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

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.

💡 Research Summary

This paper extends the greedy permanent‑magnet optimization (GPMO) framework by incorporating a block‑level macromagnetic model that accounts for finite magnetic permeability and demagnetizing interactions, resulting in a new algorithm called GPMO with macromagnetic refinement (GPMOmr). The authors first develop a fast, device‑scale macromagnetic solver. Each magnet block is treated as a hard‑magnet with a fixed crystallographic easy axis; the magnetization deviates from its remanent value through a small anisotropic susceptibility tensor (χ∥, χ⊥). The demagnetization field is expressed via an analytically derived tensor Nij for rectangular prisms, and the dense coupling matrix Aij = δij I + χi Nij is solved with Krylov methods (GMRES/biCGStab), preserving O(N²) scaling while handling the non‑symmetry introduced by anisotropic χ.

The macromagnetic model is then embedded directly into the greedy loop. In the original GPMO, candidate sites are scored using a discrete set of allowed dipole directions; the ArbVec variant already permits a continuous linear combination of these directions. GPMOmr adds a refinement step after each ArbVec selection, recomputing the block magnetizations under the finite‑µ model before proceeding to the next greedy iteration. This makes the optimization aware of the coupled magnet‑magnet and magnet‑coil feedback that would otherwise be ignored.

Applying the method to the published MUSE stellarator magnet grid, the authors perform a post‑analysis that quantifies the impact of macromagnetic effects. Finite‑µ corrections tilt individual magnets by a few degrees and change their effective remanence by a few percent. These local changes alter the surface‑normal field B·n on the plasma boundary by less than 1 % but double the global squared‑flux objective fB (the integral of (B·n)²). When the macromagnetic model is used inside the greedy loop (GPMOmr), the fB convergence history and final B·n errors are within a few percent of the classical GPMO results, yet the resulting magnetization pattern is noticeably more non‑uniform, with some blocks exhibiting larger tilts and others reduced magnitude.

The study demonstrates that, even for modest field strengths (local fields ≈0.2–0.3 T in MUSE), finite‑permeability effects can significantly affect global optimization metrics while only modestly perturbing surface‑field errors. This insight is crucial for scaling permanent‑magnet stellarators to higher fields or larger arrays, where demagnetizing fields may approach material limits and cause partial or full demagnetization.

An open‑source implementation of both the macromagnetic solver and the GPMOmr algorithm is provided in the SIMSOPT code base, enabling rapid re‑optimization of large magnet arrays with realistic material physics. The authors outline future extensions, including nonlinear saturation models for stronger fields, exploration of alternative magnetic materials with different χ⊥ values, and integration with real‑time experimental feedback.

In summary, the paper delivers a practical, scalable tool that bridges the gap between idealized permanent‑magnet design and the physical realities of finite permeability and demagnetization, paving the way for more robust and higher‑performance permanent‑magnet stellarator concepts.