How to Homogenize an Symmetric Cell

The following describes how to use the JHomogenizer tool to homogenize a region that has a periodically repeated cell as shown below.

One should think of the single cell on the left periodically repeated a number of times as shown on the right. The theory requires that the size of the cell approach zero relative to the size of the domain to obtain convergence. In practice the homogenization can only be applied to a domain with a finite number of cells. However, the hope is that the size of the single cell is small enough that the approximation will be reasonable.

• If the JHomogenizer interface is not running, start up the interface.
• Once the interface is up and running, you will need to start up the MapEditor by clicking on the Show MapEditor button on the interface.
• Once the MapEditor is up and running you will want to build a simple map. To do this:
  • Click on the Build A Map button in the MapEditor interface. A small dialog box will pop up.
  • In the dialog box enter the dimensions 4, 4, and 1 and a background value of 10.0.
  • Click on the Okay button to complete the basic definition.
• A small 4 by 4 map will pop up. This is a bit too hard to work with. So click on the zoom up button 2 or 3 times or until the cursor will fit entirely into the individual blocks or you feel comfortable with the size.
• Next to set the contrasting L shaped cells, click on the Set Map Values button. Another frame will pop up that lets you set a value that will be used to modify the map. In the Set Tensor Values Frame set the txx, tyy, and tzz values to 1.0.
• Now we can set the cells in the symmetric cell pattern to 1.0. To do this click on the appropriate cells in the 4 by 4 map in the MapEditor tool. That is, click on the (2,2), (2,3), (3,2), and (3,3) blocks in the map where (1,1) is the lower left hand corner and (4,4) the upper right hand corner. As you do this you will see the cells change color.
• Click on the Quit Setting Values button. This is the same button used to start setting values, but the label has changed. The label will change again and the Set Tensor Values Frame will disappear.
• Now we can pull the data back into the JHomogenizer interface.
  • Click on the Get All Map Data button in the JHomogenizer interface. As the data is pull out of the MapEditor tool a message will be updated in the status message area at the bottom.
  • You will be prompted for a name for the map. You might use symmetric_cell as a name to type in.
  • Either end with a carriage return or click on the Okay button.
• To get rid of the MapEditor tool, click on the Kill MapEditor button.
• To compute the homogenized value, choose a homogenization method from the Homogenization Method List in the JHomogenizer interface. For example, click on the HomCode choice in the list.
• Finally, click on the Homogenize! button. The interface will start the calculations and the status will be shown in the Status: line near the bottom of the interface. You will be prompted for a map name for the result of the homogenization process. Enter the name in the dialog and type a carriage return or click on the Okay button.
• You can display the values by clicking on the map in the map list and then you click on Display Values button. The values will be displayed in the text window. The values from the HomCode choice should be the following.
  
  

  6.6514	0.0000	0.0000
  0.0000	6.6514	0.0000
  0.0000	0.0000	7.7500

• Once this is done, other methods can be applied to the initial map by clicking on the map in the map list. In our example, click on symmetric_cell. Then click on another homogenization method choice and finally click on the Homogenize! button. To display the new values, click on the Display Values button.

We can also create an output image of the map using the MapEditor tool. To do this do the following.

• Start up the MapEditor tool.
• Once the map is loaded into the display area, click on the Create PPM File. The interface will write a ppm file that contains an image of the map. The name of the file is map.ppm.
• The ppm file can easily be translated into other graphics formats using xview or other image editing software.

Finally, you can save the map data and all of the homogenized results by clicking on the Save Maps button. The data will be stored in a folder with the same name as the name of the file. All of the homogenized results will be stored in subfolders of the original folder.

To reload the data you can use the instructions in some of the other examples. These instructions will not be repeated here.