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<html>
<head>
<title>
TRIANGULATION_CORNER - Handle triangulation corners.
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION_CORNER <br> Handle triangulation corners.
</h1>
<hr>
<p>
<b>TRIANGULATION_CORNER</b>
is a FORTRAN90 program which
tries to correct situations in which a triangulation includes
corner triangles, that is, triangles which have two sides on boundary.
</p>
<p>
Especially in finite element settings, such a triangle is considered
undesirable. Especially in the case when a linear (order 3) finite
element is involved, the function approximation in the interior of
the triangle will be entirely determined by boundary data. If, for
instance, the solution is constrained to be zero on the boundary,
then the finite element function will be zero throughout the interior
of the corner triangle.
</p>
<p>
Presumably, the triangle has a neighbor triangle on the one non-boundary
side. It is generally possible and reasonable to replace these two triangles
but another two which fill the same quadrilateral, but which are separated
by the other diagonal of the quadrilateral.
</p>
<p>
The initial situation at the corner is suggest by the following diagram:
<pre>
|/ |/
A---C--
|\ |\
| \ |
| \|
B---D--
</pre>
<p>
<p>
By rearranging the corner triangle and its neighbor, we have the more
satisfactory triangulation:
<pre>
|/ |/
A---C--
| /|\
| / |
|/ |
B---D--
</pre>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>triangulation_corner</b> <i>prefix</i>
</blockquote>
where <i>prefix</i> is the common filename prefix:
<ul>
<li>
<i>prefix</i>_nodes.txt contains the node coordinates,
</li>
<li>
<i>prefix</i>_elements.txt contains the element definitions.
</li>
<li>
<i>prefix</i>_corner_nodes.txt will contain the revised node coordinates,
</li>
<li>
<i>prefix</i>_corner_elements.txt will contain the revised element definitions.
</li>
</ul>
</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">
Languages:
</h3>
<p>
<b>TRIANGULATION_CORNER</b> is available in
<a href = "../../cpp_src/triangulation_corner/triangulation_corner.html">a C++ version</a> and
<a href = "../../f_src/triangulation_corner/triangulation_corner.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangulation_corner/triangulation_corner.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/triangulation/triangulation.html">
TRIANGULATION</a>,
a FORTRAN90 library which
performs various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.
</p>
<p>
<a href = "../../f_src/triangulation_boundary_nodes/triangulation_boundary_nodes.html">
TRIANGULATION_BOUNDARY_NODES</a>,
a FORTRAN90 program which
reads data defining a triangulation, determines which nodes
lie on the boundary, and writes their coordinates to a file.
</p>
<p>
<a href = "../../f_src/triangulation_delaunay_discrepancy/triangulation_delaunay_discrepancy.html">
TRIANGULATION_DELAUNAY_DISCREPANCY</a>,
a FORTRAN90 program which
measures the amount by which a triangulation fails the local Delaunay test;
</p>
<p>
<a href = "../../cpp_src/triangulation_display_opengl/triangulation_display_opengl.html">
TRIANGULATION_DISPLAY_OPENGL</a>,
a C++ program which
reads files defining a triangulation and displays an image
using Open GL.
</p>
<p>
<a href = "../../f_src/triangulation_histogram/triangulation_histogram.html">
TRIANGULATION_HISTOGRAM</a>,
a FORTRAN90 program which
computes histograms of data over a triangulation.
</p>
<p>
<a href = "../../f_src/triangulation_l2q/triangulation_l2q.html">
TRIANGULATION_L2Q</a>,
a FORTRAN90 program which
reads information about a 3-node (linear) triangulation and creates data
defining a corresponding 6-node (quadratic) triangulation;
</p>
<p>
<a href = "../../f_src/triangulation_mask/triangulation_mask.html">
TRIANGULATION_MASK</a>,
a FORTRAN90 program which
reads a triangulation and calls
a user-supplied routine to consider each triangle for deletion;
</p>
<p>
<a href = "../../data/triangulation_order3/triangulation_order3.html">
TRIANGULATION_ORDER3</a>,
a data directory which
describes TRIANGULATION_ORDER3 files,
description of a linear triangulation
of a set of 2D points, using a pair of files to list the node
coordinates and the 3 nodes that make up each triangle;
</p>
<p>
<a href = "../../data/triangulation_order6/triangulation_order6.html">
TRIANGULATION_ORDER6</a>,
a data directory which
describes TRIANGULATION_ORDER6 files,
a description of a quadratic triangulation
of a set of 2D points, using a pair of files to list the node
coordinates and the 6 nodes that make up each triangle.
