COMSOL 6.4 - Permanent Magnet

This example shows how to model the magnetic field surrounding a permanent magnet. It also computes the force with which it acts on a nearby iron rod. Thanks to the symmetry of the geometry and the antisymmetry of the magnetic field, only one fourth of the geometry needs to be modeled.

Figure 1: A full 3D view of the geometry. Left-right and top-down symmetry is used to minimize the problem size.

Model Definition

In a current free region, where

it is possible to define the scalar magnetic potential, Vm, from the relation

This is analogous to the definition of the electric potential for static electric fields.

Using the constitutive relation between the magnetic flux density and magnetic field

where Br is the remanent flux density, and together with the equation

you can derive an equation for Vm,

The model uses this equation by selecting the Magnetic Fields, No Currents interface from the AC/DC Module.

Boundary Conditions

The magnetic field is symmetric with respect to the xy-plane and antisymmetric with respect to the xz-plane. These planes therefore serve as exterior boundaries to the geometry.

On the symmetry plane, the magnetic field is tangential to the boundary. This is described by the Magnetic Insulation condition:

On the antisymmetry plane, the magnetic field is perpendicular to the boundary. This condition is represented by a constant magnetic scalar potential. The model uses the Zero Magnetic Scalar Potential condition.

If the air box is sufficiently large, the boundary condition used on its remaining exterior boundaries has little influence on the field in the vicinity of the magnet. Although an infinite element domain would give the very best results, this model uses the magnetic insulation condition for convenience.

Results and Discussion

The force on the rod is calculated internally as an integral of the surface stress tensor over all boundaries of the rod. The expression for the stress tensor reads

where n1 is the boundary normal pointing out from the rod and T2 the stress tensor of air. The integration gives 3.64 N, which corresponds to one quarter of the rod. The actual force on the rod is therefore four times this value, or 14.6 N.

ACDC_Module/Introductory_Magnetostatics/permanent_magnet

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Geometry 1

Import 1 (imp1)

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file permanent_magnet.mphbin.

Click

The imported geometry contains the permanent magnet and the rod that it is acting on. The following instructions show you how to create the air box and delete the part of the geometry that you do not want to include in the model.

Block 1 (blk1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.25.

In the text field, type 0.1.

Locate the section. In the text field, type -0.1.

Right-click and choose .

The air box now covers only the parts of the magnet and the rod that you want to include in the model. Perform a Boolean geometry operation to get rid of the superfluous parts.

Compose 1 (co1)

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select both objects.

In the window for , locate the section.

In the text field, type blk1+imp1*blk1.

Click

Click the

button in the toolbar.

The geometry now contains the air volume and one fourth of the imported objects.

Magnetic Fields, No Currents (mfnc)

Magnetic Flux Conservation in Solids 1

In the toolbar, click

and choose .

Select Domains 2–4 only.

Materials

Iron

In the window, under right-click and choose .

In the window for , type Iron in the text field.

Select Domains 2 and 4 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	4000	1	Basic

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > > > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Materials

N54 (Sintered NdFeB) (mat2)

In the window for , locate the section.

Click

In the dialog, type 3 in the text field.

Click .

Magnetic Fields, No Currents (mfnc)

Magnet 1

In the toolbar, click

and choose .

Select Domain 3 only.

North 1

In the window, click .

Select Boundary 17 only.

South 1

In the window, click .

Select Boundary 12 only.

All exterior boundaries are magnetically insulated by default. Use the condition on those boundaries where antisymmetry holds.

Symmetry Plane 1

In the toolbar, click

and choose .

In the window, click .

Select Boundaries 2, 8, and 24 only.

In the window for , locate the section.

From the list, choose .

Next, add a force computation on the rod.

Force Calculation 1

In the toolbar, click

and choose .

Select Domain 2 only.

In the window for , locate the section.

In the text field, type rod.

Study 1

Default mesh is sufficient in this case where the force is to be computed only on the rod, and we can directly go to the solution step.

In the window, click .

In the window for , locate the section.

Clear the checkbox.

