Force and Torque Probes¶

Magnetic machines often require estimates of the mechanical force or torque that the air-gap fields apply to a rotor. mag_sim now exposes a stress tensor probe that integrates the Maxwell stress tensor along a closed contour, giving per-unit-length force and torque values directly from the solved field map.

Maxwell Stress Tensor refresher¶

For magnetostatics the Maxwell stress tensor in Cartesian coordinates is

\[ T_{ij} = \frac{1}{\mu_0} \left(B_i B_j - \tfrac{1}{2} \delta_{ij} B^2\right), \]

with the traction on a surface of unit normal n given by

\[ \mathbf{f} = T \cdot \hat{\mathbf{n}}. \]

Integrating the traction around a closed 2D loop yields the net force per unit length, while the torque about the out-of-plane axis follows from

\[ \tau_z = \oint (x f_y - y f_x)\,\mathrm{d}s. \]

The solver evaluates these expressions numerically using midpoint sampling on polygon edges. Bilinear interpolation recovers B at arbitrary contour points so probes are not restricted to grid-aligned loops.

JSON schema¶

Declare probes inside the scenario outputs array:

{
  "type": "probe",
  "id": "rotor_mst",
  "probe_type": "force_and_torque",
  "method": "stress_tensor",
  "loop": {
    "type": "polygon",
    "vertices": [
      [-0.03, -0.02],
      [0.03, -0.02],
      [0.03, 0.02],
      [-0.03, 0.02]
    ]
  },
  "path": "outputs/rotor_mst.csv"
}

probe_type chooses which quantities are of interest: "force", "torque", or "force_and_torque" (all three values are written either way).
method currently supports only "stress_tensor".
loop supplies at least three vertices describing the closed contour. Provide either an array of [x, y] pairs or an object with type="polygon" and a vertices array as shown above.

The solver writes a CSV with a single row:

Fx,Fy,Tz,CoEnergy
-1.234567890123e+02,5.678901234567e+01,-2.468013579240e-03,1.234500000000e-01

Values represent force (newtons per metre) and torque (newton-metres per metre) about the global origin in SI units. The optional CoEnergy column captures the magnetic co-energy integral for the entire slice, enabling finite-difference virtual-work checks without recomputing the field.

When timelines are active the solver still emits per-frame CSVs with the _frame_### suffix for detailed inspection, but it also aggregates the samples into the requested base path (for example outputs/dc_motor_torque.csv). The timeline CSV adds time_s and frame_index columns ahead of the stress tensor values so downstream scripts can correlate torque with the simulation clock.

Usage tips¶

Keep probe contours well inside the simulation domain so bilinear interpolation never samples outside the grid.
Dense grids yield smoother estimates. For coarse meshes consider slightly inflating the loop to avoid sampling immediately adjacent to discretisation artefacts.
Combine with timeline frames to capture torque ripple across electrical angles or to cross-check against virtual-work calculations using the magnetic co-energy helper described below.

Virtual-work cross-check¶

The solver now exposes motorsim::compute_magnetic_coenergy, which evaluates the magnetic co-energy (including permanent-magnet contributions),

\[ W_m = \tfrac{1}{2}\int_{A} \mathbf{B} \cdot (\mathbf{H} + \mathbf{M}) \, \mathrm{d}A, \]

over the 2D slice. When magnetisation is zero this reduces to the familiar \(\tfrac{1}{2}\int \mathbf{B}\cdot\mathbf{H}\,\mathrm{d}A\). Combined with timeline frames at neighbouring rotor angles the virtual-work estimate follows directly from a finite difference,

\[ \tau_z \approx \frac{W_m(\theta + \Delta\theta) - W_m(\theta - \Delta\theta)}{2\,\Delta\theta}. \]

tests/torque_validation_test.cpp exercises the full pipeline by solving the rotor dipole scenario at ±5° offsets, evaluating both the Maxwell stress torque and the co-energy difference, and enforcing a ≤10 % agreement on the CI grid. Any probe requesting torque automatically triggers computeH() and records the co-energy alongside the stress-tensor integral, making the diagnostic available for future report/CSV exports and the three-phase PM motor walkthrough.

The tests/probe_output_test.cpp fixture exercises the ingestion and evaluation path with a synthetic field that generates a known downward force, providing a regression guard for the new feature.

Open in GUI