</p>
<p>
<a href = "../../f_src/triangulation_orient/triangulation_orient.html">
TRIANGULATION_ORIENT</a>,
a FORTRAN90 program which
ensures that the triangles in
an order 3 or order 6 triangulation have positive orientation;
</p>
<p>
<a href = "../../f_src/triangulation_plot/triangulation_plot.html">
TRIANGULATION_PLOT</a>,
a FORTRAN90 program which
makes a PostScript image of a triangulation of points.
</p>
<p>
<a href = "../../f_src/triangulation_quad/triangulation_quad.html">
TRIANGULATION_QUAD</a>,
a FORTRAN90 program which
estimates the integral of a function over a triangulated region.
</p>
<p>
<a href = "../../f_src/triangulation_refine/triangulation_refine.html">
TRIANGULATION_REFINE</a>,
a FORTRAN90 program which
can refine a triangulation.
</p>
<p>
<a href = "../../f_src/triangulation_triangle_neighbors/triangulation_triangle_neighbors.html">
TRIANGULATION_TRIANGLE_NEIGHBORS</a>,
a FORTRAN90 program which
reads data defining a triangulation, determines the neighboring
triangles of each triangle, and writes that information to a file.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Joseph ORourke,<br>
Computational Geometry,<br>
Second Edition,<br>
Cambridge, 1998,<br>
ISBN: 0521649765,<br>
LC: QA448.D38.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "triangulation_corner.f90">triangulation_corner.f90</a>, the source code.
</li>
<li>
<a href = "triangulation_corner.sh">triangulation_corner.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>P03</b> is the square with a hole. The initial mesh has 1 corner triangle.
<ul>
<li>
<a href = "p03_nodes.txt">p03_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p03_elements.txt">p03_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p03.png">p03.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p02_corner_elements.txt">p02_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p02_corner.png">p02_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P05</b> is the horn. The initial mesh has 4 corner triangles.
<ul>
<li>
<a href = "p05_nodes.txt">p05_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p05_elements.txt">p05_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p05.png">p05.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p05_corner_elements.txt">p05_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p05_corner.png">p05_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P08</b> is the holey pie slice. The initial mesh has 2 corner triangles.
<ul>
<li>
<a href = "p08_nodes.txt">p08_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p08_elements.txt">p08_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p08.png">p08.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p08_corner_elements.txt">p08_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p08_corner.png">p08_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P09</b> is Jeff Borggaard's square with two hexagonal holes.
The initial mesh has 3 corner triangles.
<ul>
<li>
<a href = "p09_nodes.txt">p09_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p09_elements.txt">p09_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p09.png">p09.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p09_corner_elements.txt">p09_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p09_corner.png">p09_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P10</b> is the unit square. The initial mesh has 4 corner triangles.
<ul>
<li>
<a href = "p10_nodes.txt">p10_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p10_elements.txt">p10_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p10.png">p10.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p10_corner_elements.txt">p10_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p10_corner.png">p10_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P11</b> is the L-shaped region. The initial mesh has 4 corner triangles.
<ul>
<li>
<a href = "p11_nodes.txt">p11_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p11_elements.txt">p11_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p11.png">p11.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p11_corner_elements.txt">p11_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p11_corner.png">p11_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>Q11</b> is the L-shaped region, but with quadratric (6 node) triangles.
The initial mesh has 4 corner triangles. Because we are using quadratic
triangles, when we merge two triangles and swap the diagonal, the midside
node on the common diagonal will no longer lie on a line, midway between the
two vertices. <b>TRIANGULATION_CORNER</b> assumes this is undesirable,
and recomputes the coordinates of such midside nodes, and outputs
a modified node coordinate file as well as the new triangle file.
<ul>
<li>
<a href = "q11_nodes.txt">q11_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "q11_elements.txt">q11_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "q11.png">q11.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "q11_corner_nodes.txt">q11_nodes.corner.txt</a>
the node coordinate file after corner triangles have been repaired,
which, for quadratric triangles, requires adjusting some midside nodes.
</li>
<li>
<a href = "q11_corner_elements.txt">q11_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "q11_corner.png">q11_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P12</b> is the John Shadid's H-shaped region. The initial mesh has 7 corner triangles.
<ul>
<li>
<a href = "p12_nodes.txt">p12_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p12_elements.txt">p12_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p12.png">p12.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p12_corner_elements.txt">p12_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p12_corner.png">p12_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P13</b> is the Sandia fork. The initial mesh has 6 corner triangles.
