From Filaments to Finite Elements: Parametric CAD Generation and Three-Method Magnetic Field Comparison for Stellarator Coils

Abstract

Stellarators employ complex three-dimensional coil structures to produce the magnetic fields required for plasma confinement. In typical design workflows, the coil structures are modeled as infinitesimally thin filaments, reducing the source integrals in the Biot-Savart law from three-dimensional volume integrals to one-dimensional line integrals, enabling quick and efficient magnetic field computation. However, this approach does not account for the cross-sectional area and internal structure of the conductors, making it difficult to apply the method in the context of engineering analyses such as structural stress and deformation, thermal management and heat dissipation, electromagnetic force distribution, and current density effects, and leading to inaccuracies in the near-field magnetic behavior. This thesis presents a computational framework for constructing detailed three-dimensional CAD models of stellarator coils from their mathematical representations as Fourier-defined space curves. This is done using the CadQuery library, an open source Python library used to construct 3D CAD models through scripting. We combine coil center-line curves with parametric definitions of the cross-sectional areas. These cross-sections are then swept along the curves, using rotation-minimizing frames to produce three-dimensional geometries that are closed and continuous solids with no gaps or self-intersections. These geometries can then be meshed using Netgen and used to calculate the magnetic fields from the volumetric current distributions. These calculated magnetic fields can then be compared with standard filamentary and volumetric models, which allow for the quantification of the differences introduced by the cross-sectional conductor areas. We found that all three models (filamentary Biot-Savart, volumetric Biot-Savart, and Finite Element Method) converged to within 4-9% of one another in the far field (>10 mm from the coil centerline), with the filamentary and volumetric Biot-Savart results becoming indistinguishable (Fil-Vol errors below 0.05%) beyond roughly 2-5 mm from the conductor surface. This establishes that the standard filamentary approximation is accurate for stellarator coil design outside the conductor volume, regardless of conductor architecture. Inside the conductor and near its surface, however, the three methods diverge sharply: the filamentary Biot-Savart is formally divergent at the centerline and still produces errors of 53-90% at 5 mm, while the volumetric Biot-Savart and FEM agreed to within 16-20% at the centerline and within 2-12% by 5 mm. The residual FEM-volumetric gap reflects the finite-element discretization floor rather than a physical disagreement, so the two volumetric methods cross-validate one another while the filamentary model fails in precisely the regime where engineering constraints, mechanical stress, peak-field limits, and thermal loading are most sensitive. This thesis provides the foundation for the integration of CAD modeling, finite element analysis, and validation of the magnetic fields into the stellarator design process.