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<html>
<head>
<title>
TET_MESH_L2Q - 10-Node Tet Mesh from 4-Node Tet Mesh
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TET_MESH_L2Q <br> 10-Node Tet Mesh <br> from <br> 4-Node Tet Mesh
</h1>
<hr>
<p>
<b>TET_MESH_L2Q</b>
is a MATLAB program which
converts a 4-node tetrahedral mesh ("tet mesh") into a 10-node tet mesh.
</p>
<p>
In particular, <b>TET_MESH_L2Q</b> reads information
describing a set of 3D points, and a 4 node ("linear")
tet mesh of those points. It creates a "quadratic" tet
mesh, which has the same number of tetrahedrons, but in
which each tetrahedron is defined by 10 nodes. In the process,
it adds just enough nodes, at the midsides of the edges of
the original mesh, to define the new mesh. The data defining
the new mesh is written out to files for further use.
</p>
<p>
The refinement of a single tetrahedron is easy. Every pair
of nodes generates a new node whose location is the average of
the locations of the original two nodes. The problem is that
many tetrahedrons may share the same edge. The new node needs
to be generated exactly once, and assigned a unique index, which
all the tetrahedrons will share. The tricky part of this calculation
is thus figuring out whether it is time to generate a new node, or
whether the node has already been generated, in which case the
appropriate index must be retrieved.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>tet_mesh_l2q</b> ( <i>'prefix'</i> )
</blockquote>
where
<ul>
<li>
<i>'prefix'</i>_nodes.txt contains the node definitions;
</li>
<li>
<i>'prefix'</i>_elements.txt contains the element definitions;
</li>
<li>
<i>'prefix'</i>_l2q_nodes.txt contains the new node definitions;
</li>
<li>
<i>'prefix'</i>_l2q_elements.txt contains the new element definitions;
</li>
</ul>
</p>
<p>
The input and output files use the simple TABLE format;
comment lines begin with a "#" character. Otherwise, each line
of the file contains one set of information, either the coordinates
of a node (for a node file), or the indices of nodes that make up
a tetrahedron, (for a tetrahedron file).
</p>
<p>
The input file <i>prefix_nodes.txt</i> contains the coordinate information
for the 4-node tet mesh. Each data line contains the X and
Y coordinates of a single node.
</p>
<p>
The input file <i>prefix_elements.txt</i> contains the tetrahedron information
for the 4-node tet mesh. Each data line contains the indices
of four nodes that form a tetrahedron.
</p>
<p>
The output file <i>prefix_l2q_nodes.txt</i> contains the coordinate information
for the 10-node tet mesh. The data lines begin with the node
information from <i>nodes.txt</i>, followed by the coordinates of
the new nodes.
</p>
<p>
The output file <i>prefix_l2q_elements.txt</i> contains the tetrahedron
information for the 10-node tet mesh. There are exactly
as many tetrahedrons as before, but now each tetrahedron uses 10 nodes.
Each line of the file contains the indices of 10 nodes that form the
tetrahedron, listed in a particular order. The first four
node indices are the same as for the linear tet mesh. The next six
node indices may be thought of as being obtained as averages of
pairs of the first four nodes. Symbolically, this may be thought of as:
<pre>
1 2 3 4 (1+2) (1+3) (1+4) (2+3) (2+4) (3+4)
</pre>
</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>TET_MESH_L2Q</b> is available in
<a href = "../../cpp_src/tet_mesh_l2q/tet_mesh_l2q.html">a C++ version</a> and
<a href = "../../f_src/tet_mesh_l2q/tet_mesh_l2q.html">a FORTRAN90 version</a> and
<a href = "../../m_src/tet_mesh_l2q/tet_mesh_l2q.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../f_src/cvt_tet_mesh/cvt_tet_mesh.html">
CVT_TET_MESH</a>,
a FORTRAN90 library which
uses CVT methods to compute a tet mesh in a region.
</p>
<p>
<a href = "../../f_src/table_tet_mesh/table_tet_mesh.html">
TABLE_TET_MESH</a>,
a FORTRAN90 program which
can compute the tet mesh
for a given set of points.
</p>
<p>
<a href = "../../f_src/test_tet_mesh/test_tet_mesh.html">
TEST_TET_MESH</a>,
a FORTRAN90 library which
defines a few test regions for the generation
of a tet mesh.
</p>
<p>
<a href = "../../m_src/tet_mesh/tet_mesh.html">
TET_MESH</a>,
a MATLAB library which
includes a variety of routines for working
with tetrahedral meshes.
