TRIANGULATION_QUAD
Integral Estimate Over a Triangulated Region

TRIANGULATION_QUAD is a MATLAB 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:

quad = 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.

Licensing:

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

Languages:

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

Related Data and Programs:

QUADRULE, a MATLAB library which defines quadrature rules for 1D domains.

STROUD, a MATLAB library which defines quadrature rules for a variety of multidimensional reqions.

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

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

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

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

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

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

TRIANGULATION_DELAUNAY_DISCREPANCY, a MATLAB program which measures the amount by which a triangulation fails the local Delaunay test;

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

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

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

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

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

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

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

TRIANGULATION_ORIENT, a MATLAB 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, a MATLAB program which reads data defining a triangulation and creates a PostScript image of the nodes and triangles.

TRIANGULATION_Q2L, a MATLAB 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_QUALITY, a MATLAB program which computes quality measures of a triangulation.

TRIANGULATION_RCM, a MATLAB 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, a MATLAB 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, a MATLAB 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.m, the program.

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' )

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