# HW2: Nonlinear Magentic

### Description

Consider an axi-symmetric model of a coil and a plate as depicted in the figure. The inner and outer radii of the coil, r_1 and r_2 , are 5 mm and 10 mm, respectively. The length of the coil, l_1 , is 8 mm. The plate thickness, l_2 , is 3 mm. The coil has 200 turns and is fed by a constant current of I = 20 A. The plate is made out of iron. Iron in general exhibits a non-linear magnetic behavior, which is assumed to be defined by \begin{equation} B(H) = \frac{H}{c_1 H + c_2} + \mu_0H \, , \end{equation} where B and H denote the magnetic flux density and field strength, respectively, and \mu_0 the vacuum permeability. The parameters are given by c_1 = 0.56 Am/N and c_2 = 370 \rm{A^2/N}. Tabulated values of the given function are included in the template files (bh.fnc). For the air domain and the coil the free space permeability can be assumed.

In a first step consider only linearized material behavior and compare the cases with air and iron core. Then, specify a non-linear permeability for the iron core and compare to the linearized case.

### Tasks

- Create a regular hexaedral mesh using Trelis.
**(2 Points)** - Create input-xml files for the required CFS++ simulations.
**(2 Points)** - Document your results by answering the questions below.
**(8 Points)** - Answer the theoretical question.
**(3 Points)**

### Hints

- Use the provided templates
`coil.xml`

and`mat.xml`

. - Adapt the Trelis input
`geometry-ue.jou`

to include the ring. - Look into the *.info.xml to get information about non-linear convergence.
- The workflow for the example is included in the shell script
`run.sh`

. Before you can successfully run it, you have to:- complete the trelis input
`geometry-ue.jou`

(choose a appropriate mesh size) - complete the CFS input
`coil.xml`

- delete the "#" symbol in
`run.sh`

in the necessary lines to execute cfs and trelis (`#trelis...`

and`#cfs...`

)

- complete the trelis input

### Submission

Submit all your input files (`*.jou`

, `*.xml`

) as well as a concise PDF report of your results. Name the archive according to the format `HWn_eNNNNNNN_Lastname.zip`

.

### Material Behavior

Plot the given material behavior B ( H ) for values up to 10 kA/m. Evaluate \frac{\rm d B}{\rm d H}(0)= \mu_{\rm lin} to obtain the linearized material behavior for iron. Also include the linear approximation and the tabulated values (from file `bh.fnc`

) in the plot. **(3 Points)**

### Linear versus non-linear iron core

First start with a linear simulation.

- Plot the magnitude of the magnetic flux density as well as the vectors
**(2 Points)**

After that, perform a nonlinear analysis. Therefore, use the tabulated data to define the input for a non-linear permeability in the iron core.

- How many non-linear iterations were done to solve the problem?
**(1 Point)** - Compare the magnitude of the magnetic flux density between linear and nonlinear result.
What do you observe? Is the linear approximation good?
**(2 Point)**

### Theoretical part

How does coil modeling work in the 2D plane and axisymmetric case? How does the boundary conditions look like?

You can download the templates linked on this page individually by *right click --> save target as*.
Alternativey, head over to the git repository and use the *download button* on the top right:
Here is a direct link to the **zip archive**.