Heating Element Design Test Problem for Numerical Homogenization Methods

Contents:
Map Image of the well.
Problem Description
This page describes the heating element design test problem for homogenization tests. This test problem is based on an actual problem proposed by David Eyre at Idaho Technologies in Salt Lake City, Utah. The data file contains diffusivity coefficients defined on a 128 by 128 grid. The test involves the creation of coarser homogenized grids for the determination of temperature distribution within the well. Cases shown below will present coarser map data that can be used as input to codes that solve the heat equation in the well.

The initial work on this problem was done by Daniel Balls, a graduate student in the Department of Mathematics and Statistics at Utah State University (and now on the faculty at BYU-Idaho) along with his advisor, Jim Powell. The goal of their work was to solve an optimization problem to design a heating element that would provide the best heat distribution within the well. The single well is one of a large number of wells in a microarray used in the PCR process developed by Idaho Technology. The goal of the work was to design a heating element that created a uniform heating of each well.

The optimization problem requires the solution of the heat equation many times for the microarray. Basically, the boundary condition along the bottom of the well is changed to reflect different heating conditions along the bottom of the well. The amount of computation even on a 128 by 128 grid is expensive. The idea of using homogenization was to try to estimate the optimal heating design on a coarser grid. Once a coarse estimate is determined this may be used as an initial guess on the fine grid simulations to further refine the location of the optimizer.

As will be seen homogenization of the entire region will not produce reasonable results. So, we show results on 8 by 8 and 16 by 16 grids for comparison. The results of the work for Idaho Technology are not presented here and the interested reader is encouraged to contact Daniel Balls or Jim Powell for more details.

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 heat equation solver or 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 reservoir data. In this example problem the results on an 8 by 8 grid will be shown on this page. The 16 by 16 coarse grid maps will not be shown due to the size of the data sets.

Arithmetic Average:

* 1 2 3 4 5 6 7 8
8
0.55147 0.0 0.0
0.0 0.55147 0.0
0.0 0.0 0.55147
0.08698 0.0 0.0
0.0 0.08698 0.0
0.0 0.0 0.08698
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.07371 0.0 0.0
0.0 0.07371 0.0
0.0 0.0 0.07371
0.59697 0.0 0.0
0.0 0.59697 0.0
0.0 0.0 0.59697
7
0.96104 0.0 0.0
0.0 0.96104 0.0
0.0 0.0 0.96104
0.11723 0.0 0.0
0.0 0.11723 0.0
0.0 0.0 0.11723
0.20721 0.0 0.0
0.0 0.20721 0.0
0.0 0.0 0.20721
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.20058 0.0 0.0
0.0 0.20058 0.0
0.0 0.0 0.20058
0.15990 0.0 0.0
0.0 0.15990 0.0
0.0 0.0 0.15990
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
6
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.45666 0.0 0.0
0.0 0.45666 0.0
0.0 0.0 0.45666
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 0.00154
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 0.00154
0.51733 0.0 0.0
0.0 0.51733 0.0
0.0 0.0 0.51733
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
5
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.82831 0.0 0.0
0.0 0.82831 0.0
0.0 0.0 0.82831
0.00156 0.0 0.0
0.0 0.00156 0.0
0.0 0.0 0.00156
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.01294 0.0 0.0
0.0 0.01294 0.0
0.0 0.0 0.01294
0.87761 0.0 0.0
0.0 0.87761 0.0
0.0 0.0 0.87761
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
4
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.22532 0.0 0.0
0.0 0.22532 0.0
0.0 0.0 0.22532
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00153 0.0 0.0
0.0 0.00153 0.0
0.0 0.0 0.00153
0.28600 0.0 0.0
0.0 0.28600 0.0
0.0 0.0 0.28600
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
3
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.59318 0.0 0.0
0.0 0.59318 0.0
0.0 0.0 0.59318
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 0.00154
0.00155 0.0 0.0
0.0 0.00155 0.0
0.0 0.0 0.00155
0.65386 0.0 0.0
0.0 0.65386 0.0
0.0 0.0 0.65386
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
2
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.91933 0.0 0.0
0.0 0.91933 0.0
0.0 0.0 0.91933
0.25186 0.0 0.0
0.0 0.25186 0.0
0.0 0.0 0.25186
0.27083 0.0 0.0
0.0 0.27083 0.0
0.0 0.0 0.27083
0.94587 0.0 0.0
0.0 0.94587 0.0
0.0 0.0 0.94587
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
1
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 0.98316

Geometric Average:

