TRIANGULATION_ORDER3
Examples of Order 3 Triangulations

TRIANGULATION_ORDER3 is a dataset directory which contains examples of triangulation data of order 3.

Defining a triangulation requires two files:

• the node file lists the coordinates of a set of points.
• the element file lists triples of node indices, forming triangles.

Other files that may be supplied include:

• the element neighbor file. ELEMENT_NEIGHBOR(I,J) lists the index of the element that is a neighbor along the element edge that is opposite to node ELEMENT_NODE(I,J), or -1 if no such neighbor exists.
• the boundarynode file lists a 0 for each interior node, and a 1 for each node on the boundary of the region.

For details of this format, go to ../../data/triangulation_order3/triangulation_order3.html

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:

FEM_BASIS_T3_DISPLAY, a MATLAB program which displays a basis function associated with a 3-node triangle "T3" mesh.

MESH_BANDWIDTH, a C++ program which returns the geometric bandwidth associated with a mesh of elements of any order and in a space of arbitrary dimension.

TRIANGULATE, a C program which triangulates a (possibly nonconvex) polygon.

TRIANGULATION, a C++ library which performs 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_DISPLAY_OPENGL, a C++ program which reads files defining a 2D triangulation and displays an image using OpenGL.

TRIANGULATION_ORDER3, a data directory which defines TRIANGULATION_ORDER3 files, a description of a linear triangulation of a set of 2D points, using a pair of files to list the node coordinates and the 3 nodes that make up each triangle;

TRIANGULATION_ORDER4, a dataset directory which contains examples of TRIANGULATION_ORDER4 files, a description of a triangulation of a set of 2D points, using a pair of files to list the node coordinates and the 4 nodes that make up each triangle.

TRIANGULATION_ORDER6, a dataset directory which contains examples of TRIANGULATION_ORDER6 files, a description of a quadratic triangulation of a set of 2D points, using a pair of files to list the node coordinates and the 6 nodes that make up each triangle.

TRIANGULATION_ORIENT, a C++ program which ensures that the triangles in an order 3 or order 6 triangulation have positive orientation;

TRIANGULATION_PLOT, a C++ program which makes a PostScript image of a triangulation of points.

TRIANGULATION_QUALITY, a C++ program which computes quality measures of a triangulation.

Sample Files:

ANNULUS is an annulus, a circular region with a circular hole.
There are 208 nodes and 384 elements and 1 hole.

• annulus_nodes.txt, the node coordinates.
• annulus_elements.txt, the triangulation;
• annulus_elements.png, a PNG image of the triangulation.
• annulus_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

BIG_CAVITY is a triangulation using 8,185 nodes and 4,000 elements.

• big_cavity_nodes.txt, the node coordinates.
• big_cavity_elements.txt, the elements;
• big_cavity_elements.png, a PNG image of the elements;
• big_cavity_element_neighbors.txt, the neighbor triangles.
• big_cavity_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

BOX3 is a rectangular region.
The smallest angle is 45 degrees.
There are 20 nodes and 24 elements.

• box3_nodes.txt, the node coordinates.
• box3_elements.txt, the elements;
• box3_elements.png, a PNG image of the elements.
• box3_element_neighbors.txt, the neighbor triangles.
• box3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

BOX_HOLE is a square with a circular hole.
Some of the elements are degenerate, and could be discarded.
There are 40,516 nodes and 79,921 elements.

• box_hole_nodes.txt, the node coordinates.
• box_hole_elements.txt, the elements;
• box_hole_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

BOXY is a simple non-Delaunay triangulation of 13 nodes using 16 elements.

• boxy_nodes.txt, the node coordinates.
• boxy_elements.txt, the triangulation;
• boxy_elements.png, a PNG image of the triangulation;
• boxy_element_neighbors.txt, the neighbor triangles.

CHANNEL3 is a rectangular region 1 unit high and 3 units wide.
The smallest angle is 45 degrees.
There are 65 nodes and 96 elements.

• channel3_nodes.txt, the node coordinates.
• channel3_elements.txt, the triangulation;
• channel3_elements.png, a PNG image of the triangulation.
• channel3_element_neighbors.txt, the neighbor triangles.
• channel3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

CIRCLE_HOLE3 is from a mesh for fluid flow past a circular obstacle. The triangulation of the nodes was created by the TRIANGLE program. This data was supplied by Professor Hyung-Chun Lee.
The smallest angle is 28.79 degrees.
There are 968 nodes and 1783 elements and 1 hole.

