Coil-FEM: Differentiable FEA for Stellarator Design

• The paper introduces an integrated differentiable FEA tool that co-optimizes coil geometry and support placements for improved structural performance.
• The methodology employs smooth spring-foundation models and curved tetrahedral meshing to accurately capture stress concentrations near clamps.
• In a W7-X case study, the approach achieved a 2.4× reduction in RMS von Mises stress while maintaining similar magnetic field quality.

Searching arXiv for the specified paper and closely related work on differentiable FEA and coil optimization. coil-fem is an open-source, differentiable finite-element tool for stellarator design that integrates support differentiable FEA into the stellarator coil optimization loop, enabling the joint optimization of coil geometry and support clamp locations for fusion reactors (Fu et al., 7 Jul 2026). It was introduced to address a standard sequential workflow in which filamentary coils are optimized for magnetic performance, support structures are then placed by hand through repeated finite element analyses, and deformation-induced field errors are only confronted after the coil geometry has largely been fixed. Within coil-fem, coil bodies are generated from filamentary centerlines, meshed with curved tetrahedral elements, loaded by gravity and electromagnetic body forces, and constrained by a differentiable spring-foundation model for cage-type supports. The resulting structural problem is solved in-loop, and stress- and deformation-based penalties are differentiated with respect to both coil-shape parameters and clamp positions. In the reported W7-X proof-of-concept, this integrated approach produced a coil set with $2.4\times$ lower RMS von Mises stress and similar field error compared to an unoptimized baseline (Fu et al., 7 Jul 2026).

1. Design role and problem setting

coil-fem was developed for a design regime in which the support structure is an integral part of the reactor and must be co-designed with the magnetic system rather than treated as a downstream engineering detail. The motivating application is the three-dimensional stellarator, for which a practical reactor’s coils and support structures must simultaneously provide an accurate magnetic field for good confinement, sufficient rigidity to protect brittle high-temperature superconductor from damage, and a simple geometry for low-cost construction (Fu et al., 7 Jul 2026).

The conventional workflow described in the underlying study is explicitly sequential. First, filamentary coils and winding surfaces are optimized to achieve the desired magnetic field. Second, the coil geometry is frozen. Third, support structures are designed manually by placing clamps, shells, or frames “by hand” and repeating FEA until stress and displacement limits are met. This process is characterized as lengthy and labor-intensive, and it also leaves limited opportunity to modify the coil geometry after structural effects have been discovered (Fu et al., 7 Jul 2026).

coil-fem targets that gap by moving structural mechanics into the optimization loop itself. It provides a differentiable mechanical load penalty that can be combined with magnetic field quality, coil length, curvature, and spacing penalties in a single objective function. This makes it possible to search automatically over both coil geometry and support clamp placement, instead of optimizing the magnetic design and the structural design in separate stages (Fu et al., 7 Jul 2026).

A common misconception in this area is that analytic force proxies are sufficient substitutes for structural analysis. The case study associated with coil-fem argues otherwise: reducing a Lorentz-force-based proxy did not necessarily reduce von Mises stress, especially near clamps, where local stress concentrations dominate. This suggests that centerline force reduction and body-stress reduction are not interchangeable objectives (Fu et al., 7 Jul 2026).

2. Coil geometry and differentiable support representation

coil-fem starts from filamentary coil centerlines $\mathbf{r}_c(\phi)$ and produces finite-build coil bodies by sweeping a rectangular cross-section along each centerline using a moving frame $\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$. The coil-body parameterization is

$$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$

with $\phi \in [0, 2\pi]$, $u, v \in [-1, 1]$, and cross-section widths $w_1, w_2$ (Fu et al., 7 Jul 2026). The moving frame is governed by

$$\frac{d}{d\phi} \begin{pmatrix} \mathbf{t} & \mathbf{p} & \mathbf{q} \end{pmatrix} = \left|\frac{d\mathbf{r}_c}{d\phi}\right| \begin{pmatrix} 0 & \kappa_1 & \kappa_2 \ - \kappa_1 & 0 & \kappa_3 \ - \kappa_2 & -\kappa_3 & 0 \end{pmatrix} \begin{pmatrix} \mathbf{t} & \mathbf{p} & \mathbf{q} \end{pmatrix},$$

where $\kappa_3 = d\alpha/dl$ is the twist rate; the implementation reported in the paper uses a rotation-minimizing frame to avoid excessive twist (Fu et al., 7 Jul 2026).

