Symmetric Cell Test Problem for Numerical Homogenization Methods

Contents:
Map Image of Symmetric Cell
Problem Description
This page describes the symmetric cell periodic test problem for homogenization tests. This test problem is about the simplest periodic pattern that one can imagine. As in the inverted-L test problem, the cell is based on a 4 by 4 grid. However, this time the pattern of cells is symmetric.
Problem/Data Links:
The following is a list of raw data sets that can be downloaded and used as input for your own homogenization codes for testing, input to reservoir simulators and other applications. These maps can also be downloaded and used as input for the JHomogenizer application.
Homogenization Results:
The following documents the results for various homogenization methods applied to the symmetric cell pattern. The results are for various ratios of coefficients for the interior portions of the pattern to the boundary cells of the pattern. The values used are given at the top of the table along with the output tensor values in the last column. These results are used to generate some of the data sets above that can be used to compare simulation results on the heterogeneous data sets.

Coefficient Ratio: Homogenization Method: Homogenization Tensor:
10:1    
  Arithmetic Average:
7.7500 0.0000 0.0000
0.0000 7.7500 0.0000
0.0000 0.0000 7.7500
  Geometric Average:
5.6234 0.0000 0.0000
0.0000 5.6234 0.0000
0.0000 0.0000 5.6234
  Harmonic Average:
3.0769 0.0000 0.0000
0.0000 3.0769 0.0000
0.0000 0.0000 3.0769
  HomCode Average:
6.6514 0.0000 0.0000
0.0000 6.6514 0.0000
0.0000 0.0000 7.7500
  Linear Boundary Condition Average:
6.6682 0.0000 0.0000
0.0000 6.6682 0.0000
0.0000 0.0000 7.7500
  Fractured Media Tensor Average: Not Applicable
  Linear Boundary Condition Wavelet Average:
7.5050 0.0000 0.0000
0.0000 7.5050 0.0000
0.0000 0.0000 7.7500
  Periodic Wavelet Average:
7.09677 0.0000 0.0000
0.0000 7.09677 0.0000
0.0000 0.0000 7.7500
100:1    
  Arithmetic Average:
75.2500 0.0000 0.0000
0.0000 75.2500 0.0000
0.0000 0.0000 75.2500
  Geometric Average:
31.6227 0.0000 0.0000
0.0000 31.6227 0.0000
0.0000 0.0000 75.2500
  Harmonic Average:
3.8835 0.0000 0.0000
0.0000 3.8835 0.0000
0.0000 0.0000 3.8835
  HomCode Average:
61.0648 0.0000 0.0000
0.0000 61.0648 0.0000
0.0000 0.0000 75.2500
  Linear Boundary Condition Average:
61.2806 0.0000 0.0000
0.0000 61.2806 0.0000
0.0000 0.0000 7.7500
  Fractured Media Tensor Average: Not Applicable
  Periodic Wavelet Average:
67.1096 0.0000 0.0000
0.0000 67.1096 0.0000
0.0000 0.0000 75.2500
  Linear Boundary Condition Wavelet Average:
72.1973 0.0000 0.0000
0.0000 72.1973 0.0000
0.0000 0.0000 75.2500
1000:1    
  Arithmetic Average:
750.25 0.0000 0.0000
0.0000 750.25 0.0000
0.0000 0.0000 750.25
  Geometric Average:
177.83 0.0000 0.0000
0.0000 177.83 0.0000
0.0000 0.0000 750.25
  Harmonic Average:
3.9880 0.0000 0.0000
0.0000 3.9880 0.0000
0.0000 0.0000 3.9880
  HomCode Average:
604.818 0.0000 0.0000
0.0000 604.818 0.0000
0.0000 0.0000 750.25
  Linear Boundary Condition Average:
607.029 0.0000 0.0000
0.0000 607.029 0.0000
0.0000 0.0000 7.7500
  Fractured Media Tensor Average: Not Applicable
  Periodic Wavelet Average:
667.110 0.0000 0.0000
0.0000 667.110 0.0000
0.0000 0.0000 750.25
  Linear Boundary Condition Wavelet Average:
719.072 0.0000 0.0000
0.0000 719.072 0.0000
0.0000 0.0000 750.25
Simulation Results:
The following graphics show results for simulations performed on a heterogeneous maps and the homogenized maps that result from applying various methods of averaging. The heterogeneous map used in the generation of the flow simulations looks like the map below. The simulation results shown below were obtained using maps generated by the JHomogenizer tool. The graphics below were generated using the JHomogenizer tool.

Example 1: 8x8 Cell Pattern - 32x32 blocks:

To show how using homogenization results effect the solution of the elliptic problem the following case is considered. The symmetric cell heterogeneous used in the simulations is shown below.



In this example the symmetric cell is repeated in two dimensions a total of 8 times in each of the coordinate directions. This means that there are a total of 32 small blocks blocks in the region on which the simulations are performed. The set of figures below show the saturation from the solution of a simple two phase flow equation. The fluxes are determined from a solution of an elliptic equation for the head/pressure variable. The use of the lowest order mixed finite element method gives the approximations of the fluxes that are used as input into a simple upwind method for approximately solving the flow equation.

Solutions below are shown for the heterogeneous data set and data sets that result from the homogenization methods.

Heterogeneous Simulation Results
Elliptic Solution Flux Solution Flow Solution 1 Flow Solution 2
Homogenized Simulation Results

Arithmetic Average

Elliptic Solution Flux Solution Flow Solution 1 Flow Solution 2

Geometric Average

Elliptic Solution Flux Solution Flow Solution 1 Flow Solution 2

HomCode Average

Elliptic Solution Flux Solution Flow Solution 1 Flow Solution 2

Harmonic Average

Elliptic Solution Flux Solution Flow Solution 1 Flow Solution 2

Steps to use JHomogenizer to create and work with the symmetric cell:
To create the results included on these pages you can use the JHomogenizer tool. The instructions for creating an symmetric cell and the homogenization results are in the following how to file.