Permanent magnet optimization of stellarators with coupling from finite permeability and demagnetization effects

TL;DR

The paper develops a device-scale macromagnetic framework to quantify finite-permeability and demagnetization effects in permanent-magnet stellarator design, formulating a linear equilibrium problem $Aoldsymbol M=oldsymbol b$ that accounts for anisotropic susceptibility and demagnetizing interactions. It shows that, for MUSE-like hard magnets, these corrections induce percent-level changes in local magnetization and a ~1% perturbation in the surface normal field, while the global squared-flux objective $f_B$ can double when macromagnetics are included in postprocessing. Embedding the macromagnetic solve into the greedy PM optimization (GPMO) loop yields GPMOmr, whose $f_B$ histories closely track classical GPMO with only modest differences in magnetization patterns, demonstrating robustness of the discrete layout under macromagnetic feedback. The work also explores higher-field and demagnetization scenarios (GB50UH and AlNiCo) to identify regimes where macromagnetic coupling becomes critical, highlighting the potential need for expanded degrees of freedom or nonlinear material models for future PM stellarator designs. Together with open-source tooling in SIMSOPT, this framework bridges idealized design and device reality, guiding material choices and layout strategies for scalable PM stellarator optimization.

Abstract

Permanent magnets provide an attractive path for shaping university-scale stellarator magnetic fields. Previous work has shown that greedy permanent magnet optimization (GPMO) can produce sparse, grid-aligned arrays that match target surfaces with high accuracy under an ideal rigid-remanence model. Here we extend this approach to a greedy permanent magnet optimization with macromagnetic refinement (GPMOmr) by introducing a block-level macromagnetic model that accounts for magnet-magnet and magnet-coil coupling from finite permeability and demagnetizing interactions, and apply it to the published magnet grid from the MUSE stellarator design. Finite-permeability effects produce degree-scale tilts and few-percent magnitude changes in individual magnets and modify the surface-normal field $\mathbf B\cdot\mathbf n$ only at the percent level, yet for a fixed layout they increase the standard squared-flux objective by more than a factor of two. When the same model is embedded in the greedy loop, GPMOmr achieves $f_B$ histories and final errors within a few percent of classical GPMO while producing visibly more nonuniform magnetization patterns. Our formulation provides a fast and practical tool for quantifying and incorporating finite-permeability effects in permanent-magnet stellarator designs, and offers a framework for extending permanent-magnet optimization to higher field strengths and to materials with stronger macromagnetic coupling.

Figures (13)

• Figure 1: Difference in surface normal field $\Delta(\mathbf B\cdot\mathbf n)$ between the uncoupled rigid-remanence solution and the fully coupled macromagnetic solution on the MUSE plasma boundary. The color bar spans approximately $[-1.5,1.3]\times10^{-3}$ T, corresponding to a peak fractional change of $1.00\%$ relative to the $0.15$ T target field. Small asymmetries in the extrema reflect discretization-induced violations of stellarator symmetry in the PM grid rather than any large macromagnetic effect.
• Figure 2: Surface $\mathbf B\cdot\mathbf n$ residual on the MUSE plasma boundary for three postprocessed fields: (left) uncoupled rigid-remanence solution, (center) macromagnetic solution with magnet--magnet coupling only, and (right) macromagnetic solution with both magnet--magnet and coil coupling. The patterns are qualitatively similar in all cases; macromagnetic and coil coupling primarily modulate the amplitude of existing structures rather than creating new defects. The color scale spans $[-2.6,3.0]\times10^{-3}$ T.
• Figure 3: Squared-flux error $f_B$ versus greedy iteration $K$ for classical GPMO (uncoupled) and GPMOmr on the MUSE PM grid (no backtracking). The three GPMOmr curves correspond to macromagnetic refinement intervals $k_{\rm mm}=1,25,50$ and lie almost on top of one another, indicating that the choice of $k_{\rm mm}$ has little effect on the $f_B(K)$ history while strongly affecting the number of macromagnetic solves.
• Figure 4: Glyph plots of the final magnetization patterns on the MUSE PM grid without backtracking. Left: GPMOmr, including finite-$\mu$ corrections through the macromagnetic solve. Right: classical uncoupled GPMO. The uncoupled solution exhibits nearly uniform magnitudes, while the macromag solution shows substantial spatial variation driven by local demagnetizing fields.
• Figure 5: Histograms of the pointwise dipole-moment difference $\Delta M_i$ between the macromagnetic-refinement and classical runs on the MUSE grid (no backtracking). Four clear peaks appear: near zero (almost identical blocks), near the single-dipole cap $m_{\max}$ (blocks active in only one run), near $\sqrt{2}\,m_{\max}$ (approximately orthogonal placements), and near $2m_{\max}$ (rare anti-parallel cases).