# TRIANGULATION_ORDER6 Pairs of Files Defining an Order 6 Triangulation

TRIANGULATION_ORDER6 is a data directory which defines the format of files for storing triangulation data of order 6.

Defining a triangulation requires two files:

• the node file lists the coordinates of a set of points.
• the element file lists sets of six node indices, which are the vertices and side nodes.

The six node triangulation, sometimes called a quadratic triangulation, includes the extra three nodes to allow for a higher degree of approximation when a finite element method is used. When listing the six nodes for a given triangle, the first three items are the vertices, in counterclockwise order. The fourth item is the side node between nodes 1 and 2, the fifth the side node between nodes 2 and 3, and the sixth the side node between nodes 3 and 1.

The point sets are distinguished by the values of the following parameters:

• M, the spatial dimension;
• N, the number of points;
The values of M and N are specified in the dataset file names.

At the moment, no facility is provided for allowing the specification of constrained edges, or the existence of holes in the region, both of which are of some interest.

### Example of a node file:

A node file can come from anywhere. In most cases, you generate this data by observation or the characteristics of a particular problem or set of data you are working on. In some cases, you start only with a region, and must fill the region with random points, or with well separated points.

As a simple example, suppose we had the following set of points:

```        11 12 13 14 15

6  7  8  9 10

1  2  3  4  5
```
then the node file might look like this:
```        #  Node file for simple example.
#
0.0  0.0
1.0  0.0
2.0  0.0
3.0  0.0
4.0  0.0
0.0  1.0
1.0  1.0
2.0  1.0
3.0  1.0
4.0  1.0
0.0  2.0
1.0  2.0
2.0  2.0
3.0  2.0
4.0  2.0
```

### Example of a triangle file:

A triangulation file describes how triangles are formed from the nodes. In an order 3 triangulation, each triangle is described by just 3 nodes. (An order 6 triangulation includes an extra node along the middle of each side). If you have nodes, but no triangulation of them, then there are programs available which can form a Delaunay triangulation of the nodes.

This triangulation will be order 3, so it is then necessary to convert the triangulation to order 6 by adding the midside nodes of each triangle.

An order 6 triangulation of these nodes is:

```        11-12-13-14-15
|\    |\    |
| \   | \   |
6  7  8  9 10
|   \ |   \ |
|    \|    \|
1--2--3--4--5
```
in which case the triangle file would look like this:
```        #  Triangle file for simple example.
#
1  3 11  2  7  6
13 11  3 12  7  8
3  5 13  4  9  8
15 13  5 14  9 10
```
The triangles could be listed in any order.

### Related Data and Programs:

FEM2D, a data directory which contains examples of 2D FEM files, three text files that can be used to describe many finite element models;

FEM_BASIS_T6_DISPLAY, a MATLAB program which displays a basis function associated with a 6-node triangle "T6" 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.

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

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 contains examples of 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 data directory which defines 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 (3 vertices and the centroid);

TRIANGULATION_ORDER6, a dataset directory which contains examples of TRIANGULATION_ORDER63 files, 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_PLOT, a C++ program which makes a PostScript image of a triangulation of points.

### Sample Files:

BOX6 is a rectangular region.
The smallest angle is 45 degrees.
There are 63 nodes and 24 elements.

You can go up one level to the DATA page.

Last revised on 03 December 2010.