In the toolbar, click

Results

3D Plot Group 1

In the toolbar, click

Slice 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose > > > .

Locate the section. From the list, choose .

From the list, choose .

In the text field, type 0.005.

Locate the section. From the list, choose .

In the toolbar, click

The plot shows the magnitude of the flux density just above the symmetry plane. Add an arrow plot to see its direction.

Magnetic Flux Density

In the window, under click .

In the window for , type Magnetic Flux Density in the text field.

Arrow Volume 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose > > > .

Locate the section. Find the subsection. In the text field, type 100.

Find the subsection. In the text field, type 50.

Find the subsection. From the list, choose .

In the text field, type 0.0051.

In the toolbar, click

The plot shows only the quarter of the geometry used for the computation.

Introduce additional mirror datasets to plot the solution in the complete geometry.

Symmetry Condition

In the toolbar, click

and choose .

In the window for , type Symmetry Condition in the text field.

Locate the section. From the list, choose .

Antisymmetry Condition

In the toolbar, click

and choose .

In the window for , type Antisymmetry Condition in the text field.

Locate the section. From the list, choose .

Click to expand the section. Find the subsection. From the list, choose .

Magnetic Flux Density

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Arrow Volume 1

In the window, click .

In the window for , click in the upper-right corner of the section. From the menu, choose > > > .

In the toolbar, click

Add Global Evaluation under Derived Values to statically store the force on the rod for the default mesh.

Force on Rod (Default Mesh)

In the toolbar, click

In the window for , click in the upper-right corner of the section. From the menu, choose > > > > .

The variable containing the force on a quarter of the rod will be added to the table. Multiply this value by 4 to compute the total force on the rod.

In the column, change the expression to mfnc.Forcex_rod*4.

In the column, type Total force on the rod.

In the text field, type Force on Rod (Default Mesh).

Click

The total force on the rod evaluates to 15 N.

Now compute the force also on the horse-shoe magnet.

Magnetic Fields, No Currents (mfnc)

Force Calculation 2

In the toolbar, click

and choose .

Select Domains 3 and 4 only.

In the window for , locate the section.

In the text field, type horseShoeMagnet.

Boundary Force Calculation 1

In the toolbar, click

and choose .

Select Boundaries 1, 4, 5, and 25 only.

In the window for , locate the section.

In the text field, type exteriorBoundaries.

Mesh 1

Build the default mesh.

In the window, under right-click and choose .

For the magnet we need a finer mesh near to its sharp corners. Such a mesh can be specified by editing the default mesh.

Right-click > and choose .

Size 2

Click the

button in the toolbar.

In the window, under > click .

In the window for , locate the section.

Click

Select Edges 21, 24–26, and 28 only.

Click to expand the section. Locate the section. Click the button.

Locate the section.

Select the checkbox. In the associated text field, type 0.3[mm].

Size 1

In the window, click .

In the window for , locate the section.

Select the checkbox. In the associated text field, type 1.1.

Boundary Layer Properties 2

In the window, expand the > > node, then click .

Select Boundaries 6, 9–13, 15, 16, 18, and 20–23 only.

In the window for , locate the section.

In the text field, type 1.

Study 1

In the toolbar, click

Results

Force Calculation Results

In the toolbar, click

In the window for , type Force Calculation Results in the text field.

Global Evaluation 1

Right-click and choose .

Add the forces computed from the Force Calculation features, multiply the relevant ones by 4 to account for symmetries, and compute the forces on the rod and the horse-shoe magnet.

In the window for , click in the upper-right corner of the section. From the menu, choose > > > > .

Click in the upper-right corner of the section. From the menu, choose > > > > .

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
4*mfnc.Forcex_rod	N	Total electromagnetic force, rod
4*mfnc.Forcex_horseShoeMagnet	N	Total electromagnetic force, horse shoe magnet

Click in the upper-right corner of the section. From the menu, choose > > > > .

In the toolbar, click

Appreciate that the force on the rod is changing just slightly, that the force on the horse-shoe magnet is approximately minus the force on the rod, and that the effect of exterior boundaries is negligible.

Force Calculation Results

Go to the window.