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<html>
<head>
<title>
QUAD_MESH - Quadrilateral Meshes
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
QUAD_MESH <br> Quadrilateral Meshes
</h1>
<hr>
<p>
<b>QUAD_MESH</b>
is a MATLAB library which
carries out operations involving meshes of quadrilaterals.
</p>
<p>
The mesh is the collection of quadrilaterals. Each quadrilateral
is termed an "element". The points used to define the shape of the
quadrilateral (the corners, and sometimes a few more points) are called
the "nodes".
</p>
<p>
Routines are available to:
<ul>
<li>
evaluate "quality measures" for the mesh;
</li>
<li>
create a "node neighbor array" for each node;
</li>
<li>
create an "element neighbor array" for each element;
</li>
<li>
estimate the integral of a function over the region covered by the mesh;
</li>
<li>
plot the nodes and elements of a mesh;
</li>
<li>
determine the parts of the mesh that lie on the boundary;
</li>
<li>
sample points at random from the region covered by the mesh;
</li>
<li>
search a mesh to determine which element contains a point.
</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>QUAD_MESH</b> is available in
<a href = "../../cpp_src/quad_mesh/quad_mesh.html">a C++ version</a> and
<a href = "../../f_src/quad_mesh/quad_mesh.html">a FORTRAN90 version</a> and
<a href = "../../m_src/quad_mesh/quad_mesh.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../data/fem2d/fem2d.html">
FEM2D</a>,
a data directory which
contains a description and examples of files that describe a 2D finite element model.
</p>
<p>
<a href = "../../m_src/fem_io/fem_io.html">
FEM_IO</a>,
a MATLAB library which
reads or writes node, element and data files defining a finite element model.
</p>
<p>
<a href = "../../m_src/fem2d_pack/fem2d_pack.html">
FEM2D_PACK</a>,
a MATLAB library which
handles various tasks associated with finite element meshes.
</p>
<p>
<a href = "../../m_src/mesh_bandwidth/mesh_bandwidth.html">
MESH_BANDWIDTH</a>,
a MATLAB program which
returns the geometric bandwidth associated with a mesh of
elements of any order and in a space of arbitrary dimension.
</p>
<p>
<a href = "../../m_src/mesh_display/mesh_display.html">
MESH_DISPLAY</a>,
a MATLAB program which
reads data defining a polygonal mesh and displays it, with optional numbering.
</p>
<p>
<a href = "../../cpp_src/mesh_display_opengl/mesh_display_opengl.html">
MESH_DISPLAY_OPENGL</a>,
a C++ program which
reads files defining a polygonal mesh and displays an image using OpenGL.
</p>
<p>
<a href = "../../data/quad_mesh/quad_mesh.html">
QUAD_MESH</a>,
a data directory which
defines a format for storing meshes of quadrilaterals over a 2D region.
</p>
<p>
<a href = "../../m_src/quad_mesh_order1_display/quad_mesh_order1_display.html">
QUAD_MESH_ORDER1_DISPLAY</a>,
a MATLAB program which
plots piecewise constant data associated with a mesh of quadrilaterals;
</p>
<p>
<a href = "../../m_src/quad_mesh_rcm/quad_mesh_rcm.html">
QUAD_MESH_RCM</a>,
a MATLAB program which
computes the reverse Cuthill-McKee (RCM) reordering for nodes in a mesh of 4-node
quadrilaterals.
</p>
<p>
<a href = "../../m_src/triangulation/triangulation.html">
TRIANGULATION</a>,
a MATLAB library which
carries out operations involving meshes of triangular elements.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Hans Rudolf Schwarz,<br>
Methode der Finiten Elemente,<br>
Teubner Studienbuecher, 1980,<br>
ISBN: 3-519-02349-0.
</li>
<li>
Gilbert Strang, George Fix,<br>
An Analysis of the Finite Element Method,<br>
Cambridge, 1973,<br>
ISBN: 096140888X,<br>
LC: TA335.S77.
</li>
<li>
Olgierd Zienkiewicz,<br>
The Finite Element Method,<br>
Sixth Edition,<br>
Butterworth-Heinemann, 2005,<br>
ISBN: 0750663200,<br>
TA640.2.Z54
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "adj_bandwidth.m">
adj_bandwidth.m</a>,
omputes the bandwidth of an adjacency matrix.
</li>
<li>
<a href = "adj_perm_bandwidth.m">
adj_perm_bandwidth.m</a>,
omputes the bandwidth of a permuted adjacency matrix.
</li>
<li>
<a href = "adj_set_q4_mesh.m">
adj_set_q4_mesh.m</a>,
sets the adjacencies in a Q4 mesh.
</li>
<li>
<a href = "adj_size_q4_mesh.m">
adj_size_q4_mesh.m</a>,
counts adjacencies in a Q4 mesh.
