In the development and implementation of homogenization methods it is important to have a suite of test problems to verify that the method is producing reasonable results. Over the past decade a number of standard problems have been reported in the literature (e.g., the inverted L test) where results have been computed and documented. The links on this page lead to a variety of these and other problems that show how methods perform relative to each other.

In addition to the output and documentation of the examples links to input data sets are given. So, the reader can download the data and use this in their own methods. Then the output can be compared to the results shown on these pages.

The results that are available on these pages have been obtained with the JHomogenizer tool that can be found on this page. So, the interested person can reproduce these results using the JHomogenizer tool. The JHomogenizer Tool can be downloaded from here.

The Benchmark Problems:

• The Periodic Symmetric Cell Problem: This test problem is based on a cell that is on a 4 by 4 grid. The values in the cells on the boundary are assumed to have a coefficient value that is larger than the value interior to the cell. The cell is then repeated periodically on domain of interest. The local problem is defined and solved on this cell.
• The Periodic Inverted L Cell Problem: This test problem is about the simplest aperiodic pattern that one can imagine. As in the symmetric test problem above, the cell is based on a 4 by 4 grid. However, this time one of the interior cells is set to the same value as the border cells and one of the border cells is set to the interior value. The shape of the low value region looks like an upside down or inverted L.
• Basic Fracture Data Problem: In this case, the cell looks like the cell pattern in the symmetric test problem above. However, the border of high values is allowed to vary in size and in particular, to model a fracture, the border value is assumed to be very small.
• Bourgeat Fracture Data Problem: The data in this problem is based on a problem related to modeling a real fracture in a matrix. The data uses realistic coefficient values in an idealized model of a fracture. A rectangular region is split vertically by a thin fracture with a region on either side with coefficient values that are very small.
• ELF Data Problem: In this problem an idealized version of the cross section of a porous media is given. A distinct region of higher coefficient values occurs in such a manner that fluid flowing through the porous media will finger prefentially through the region. As in the fracture problem proposed by Bourgeat described above, the constrast in coefficient values chosen is realistic. There are three distinct coefficient values present in this example.
• Nuclear Waste Repository Problem: In an attempt to apply homogenization methods to nuclear waste storage modeling in porous media this example looks at a single storage area and the coefficient values in the region near the storage area.
• Heating Element Design Problem: Idaho Technologies in Salt Lake City, UT proposed a heting element design problem where coefficient values were specified inside a device for heating a number of samples. The data is real and represents a reasonably tough "periodic" looking problem tied to an optimization problem.
• Stratified Porous Medium Problem: This problem provides a sanity check in multiple dimensions. The values returned by averaging methods should agree with the arithmetic average parallel to the stratification and agree with the harmonic average perpendicular to the stratification. This problem is included as a base line test for methods.