* 1 2 3 4 5 6 7 8
8
0.05997 0.0 0.0
0.0 0.05997 0.0
0.0 0.0 0.05997
0.01136 0.0 0.0
0.0 0.01136 0.0
0.0 0.0 0.01136
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.00836 0.0 0.0
0.0 0.00836 0.0
0.0 0.0 0.00836
0.08105 0.0 0.0
0.0 0.08105 0.0
0.0 0.0 0.08105
7
0.90189 0.0 0.0
0.0 0.90189 0.0
0.0 0.0 0.90189
0.00421 0.0 0.0
0.0 0.00421 0.0
0.0 0.0 0.00421
0.18342 0.0 0.0
0.0 0.18342 0.0
0.0 0.0 0.18342
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.15733 0.0 0.0
0.0 0.15733 0.0
0.0 0.0 0.15733
0.00501 0.0 0.0
0.0 0.00501 0.0
0.0 0.0 0.00501
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
6
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.03202 0.0 0.0
0.0 0.03202 0.0
0.0 0.0 0.03202
0.00153 0.0 0.0
0.0 0.00153 0.0
0.0 0.0 0.00153
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 0.00154
0.04785 0.0 0.0
0.0 0.04785 0.0
0.0 0.0 0.04785
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
5
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.37468 0.0 0.0
0.0 0.37468 0.0
0.0 0.0 0.37468
0.00156 0.0 0.0
0.0 0.00156 0.0
0.0 0.0 0.00156
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00168 0.0 0.0
0.0 0.00168 0.0
0.0 0.0 0.00168
0.51923 0.0 0.0
0.0 0.51923 0.0
0.0 0.0 0.51923
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
4
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.00692 0.0 0.0
0.0 0.00692 0.0
0.0 0.0 0.00692
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00153 0.0 0.0
0.0 0.00153 0.0
0.0 0.0 0.00153
0.01035 0.0 0.0
0.0 0.01035 0.0
0.0 0.0 0.01035
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
3
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.07904 0.0 0.0
0.0 0.07904 0.0
0.0 0.0 0.07904
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 0.00154
0.00155 0.0 0.0
0.0 0.00155 0.0
0.0 0.0 0.00155
0.11810 0.0 0.0
0.0 0.11810 0.0
0.0 0.0 0.11810
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
2
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.68432 0.0 0.0
0.0 0.68432 0.0
0.0 0.0 0.68432
0.00822 0.0 0.0
0.0 0.00822 0.0
0.0 0.0 0.00822
0.00933 0.0 0.0
0.0 0.00933 0.0
0.0 0.0 0.00933
0.81575 0.0 0.0
0.0 0.81575 0.0
0.0 0.0 0.81575
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
1
0.98298 0.0 0.0
0.0 0.98298 0.0
0.0 0.0 0.98298
0.98298 0.0 0.0
0.0 0.98298 0.0
0.0 0.0 0.98298
0.98298 0.0 0.0
0.0 0.98298 0.0
0.0 0.0 0.98298
0.98298 0.0 0.0
0.0 0.98298 0.0
0.0 0.0 0.98298
0.98298 0.0 0.0
0.0 0.98298 0.0
0.0 0.0 0.98298
0.98298 0.0 0.0
0.0 0.98298 0.0
0.0 0.0 0.98298
0.98298 0.0 0.0
0.0 0.98298 0.0
0.0 0.0 0.98298
0.98298 0.0 0.0
0.0 0.98298 0.0
0.0 0.0 0.98298

HomCode Average:

