Magnetically Permeable Sphere in a Static Magnetic Field

A sphere of relative permeability greater than unity is exposed to a spatially uniform static background magnetic field. Two formulations are used to solve this problem, and the differences between these are discussed. The field strength inside the sphere is computed and compared against the analytic solution.

Figure 1: A magnetically permeable sphere in a spatially uniform, static background magnetic field. The sphere at the center is surrounded by air, and enclosed in a region of Infinite Elements.

Model Definition

Figure 1 shows the model setup, with a 0.25 mm diameter sphere placed in a spatially uniform background magnetic field of strength 1 mT. The computational model consists of three concentric spheres. The innermost is the permeable sphere, the surrounding spherical shell volume represents free space, and the outside shell volume represents a region extending to infinity, modeled with an Infinite Element Domain. When using Infinite Element Domain features, the boundary condition on the outside of the modeling domain does not affect the solution.

The relative permeability of the sphere is varied from μr = 2 to μr = 1000. The analytic solution for the field inside a permeable sphere exposed to a uniform magnetic field is:

Where B0 is the background magnetic field.

There are two ways in which this problem can be formulated. The scalar potential formulation, used in the Magnetic Fields, No Currents interface, solves the magnetic flux conservation equation:

a partial differential equation for the magnetic scalar potential field, Vm:

where the background field is specified in terms of the H-field, Hb. The B-field is then computed from the H-field: B = μrμ0H. The magnetic field is in turn computed from the gradient of the magnetic scalar potential. Because the governing equation evaluates the gradients of a scalar field, the Lagrange element formulation is used. In this formulation, the background field and boundary conditions for this problem are specified purely in terms of derivatives of the Vm field, and the solution is unique up to a constant. To remove this indeterminacy, the value of the magnetic scalar potential must be constrained at one point in the model, to fix the value of the constant.

The vector potential formulation, used in the Magnetic Fields interface, solves an equation for the magnetic vector potential, A:

Where the B-field is the curl of the (A + Ab) field. In this approach, the background field and boundary conditions are specified directly in terms of the A-field. Here, the governing equation takes the curl of a vector valued field, and this problem is solved using a Curl element formulation. This formulation does not require as fine of a mesh as the Lagrange element formulation to achieve the same accuracy.

Results and Discussion

Figure 2 plots the magnetic field for the Magnetic Fields, No Currents interface formulation, and Figure 3 shows the results computed using the Magnetic Fields interface formulation, both for the μr = 1000 case. The fields in the Infinite Element region are not plotted, as these do not have any physical significance.

Figure 4 shows the field enhancement versus the permeability for both cases, along with the analytic solution. The relative difference is plotted in Figure 5. In the limit as the mesh is refined the solutions agree within numerical precision.

There are some differences between the two formulations. In this case, the Magnetic Fields interface slightly underestimates the field strength, while the Magnetic Fields, No Current interface overestimates it. The agreement with the analytic solution for both formulations improves with increasing mesh refinement. Although the Magnetic Fields, No Currents interface require a finer mesh for approximately the same level of accuracy, it does use less total memory. Its drawback is that it cannot be used to model situations where there is any current flowing in the model, or any variation with respect to time.

Figure 2: The magnetic field for the Magnetic Fields, No Currents interface.

Figure 3: The magnetic field for the Magnetic Fields interface.

Figure 4: Comparison of numerical results to analytic result.

Figure 5: Relative difference compared to the analytic solution.

ACDC_Module/Tutorials/permeable_sphere_static_bfield

Modeling Instructions

From the menu, choose .

New

In the window, click .

Model Wizard

In the window, click .

In the tree, select .

Click .

In the tree, select .

Click .

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
r0	0.125[mm]	1.25E-4 m	Radius, magnetically permeable sphere
mu_r	1000	1000	Relative permeability, magnetically permeable sphere
B0	1[mT]	0.001 T	Background magnetic fields
B_analytic	((3mu_r)/(mu_r+2))B0	0.002994 T	Analytic solution for the field inside the permeable sphere

Geometry 1

Sphere 1 (sph1)

Create a sphere with two layers plus an inner core. The outermost layer represents the exterior air region, scaled using the Infinite Element Domain, the median layer is the unscaled air domain, and the core represents the permeable sphere.

In the toolbar, click .

In the window for , locate the section.

In the text field, type r0*3.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (mm)
Layer 1	r0
Layer 2	r0

Click .

Click the button in the toolbar.

Definitions

Create a set of selections before setting up the physics. First, create a selection for the Infinite Element Domain feature.

Explicit 1

In the toolbar, click .

Select Domains 1–4, 10, 11, 14, and 17 only.

Right-click and choose .

In the dialog box, type Infinite Element domains in the text field.

Click .

Add a selection for the analysis domains. This is the complement of the selection.

Complement 1

In the toolbar, click .

In the window for , locate the section.

Under , click .

In the dialog box, select in the list.

Click .

Right-click and choose .

In the dialog box, type Analysis domain in the text field.

Click .

Infinite Element Domain 1 (ie1)

In the toolbar, click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Suppress some domains to get a better view when setting up the physics and reviewing the meshed results.

Hide for Physics 1

In the window, right-click and choose .

Select Domains 2, 6, 11, and 13 only.

Magnetic Fields, No Currents (mfnc)

Set up the first physics — . Begin by specifying the background magnetic field.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Specify the Hb vector as


B0/mu0_const	x
0	y
0	z

Add a constraint point for the magnetic scalar potential.

Zero Magnetic Scalar Potential 1

In the toolbar, click and choose .

Select Point 8 only.

Materials

Assign the material properties. First, use air for all domains.

Add Material

In the toolbar, click to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click to close the window.

Materials

Override the core sphere with a permeable material.

Material 2 (mat2)

In the window, under right-click and choose .

Select Domain 9 only.

In the window for , 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	mu_r	1	Basic

The setting creates a swept mesh for the Infinite Element Domain.

Mesh 1

In the window, under click .

In the window for , click .

Study 1

Step 1: Stationary

In the window, under click .

In the window for , click to expand the section.

Select the check box.

Click .

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
mu_r (Relative permeability, magnetically permeable sphere)	2 4 10 20 40 100 200 400 1000

In the toolbar, click .

Results

Datasets

The default plot shows the magnetic flux density for all domains. Add a selection to the current solution to visualize only the Analysis domain.

Selection

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Magnetic Flux Density Norm (mfnc)

Add an arrow plot showing the direction of the magnetic flux density.

Arrow Volume 1

In the window, right-click and choose .

In the window for , locate the section.

Find the subsection. In the text field, type 20.

Find the subsection. In the text field, type 1.

In the toolbar, click .

Compare the plot with Figure 2.

Repeat the analysis with the interface.

Add Physics

In the toolbar, click to open the window.

Go to the window.

In the tree, select .

Find the subsection. In the table, clear the check box for .

Click in the window toolbar.

In the toolbar, click to close the window.

Root

In the window, click the root node.

Add Study

In the toolbar, click to open the window.

Go to the window.

Find the subsection. In the tree, select .

Find the subsection. In the table, clear the check box for .


Physics	Solve
Magnetic Fields, No Currents (mfnc)

Click in the window toolbar.

In the toolbar, click to close the window.

Study 2

Step 1: Stationary

Set up the Magnetic Fields interface. Specify the background field in terms of a magnetic vector potential.

Magnetic Fields (mf)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Specify the Ab vector as


0	x
0	y
B0*y	z

Materials

Material 2 (mat2)

Assign the material properties to the permeable sphere.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	mu_r	1	Basic