Inverted L Test Problem for Numerical Homogenization Methods

Contents:
Map Image of Inverted L
Problem Description
This page describes the inverted-L periodic test problem for homogenization tests. This test problem is about the simplest aperiodic pattern that one can imagine. As in the symmetric test problem, 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.
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 inverted-L pattern. The results are for various ratios of coefficients for the L portion of the pattern and for the complement of the region. 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:
5.5741 -0.3482 0.0000
-0.3482 6.9488 0.0000
0.0000 0.0000 7.7500
  Linear Boundary Condition Average:
5.5752 0.0000 0.0000
0.0000 6.8340 0.0000
0.0000 0.0000 7.7500
  Fractured Media Tensor Average: Not Applicable
  Linear Boundary Condition Wavelet Average:
7.5647 -0.2046 0.0000
-0.2046 7.1750 0.0000
0.0000 0.0000 7.7500
  Periodic Wavelet Average:
5.8595 0.0000 0.0000
0.0000 7.4090 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:
43.5643 -5.8841 0.0000
-5.8841 63.9012 0.0000
0.0000 0.0000 75.2500
  Linear Boundary Condition Average:
49.4521 -6.18236 0.0000
-6.18236 64.2331 0.0000
0.0000 0.0000 7.7500
  Fractured Media Tensor Average: Not Applicable
  Periodic Wavelet
48.8806 0.0000 0.0000
0.0000 70.9470 0.0000
0.0000 0.0000 75.2500
  Linear Boundary Condition Wavelet Average:
72.9334 -2.6070 0.0000
-2.6070 67.6969 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:
420.489 -65.3553 0.0000
-65.3553 632.461 0.0000
0.0000 0.0000 750.25
  Linear Boundary Condition Average:
482.022 -65.367 0.0000
-65.367 635.799 0.0000
0.0000 0.0000 7.7500
  Fractured Media Tensor Average: Not Applicable
  Periodic Wavelet
477.471 0.0000 0.0000
0.0000 706.242 0.0000
0.0000 0.0000 750.25
  Linear Boundary Condition Wavelet Average:
726.582 -26.6913 0.0000
-26.6913 672.668 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 inverted-L cell heterogeneous used in the simulations is shown below.






In this example the inverted-L 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 Results

Elliptic Solution Flux Solution Flow Solution 1 Flow Solution 2

Geometric Average Results

Elliptic Solution Flux Solution Flow Solution 1 Flow Solution 2

HomCode Average Results

Elliptic Solution Flux Solution Flow Solution 1 Flow Solution 2

Harmonic Average Results

Elliptic Solution Flux Solution Flow Solution 1 Flow Solution 2

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