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.