</li>
<li>
<a href = "area_q4_mesh.m">
area_q4_mesh.m</a>,
computes areas of elements in a Q4 mesh;
</li>
<li>
<a href = "area_quad.m">
area_quad.m</a>,
return the area of a quadrilateral;
</li>
<li>
<a href = "bandwidth.m">
bandwidth.m</a>,
determines the bandwidth associated with the finite element mesh;
</li>
<li>
<a href = "boundary_edge_count_q4_mesh.m">
boundary_edge_count_q4_mesh.m</a>,
counts boundary edges.
</li>
<li>
<a href = "boundary_edge_count_euler_q4_mesh.m">
boundary_edge_count_euler_q4_mesh.m</a>,
counts boundary edges using Euler's formula.
</li>
<li>
<a href = "example1_q4_mesh.m">
example1_q4_mesh.m</a>,
sets up example #1 Q4 mesh.
</li>
<li>
<a href = "example1_q4_mesh_size.m">
example1_q4_mesh_size.m</a>,
sets sizes for example #1 Q4 mesh.
</li>
<li>
<a href = "example2_q4_mesh.m">
example2_q4_mesh.m</a>,
sets up example #2 Q4 mesh.
</li>
<li>
<a href = "example2_q4_mesh_size.m">
example2_q4_mesh_size.m</a>,
sets sizes for example #2 Q4 mesh.
</li>
<li>
<a href = "file_column_count.m">file_column_count.m</a>,
counts the number of columns in the first line of a file.
</li>
<li>
<a href = "file_row_count.m">file_row_count.m</a>,
counts the number of row records in a file.
</li>
<li>
<a href = "i4_modp.m">i4_modp.m</a>,
computes the positive modulus of an I4.
</li>
<li>
<a href = "i4_wrap.m">i4_wrap.m</a>,
wraps an I4.
</li>
<li>
<a href = "i4col_compare.m">i4col_compare.m</a>,
compares two columns of an I4COL;
</li>
<li>
<a href = "i4col_sort_a.m">i4col_sort_a.m</a>,
ascending sorts the columns of an I4COL;
</li>
<li>
<a href = "i4col_sorted_unique_count.m">i4col_sorted_unique_count.m</a>,
counts the number of unique elements in a sorted I4COL;
</li>
<li>
<a href = "i4col_swap.m">
i4col_swap.m</a>,
swaps two columns of an I4COL;
</li>
<li>
<a href = "i4mat_data_read.m">i4mat_data_read.m</a>,
reads data from an I4MAT file..
</li>
<li>
<a href = "i4mat_header_read.m">i4mat_header_read.m</a>,
reads the header from an I4MAT file.
</li>
<li>
<a href = "i4mat_transpose_print.m">i4mat_transpose_print.m</a>,
prints the transpose of an I4MAT.
</li>
<li>
<a href = "i4mat_transpose_print_some.m">i4mat_transpose_print_some.m</a>,
prints some of the transpose of an I4MAT.
</li>
<li>
<a href = "i4mat_write.m">i4mat_write.m</a>,
writes an I4MAT file.
</li>
<li>
<a href = "i4vec_print.m">i4vec_print.m</a>,
prints an I4VEC.
</li>
<li>
<a href = "mesh_base_one.m">mesh_base_one.m</a>
detects and corrects 0-based indexing.
</li>
<li>
<a href = "node_order_q4_mesh.m">node_order_q4_mesh.m</a>,
determines the order of nodes in a Q4 mesh.
</li>
<li>
<a href = "neighbor_elements_q4_mesh.m">neighbor_elements_q4_mesh.m</a>,
determines element neighbors in a Q4 mesh.
</li>
<li>
<a href = "plot_q4_mesh.m">plot_q4_mesh.m</a>,
plots a Q4 mesh.
</li>
<li>
<a href = "r8_uniform_01.m">r8_uniform_01.m</a>,
returns a unit pseudorandom R8.
</li>
<li>
<a href = "r8mat_data_read.m">r8mat_data_read.m</a>,
reads a table from an R8MAT file.
</li>
<li>
<a href = "r8mat_header_read.m">r8mat_header_read.m</a>,
reads a table header from an R8MAT file.
</li>
<li>
<a href = "r8mat_transpose_print.m">r8mat_transpose_print.m</a>,
prints the transpose of an R8MAT;
</li>
<li>
<a href = "r8mat_transpose_print_some.m">
r8mat_transpose_print_some.m</a>,
prints some of the transpose of an R8MAT;
</li>
<li>
<a href = "r8mat_write.m">r8mat_write.m</a>,
writes an R8MAT file.
</li>
<li>
<a href = "r8vec_print.m">r8vec_print.m</a>,
prints an R8VEC.
</li>
<li>
<a href = "r8vec_uniform_01.m">r8vec_uniform_01.m</a>,
returns a unit pseudorandom R8VEC.
</li>
<li>
<a href = "reference_to_physical_q4.m">reference_to_physical_q4.m</a>
maps points in a Q4 reference triangle into physical space.
</li>
<li>
<a href = "s_len_trim.m">s_len_trim.m</a>,
returns the length of a character string to the last nonblank.
</li>
<li>
<a href = "s_word_count.m">s_word_count.m</a>,
counts the number of words in a string.
