Magpylib Force v0.1.10#

The package magpylib-force provides an addon for magnetic force and torque computation between magpylib source objects.

Installation#

Install magpylib-force with pip:

pip install magpylib-force

API#

The package provides only a single top-level function getFT() for computing force and torque.

import magpylib_force as mforce
mforce.getFT(sources, targets, anchor, eps=1e-5, squeeze=True)

Here sources are Magpylib source objects that generate the magnetic field. The targets are the objects on which the magnetic field of the sources acts to generate force and torque. With current version 0.1.10 only Cuboid, Dipole, and Polyline objects can be targets. The anchor denotes an anchor point which is the barycenter of the target. If no barycenter is given, homogeneous mass density is assumed and the geometric center of the target is also it’s barycenter. eps refers to the finite difference length when computing the magnetic field gradient and should be adjusted to be much smaller than size of the system. squeeze can be used to squeeze the output array dimensions as in Magpylib’s getBHJM.

The computation is based on integrating the magnetic field generated by the sources over the targets, see here for more details. This requires that each target has a meshing directive, which must be provided via an attribute to the object. How meshing is defined:

Cuboid: a 3-vector which denotes the number of equidistant splits along each axis resulting in a rectangular regular grid.
Dipole: no meshing required
PolyLine: a scalar which denotes equidistant splits of each PolyLine segment.

The function getFT() returns force and torque as np.ndarray of shape (2,3), or (t,2,3) when t targets are given.

The following example code computes the force acting on a cuboid magnet, generated by a current loop.

import magpylib as magpy
import magpylib_force as mforce

# create source and target objects
loop = magpy.current.Circle(diameter=2e-3, current=10, position=(0,0,-1e-3))
cube = magpy.magnet.Cuboid(dimension=(1e-3,1e-3,1e-3), polarization=(1,0,0))

# provide meshing for target object
cube.meshing = (5,5,5)

# compute force and torque
FT = mforce.getFT(loop, cube)
print(FT)
# [[ 1.36304272e-03  6.35274710e-22  6.18334051e-20]  # force in N
#  [-0.00000000e+00 -1.77583097e-06  1.69572026e-23]] # torque in Nm

Warning

Scaling invariance does not hold for force computations! Be careful to provide the inputs in the correct units!

Computation details#

The force \(\vec{F}_m\) acting on a magnetization distribution \(\vec{M}\) in a magnetic field \(\vec{B}\) is given by

\[\vec{F}_m = \int \nabla (\vec{M}\cdot\vec{B}) \ dV.\]

The torque \(\vec{T}_m\) which acts on the magnetization distribution is

\[\vec{T}_m = \int \vec{M} \times \vec{B} \ dV.\]

The force \(\vec{F}_c\) which acts on a current distribution \(\vec{j}\) in a magnetic field is

\[\vec{F}_c = \int \vec{j}\times \vec{B} \ dV.\]

And there is no torque. However, one must not forget that a force, when applied off-center, adds to the torque as

\[\vec{T}' = \int \vec{r} \times \vec{F} \ dV,\]

where \(\vec{r}\) points from the body barycenter to the position where the force is applied.

The idea behind magplyib-force is to compute the above integrals by discretization. For this purpose, the target body is split up into small cells using the object meshing attribute. The computation force/torque computation of all cells relies on Magpylib field computation, is itself also fully vectorized, and simply returns the sum of all cells.