test_triangulation

test_triangulation, a FORTRAN90 code which sets up a number of triangulation test problems.

As far as possible, a uniform and abstract approach has been used. For each test region, a number of routines are provided, via which it is possible for the user to determine many things about the region. Often, just one or two routines would be needed for a particular purpose. The names of the routines, and their purposes are:

• BOUNDARY_NEAREST: returns the nearest point on the boundary of the region to a given point or set of points;
• BOUNDARY_PROJECT: projects exterior points onto the boundary.
• BOUNDARY_SEGMENT: returns a sequence of roughly equally spaced points that lie on one particular boundary segment;
• BOUNDARY_SEGMENT_LENGTH: returns the "length" (number of nodes) of a particular boundary segment. This simply counts the number of points needed to trace or approximate the boundary segment;
• BOUNDARY_SEGMENT_NUM: returns the number of boundary segments. Simple regions have one boundary segment. A region with one hole has two, and so on;
• BOX: returns a bounding box for the region. All points in the region are within this box;
• DENSITY: the value of the mesh density function at any point in the region. If this is not constant, then high values correspond to places where many more mesh points should be placed; is always 2 for this set;
• ELEMENT_SIZE: returns a requested typical element size. This refers to the average size of the triangles formed by a triangulation of the points;
• FIXED_NUM: returns the number of points which must be included as nodes of the mesh (which may be zero);
• FIXED_POINTS: returns the coordinates of the points which must be included as nodes of the mesh;
• HEADER: prints a brief description of the problem;
• HOLE_NUM: returns the number of "holes" in the region;
• HOLE_POINT: returns the coordinates of one point in a hole (useful when TRIANGLE is to be invoked);
• INSIDE: reports which of a given set of points are inside the region.
• SAMPLE: returns a set of sample points from the region, chosen with uniform probability;
• SAMPLE_H1: returns a set of sample points from the region, after it has been enlarged by an amount H1;
• SDIST: returns the signed distance to the boundary of the region for each of a set of input points. (a positive distance means the point is outside the region, a negative distance means it is inside); (Not ready for problems 4, 5, 6, 7, 8, 9 )
• TITLE: a title for the problem;

The test problems include:

1. The unit circle;
2. The unit circle with a circular hole;
3. A square with a circular hole;
4. A hexagon with a hexagonal hole;
5. The horn;
6. The superellipse with a superelliptical hole;
7. The bicycle seat;
8. The slice of pie with a circular hole and triangular notch;
9. Jeff Borggaard's square with two hexagonal holes;
10. The unit square;
11. The L-shaped region;
12. John Shadid's H-shaped region;
13. The Sandia fork;
14. Marcus Garvie's Lake Alpha, with Beta Island;
15. Sangbum Kim's forward step;

Licensing:

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

Languages:

test_triangulation is available in a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CVT_TRIANGULATION, a FORTRAN90 code which uses routines from the TEST_TRIANGULATION library to create a CVT-based triangularization.

TABLE_IO, a FORTRAN90 code which is used to write some of the output files.

test_triangulation_test

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

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

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

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

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

TRIANGULATION ORDER3, a directory which describes the format for triangulations of order 3.

TRIANGULATION ORDER6, a directory which describes the format for triangulations of order 6.

TRIANGULATION_ORIENT, a FORTRAN90 code 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 FORTRAN90 code which reads data defining a triangulation and creates a PostScript image of the nodes and triangles.

TRIANGULATION_Q2L, a FORTRAN90 code 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 FORTRAN90 code which reads data defining a triangulation and computes a number of quality measures.

TRIANGULATION_RCM, a FORTRAN90 code 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 FORTRAN90 code 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 FORTRAN90 code 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. Barry Joe,
   GEOMPACK - a software package for the generation of meshes using geometric algorithms,
   Advances in Engineering Software,
   Volume 13, 1991, pages 325-331.
3. Joseph ORourke,
   Computational Geometry,
   Second Edition,
   Cambridge, 1998,
   ISBN: 0521649765,
   LC: QA448.D38.
4. Per-Olof Persson, Gilbert Strang,
   A Simple Mesh Generator in MATLAB,
   SIAM Review,
   Volume 46, Number 2, June 2004, pages 329-345.

Source Code:

• test_triangulation.f90, the source code.
• test_triangulation.sh, compiles the source code.