# 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 $$B(H) = \frac{H}{c_1 H + c_2} + \mu_0H \, ,$$ 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.

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 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...)

### 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

• 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?