Support structures are represented through a differentiable boundary model focused on cage-type supports. Rather than imposing hard Dirichlet clamps on a changing node set, coil-fem introduces a spatially varying spring coefficient $k(\mathbf{x})$ on the coil surface. The clamp model is

$\mathbf{r}_c(\phi)$0

with $\mathbf{r}_c(\phi)$1, $\mathbf{r}_c(\phi)$2 clamps per coil, clamp angles $\mathbf{r}_c(\phi)$3, base stiffness $\mathbf{r}_c(\phi)$4, clamp radius $\mathbf{r}_c(\phi)$5, and smoothing width $\mathbf{r}_c(\phi)$6 (Fu et al., 7 Jul 2026). Because $\mathbf{r}_c(\phi)$7 varies smoothly with the clamp angles, the structural solution remains differentiable with respect to support location.

This representation is central to the tool’s practical role. The support model is not a full beam-or-shell model of the surrounding cage; instead, it is a smooth penalty approximation that permits gradient-based optimization of support placement without changing mesh topology or boundary-node sets. A plausible implication is that coil-fem prioritizes differentiability and optimization compatibility over exact modeling of support flexibility, an explicit trade-off acknowledged in the paper’s limitations (Fu et al., 7 Jul 2026).

3. Governing equations, constitutive model, and loading

For each coil body $\mathbf{r}_c(\phi)$8, coil-fem solves a linear elasticity problem for the displacement field $\mathbf{r}_c(\phi)$9:

$\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$0

The boundary condition is a Robin-type spring foundation,

$\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$1

with outward normal $\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$2 and spring coefficient $\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$3 defined by the clamp model (Fu et al., 7 Jul 2026). The paper emphasizes that this boundary treatment approximates clamping through a large but finite stiffness and is smooth enough to remain AD-friendly.

The constitutive law is isotropic linear elasticity,

$\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$4

with Lamé parameters $\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$5. Thermal strain is included through an additive small-strain decomposition,

$\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$6

where $\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$7 is the integral thermal contraction (Fu et al., 7 Jul 2026). In the W7-X study this contraction is $\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$8, treated as uniform and isotropic.

The body-force density combines gravity and electromagnetic loading:

$\{\mathbf{t}, \mathbf{p}, \mathbf{q}\}$9

Here $$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$0 is density, $$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$1 is gravity, $$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$2 is current density, $$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$3 is the coil’s own magnetic field, and $$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$4 is the field from other coils (Fu et al., 7 Jul 2026). The self-field is computed analytically through the Landreman–Hurwitz formula for rectangular cross-section coils, while the mutual field is computed by treating the other coils as one-dimensional filaments to reduce cost.

This loading model places coil-fem in a broader methodological lineage. Reduced filamentary models were introduced precisely because conventional finite element evaluation of Lorentz forces is too time-consuming within optimization loops; in particular, regularized self-force models implemented in SIMSOPT have been used to reduce pointwise forces without volumetric meshing (Hurwitz et al., 2024). coil-fem departs from those approaches by resolving stresses in a finite-build coil body with support boundary conditions, rather than optimizing only centerline forces (Fu et al., 7 Jul 2026, Hurwitz et al., 2024).

4. Optimization problem and structural penalty

The optimization variables in coil-fem include Fourier coefficients of the coil centerlines and, in the joint-design setting, the support clamp locations $$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$5 (Fu et al., 7 Jul 2026). The overall objective reported in the paper is

$$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$6

The magnetic field quality term is

$$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$7

where $$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$8 is the plasma boundary, $$\mathbf{r}(\phi, u, v) = \mathbf{r}_c(\phi) + \frac{u w_1}{2}\,\mathbf{p}(\phi) + \frac{v w_2}{2}\,\mathbf{q}(\phi),$$9 is the normal component of the coil-generated field, and $\phi \in [0, 2\pi]$0 is the target normal field, which is zero in the vacuum configuration considered (Fu et al., 7 Jul 2026).

