TRIANGULATION_QUAD
Integral Estimate Over a Triangulated Region

TRIANGULATION_QUAD is a FORTRAN90 program which reads information defining a triangulation, and estimates the integral of a function whose values are given at the nodes.

Note that this program only expects to recieve values of the function f(x,y) at the nodes of the triangulation, that is, as a simple list of values. No formula for f is expected or used. A much better estimate for the integral might be possible if a formula for f(x,y) were available, in which case a higher order quadrature scheme could be employed.

This program should be able to compute exactly the integrals of 1, x, y, and any linear combination of these. It will only be able to approximate the integrals of other functions, and the accuracy of the approximation will depend in part on the size of the triangles in the triangulation.

Usage:

triangulation_quad prefix

• prefix_nodes.txt, the node coordinates;
• prefix_elements.txt, the elements of the triangulation.
• prefix_values.txt, the value (which may be a scalar or vector) of some function at each node.
• quad is the (scalar or vector) result of the integration estimate, which is computed and printed out by the program.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

QUADRULE is a FORTRAN90 library which defines quadrature rules for 1D domains.

STROUD is a FORTRAN90 library which defines quadrature rules for a variety of multidimensional reqions.

TABLE_DELAUNAY is a FORTRAN90 program which triangulates a set of nodes whose coordinates are stored in a file.

TOMS706 is a FORTRAN77 library which estimates the integral of a function over a triangulated region.

TRIANGLE is a C program which computes a triangulation of a geometric region.

TRIANGULATION is a FORTRAN90 library which carries out various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.

TRIANGULATION_BOUNDARY_NODES is a FORTRAN90 program which reads data defining a triangulation, determines which nodes lie on the boundary, and writes their coordinates to a file.

TRIANGULATION_CORNER, a FORTRAN90 program which patches triangulations so that no triangle has two sides on the boundary.

TRIANGULATION_DELAUNAY_DISCREPANCY is a FORTRAN90 program which measures the amount by which a triangulation fails the local Delaunay test;

TRIANGULATION_DISPLAY is a MATLAB program which displays the nodes and elements of a triangulation on the MATLAB graphics screen;

TRIANGULATION_DISPLAY_OPEN_GL is a C++ program which reads files defining a triangulation and displays an image using Open GL.

TRIANGULATION_HISTOGRAM, a FORTRAN90 program which computes histograms of data over a triangulation.

TRIANGULATION_L2Q is a FORTRAN90 program which reads data defining a 3-node triangulation and generates midside nodes and writes out the corresponding 6-node triangulation.

TRIANGULATION_MASK is a FORTRAN90 program which takes an existing triangulation and deletes triangles and their corresponding nodes as requested by the user.

TRIANGULATION_ORDER3 is a directory which contains a description and examples of order 3 triangulations.

TRIANGULATION_ORDER6 is a directory which contains a description and examples of order 6 triangulations.

TRIANGULATION_ORIENT is a FORTRAN90 program which reads data defining a triangulation, makes sure that every triangle has positive orientation, and if not, writes a corrected triangle file.

TRIANGULATION_PLOT is a FORTRAN90 program which reads data defining a triangulation and creates a PostScript image of the nodes and triangles.

TRIANGULATION_Q2L is a FORTRAN90 program which reads data defining a 6-node triangulation, and subdivides each triangle into 4 3-node triangles, writing the resulting triangulation to a file.

TRIANGULATION_QUAD is available in a C++ version and a FORTRAN90 version and a MATLAB version.

TRIANGULATION_QUAD is a FORTRAN90 program which estimates the integral of a function over a triangulated region.

TRIANGULATION_QUALITY is a FORTRAN90 program which computes quality measures of a triangulation.

TRIANGULATION_RCM is a FORTRAN90 program which reads data defining a triangulation, determines an ordering of the nodes that will reduce the bandwidth of the adjacency matrix, and writes the new triangulation information to a file.

TRIANGULATION_REFINE is a FORTRAN90 program which reads data defining a triangulation, replaces each triangle by four congruent smaller ones, and writes the new triangulation information to a file.

TRIANGULATION_TRIANGLE_NEIGHBORS is a FORTRAN90 program which reads data defining a triangulation, determines the neighboring triangles of each triangle, and writes that information to a file.

Source Code:

• triangulation_quad.f90, the program.
• triangulation_quad.csh, commands to compile the source code.

Examples and Tests:

EXAMPLE is a set of nodes in the unit square, which have been arranged into an order 3 triangulation.

• example_nodes.txt, the node file.
• example_elements.txt, the element file.
• example_values.txt, the values file, containing 4 values at each node. (These values are actually the values of 1, x, y, and x*y, which have exact integrals of 1, 1/2, 1/2, and 1/4.) Only the first three integrals will be estimated exactly.
• example_output.txt, the output file from the command
  quad = triangulation_quad example

List of Routines:

• MAIN is the main program for TRIANGULATION_QUAD.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• I4MAT_DATA_READ reads data from an I4MAT file.
• I4MAT_HEADER_READ reads the header from an I4MAT.
• I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of the transpose of an I4MAT.
• MESH_BASE_ONE ensures that the element definition is one-based.
• R8MAT_DATA_READ reads data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT transposed.
• R8VEC_PRINT prints an R8VEC.
• S_TO_I4 reads an I4 from a string.
• S_TO_I4VEC reads an I4VEC from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.