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<h2> 
Model Reduction
</h2>
<p>
The following are implementations used in the model reduction examples from</br>
<ul>
Internal Diode for Frequency Selective EIT Contrasts,
Alistair Boyle, Daniel Kyrollos, John Harvey and Andy Adler,
18th International Conference on Biomedical Applications of Electrical
Impedance Tomography, Thayer School of Engineering at Dartmouth, Hanover, New
Hampshire, USA, June 21-24, 2017.</ul></p>

<h3>
Build Models
</h3>


<p>
First we build a circular 2D Netgen model. There are two stimulus electrodes
(#1 and #2) and two measurement electrodes (#3 and #4) with locations set by
selecting angle in degrees.
In addition, we have added two interior electrodes (#5 and #6) with a particular separation (Figure 1).
<pre>
[% PROCESS "model_reduction01.m" %]
</pre>
<center>
<img width="200px" src="model_reduction01a.png">
<img width="200px" src="model_reduction01c.png">
<img width="200px" src="model_reduction01e.png">
<br><i>Figure 1:</i> three of six models, stimulus electrodes #1 and #2, measurement electrodes #3 and #4, interior electrodes #5 and #6;
<br/>40% diametre <i>(Left)</i>,
13.2% diametre <i>(Middle)</i>,
6.4% diametre <i>(Right)</i>  interior electrode separation
</center>
</p>

<h3>
Reduce Model
</h3>

<p>
The model is then reduced to a fully-connected "mesh" network of resistors (Figure 2).
Some mesh networks may be reduced to a "star" configuration, reducing the
resistors and wiring criss-cross in a physical implementation, but we do not address
that conversion here.
<pre>
[% PROCESS "model_reduction02.m" %]
</pre>
The fully-connected mesh network is illustrated with the following code, by
stripping away the FEM lines and drawing connected lines between number
electrodes.
<pre>
[% PROCESS "model_reduction02_fig.m" %]
</pre>
<center>
<img width="200px" src="model_reduction02.png">
<br><i>Figure 2:</i> reduced mesh network; resistors (blue lines) between electrodes (numbers)
</center>
</p>

<h3>
Find Appropriate Resistors
</h3>

<p>
The best standard resistors for our particular mesh can then be found by minimizing the fitting error
<pre>
[% PROCESS "model_reduction03.m" %]
</pre>
which, for the first model (40% diametre interior electrode separation) gives a
resistor configuration in the form of an upper triangular matrix
<pre>
[% PROCESS "model_reduction03.txt" %]
</pre>
where the resistor values are given by the off-diagonal values between each row
(electrode n) and column (electrode m) with resistor selection error minimized (Figure 3).
<center>
<img width="600px" src="model_reduction03.png">
<br><i>Figure 3:</i> resistor fitting error; possible resistor values (blue),
resistor tolerance (red dashed), initial resistor scaling (red circle) and
selected resistor scaling (black box)
</center>
</p>

<p>
Commonly available resistors are defined by EIA standards
<pre>
[% PROCESS "model_reduction03_stdres.m" %]
</pre>
which is used to calculate the misfit between ideal resistor values and available resistor values
<pre>
[% PROCESS "model_reduction03_calc_err.m" %]
</pre>
</p>

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<p>
<small>
    Last Modified: $Date: 2017-05-15 11:11:05 -0400 (Mon, 15 May 2017) $ by $Author: alistair_boyle $
</small>
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