For the cases that actually use coil-fem, the structural load term is based on the volume integral of squared von Mises stress:

$\phi \in [0, 2\pi]$1

The study explicitly contrasts this with a Lorentz-force-based proxy used in a comparison case:

$\phi \in [0, 2\pi]$2

That comparison is one of the central substantive results: the force proxy does not reproduce the stress distributions obtained by FEA, because support-induced local peaks in $\phi \in [0, 2\pi]$3 are not captured by a centerline force integral (Fu et al., 7 Jul 2026).

The remaining penalties encode manufacturability and configuration constraints. The length penalty is

$\phi \in [0, 2\pi]$4

The coil–coil and coil–plasma spacing penalties are defined by double integrals over centerlines and the plasma surface, enforcing minimum thresholds $\phi \in [0, 2\pi]$5 and $\phi \in [0, 2\pi]$6. Additional terms penalize nonzero linking number and curvature above a threshold $\phi \in [0, 2\pi]$7, the latter set to W7-X’s maximum curvature (Fu et al., 7 Jul 2026).

The structure of the objective function shows that coil-fem is not merely a structural post-processor. It is a source of differentiable penalties inside a single, multi-term design problem. This suggests a shift from proxy-based magnetics-first optimization toward explicitly multi-physics coil design, in which field quality and structural robustness are treated as simultaneous design objectives rather than sequential checks.

5. Numerical implementation and differentiability

coil-fem is built on JAX-FEM and uses curved-edge 10-node tetrahedral elements with quadratic Lagrange basis functions (Fu et al., 7 Jul 2026). Coil volumes are meshed from a uniform grid in $\phi \in [0, 2\pi]$8 mapped through the sweep geometry, and mesh topology is kept fixed during optimization for AD compatibility. This fixed-topology choice is operationally important because it avoids the discontinuities that would arise from remeshing or changing connectivity during gradient computation.

The software differentiates through the entire structural pipeline: geometry mapping, matrix assembly, spring-foundation boundary conditions, linear-system solution, and the evaluation of stress-derived objectives. The study highlights three implementation decisions as especially important for differentiability: fixed mesh topology, smooth spring-foundation boundary conditions instead of dynamic Dirichlet sets, and differentiable treatment of body forces and material properties (Fu et al., 7 Jul 2026).

The linear systems are solved through JAX-FEM backends. On CPU, the implementation uses PETSc sparse linear solvers, described as the same as DOLFINx. On GPU, it uses cuDSS and spineax, and the reported benchmarks use an Nvidia L40S (Fu et al., 7 Jul 2026). Optimization is performed with L-BFGS via SciPy and a three-stage multi-grid strategy in Fourier space, with increasing centerline resolution and up to 500 iterations per stage.

For the W7-X optimization with $\phi \in [0, 2\pi]$9, corresponding to 1920 mesh cells per coil, each gradient evaluation $u, v \in [-1, 1]$0 costs approximately $u, v \in [-1, 1]$1–$u, v \in [-1, 1]$2 s on a single L40S GPU (Fu et al., 7 Jul 2026). The paper compares this favorably to finite-difference plus DOLFINx in the StellCoilBench setting, which requires multiple FEA calls per gradient and about $u, v \in [-1, 1]$3 s for a simpler case.

This computational strategy distinguishes coil-fem from earlier custom coil-FEM formulations aimed at electromagnetic field computation. For example, low-frequency eddy-current solvers based on weakly coupled edge-element FEM reduce vector problems to scalar ones and avoid meshing the excitation coil in order to make very large biological meshes tractable (Yin et al., 2019). coil-fem addresses a different PDE class—linear elasticity under electromagnetic loading rather than eddy-current Maxwell solves—but shares the same design principle: embed only the necessary finite-element physics inside the loop, and simplify all other pieces enough to preserve tractability (Fu et al., 7 Jul 2026, Yin et al., 2019).