• circle_hole3_nodes.txt, the node coordinates.
• circle_hole3_elements.txt, the triangulation;
• circle_hole3_elements.png, a PNG image of the triangulation.
• circle_hole3_element_neighbors.txt, the neighbor triangles;
• circle_hole3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

DEVILLERS3 is data provided by Olivier Devillers for the demonstration of a 2D Delaunay triangulation program.
The smallest angle is 0.07 degrees.
There are 1000 nodes and 1981 elements.

• devillers3_nodes.txt, the node coordinates.
• devillers3_elements.txt, the triangulation;
• devillers3_elements.png, a PNG image of the triangulation.
• devillers3_element_neighbors.txt, the neighbor triangles;
• devillers3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

ELBOW3 is an elbow-shaped region.
The smallest angle is 39.06 degrees.
There are 320 nodes and 568 elements.

• elbow3_nodes.txt the nodes.
• elbow3_nodes.png an image of the nodes.
• elbow3_elements.txt the elements.
• elbow3_elements.png a PNG image of the elements.
• elbow3_element_neighbors.txt, the neighbor triangles;
• elbow3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

ELL3 is an L shaped region.
The smallest angle is 45 degrees.
There are 21 nodes and 24 elements.

• ell3_nodes.txt the node coordinates.
• ell3_nodes.png an image of the nodes.
• ell3_elements.txt the elements.
• ell3_elements.png a PNG image of the elements.
• ell3_element_neighbors.txt, the neighbor triangles.
• ell3_element_neighbors.png, a PNG image of the neighbor information.
• ell3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

EXAMPLE_DELAUNAY is an example in which 16 nodes are arranged in a Delaunay triangulation.

• example_delaunay_nodes.txt the node coordinates.
• example_delaunay_nodes.png an image of the nodes.
• example_delaunay_elements.txt the elements.
• example_delaunay_elements.png a PNG image of the elements.

EXAMPLE_NONDELAUNAY is an example in which the same 16 nodes used in the EXAMPLE_DELAUNAY example are arranged in a non-Delaunay triangulation. Since almost all triangulation algorithms use a Delaunay approach, it is important to see an occasional example of what a non-Delaunay triangulation looks like. Simply by looking at this mesh and the Delaunay triangulation of the same set of points, you should have some intuitive feeling that the Delaunay triangulation has better properities, even if you can't say immediately what those properties might be exactly!

• example_nondelaunay_nodes.txt the node coordinates.
• example_nondelaunay_nodes.png an image of the nodes.
• example_nondelaunay_elements.txt the elements.
• example_nondelaunay_elements.png a PNG image of the elements.

GREENLAND is a triangulation using 33,343 nodes and 64,125 elements.

• greenland_nodes.txt, the nodes.
• greenland_elements.txt, the elements;
• greenland_elements.png a PNG image of a "zoomed" portion of the elements. The whole image is difficult to view at once.
• greenland_element_neighbors.txt, the neighbor triangles.
• greenland_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

HANG3 is a rectangular region 8 units high and 8 units wide. This example is interesting because it is a triangulation which includes "hanging nodes", that is, nodes which are shared by two triangles, but which are only a vertex of one of them. Triangulations which allow hanging nodes can cause problems for many meshing algorithms. This is not a maximal triangulation, since more nodes could be connected without crossing existing edges; therefore it is not a Delaunay triangulation.
The smallest angle is 18.43 degrees.
There are 12 nodes and 11 elements.

• hang3_nodes.txt, the node coordinates.
• hang3_elements.txt, the elements;
• hang3_elements.png, a PNG image of the elements.
• hang3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

HEX_HOLES3 is a region with two hexagonal holes.
The smallest angle is 31.78 degrees.
There are 142 nodes and 237 elements and 2 holes.

• hex_holes3_nodes.txt, the node coordinates.
• hex_holes3_elements.txt, the elements;
• hex_holes3_elements.png, a PNG image of the elements.
• hex_holes3_element_neighbors.txt, the neighbor triangles.
• hex_holes3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

HOLEY_PIE is a polygonal shape suggesting a slice of pie with a circular hole and a triangular notch.

• holey_pie_nodes.txt, the node coordinates.
• holey_pie_elements.txt, the elements;
• holey_pie_elements.png, a PNG image of the elements.