</p>
<p>
<a href = "../../m_src/tet_mesh_boundary/tet_mesh_boundary.html">
TET_MESH_BOUNDARY</a>,
a MATLAB program which
returns the nodes and faces of the boundary of a tetrahedral mesh,
which themselves form a 3D triangular mesh or "TRI_SURFACE".
</p>
<p>
<a href = "../../m_src/tet_mesh_display/tet_mesh_display.html">
TET_MESH_DISPLAY</a>,
a MATLAB program which
can read in the
node and tetra files defining a tet mesh and display a wireframe
image.
</p>
<p>
<a href = "../../cpp_src/tet_mesh_display_opengl/tet_mesh_display_opengl.html">
TET_MESH_DISPLAY_OPENGL</a>,
a C++ program which
reads a tet mesh and displays the
nodes and edges using OpenGL.
</p>
<p>
<a href = "../../data/tet_mesh_order4/tet_mesh_order4.html">
TET_MESH_ORDER4</a>,
a directory which
contains a description and
examples of a tet mesh using order 4 elements.
</p>
<p>
<a href = "../../data/tet_mesh_order10/tet_mesh_order10.html">
TET_MESH_ORDER10</a>,
a directory which
contains a description and
examples of a tet mesh using order 10 elements.
</p>
<p>
<a href = "../../m_src/tet_mesh_q2l/tet_mesh_q2l.html">
TET_MESH_Q2L</a>,
a MATLAB program which
takes a 10-node tet mesh and makes
a 4-node tet mesh.
</p>
<p>
<a href = "../../m_src/tet_mesh_quality/tet_mesh_quality.html">
TET_MESH_QUALITY</a>,
a MATLAB program which
takes a 4-node tet mesh and computes
the "quality" of the mesh.
</p>
<p>
<a href = "../../m_src/tet_mesh_rcm/tet_mesh_rcm.html">
TET_MESH_RCM</a>,
a MATLAB program which
takes a tet mesh and
relabels the nodes to reduce the bandwidth of the
corresponding adjacency matrix.
</p>
<p>
<a href = "../../m_src/tet_mesh_refine/tet_mesh_refine.html">
TET_MESH_REFINE</a>,
a MATLAB program which
can refine a tet mesh.
</p>
<p>
<a href = "../../m_src/tet_mesh_tet_neighbors/tet_mesh_tet_neighbors.html">
TET_MESH_TET_NEIGHBORS</a>,
a MATLAB program which
computes the tetrahedral adjacency information.
</p>
<p>
<a href = "../../m_src/tet_mesh_volumes/tet_mesh_volumes.html">
TET_MESH_VOLUMES</a>,
a MATLAB program which
computes the volume of each tetrahedron in a tet mesh;
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Herbert Edelsbrunner,<br>
Geometry and Topology for Mesh Generation,<br>
Cambridge, 2001,<br>
ISBN: 0-521-79309-2,<br>
LC: QA377.E36.
</li>
<li>
Barry Joe,<br>
GEOMPACK - a software package for the generation of meshes
using geometric algorithms,<br>
Advances in Engineering Software,<br>
Volume 13, Number 5, 1991, pages 325-331.
</li>
<li>
Anwei Liu, Barry Joe,<br>
Quality Local Refinement of Tetrahedral Meshes Based
on 8-Subtetrahedron Subdivision,<br>
Mathematics of Computation,<br>
Volume 65, Number 215, July 1996, pages 1183-1200.
</li>
<li>
Per-Olof Persson, Gilbert Strang,<br>
A Simple Mesh Generator in MATLAB,<br>
SIAM Review,<br>
Volume 46, Number 2, June 2004, pages 329-345.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "tet_mesh_l2q.m">tet_mesh_l2q.m</a>,
is the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "mesh_nodes.txt">mesh_nodes.txt</a>,
the nodes for the 4-node tet mesh.
</li>
<li>
<a href = "mesh_elements.txt">mesh_elements.txt</a>,
the tetras for the 4-node tet mesh.
</li>
<li>
<a href = "mesh_l2q_nodes.txt">mesh_l2q_nodes.txt</a>,
the nodes for the 10-node tet mesh.
</li>
<li>
<a href = "mesh_l2q_elements.txt">mesh_l2q_elements.txt</a>,
the tetras for the 10-node tet mesh.
</li>
<li>
<a href = "mesh_output.txt">mesh_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 01 October 2009.
</i>
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