6. Case study, results, and interpretation

The proof-of-concept application is a simplified W7-X type 1 coil set modeled as a uniform 316LN body with a cage-like support represented by two clamps per coil (Fu et al., 7 Jul 2026). The material parameters are $u, v \in [-1, 1]$4, $u, v \in [-1, 1]$5, $u, v \in [-1, 1]$6, and $u, v \in [-1, 1]$7. The spring coefficient is $u, v \in [-1, 1]$8, the clamp radius is $u, v \in [-1, 1]$9, and the target magnetic configuration is the vacuum “EIM” case (Fu et al., 7 Jul 2026).

Five configurations are considered: a baseline with fixed top-and-bottom clamps; Case A with clamp optimization only; Case B with coil geometry optimization only and fixed supports; Case C with joint optimization of coils and supports; and Case D with coil geometry optimization based on the Lorentz-force proxy and fixed supports (Fu et al., 7 Jul 2026).

The central findings are structural. Cases A and C, both of which optimize clamp locations, reduce RMS von Mises stress by more than a factor of 2 relative to baseline. Case B, which optimizes only coil geometry while keeping supports fixed, yields only marginal improvement. Case C achieves the best overall result: approximately $w_1, w_2$0 reduction in RMS von Mises stress together with similar magnetic field error $w_1, w_2$1 relative to baseline, and RMS displacement is also reduced by approximately $w_1, w_2$2 (Fu et al., 7 Jul 2026).

The paper also reports a thermal-contraction comparison for Case C. With $w_1, w_2$3, RMS $w_1, w_2$4 and maximum $w_1, w_2$5. Without thermal contraction, RMS $w_1, w_2$6, corresponding to a $w_1, w_2$7 reduction, and maximum $w_1, w_2$8, corresponding to a $w_1, w_2$9 reduction (Fu et al., 7 Jul 2026). The interpretation given is that Lorentz forces dominate force density by three orders of magnitude, yet thermal effects remain non-negligible for peak stress.

These results have two direct implications. First, support placement is not a minor implementation detail: in the reported design space it is a first-order determinant of stress distribution. Second, replacing structural penalties with force proxies can be misleading when stress concentrations are support-driven. The broader relevance is that joint coil-support optimization appears to change the feasible design set itself, not merely refine a design found by magnetic optimization alone (Fu et al., 7 Jul 2026).

A plausible implication is that coil-fem is most valuable at the interface between magnetic design and engineering feasibility, where filamentary or centerline-based metrics cease to be reliable. That interpretation is consistent with related coil-optimization work, in which reduced self-force models enable large-scale search over coil shapes, but final structural realism still requires a finite-build description (Hurwitz et al., 2024).

7. Scope, limitations, and broader significance

coil-fem is presented as a proof-of-concept rather than a full reactor-grade mechanics platform. The finite-build coil is a swept rectangular cross-section with homogeneous, isotropic material properties and current density. Thermal contraction is uniform, not spatially varying. The structural model assumes linear elasticity, small strain, and body forces independent of deformation, so no geometric nonlinearity is included (Fu et al., 7 Jul 2026). The support structure is limited to cage-type supports represented by spring-foundation clamps; full beams, shells, contact, friction, and anisotropic connector stiffness are not modeled.

The paper explicitly identifies extensions toward higher-fidelity coil descriptions, multi-material winding pack and case models, full thermo-elastic equations with realistic temperature gradients, beam or rod network models for cage structures, shell-type supports, and the use of displaced coil geometries to re-optimize loaded-state magnetic performance (Fu et al., 7 Jul 2026). This suggests that the current version of coil-fem is intended as a differentiable structural kernel that can be embedded into larger stellarator design frameworks rather than as a complete end-state engineering suite.

Within the broader “coil-FEM” landscape, coil-fem occupies a distinct niche. Earlier work in coil-based electromagnetic simulation emphasized efficient field computation, reduced-order magnetic-force models, or hybrid analytic–FEM workflows for specific devices such as stellarators, biological eddy-current systems, or shielded-room coils (Hurwitz et al., 2024, Yin et al., 2019, Liu et al., 2020). coil-fem instead makes structural FEA itself differentiable and directly optimizable, with support placement as an explicit design variable (Fu et al., 7 Jul 2026). This suggests a broader redefinition of coil-FEM from “finite elements for fields around coils” to “finite elements inside the coil-design loop,” where structural mechanics participates in coil synthesis rather than in post hoc validation alone.