<ul>
<li>
<a href = "p13_nodes.txt">p13_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p13_elements.txt">p13_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p13.png">p13.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p13_corner_elements.txt">p13_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p13_corner.png">p13_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P14</b> is Marcus Garvie's Lake Alpha, with Beta Island.
The initial mesh has 5 corner triangles. (Must be small!)
<ul>
<li>
<a href = "p14_nodes.txt">p14_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p14_elements.txt">p14_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p14.png">p14.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p14_corner_elements.txt">p14_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p14_corner.png">p14_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P15</b> is Sangbum Kim's forward step region. The initial mesh has 6 corner triangles.
<ul>
<li>
<a href = "p15_nodes.txt">p15_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p15_elements.txt">p15_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p15.png">p15.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p15_corner_elements.txt">p15_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p15_corner.png">p15_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P16</b> is Kevin Pond's elbow, a quarter of a circular annulus.
The initial mesh has 1 corner triangle.
<ul>
<li>
<a href = "p16_nodes.txt">p16_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p16_elements.txt">p16_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p16.png">p16.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p16_corner_elements.txt">p16_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p16_corner.png">p16_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<p>
<b>P17</b> is the rectangular channel with a Reuleaux triangle obstacle.
The initial mesh has 4 corner triangles.
<ul>
<li>
<a href = "p17_nodes.txt">p17_nodes.txt</a>
the mesh nodes of the problem.
</li>
<li>
<a href = "p17_elements.txt">p17_elements.txt</a>
the mesh triangles of the problem.
</li>
<li>
<a href = "p17.png">p17.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the initial mesh.
</li>
<li>
<a href = "p17_corner_elements.txt">p17_corner_elements.txt</a>
the triangle file after corner triangles have been repaired.
</li>
<li>
<a href = "p17_corner.png">p17_corner.png</a>
a <a href = "../../data/png/png.html">PNG</a> image of
the repaired mesh.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TRIANGULATION_CORNER.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer ( kind = 4 ) value of a base 10 digit.
</li>
<li>
<b>DTABLE_CLOSE_WRITE</b> closes a file used to write a DTABLE.
</li>
<li>
<b>DTABLE_DATA_READ</b> reads data from a double precision table file.
</li>
<li>
<b>DTABLE_DATA_WRITE</b> writes DTABLE data to a file.
</li>
<li>
<b>DTABLE_HEADER_READ</b> reads the header from a double precision table file.
</li>
<li>
<b>DTABLE_HEADER_WRITE</b> writes the header to a DTABLE file.
</li>
<li>
<b>DTABLE_OPEN_WRITE</b> opens a file to write a DTABLE.
</li>
<li>
<b>DTABLE_WRITE</b> writes a DTABLE to a file.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_NAME_EXT_GET</b> determines the "extension" of a file name.
</li>
<li>
<b>FILE_NAME_EXT_SWAP</b> replaces the current "extension" of a file name.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of the transpose of an I4MAT.
</li>
<li>
<b>I4ROW_COMPARE</b> compares two rows of an I4ROW.
</li>
<li>
<b>I4ROW_SORT_A</b> ascending sorts the rows of an I4ROW.
</li>
<li>
<b>I4ROW_SWAP</b> swaps two rows of an I4ROW.
</li>
<li>
<b>ITABLE_DATA_READ</b> reads data from an integer ( kind = 4 ) table file.
</li>
<li>
<b>ITABLE_DATA_WRITE</b> writes data to an integer ( kind = 4 ) table file.
</li>
<li>
<b>ITABLE_HEADER_READ</b> reads the header from an integer ( kind = 4 ) table file.
</li>
<li>
<b>ITABLE_HEADER_WRITE</b> writes the header to an integer ( kind = 4 ) table file.
</li>
<li>
<b>ITABLE_WRITE</b> writes an integer ( kind = 4 ) table file.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>S_BLANK_DELETE</b> removes blanks from a string, left justifying the remainder.
</li>
<li>
<b>S_INDEX_LAST_C</b> finds the LAST occurrence of a given character.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_I4VEC</b> reads an I4VEC from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>SORT_HEAP_EXTERNAL</b> externally sorts a list of items into ascending order.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRIANGLE_AREA_2D</b> computes the area of a triangle in 2D.
</li>
<li>
<b>TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES</b> determines triangle neighbors.
</li>
<li>
<b>TRIANGULATION_ORDER6_NEIGHBOR_TRIANGLES</b> determines triangle neighbors.
</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 30 December 2010.
</i>
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