* 1 2 3 4 5 6 7 8
8
0.00742 -5.7464 0.0
-5.7464 0.00576 0.0
0.0 0.0 0.55147
0.00287 -1.0512 0.0
-1.0512 0.07371 0.0
0.0 0.0 0.08698
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.00258 9.65973 0.0
9.65973 0.06022 0.0
0.0 0.0 0.07371
0.01129 6.25015 0.0
6.25015 0.00593 0.0
0.0 0.0 0.59697
7
0.93201 0.0 0.0
0.0 0.95774 0.0
0.0 0.0 0.96104
0.00220 -2.4530 0.0
-2.4530 0.04114 0.0
0.0 0.0 0.11723
0.19052 -0.0019 0.0
-0.0019 0.20401 0.0
0.0 0.0 0.20721
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.16109 0.00583 0.0
0.00583 0.19394 0.0
0.0 0.0 0.20058
0.00223 3.18739 0.0
3.18739 0.04844 0.0
0.0 0.0 0.15990
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
6
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.00348 -2.0232 0.0
-2.0232 0.38443 0.0
0.0 0.0 0.45666
0.00153 -1.9603 0.0
-1.9603 0.00153 0.0
0.0 0.0 0.00153
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00154 1.96461 0.0
1.96461 0.00154 0.0
0.0 0.0 0.00154
0.00405 2.34707 0.0
2.34707 0.44535 0.0
0.0 0.0 0.51733
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
5
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.41424 -0.0565 0.0
-0.0565 0.76961 0.0
0.0 0.0 0.82831
0.00155 -1.9100 0.0
-1.9100 0.00156 0.0
0.0 0.0 0.00156
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00159 2.44531 0.0
2.44531 0.00164 0.0
0.0 0.0 0.01294
0.60578 0.04234 0.0
0.04234 0.83188 0.0
0.0 0.0 0.87761
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
4
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.00235 -2.2039 0.0
-2.2039 0.13705 0.0
0.0 0.0 0.22532
0.00152 -1.6840 0.0
-1.6840 0.00152 0.0
0.0 0.0 0.00152
0.00152 2.24542 0.0
2.24542 0.00153 0.0
0.0 0.0 0.00153
0.00259 2.16581 0.0
2.16581 0.20549 0.0
0.0 0.0 0.28600
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
3
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.00541 -4.3467 0.0
-4.3467 0.51585 0.0
0.0 0.0 0.59318
0.00154 -2.2699 0.0
-2.2699 0.00154 0.0
0.0 0.0 0.00154
0.00155 2.27488 0.0
2.27488 0.00155 0.0
0.0 0.0 0.00155
0.00698 5.85186 0.0
5.85186 0.57660 0.0
0.0 0.0 0.65386
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
2
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.77097 -0.0242 0.0
-0.0242 0.88969 0.0
0.0 0.0 0.91933
0.24527 1.88129 0.0
1.88129 0.00220 0.0
0.0 0.0 0.25186
0.24742 3.95452 0.0
3.95452 0.00268 0.0
0.0 0.0 0.27082
0.86430 0.01069 0.0
0.01069 0.93191 0.0
0.0 0.0 0.94587
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
1
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98316
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98316

Linear Boundary Condition Average:

* 1 2 3 4 5 6 7 8
8
0.33389 -0.1723 0.0
-0.1723 0.28004 0.0
0.0 0.0 1.00000
0.03750 -0.0141 0.0
-0.0141 0.08128 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.02816 0.01282 0.0
0.01282 0.06843 0.0
0.0 0.0 1.00000
0.37554 0.18357 0.0
0.18357 0.28503 0.0
0.0 0.0 1.00000
7
0.95005 -0.0012 0.0
-0.0012 0.96043 0.0
0.0 0.0 1.00000
0.03822 -0.0268 0.0
-0.0268 0.10210 0.0
0.0 0.0 1.00000
0.20047 -0.0021 0.0
-0.0021 0.20605 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.18255 0.00613 0.0
0.00613 0.19771 0.0
0.0 0.0 1.00000
0.05308 0.03742 0.0
0.03742 0.14011 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
6
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.23354 -0.0790 0.0
-0.0790 0.42122 0.0
0.0 0.0 1.00000
0.00153 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 1.00000
0.28550 0.08224 0.0
0.08224 0.48071 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
5
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.62276 -0.0727 0.0
-0.0727 0.79663 0.0
0.0 0.0 1.00000
0.00156 0.0 0.0
0.0 0.00156 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00404 0.00187 0.0
0.00187 0.01148 0.0
0.0 0.0 1.00000
0.74700 0.04594 0.0
0.04594 0.85697 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
4
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.08515 -0.0524 0.0
-0.0524 0.19806 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00153 0.0 0.0
0.0 0.00153 0.0
0.0 0.0 1.00000
0.11634 0.06313 0.0
0.06313 0.25482 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
3
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.35543 -0.0881 0.0
-0.0881 0.55283 0.0
0.0 0.0 1.00000
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 1.00000
0.00155 0.0 0.0
0.0 0.00155 0.0
0.0 0.0 1.00000
0.41223 0.08961 0.0
0.08961 0.61297 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
2
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.85353 -0.0241 0.0
-0.0241 0.90742 0.0
0.0 0.0 1.00000
0.24585 -0.0037 0.0
-0.0037 0.08295 0.0
0.0 0.0 1.00000
0.24973 0.01086 0.0
0.01086 0.10979 0.0
0.0 0.0 1.00000
0.91088 0.01072 0.0
0.01072 0.94073 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
1
0.98316 0.0 0.0
0.0 0.98296 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98296 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98296 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98296 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98296 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98296 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98296 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98296 0.0
0.0 0.0 1.00000

Periodic Wavelet Average:

