Efficient calculation of magnetic fields from ferromagnetic materials near strong electromagnets, and application to stellarator coil optimization

TL;DR

The paper addresses the challenge of efficiently including ferromagnetic materials near strong electromagnets in fusion devices. It introduces a dipole-magnetization model with saturated magnitude $|\mathbf{M}| = M_{sat}$ and direction aligned to the external field $\mathbf{B}^{coils}+\mathbf{B}^{plasma}$, enabling rapid, non-iterative Biot–Savart evaluations by representing the ferromagnetic region as an array of dipoles. Validation against COMSOL shows excellent agreement and orders-of-magnitude speedups, while enabling gradient-based coil optimization that accounts for steel effects with only minor coil adjustments. The method is CAD-friendly, provides force calculations, and yields derivatives that facilitate efficient optimization, making it practical for stellarator and tokamak design workflows. Overall, including ferromagnetic materials in design loops reveals small plasma-physics perturbations that can be countered by modest coil reoptimization, preserving confinement and stability with minimal layout changes.

Abstract

In fusion reactor design, steels under consideration for the blanket are ferromagnetic, so the steel's effect on the plasma physics must be examined. For efficient calculation of these fields, we can exploit the fact that the magnetic material gives a small perturbation relative to the fields from the electromagnetic coils and plasma. Moreover the magnetization is saturated due to the strong fields in typical fusion systems. These approximations significantly reduce the nonlinearity of the problem, so the magnetic materials can be described by an array of point dipoles of known magnitude, oriented in the direction of the coil and plasma field. The approach is verified by comparison to finite-element calculations with commercial software and shown to be accurate. As no linear or nonlinear solve is required, only evaluation of Biot-Savart-type integrals, the method here is significantly simpler to implement than other methods, and extremely fast. The method is compatible with arbitrary CAD geometry, and also allows rapid computation of the magnetic forces. We demonstrate adding the ferromagnetic effects to free-boundary MHD equilibrium calculations, assessing the effect on plasma properties such as confinement and stability. Moreover, it is straightforward to differentiate through the model to get the derivative of the field with respect to the electromagnet parameters. We thereby demonstrate gradient-based coil optimization for a quasi-isodynamic stellarator in which the field contribution from a ferromagnetic blanket is included. Even a significant steel volume is found to have little impact on the plasma physics properties, with the main effects being a slight destabilization of ballooning modes and a radial shift of the edge islands due to decrease in rotational transform. Both issues are corrected by minor reoptimization of the coil shapes to account for the field from the steel.

Figures (15)

• Figure 1: $M-H$ curve for EUROFER97. Figure reproduced from mergia2008structural. The x-axis range of $\pm 1000$ kA/m is equivalent to $\pm 1.26$ T.
• Figure 2: Illustration of the approximate method proposed here, eq (\ref{['eq:dipole_sum']})-(\ref{['eq:dipole_moments']}). The field from the coils is first evaluated at quadrature points in the ferromangetic material. If the plasma current is substantial, its contribution to the total field there can be added using the virtual casing method. The field at the quadrature points determines the magnetization there, and the resulting field from the magnetized material can be computed by summing the field of point dipoles at the quadrature points. This field is computed at the evaluation points of interest, where the direct field from the coils is added.
• Figure 3: Geometry for the COMSOL benchmark in section \ref{['sec:comsol']} and optimization example in section \ref{['sec:opt_example']}. Left: bird's eye view. Center: Side view showing the mesh used for the steel layers and the holes for ports. Right: Side view showing the dipole locations (quadrature points) representing the two steel layers. In the right two panels, only one half period of the steel region is shown.
• Figure 4: Comparison between our approximate model and a high-fidelity COMSOL calculation for the configuration of figure \ref{['fig:example_setup']}, showing the magnetic field on the plasma boundary. Comparing the four panels in the top-left quadrant, both COMSOL and our model agree that the change to $\mathbf{B}$ due to the steel is small compared to the field from the coils. The change to $\mathbf{B}$ from the steel (right column) agrees between the two methods. Differences between the two methods (bottom row) are comparable with and without the steel, and are small compared to the effect of the steel (right column).
• Figure 5: The difference in the magnetization vector between COMSOL and the approximate model, normalized to the magnitude of magnetization in the approximate model, is small. Data are evaluated in the middle of the inner steel layer.