TET_MESH_ORDER4
Pairs of Files Defining a 4-Node Tet Mesh

TET_MESH_ORDER4 contains examples of the organization of a scattered set of points in 3D into a set of nonintersecting tetrahedrons, with each tetrahedron defined by 4 points.

A 4-node tet mesh is called linear since it easily allows for piecewise linear interpolation of data values defined at the nodes.

The definition of an order 4 tet mesh requires two files:

the node file lists the coordinates of a set of points.
the tetra file lists quadruples of node indices, each of which forms a tetrahedron.

Related Data and Programs:

FEM is a format used to store a finite element model. It includes a node and element file, as well as a node data file. The node and triangle files described here are an example of the first two of these files.

MESH_BANDWIDTH is an interactive executable program which returns the geometric bandwidth associated with a mesh of elements of any order and in a space of arbitrary dimension. It is available in a C++ version and a FORTRAN90 version and a MATLAB version.

TABLE is a format used for both the node and triangle files.

TET_MESH_ORDER10 describes a format to be used for tetrahedral meshes of order 10.

Example of a node file:

As a very simple example, suppose we had the eight points that form the vertices of a unit cube. The node file might look like this:

        #  Node file for the vertices of a cube.
        #
        0.0  0.0  0.0
        0.0  0.0  1.0
        0.0  1.0  0.0
        0.0  1.0  1.0
        1.0  0.0  0.0
        1.0  0.0  1.0
        1.0  1.0  0.0
        1.0  1.0  1.0

Example of a tet mesh file:

A possible tet mesh of these nodes would be described by the following tetra file:

        #  Tetra file for the vertices of a cube.
        #
          4  3  5  1
          4  2  5  1
          4  7  3  5
          4  7  8  5
          4  6  2  5
          4  6  8  5

Reference:

Herbert Edelsbrunner,
Geometry and Topology for Mesh Generation,
Cambridge, 2001,
ISBN: 0-521-79309-2,
LC: QA377.E36.
Barry Joe,
GEOMPACK - a software package for the generation of meshes using geometric algorithms,
Advances in Engineering Software,
Volume 13, Number 5, 1991, pages 325-331.
Per-Olof Persson, Gilbert Strang,
A Simple Mesh Generator in MATLAB,
SIAM Review,
Volume 46, Number 2, June 2004, pages 329-345.

Programs to create a set of nodes to fill a given region:

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 a very interesting case, you specify the boundary of a region, and ask a program to fill the interior with points:

DISTMESH can take a description of a 3D region, and fill it with grid points that are well distributed, and include the boundary.
DISTMESH_3D is a subset of DISTMESH that is just for 3D problems.

Programs to generate a set of nodes:

If you just need some artificial set of sample node data, there are programs available to generate it:

CVT_DATASET, available in a C++ version, a FORTRAN90 version, and a MATLAB version, produces a TABLE file of N points in M dimensions, as a centroidal Voronoi tessellation.
CVT_MOD_DATASET produces a TABLE file of N points in M dimensions, as a centroidal Voronoi tessellation on a unit hypercube with modular arithmetic.
FAURE_DATASET produces a TABLE file of N points in M dimensions, from a Faure sequence.
GRID_DATASET produces a TABLE file of N points in M dimensions, as a regular grid.
HALTON_DATASET, available in a C++ version and a FORTRAN90 version, and a MATLAB version, produces a TABLE file of N points in M dimensions, from a Halton sequence.
HAMMERSLEY_DATASET, available in a C++ version and a FORTRAN90 version, and a MATLAB version, produces a TABLE file of N points in M dimensions, from a Hammersley sequence.
IHS_DATASET produces a TABLE file of N points in M dimensions, from an Improved Hypercube Sampling pointset.
LATIN_CENTER_DATASET produces a TABLE file of N points in M dimensions, from a Latin hypercube pointset with centering.
LATIN_EDGE_DATASET produces a TABLE file of N points in M dimensions, from a Latin hypercube pointset with edge "centering".
LATIN_RANDOM_DATASET produces a TABLE file of N points in M dimensions, from a Latin hypercube pointset with random "centering".
LCVT_DATASET produces a TABLE file of N points in M dimensions, containing a Latin hypercube dataset derived from a centroidal Voronoi tessellation.
NIEDERREITER2_DATASET produces a TABLE file of N points in M dimensions, from a Niederreiter sequence base 2.
RANDOM_DATA can produce a set of points that are randomly sampled from a given 2D or 3D geometric shape.
RBOX will produce a random sample of points according to a given distribution, and over a given geometric shape. The resulting output file is ALMOST a TABLE file (you just need to delete the two initial records that list the spatial dimension and number of points.)
RSITES will produce a set of N points in M dimensional space, with the user allowed to specify the SEED to be passed to the C++ RAND routine. Point coordinates are integers between 0 and 999999.
SOBOL_DATASET produces a TABLE file of N points in M dimensions, from a Sobol sequence.
UNIFORM_DATASET, available in a C++ version or a FORTRAN90 version, or a MATLAB version, produces a TABLE file of N points in M dimensions, containing a uniform pseudorandom sequence.

