How to Homogenize an Inverted-L
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.
The Steps:
In the following you will start from scratch. You will generate the cell using
the MapEditor, apply a few homogenization methods and then simulate flow through
the domain to compare the methods.
- 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 inverted-L 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 (1,3), (2,3), (3,3), and (3,2) 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
invertedL 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.
5.5741 |
-0.3482 |
0.0000 |
-0.3482 |
6.9488 |
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 invertedL.
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.