# HW1: Linear Magentic

### Description

We consider the axi-symmetric geometry (2D model) of a ring-shaped permanent magnet as depicted in the sketch. The inner radius of the ring $r_1$ is 5 mm, the outer radius $r_2$ is 10 mm, and the height of the ring $l_1$ is 8 mm. The size of the simulation domain should be chosen sufficiently large, such that the solution is not significantly influenced by the boundary conditions at the outer edge of the domain. The ring-shaped permanent magnet is assumed to show a perfect magnetization $M$ of 1.2 T in z-direction. The core of the ring is either air, copper, aluminum or iron.

Use the material air for the domain of the permanent magnet. In this homework we assume linear material behavior. For the air domain and the magnet the free space magnetic permeability $\mu_0$ can be assumed.

1. Create a regular hexaedral mesh using Trelis. (2 Points)
2. Create input-xml files for the required CFS++ simulations. (2 Points)
4. Answer the theoretical question. (3 Points)

### Hints

• Use the provided templates: simulation input magnet.xml and material file mat.xml
• Adapt the Trelis input geometry-ue.jou to include the ring.
• The complete workflow for the example is included in the shell script run.sh, uncomment the necessary lines (remove leading #) to execute cfs and trelis (#trelis ... and #cfs ... and #sed ...)
• A distance $d \approx 2r_2$ from the ring to the outer boundary should be sufficient.

### Submission

Submit all your input files (*.jou, *.xml) as well as a concise PDF report of your results.

### Modeling Assumptions

What are the modeling assumptions in the analyses? Consider the used PDE (and the assumptions necessary to derive it) as well as material behavior and boundary conditions. How is the free space magnetic permeability defined? (2 Points)

### Material Behavior

What is the difference between the materials iron, copper and aluminum in magnetics? (1 Point)

### Material Influence

Perform two different linear analyses using iron and air for the material in the core region.

• Plot the magnetic vector potential (1 Points)
• Plot the magnetic field lines and the vectors of the magnetic flux density (2 Points)

What are the differences between iron and air? (1 Point) Now use the materials aluminum and copper for the core and compare them. Is there a difference in the result? Give a short explanation if there is one or not? (1 Point)

### Theoretical part

Describe the PDE for solving electromagnetics, including the eddy current parts. How is the magnetic vector potential introduced?