* 1 2 3 4 5 6 7 8
8
0.00708 0.0 0.0
0.0 0.00625 0.0
0.0 0.0 1.00000
0.00310 0.0 0.0
0.0 0.07900 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.00257 0.0 0.0
0.0 0.06508 0.0
0.0 0.0 1.00000
0.01087 0.0 0.0
0.0 0.00626 0.0
0.0 0.0 1.00000
7
0.93659 0.0 0.0
0.0 0.95875 0.0
0.0 0.0 1.00000
0.00216 0.0 0.0
0.0 0.03943 0.0
0.0 0.0 1.00000
0.19241 0.0 0.0
0.0 0.20504 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.14652 0.0 0.0
0.0 0.19748 0.0
0.0 0.0 1.00000
0.00215 0.0 0.0
0.0 0.00665 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
6
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.00350 0.0 0.0
0.0 0.40605 0.0
0.0 0.0 1.00000
0.00153 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 1.00000
0.00392 0.0 0.0
0.0 0.48037 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
5
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.46652 0.0 0.0
0.0 0.79835 0.0
0.0 0.0 1.00000
0.00156 0.0 0.0
0.0 0.00156 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00159 0.0 0.0
0.0 0.00168 0.0
0.0 0.0 1.00000
0.61033 0.0 0.0
0.0 0.85264 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
4
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.00232 0.0 0.0
0.0 0.15425 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00153 0.0 0.0
0.0 0.00153 0.0
0.0 0.0 1.00000
0.00312 0.0 0.0
0.0 0.23677 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
3
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.00532 0.0 0.0
0.0 0.54941 0.0
0.0 0.0 1.00000
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 1.00000
0.00155 0.0 0.0
0.0 0.00155 0.0
0.0 0.0 1.00000
0.00680 0.0 0.0
0.0 0.57926 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
2
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.80199 0.0 0.0
0.0 0.90288 0.0
0.0 0.0 1.00000
0.24430 0.0 0.0
0.0 0.00223 0.0
0.0 0.0 1.00000
0.24439 0.0 0.0
0.0 0.00317 0.0
0.0 0.0 1.00000
0.87253 0.0 0.0
0.0 0.93994 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
1
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 1.00000

Linear Boundary Condition Wavelet Average:

* 1 2 3 4 5 6 7 8
8
0.53085 -0.0121 0.0
-0.0121 0.45202 0.0
0.0 0.0 1.00000
0.08417 0.00611 0.0
0.00611 0.05110 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.07123 0.00569 0.0
0.00569 0.03914 0.0
0.0 0.0 1.00000
0.57375 -0.0130 0.0
-0.0130 0.52540 0.0
0.0 0.0 1.00000
7
0.96011 0.00135 0.0
0.00135 0.95554 0.0
0.0 0.0 1.00000
0.07731 -0.0269 0.0
-0.0269 0.05336 0.0
0.0 0.0 1.00000
0.20646 -4.2050 0.0
-4.2050 0.20375 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 1.00000
0.19948 -0.0010 0.0
-0.0010 0.19315 0.0
0.0 0.0 1.00000
0.12296 -0.0354 0.0
-0.0354 0.06647 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
6
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.44178 -0.0256 0.0
-0.0256 0.27369 0.0
0.0 0.0 1.00000
0.00154 -1.9775 0.0
-1.9775 0.00154 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00154 -1.4564 0.0
-1.4564 0.00154 0.0
0.0 0.0 1.00000
0.50879 -0.0026 0.0
-0.0026 0.33570 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
5
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.81988 0.01009 0.0
0.01009 0.74957 0.0
0.0 0.0 1.00000
0.00156 1.64127 0.0
1.64127 0.00156 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 1.00000
0.00710 -0.0036 0.0
-0.0036 0.00577 0.0
0.0 0.0 1.00000
0.86873 0.00720 0.0
0.00720 0.82539 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
4
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.20066 -0.0342 0.0
-0.0342 0.10277 0.0
0.0 0.0 1.00000
0.00152 -1.1125 0.0
-1.1125 0.00152 0.0
0.0 0.0 1.00000
0.00153 -1.3550 0.0
-1.3550 0.00153 0.0
0.0 0.0 1.00000
0.26718 -0.0348 0.0
-0.0348 0.14839 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
3
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.57844 0.00677 0.0
0.00677 0.42818 0.0
0.0 0.0 1.00000
0.00154 -5.0902 0.0
-5.0902 0.00154 0.0
0.0 0.0 1.00000
0.00155 9.16194 0.0
9.16194 0.00155 0.0
0.0 0.0 1.00000
0.63524 0.00560 0.0
0.00560 0.49834 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
2
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.91389 0.00158 0.0
0.00158 0.89507 0.0
0.0 0.0 1.00000
0.24830 -0.0020 0.0
-0.0020 0.24758 0.0
0.0 0.0 1.00000
0.26620 -0.0036 0.0
-0.0036 0.25838 0.0
0.0 0.0 1.00000
0.94340 0.00464 0.0
0.00464 0.93236 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 1.00000
1
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 1.00000
0.98316 0.0 0.0
0.0 0.98316 0.0
0.0 0.0 1.00000

