MIME-VER-110 — Permanent-Magnet Field vs Analytical

Date: 2026-04-30 Node under test: mime.nodes.actuation.permanent_magnet.PermanentMagnetNode Algorithm ID: MIME-NODE-101 Benchmark type: Analytical (Mode 2 independent) Test file: tests/verification/test_permanent_magnet.py::test_ver110_dipole_field_far_field Acceptance: \(|B_{\text{node}} - B_{\text{analytic}}| / |B_{\text{analytic}}| < 10^{-4}\)

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Goal

Verify that each of the three field models (point_dipole, current_loop, coulombian_poles) returns the B field at a target point in agreement with its own analytical reference, in the far-field regime \(r/R_{\text{magnet}} \in \{10, 20, 50\}\).

This benchmark validates the value output of the node. The companion benchmarks MIME-VER-111 and MIME-VER-112 cover the gradient and the Earth-field superposition respectively.

Configuration

Parameter	Value
dipole_moment_a_m2	\(1.0\) A·m²
magnet_radius_m	\(10^{-3}\) m
magnet_length_m	\(2 \times 10^{-3}\) m
magnetization_axis_in_body	\((0,0,1)\)
earth_field_world_t	\((0,0,0)\) — magnet-only check
Magnet pose	identity quaternion at origin
Target points	\((0,0,z)\) with \(z \in \{10, 20, 50\}\,R_{\text{magnet}}\)
JAX precision	x64 (enabled at module load)

Analytical references (numpy, double precision)

Point dipole (any displacement \(\mathbf{r}\)):

\[ \mathbf{B}_{\text{dipole}} = \frac{\mu_0}{4\pi} \cdot \frac{3(\mathbf{m}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}} - \mathbf{m}}{r^3}. \]

Current loop, on-axis:

\[ B_z = \frac{\mu_0 I R^2}{2(R^2+z^2)^{3/2}},\quad I_{\text{eff}} = \frac{|\mathbf{m}|}{\pi R^2}. \]

Coulombian poles, on-axis (\(M = |\mathbf{m}| / (\pi R^2 L)\)):

\[ B_z = \frac{\mu_0 M}{2}\left[ \frac{z+L/2}{\sqrt{R^2+(z+L/2)^2}} - \frac{z-L/2}{\sqrt{R^2+(z-L/2)^2}} \right]. \]

Method

1. Construct PermanentMagnetNode with the model under test and earth_field_world_t = (0, 0, 0).

2. Call update(state, boundary_inputs, dt) with the magnet pose and target. Read state["field_vector"].

3. Compute the analytical reference in float64 numpy.

4. Compare per-point: \(\|B_{\text{node}} - B_{\text{ref}}\| / \|B_{\text{ref}}\| < 10^{-4}\).

A second test (test_ver110_dipole_field_off_axis) checks the point_dipole model at three off-axis configurations (\(r \in \{(3,0,5), (3,4,5), (0,50,0)\}\) mm) — these are 3D configurations where coordinate-frame errors would manifest as cross-component leakage.

Results

Model	\(z = 10\,R\)	\(z = 20\,R\)	\(z = 50\,R\)
point_dipole	\(< 10^{-4}\)	\(< 10^{-4}\)	\(< 10^{-4}\)
current_loop	\(< 10^{-4}\)	\(< 10^{-4}\)	\(< 10^{-4}\)
coulombian_poles	\(< 10^{-4}\)	\(< 10^{-4}\)	\(< 10^{-4}\)

Off-axis (point_dipole only):

Target \((x,y,z)\) mm	Relative error
\((3, 0, 5)\)	\(< 10^{-4}\)
\((3, 4, 5)\)	\(< 10^{-4}\)
\((0, 50, 0)\)	\(< 10^{-4}\)

All eight configurations pass the \(10^{-4}\) acceptance threshold.

Verdict: PASS

The three field models match their analytical references to better than \(10^{-4}\) relative error in the far field, on-axis. The point-dipole model also matches off-axis to the same tolerance. The v1 implementation is faithful in its declared validity envelope.

Related notes

• Float32 is insufficient for this benchmark; the coulombian_poles model differs from its float64 reference by \(3.6 \times 10^{-4}\) at \(z = 20\,R_{\text{magnet}}\) in float32. The test module enables jax_enable_x64 = True at import time.

• The current_loop and coulombian_poles models reduce to dipole + near-field correction off axis; this benchmark only exercises the on-axis closed forms for those two models.

• For near-field behaviour (z < 5 R), the point-dipole formula diverges from the finite-magnet truth — see docs/deliverables/dejongh_benchmark_summary.md lines 393–413 for the calibrating numbers (~ 4 % at z = 3 R, < 1 % at z = 10 R). This benchmark therefore exercises only z ≥ 10 R, where all three models agree with their respective analytical truths.