HOT_PIPE3 is a sample region for the MATLAB program FEM_50_HEAT, defined on a 13x13 grid in the unit square.
The smallest angle is 45 degrees.
There are 169 nodes and 288 elements.

• hot_pipe3_nodes.txt the node coordinates.
• hot_pipe3_elements.txt the elements;
• hot_pipe3_elements.png a PNG image of the elements
• hot_pipe3_element_neighbors.txt, the neighbor triangles.
• hot_pipe3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

HOUSE3 is a child's image of a house using an order 3 triangulation.
The smallest angle is 45 degrees.
There are 5 nodes and 3 elements.

• house3_nodes.txt, the node coordinates.
• house3_elements.txt, the elements;
• house3_elements.png, a PNG image of the elements.
• house3_element_neighbors.txt, the neighbor triangles.
• house3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

LAKE_COARSE is an irregular lake with an island.
There are 621 nodes and 974 elements and 1 hole.

• lake_coarse_nodes.txt, the node coordinates.
• lake_coarse_elements.txt, the elements;
• lake_coarse_elements.png, a PNG image of the elements.
• lake_coarse_element_neighbors.txt, the neighbor triangles.
• lake_coarse_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.
• lake_coarse_boundarynodes.png, a plot of the boundary nodes.
• lake_coarse_triangulation.png, a plot of the Delaunay triangulation of the nodes.

LAKE3 is an irregular lake with an island.
The smallest angle is 23.46 degrees.
There are 1552 nodes and 2820 elements and 1 hole.

• lake3_nodes.txt, the node coordinates.
• lake3_elements.txt, the elements;
• lake3_elements.png, a PNG image of the elements.
• lake3_element_neighbors.txt, the neighbor triangles.
• lake3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

SHRINK3 is a 16x15 rectangle which uses a "shrinking" grid in which the triangles get smaller as they rise.
The smallest angle is 45 degrees.
There are 36 nodes and 45 elements.

• shrink3_nodes.txt, the node coordinates.
• shrink3_elements.txt, the elements;
• shrink3_elements.png, a PNG image of the elements.
• shrink3_element_neighbors.txt, the neighbor triangles.
• shrink3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

SLIVER3 is rectangle divided into triangles, some of which are long and thin. Such a triangulation often causes numerical errors.
The smallest angle is 0.42 degrees.
There are 14 nodes and 12 elements.

• sliver3_nodes.txt the node coordinates.
• sliver3_elements.txt the elements.
• sliver3_elements.png a PNG image of the elements.
• sliver3_element_neighbors.txt, the neighbor triangles.
• sliver3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

SMALL3 is a square with a 5 by 5 grid of nodes.
The smallest angle is 45 degrees.
There are 25 nodes and 32 elements.

• small3_nodes.txt, the node coordinates.
• small3_elements.txt, the elements;
• small3_elements.png, a PNG image of the elements.
• small3_element_neighbors.txt, the neighbor triangles.
• small3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

TED3 is a triangulation which is a Delaunay triangulation. Compare this with TEN3 which is a non-Delaunay triangulation of the same set of nodes.
The smallest angle is 26.99 degrees.
There are 10 nodes and 10 elements.

• ted3_nodes.txt, the node coordinates.
• ted3_elements.txt, the elements;
• ted3_elements.png, a PNG image of the elements.
• ted3_element_neighbors.txt, the neighbor triangles.
• ted3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

TEN3 is a triangulation which is NOT a Delaunay triangulation. Compare this with TED3 which is a Delaunay triangulation of the same set of nodes.
The smallest angle is 9.97 degrees.
There are 10 nodes and 10 elements.

• ten3_nodes.txt, the node coordinates.
• ten3_elements.txt, the elements;
• ten3_elements.png, a PNG image of the elements.
• ten3_element_neighbors.txt, the neighbor triangles.
• ten3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

TRIANGLE3 is a single triangle.
The smallest angle is 60 degrees.
There are 3 nodes and 1 element.

• triangle3_nodes.txt, the node coordinates.
• triangle3_elements.txt, the elements;
• triangle3_elements.png, a PNG image of the elements.
• triangle3_element_neighbors.txt, the neighbor triangles.
• triangle3_boundarynodes.txt, 0 for interior nodes, 1 for boundary nodes.

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