TRIANGULATION_REFINE
Refine a Triangulation

TRIANGULATION_REFINE is a C++ program which reads information describing a triangulation of a set of points and creates a refined triangulation.

The refined triangulation is created by dividing each triangle into four similar subtriangles. The mesh size parameter h will be halved by such a procedure. If the input triangulation is Delaunay, then so will be the output triangulation.

The program can handle triangulations that use 3 or 6 nodes per triangle.

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 triangle, (for a triangle file).

The input file prefix_nodes.txt contains the node information for the triangulation. Each data line contains the X and Y coordinates of a single node.

The input file prefix_elements.txt contains the triangle information for the triangulation. Each line contains the indices of 3 or 6 nodes that form a triangle.

Usage:

triangulation_refine prefix

• prefix_nodes.txt contains the node coordinates,
• prefix_elements.txt contains the element definitions.
• prefix_ref_nodes.txt is the refined node file;
• prefix_ref_elements.txt is the refined element fle.

Licensing:

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

Languages:

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

Related Data and Programs:

TABLE, a data format which is used for the input and output files.

TABLE_DELAUNAY, a C++ program which can read a file of point coordinates and construct the Delaunay triangulation for that set of points.

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

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

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

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

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

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

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

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

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

TRIANGULATION_NODE_TO_ELEMENT, a C++ program which reads files describing a set of nodes, their triangulation, and the value of one or more quantities at each node, and outputs a file that averages the quantities for each element. This operation in effect creates an "order1" finite element model of the data.

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 C++ 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 C++ program which reads data defining a triangulation and creates a PostScript image of the nodes and triangles.

TRIANGULATION_Q2L, a C++ 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, a C++ program which estimates the integral of a function over a triangulated region.

TRIANGULATION_QUALITY, a C++ program which reads data defining a triangulation and computes a number of quality measures.

TRIANGULATION_RCM, a C++ 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_TRIANGLE_NEIGHBORS, a C++ program which reads data defining a triangulation, determines the neighboring triangles of each triangle, and writes that information to a file.

Reference:

1. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
   Computational Geometry,
   Springer, 2000,
   ISBN: 3-540-65620-0.
2. Joseph ORourke,
   Computational Geometry,
   Second Edition,
   Cambridge, 1998,
   ISBN: 0521649765,
   LC: QA448.D38.

Source Code:

• triangulation_refine.cpp, the source code.

Examples and Tests:

BOX3 is an example of the refinement of a 3-node triangulation of a rectangular box.

• box3_nodes.txt, the nodes for a 3-node triangulation.
• box3_elements.txt, the triangles for an order 3 triangulation.
• box3.png, a PNG image of the triangulation, created by TRIANGULATION_PLOT.
• box3_ref_nodes.txt, the nodes for the refined triangulation.
• box3_ref_elements.txt, the triangles for the refined triangulation.
• box3_ref.png, a PNG image of the refined triangulation, created by TRIANGULATION_PLOT.
• box3_output.txt, output printed by the program.

IRREG6 is an example of the refinement of a 6-node triangulation of an irregular region with a hole:

• irreg6_nodes.txt, the nodes for a 6-node triangulation.
• irreg6_elements.txt, the triangles for an order 6 triangulation.
• irreg6.png, a PNG image of the triangulation, created by TRIANGULATION_PLOT.
• irreg6_ref_nodes.txt, the nodes for the refined triangulation.
• irreg6_ref_elements.txt, the triangles for the refined triangulation.
• irreg6_ref.png, a PNG image of the refined triangulation, created by TRIANGULATION_PLOT.
• irreg6_output.txt, output printed by the program.

HOUSE6 is an example of the refinement of a simple 6-node triangulation of a "house" diagram that involved 3 triangles.

• house6_nodes.txt, the nodes for a 6-node triangulation.
• house6_elements.txt, the triangles for an order 6 triangulation.
• house6.png, a PNG image of the triangulation, created by TRIANGULATION_PLOT.
• house6_ref_nodes.txt, the nodes for the refined triangulation.
• house6_ref_elements.txt, the triangles for the refined triangulation.
• house6_ref.png, a PNG image of the refined triangulation, created by TRIANGULATION_PLOT.
• house6_output.txt, output printed by the program.

List of Routines:

• MAIN is the main program for TRIANGULATION_REFINE.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• FILE_COLUMN_COUNT counts the columns in the first line of a file.
• FILE_EXIST reports whether a file exists.
• FILE_ROW_COUNT counts the number of row records in a file.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• I4_MODP returns the nonnegative remainder of I4 division.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• I4COL_COMPARE compares columns I and J of an I4COL.
• I4COL_SORT_A ascending sorts the columns of an I4COL.
• I4COL_SORTED_UNIQUE_COUNT counts unique elements in an I4COL.
• I4COL_SWAP swaps two columns of an I4COL.
• I4I4_SORT_A ascending sorts a pair of I4's.
• I4MAT_DATA_READ reads data from an I4MAT file.
• I4MAT_HEADER_READ reads the header from an I4MAT file.
• I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
• I4MAT_WRITE writes an I4MAT file with no header.
• R8MAT_DATA_READ reads the 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.
• R8MAT_WRITE writes an R8MAT file with no header.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• 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.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGULATION_ORDER3_REFINE_COMPUTE computes a refined order 3 triangulation.
• TRIANGULATION_ORDER3_REFINE_SIZE sizes a refined order 3 triangulation.
• TRIANGULATION_ORDER6_REFINE_COMPUTE computes a refined order 6 triangulation.
• TRIANGULATION_ORDER6_REFINE_SIZE sizes a refined order 6 triangulation.

You can go up one level to the C++ source codes.