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<html>
<head>
<title>
CCVT_REFLECT - Interior + Boundary CVT
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CCVT_REFLECT <br> Interior + Boundary CVT
</h1>
<hr>
<p>
<b>CCVT_REFLECT</b>
is a FORTRAN90 program which
creates a Centroidal Voronoi Tessellation
of N points in an M-dimensional region, in which points are
also placed along the boundary.
</p>
<p>
The standard CVT algorithm will tend to place points uniformly
inside the region, but will never place points <i>on</i> the boundary.
However, in many cases, it would be very desirable to smoothly
modify the set of points so that some of them fall on the boundary.
This is the case, for instance, when the points are to be used to
triangulate the region and define a finite element grid.
</p>
<p>
The method for doing this relies on the observation that the
generator points become uniformly distributed because they
"push" each other away, by grabbing sample points. The generator
points do not approach the boundary too closely, because there
are no sample points on the other side of the boundary. In essence,
the boundary also "pushes" the generator points away.
By making "reflected" sample points whenever a generator is near
the boundary, we essentially neutralize the boundary effect, allowing
the generator points in the interior to push a layer of generators
onto the boundary. In most cases, once a point hits the boundary,
it will not leave, although it may continue to adjust its position
on the boundary itself.
</p>
<p>
<i>
This program was a "work in progress", until progress was halted.
So a number of ideas were started, and only some completed.
For the program as it stands now, only a simple 2D box region
has been examined. A next logical step would be to work on
more general 2D regions; then to make the natural extension to
3D or arbitrary dimension.
</i>
</p>
<p>
<b>CCVT_REFLECT</b> is an experimental code; so far, the experiment
is not doing well. I haven't figured out yet how to make the
points behave as well as they do for <b>CCVT_BOX</b>. The idea is,
though, that the method of pulling points to the boundary seems
fairly natural and flexible to me, so maybe I just need to
find the right way to implement it.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/ccvt_box/ccvt_box.html">
CCVT_BOX</a>,
a FORTRAN90 program which
is similar to CCVT_REFLECT, but which works in a 2D box.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Franz Aurenhammer,<br>
Voronoi diagrams -
a study of a fundamental geometric data structure,<br>
ACM Computing Surveys,<br>
Volume 23, Number 3, pages 345-405, September 1991.
</li>
<li>
John Burkardt, Max Gunzburger, Janet Peterson, Rebecca Brannon,<br>
User Manual and Supporting Information for Library of Codes
for Centroidal Voronoi Placement and Associated Zeroth,
First, and Second Moment Determination,<br>
Sandia National Laboratories Technical Report SAND2002-0099,<br>
February 2002.
</li>
<li>
Qiang Du, Vance Faber, Max Gunzburger,<br>
Centroidal Voronoi Tessellations: Applications and Algorithms,<br>
SIAM Review, Volume 41, 1999, pages 637-676.
</li>
<li>
Qiang Du, Max Gunzburger, Lili Ju,<br>
Meshfree, Probabilistic Determination of Point Sets and Support
Regions for Meshfree Computing,<br>
Computer Methods in Applied Mechanics in Engineering,<br>
Volume 191, 2002, pages 1349-1366;
</li>
<li>
Lili Ju, Qiang Du, Max Gunzburger,<br>
Probabilistic Methods for Centroidal Voronoi Tessellations and
their Parallel Implementations,<br>
Parallel Computing, Volume 28, 2002, pages 1477-1500.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "ccvt_reflect.f90">ccvt_reflect.f90</a>, the source code;
</li>
<li>
<a href = "ccvt_reflect.sh">ccvt_reflect.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "ccvt_reflect_input.txt">ccvt_reflect_input.txt</a>,
input that would normally be entered
interactively by the user.
</li>
<li>
<a href = "ccvt_reflect_output.txt">ccvt_reflect_output.txt</a>,
printed output from a run of the program.
</li>
<li>
<a href = "initial.txt">initial.txt</a>,
a <a href = "../../data/table/table.html">TABLE file</a>
containing the initial CVT generators.
</li>
<li>
<a href = "initial.png">initial.png</a>,
a <a href = "../../data/png/png.html">PNG</a>
image of the initial CVT generators.
</li>
<li>
<a href = "final.txt">final.txt</a>,
a <a href = "../../data/table/table.html">TABLE file</a>
containing the final CVT generators.
</li>
<li>
<a href = "final.png">final.png</a>,
a <a href = "../../data/png/png.html">PNG</a>
image of the final CVT generators.
</li>
<li>
<a href = "projected.txt">projected.txt</a>,
a <a href = "../../data/table/table.html">TABLE file</a>
containing the CVT dataset, with points near the boundary
projected onto the boundary.
</li>
<li>
<a href = "projected.png">projected.png</a>,
a <a href = "../../data/png/png.html">PNG</a>
image of the CVT generators with projection.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for CCVT_REFLECT.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CVT_ENERGY</b> computes the CVT energy of a dataset.
</li>
<li>
<b>CVT_SAMPLE</b> returns sample points.
</li>
<li>
<b>CVT_WRITE</b> writes a CVT dataset to a file.
</li>
<li>
<b>FIND_CLOSEST</b> finds the nearest R point to each S point.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>HALHAM_LEAP_CHECK</b> checks LEAP for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_N_CHECK</b> checks N for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_DIM_NUM_CHECK</b> checks DIM_NUM for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_SEED_CHECK</b> checks SEED for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_STEP_CHECK</b> checks STEP for a Halton or Hammersley sequence.
</li>
<li>
<b>HALTON_BASE_CHECK</b> checks BASE for a Halton sequence.
</li>
<li>
<b>I4_TO_HALTON_SEQUENCE</b> computes N elements of a leaped Halton subsequence.
</li>
<li>
<b>I4VEC_TRANSPOSE_PRINT</b> prints an I4VEC "transposed".
</li>
<li>
<b>MPB</b> projects generators onto the boundary of the region.
</li>
<li>
<b>POINTS_EPS</b> creates an EPS file image of a set of points.
</li>
<li>
<b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<b>R8MAT_UNIFORM_01</b> returns a unit pseudorandom R8MAT.
</li>
<li>
<b>RANDOM_INITIALIZE</b> initializes the FORTRAN90 random number seed.
</li>
<li>
<b>S_EQI</b> is a case insensitive comparison of two strings for equality.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TIMESTRING</b> writes the current YMDHMS date into a string.
</li>
<li>
<b>TUPLE_NEXT_FAST</b> computes the next element of a tuple space, "fast".
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 26 November 2006.
</i>
<!-- John Burkardt -->
</body>
</html>