</li>
<li>
<a href = "sample_q4_mesh.m">
sample_q4_mesh.m</a>,
randomly samples a point from a Q4 mesh;
</li>
<li>
<a href = "sample_quad.m">
sample_quad.m</a>,
randomly samples a point from a quadrilateral;
</li>
<li>
<a href = "sort_heap_external.m">
sort_heap_external.m</a>,
externally sorts a list of values into ascending order;
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a timestamp.
</li>
<li>
<a href = "triangle_area.m">
triangle_area.m</a>,
returns the area of a triangle;
</li>
<li>
<a href = "triangle_sample.m">
triangle_sample.m</a>,
randomly samples a point from a triangle;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "quad_mesh_test.m">quad_mesh_test.m</a>,
a sample calling program.
</li>
<li>
<a href = "quad_mesh_test01.m">quad_mesh_test01.m</a>,
tests ADJ_SIZE_Q4_MESH.
</li>
<li>
<a href = "quad_mesh_test02.m">quad_mesh_test02.m</a>,
tests AREA_Q4_MESH.
</li>
<li>
<a href = "quad_mesh_test03.m">quad_mesh_test03.m</a>,
tests AREA_QUAD.
</li>
<li>
<a href = "quad_mesh_test04.m">quad_mesh_test04.m</a>,
tests BOUNDARY_EDGE_COUNT_Q4_MESH.
</li>
<li>
<a href = "quad_mesh_test05.m">quad_mesh_test05.m</a>,
tests BOUNDARY_EDGE_COUNT_EULER_Q4_MESH.
</li>
<li>
<a href = "quad_mesh_test06.m">quad_mesh_test06.m</a>,
tests EXAMPLE1_Q4_MESH.
</li>
<li>
<a href = "quad_mesh_test07.m">quad_mesh_test07.m</a>,
tests EXAMPLE2_Q4_MESH.
</li>
<li>
<a href = "quad_mesh_test08.m">quad_mesh_test08.m</a>,
tests NEIGHBOR_ELEMENTS_Q4_MESH.
</li>
<li>
<a href = "quad_mesh_test09.m">quad_mesh_test09.m</a>,
writes data to files.
</li>
<li>
<a href = "quad_mesh_test10.m">quad_mesh_test10.m</a>,
tests NODE_ORDER_Q4_MESH.
</li>
<li>
<a href = "quad_mesh_test105.m">quad_mesh_test105.m</a>,
tests SAMPLE_Q4_MESH.
</li>
<li>
<a href = "quad_mesh_test11.m">quad_mesh_test11.m</a>,
tests SAMPLE_QUAD.
</li>
<li>
<a href = "quad_mesh_test_output.txt">quad_mesh_test_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>EXAMPLE1</b> is a mesh made up of squares of uniform size. The squares
are arranged in a grid with a few irregularities, however. In particular,
there is an interior hole.
<ul>
<li>
<a href = "q4_mesh_ex1_element_neighbors.txt">q4_mesh_ex1_element_neighbors.txt</a>,
element neighbors for example 1 Q4 mesh.
</li>
<li>
<a href = "q4_mesh_ex1_elements.txt">q4_mesh_ex1_elements.txt</a>,
elements for example 1 Q4 mesh.
</li>
<li>
<a href = "q4_mesh_ex1_xy.txt">q4_mesh_ex1_xy.txt</a>,
node coordinates for example 1 Q4 mesh.
</li>
<li>
<a href = "q4_mesh_ex1.png">q4_mesh_ex1.png</a>,
a PNG image of example 1 Q4 mesh.
</li>
</ul>
</p>
<p>
<b>EXAMPLE2</b> is a mesh made up of quadrilaterals that are "trying" to tile a
semicircle. The row of elements that touches the origin is degenerate, and
are essentially triangles. The elements vary in area.
<ul>
<li>
<a href = "q4_mesh_ex2_element_neighbors.txt">q4_mesh_ex2_element_neighbors.txt</a>,
element neighbors for example 2 Q4 mesh.
</li>
<li>
<a href = "q4_mesh_ex2_elements.txt">q4_mesh_ex2_elements.txt</a>,
elements for example 2 Q4 mesh.
</li>
<li>
<a href = "q4_mesh_ex2_xy.txt">q4_mesh_ex2_xy.txt</a>,
node coordinates for example 2 Q4 mesh.
</li>
<li>
<a href = "q4_mesh_ex2.png">q4_mesh_ex2.png</a>,
a PNG image of example 2 Q4 mesh.
</li>
</ul>
</p>
<p>
<b>SAMPLE_QUAD</b> demonstrates how a single quadrilateral can be randomly sampled.
<ul>
<li>
<a href = "sample_quad.txt">sample_quad.txt</a>,
node coordinates for random sample points in a quadrilateral.
</li>
<li>
<a href = "sample_quad.png">sample_quad.png</a>,
a PNG image of the random sample points in a quadrilateral.
</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 21 September 2009.
</i>
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