## How to model Magnetism in Abaqus

In December 2020, I shared a blog article on how I used Abaqus and 3D Experience to model ice solidification in a skating ring. I thought of sharing something different and exciting this year as well. Very few users think of modeling magnets in Abaqus. However, it is correct that Abaqus can be used to model both electromagnets as well permanent magnets. In this blog we will discuss few methods to model permanent magnets. Some of them are straightforward and easy.

**Subroutines to model permanent magnets**

Magnetic field arises due to motion of charged particles. This is evident by the fact that any current carrying coil creates a magnetic field pattern around it. Most of us have read it in engineering and the application lies in electric motors and electric drive vehicles. But permanent magnets don’t carry current. Then what is the source of magnetic field in such magnets! Well, in reality current exists in permanent magnets at atomic level due to spin motion of electrons thus giving rise to magnetic fields.

In subroutine approach, we define the magnitude and direction of such currents and then use the UDECURRENT subroutine of Abaqus to model such currents. We demonstrate this using the horse shoe magnet example. This magnet has a magnetic flux density (B) of 1.3 Tesla and is directed along its longest edge as shown.

The remnant magnetization can be defined as follows for the three sections of this horseshoe magnet.

This vector is expressed in terms of cylindrical coordinate system with origin at the center of circular section of the magnet. Mr is another physical term called as remanence and it can be expressed in terms of magnetic field intensity as

Mr = B/(4*Pi*1e-7)

The equivalent surface and volume current densities are given by

Where n is the surface normal unit vector for each face of the two straight sections of horseshoe magnet. There are 12 such sections in total. The surface current densities on these faces is a single numerical value that can be directly given as input to abaqus. The volume current density exists only in the circular section of the magnet. It is a radially varying non uniform current density that requires the UDECURRENT routine.

Once all these current densities are defined in the magnetostatic step of Abaqus, the model is ready for submission. The output of this model is as follows. The magnetic flux density (EMB) output matches with the actual value of this horseshoe magnet.

This approach works well for magnets of simple geometry. The subroutine is not required for magnets without any curvature as the current densities for such shares is a single numerical value to be entered directly to Abaqus. Sounds complicated for complex shapes! There are other approaches as well.

The good news is that it is possible to compute the magnetic flux density in a magnetostatic step if the permeability of magnet is known. The subroutine-based approach is needed only if magnetic field is to be computed from associated currents.

**Plug in approach to compute magnetic force**

Interested in computing the force between two magnets! Abaqus CAE has a plug-in to do this. Whatever way the magnetic field is defined, this plugin works to compute force and torque between two magnets. The plug-in works by integrating the Maxwell stress tensor over the wrapped surface of magnet. The Maxwell tensor is defined as follows and it requires both the magnetic flux density and magnetic field intensity as inputs.

Plug in is placed in the plug in folder of installation directory. It should be launched when odb is loaded. The plug-in asks for odb file, step and frame, instance on which force or torque is needed and name of element based wrapped surface over magnet.

The user should take caution that direction of wrapped surface points inwards towards the magnet. It is also worth mentioning that keeping other properties constant, the force between two magnets depends on the medium between them. Hence the surrounding air medium should be well defined with its associated mesh and permeability.