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<div class="moz-cite-prefix">On 04/22/2018 06:20 PM, Andrew Devin
Mcbryde wrote:<br>
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cite="mid:0e49dd2823fc4aa1ba792cd1dd731785@spiderman.MercerU.local">
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<p style="margin-top:0; margin-bottom:0">My group and I are
having some difficulty in understanding your expectation using
the Jacobi algorithm to calculate the iterations that the
simulation code uses. We do not understand what the Jacobi
algorithm actually is and how it can be applied to the problem
we are currently facing. Most of the information we find
regarding the function references how the memory is referenced
and not the actual math that calculates the final values.</p>
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<p><br>
</p>
<p>Let's see if we can break this down a little... equation (2) on
the group project sheet (which shows how the concentrations of
each cell change with time) should be replaced with something
like equation 3.7 in the text. In equation 3.7 the <img
alt="$\Phi$" style="vertical-align: middle;"
src="cid:part1.17233F9E.EC559767@mercer.edu"> values refer to
the quantity changing (in your case it is concentrations). <br>
</p>
<p><br>
</p>
<p>You will, however, need to convert equation 3.7 to work in 3D.
To do this you will need to add a <img alt="$z$"
style="vertical-align: middle;"
src="cid:part2.8133B7BC.099788E9@mercer.edu"> component to each
concentration cell, i.e. - <img alt="$\Phi(x,y,z)$"
style="vertical-align: middle;"
src="cid:part3.92097864.CBEACE4F@mercer.edu">. You will also
need to add an additional term to account for the change in the <img
alt="$z$" style="vertical-align: middle;"
src="cid:part2.8133B7BC.099788E9@mercer.edu"> dimension. <br>
</p>
<p><br>
</p>
<p>Since the diffusion is taking place through the faces only
(areas), the denominators in your 3D equation will remain squared
terms, and note cubed terms.<br>
</p>
<p><br>
</p>
<p>Now think about this -- your modified equation 3.7 will let you
determine how much any cell in your cube should change with a VERY
significant new feature -- you have BROKEN all data dependencies
that would inhibit parallelism (and in many cases those that would
inhibit vectorization). Let's call <img alt="$\Phi_c$"
style="vertical-align: middle;"
src="cid:part5.6D4DC66B.9F295B95@mercer.edu"> the copy of our
diffusion cube. I can calculate the entities of <img
alt="$\Phi_c$" style="vertical-align: middle;"
src="cid:part5.6D4DC66B.9F295B95@mercer.edu"> as...</p>
<p><br>
</p>
<p><img alt="$\Phi_c(x,y,z) = \Phi(x,y,z) +
\frac{\delta\Phi(x,y,z)}{\delta t}$" style="vertical-align:
middle;" src="cid:part7.0028A2EB.6E7A958E@mercer.edu"></p>
<p><br>
</p>
<p>With this technique you can calculate ALL of the elements of <img
alt="$\Phi_c$" style="vertical-align: middle;"
src="cid:part5.6D4DC66B.9F295B95@mercer.edu"> independently and
then, when done, copy all of the elements of <img alt="$\Phi_c
\rightarrow \Phi$" style="vertical-align: middle;"
src="cid:part9.2B02C603.7943D6D8@mercer.edu">, and then repeat
the process until you reach equilibrium. <br>
</p>
<p><br>
</p>
<p>Once you develop a new algorithm, test the serial version and
try to obtain the same results you got on your CSC 330 projects. <br>
</p>
<p><br>
</p>
<p>I STRONGLY encourage you to get the serial version working on
hammer first. <br>
</p>
<p><br>
</p>
<p><br>
</p>
<p><br>
</p>
<pre class="moz-signature" cols="72">--
Andrew J. Pounds, Ph.D. (<a class="moz-txt-link-abbreviated" href="mailto:pounds_aj@mercer.edu">pounds_aj@mercer.edu</a>)
Professor of Chemistry and Computer Science
Mercer University, Macon, GA 31207 (478) 301-5627
<a class="moz-txt-link-freetext" href="http://faculty.mercer.edu/pounds_aj">http://faculty.mercer.edu/pounds_aj</a>
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