Harmonic Average:

* 1 2 3 4 5 6 7 8
8
0.00362 0.0 0.0
0.0 0.00362 0.0
0.0 0.0 0.00362
0.00262 0.0 0.0
0.0 0.00262 0.0
0.0 0.0 0.00262
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.00237 0.0 0.0
0.0 0.00237 0.0
0.0 0.0 0.00237
0.00406 0.0 0.0
0.0 0.00406 0.0
0.0 0.0 0.00406
7
0.11830 0.0 0.0
0.0 0.11830 0.0
0.0 0.0 0.11830
0.00189 0.0 0.0
0.0 0.00189 0.0
0.0 0.0 0.00189
0.04105 0.0 0.0
0.0 0.04105 0.0
0.0 0.0 0.04105
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.21384 0.0 0.0
0.0 0.21384 0.0
0.0 0.0 0.21384
0.02270 0.0 0.0
0.0 0.02270 0.0
0.0 0.0 0.02270
0.00193 0.0 0.0
0.0 0.00193 0.0
0.0 0.0 0.00193
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
6
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.00296 0.0 0.0
0.0 0.00296 0.0
0.0 0.0 0.00296
0.00153 0.0 0.0
0.0 0.00153 0.0
0.0 0.0 0.00153
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 0.00154
0.00335 0.0 0.0
0.0 0.00335 0.0
0.0 0.0 0.00335
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
5
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.01051 0.0 0.0
0.0 0.01051 0.0
0.0 0.0 0.01051
0.00156 0.0 0.0
0.0 0.00156 0.0
0.0 0.0 0.00156
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00158 0.0 0.0
0.0 0.00158 0.0
0.0 0.0 0.00158
0.01589 0.0 0.0
0.0 0.01589 0.0
0.0 0.0 0.01589
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
4
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.00204 0.0 0.0
0.0 0.00204 0.0
0.0 0.0 0.00204
0.00152 0.0 0.0
0.0 0.00152 0.0
0.0 0.0 0.00152
0.00153 0.0 0.0
0.0 0.00153 0.0
0.0 0.0 0.00153
0.00222 0.0 0.0
0.0 0.00222 0.0
0.0 0.0 0.00222
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
3
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.00402 0.0 0.0
0.0 0.00402 0.0
0.0 0.0 0.00402
0.00154 0.0 0.0
0.0 0.00154 0.0
0.0 0.0 0.00154
0.00155 0.0 0.0
0.0 0.00155 0.0
0.0 0.0 0.00155
0.00478 0.0 0.0
0.0 0.00478 0.0
0.0 0.0 0.00478
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
2
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.02803 0.0 0.0
0.0 0.02803 0.0
0.0 0.0 0.02803
0.00211 0.0 0.0
0.0 0.00211 0.0
0.0 0.0 0.00211
0.00217 0.0 0.0
0.0 0.00217 0.0
0.0 0.0 0.00217
0.05449 0.0 0.0
0.0 0.05449 0.0
0.0 0.0 0.05449
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
0.97242 0.0 0.0
0.0 0.97242 0.0
0.0 0.0 0.97242
1
0.98281 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98281
0.98281 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98281
0.98281 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98281
0.98281 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98281
0.98281 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98281
0.98281 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98281
0.98281 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98281
0.98281 0.0 0.0
0.0 0.98281 0.0
0.0 0.0 0.98281

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.
Heterogeneous Simulation Results
The results below show the solution of the elliptic equation for the original heterogeneous map provided in the heating element data set. These are shown to compare with the results with solutions computed on homogenized versions of the heating element data.

Heterogeneous Simulation Results

Elliptic Solution

Homogenized Simulation Results
8x8 Simulation Results

Arithmetic Average Geometric Average HomCode Average Linear Boundary Condition Average Periodic Wavelet Average Linear Boundary Condition Wavelet Average Harmonic Average

16x16 Simulation Results

Arithmetic Average Geometric Average HomCode Average Linear Boundary Condition Average Periodic Wavelet Average Linear Boundary Condition Wavelet Average Harmonic Average

Steps to use JHomogenizer to create and work with the heating element map:
To create the results included on these pages and to get started working on other applications you can use the JHomogenizer tool.