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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 C++ 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 = "../../cpp_src/fem_io/fem_io.html">
FEM_IO</a>,
a C++ library which
reads or writes node, element and data files defining a finite element model.
</p>
<p>
<a href = "../../cpp_src/fem2d_pack/fem2d_pack.html">
FEM2D_PACK</a>,
a C++ library which
handles various tasks associated with finite element meshes.
</p>
<p>
<a href = "../../cpp_src/mesh_bandwidth/mesh_bandwidth.html">
MESH_BANDWIDTH</a>,
a C++ 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 = "../../cpp_src/quad_mesh_rcm/quad_mesh_rcm.html">
QUAD_MESH_RCM</a>,
a C++ program which
computes the reverse Cuthill-McKee (RCM) reordering for nodes in a mesh of 4-node
quadrilaterals.
</p>
<p>
<a href = "../../cpp_src/triangulation/triangulation.html">
TRIANGULATION</a>,
a C++ 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 = "quad_mesh.cpp">quad_mesh.cpp</a>, the source code.
</li>
<li>
<a href = "quad_mesh.hpp">quad_mesh.hpp</a>, the include file.
</li>
<li>
<a href = "quad_mesh.sh">quad_mesh.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "quad_mesh_prb.cpp">quad_mesh_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "quad_mesh_prb.sh">quad_mesh_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "quad_mesh_prb_output.txt">quad_mesh_prb_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>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>_ADJ_SET_Q4_MESH</b> sets adjacencies in a triangulation.
</li>
<li>
<b>ADJ_SIZE_Q4_MESH</b> counts adjacencies in a Q4 mesh.
</li>
<li>
<b>AREA_Q4_MESH</b> computes areas of elements in a Q4 mesh.
</li>
<li>
<b>AREA_QUAD</b> returns the area of a quadrilateral.
</li>
<li>
<b>BANDWIDTH</b> determines the bandwidth associated with the finite element mesh.
</li>
<li>
<b>BOUNDARY_EDGE_COUNT_Q4_MESH</b> counts the boundary edges.
</li>
<li>
<b>BOUNDARY_EDGE_COUNT_EULER_Q4_MESH</b> counts boundary edges.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is true if two characters are equal, disregarding case.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>EXAMPLE1_Q4_MESH</b> sets up example #1 Q4 mesh.
</li>
<li>
<b>EXAMPLE1_Q4_MESH_SIZE</b> sets sizes for example #1 Q4 mesh
</li>
<li>
<b>EXAMPLE2_Q4_MESH</b> sets up example #2 Q4 mesh.
</li>
<li>
<b>EXAMPLE2_Q4_MESH_SIZE</b> sets sizes for example #2 Q4 mesh
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the minimum of two I4's.
</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>I4COL_COMPARE</b> compares columns I and J of an I4COL.
</li>
<li>
<b>I4COL_SORT_A</b> ascending sorts the columns of an I4COL.
</li>
<li>
<b>I4COL_SORTED_UNIQUE_COUNT</b> counts unique elements in an I4COL.
</li>
<li>
<b>I4COL_SWAP</b> swaps two columns of an I4COL.
</li>
<li>
<b>I4MAT_COPY</b> copies one I4MAT to another.
</li>
<li>
<b>I4MAT_DATA_READ</b> reads data from an I4MAT file.
</li>
<li>
<b>I4MAT_HEADER_READ</b> reads the header from an I4MAT file.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT</b> prints an I4MAT, transposed.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of an I4MAT, transposed.
</li>
<li>
<b>I4MAT_WRITE</b> writes an I4MAT file with no header.
</li>
<li>
<b>I4ROW_COMPARE</b> compares two rows of a integer array.
</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>I4VEC_HEAP_D</b> reorders an I4VEC into a descending heap.
</li>
<li>
<b>I4VEC_PRINT</b> prints an I4VEC.
</li>
<li>
<b>I4VEC_SORT_HEAP_A</b> ascending sorts an I4VEC using heap sort.
</li>
<li>
<b>I4VEC_ZERO_NEW</b> creates and zeroes an I4VEC.
</li>
<li>
<b>MESH_BASE_ZERO</b> ensures that the element definition is zero-based.
</li>
<li>
<b>NEIGHBOR_ELEMENTS_Q4_MESH</b> determines element neighbors in a Q4 mesh.
</li>
<li>
<b>NODE_ORDER_Q4_MESH</b> determines the order of nodes in a Q4 mesh.
</li>
<li>
<b>PLOT_Q4_MESH</b> plots a Q4 mesh.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to an R8.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8MAT_COPY</b> copies one R8MAT to another.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads the data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>R8MAT_MM</b> multiplies two matrices.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file with no header.
</li>
<li>
<b>R8VEC_BRACKET</b> searches a sorted array for successive brackets of a value.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_SUM</b> returns the sum of an R8VEC.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_Q4</b> maps Q4 reference points to physical points.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</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>SAMPLE_Q4_MESH</b> returns random points in a Q4 mesh.
</li>
<li>
<b>SAMPLE_QUAD</b> returns random points in a quadrilateral.
</li>
<li>
<b>SAMPLE_QUAD_NEW</b> returns random points in a quadrilateral.
</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</b> computes the area of a triangle in 2D.
</li>
<li>
<b>TRIANGLE_SAMPLE</b> returns random points in a triangle.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 29 September 2009.
</i>
<!-- John Burkardt -->
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