Programs to view a node file:

Once you have generated a node file, it is possible to try to view it in 3D. Programs to visualize a set of points in 3D include:

MATLAB includes a function scatter3 which can display a set of points, allowing the user to turn or zoom in on the image.

Programs to create a tet mesh from a set of nodes:

Once you have generated a node file, it is possible to generate a tet mesh of those nodes. Programs and routines to create a tet mesh include:

DISTMESH generates a tet mesh as part of its work, if the dataset is 3D.
DISTMESH_3D generates a tet mesh as part of its work.
The GEOMPACK3 includes the routine DTRIS3 which can generate a tet mesh of a set of points.
MATLAB includes a function delaunay3 which can return the tet mesh of a set of points by the command
tet = delaunay3 ( x, y, z );

Programs to manipulate an order 4 mesh:

TET_MESH_L2Q generates a 10-node tet mesh from a 4-node tet mesh.
TET_MESH_NEIGHBORS determines, for each tetrahedron in a tet mesh, the indices of its four tetrahedral neighbors.

Programs to view a tet mesh:

Once you have generated a node file and a tetrahedron file, it is possible (although somewhat difficult!) to try to view it in 3D. Programs to visualize a tet mesh in 3D:

DISTMESH_3D includes a routine called simp_plot_3d which is specially designed to display all or part of a tet mesh.
TET_MESH_DISPLAY_MATLAB is a MATLAB program which can read in the node and tetra files defining a tet mesh and display a wireframe image.
TET_MESH_DISPLAY_OPEN_GL is a C++ executable program which reads a tet mesh and displays the nodes and edges using OpenGL.

Sample Tet Mesh Datasets:

CUBE_ORDER4 is the 8 vertices of a cube, defining 6 tetrahedrons.

cube_order4_nodes.txt, the node file;
cube_order4_tetras.txt, the tetra file.
cube_order4.png, a PNG image of the tetrahedral mesh.

P01_00584 is the 3x1x1 channel defined by 584 nodes.

p01_00584_nodes.txt, the coordinates of the nodes;
p01_00584_tetras.txt, the tetrahedral mesh (2482 tetrahedrons).
p01_00584.png, a PNG image of the tetrahedral mesh.

P02_00588 is the vertical cylinder defined by 588 nodes.

p02_00588_nodes.txt, the coordinates of the nodes;
p02_00588_tetras.txt, the tetrahedral mesh (2373 tetrahedrons).
p02_00588.png, a PNG image of the tetrahedral mesh.

P03_00008 is the unit cube defined by 8 nodes.

p03_00008_nodes.txt, the coordinates of 8 nodes;
p03_00008_tetras.txt, the tetrahedral mesh (6 tetrahedrons).
p03_00008.png, a PNG image of the tetrahedral mesh.

P03_00224 is the unit cube defined by 224 nodes.

p03_00224_nodes.txt, the coordinates of 8 nodes;
p03_00224_tetras.txt, the tetrahedral mesh (798 tetrahedrons).
p03_00224.png, a PNG image of the tetrahedral mesh.

P04_00587 is the unit sphere defined by 587 nodes.

p04_00587_nodes.txt, the coordinates of the nodes;
p04_00587_tetras.txt, the tetrahedral mesh (2775 tetrahedrons).
p04_00587.png, a PNG image of the tetrahedral mesh.

P05_01084 is the cylinder with a spherical hole defined by 1084 nodes.

p05_01084_nodes.txt, the coordinates of the nodes;
p05_01084_tetras.txt, the tetrahedral mesh (4738 tetrahedrons).
p05_01084.png, a PNG image of the tetrahedral mesh.

TETRA_RHOMBIC_ORDER4 is the 4 vertices of a tetrahedron, plus the 10 midside points, making a mesh of 8 tetrahedron of order 4. This tetrahedron has some nice properties under a subdivision algorithm.

tetra_rhombic_order4_nodes.txt, the node file;
tetra_rhombic_order4_tetras.txt, the tetra file;

TWENTY_ORDER4 is 20 random nodes, defining 70 tetrahedrons.

twenty_order4_nodes.txt, the node file;
twenty_order4_tetras.txt, the tetra file.

You can go up one level to the DATA page.

Last revised on 18 January 2007.

TET_MESH_ORDER4 Pairs of Files Defining a 4-Node Tet Mesh