two regions and exploiting the unique assumptions of each region. We demonstrate that these formulae show good analytic and numerical agreement to high-fidelity calculations yet evaluate ~250x faster than finite element analysis software. Taking advantage of the numerical efficiency of the reduced formulae, we optimize stellarator coils for small Lorentz forces and demonstrate reductions in the maximum force by up to ~50% with flux surfaces that qualitatively match the stage I target.