TET_MESH_ORDER10
Quadratic Tet Mesh Definition

TET_MESH_ORDER10 contains examples of 10-node "tet meshes" (meshes of tetrahedrons). In such a mesh, the fundamental shape is a tetrahedron, but each tetrahedron is described by 10 nodes, namely, the four vertices, and the six midside nodes.

Typically, such a mesh is not generated directly from arbitrary data, since the midside nodes are expected to be the averages of pairs of vertices, or nearly so. It is more typical that a linear mesh, using 4-node tetrahedrons, is generated, and then a quadratic mesh is derived from the linear mesh.

A 10-node tet mesh is called quadratic since it easily allows for piecewise quadratic interpolation of data values defined at the nodes.

The definition of an order 10 tet mesh requires two files:

• the node file lists the coordinates of a set of points.
• the tet file lists groups of 10 node indices, which form each 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 the format used for both the node and tet files.

TET_MESH_ORDER4 is a data directory containing a description and examples of order 4 tet meshes.

Example of a node file:

Here is an example of a node file:

        #  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    
        0.0  0.0  0.5    
        0.0  0.5  0.0    
        0.0  0.5  0.5   
        0.5  0.0  0.0    
        0.0  0.5  1.0    
        0.5  0.0  0.5   
        0.5  0.0  1.0    
        0.0  1.0  0.5   
        0.5  0.5  0.0    
        0.5  1.0  0.0    
        0.5  0.5  0.5 
        0.5  0.5  1.0    
        0.5  1.0  0.5    
        0.5  1.0  1.0    
        1.0  0.0  0.5    
        1.0  0.5  0.0    
        1.0  0.5  0.5   
        1.0  0.5  1.0   
        1.0  1.0  0.5    
      

Example of a tet mesh file:

Each tetrahedron will be described by a list of 10 nodes. An arbitrary ordering for these nodes must be imposed. It is natural to list the four vertices first, but the 6 midside nodes have no natural ordering. Here, then, is one ordering that we use, where we have indicated the position of the midside nodes by listing the two "parent" vertices:

        1, 2, 3, 4, (1+2), (1+3), (1+4), (2+3), (2+4), (3+4)
      

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

       4   3   5   1  16  19  11  17   10  12
       4   2   5   1  13  19  11  14    9  12
       4   7   3   5  21  16  19  18   24  17
       4   7   8   5  21  22  19  27   24  25
       4   6   2   5  20  13  19  15   23  14
       4   6   8   5  20  22  19  26   23  25
      

Reference:

1. Herbert Edelsbrunner,
   Geometry and Topology for Mesh Generation,
   Cambridge, 2001,
   ISBN: 0-521-79309-2,
   LC: QA377.E36.
2. 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.
3. 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 linear tet mesh of those nodes. Programs and routines to create a linear tet mesh include:

• DISTMESH generates a linear tet mesh as part of its work, if the dataset is 3D.
• DISTMESH_3D generates a linear tet mesh as part of its work.
• The GEOMPACK3 includes the routine DTRIS3 which can generate a linear tet mesh of a set of points.
• MATLAB includes a function delaunay3 which can return the linear tet mesh of a set of points.

Programs to manipulate an order 10 mesh:

• TET_MESH_L2Q generates a quadratic tet mesh from a linear one.
• 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 linear tet mesh, it is possible (although somewhat difficult!) to try to view it in 3D. Programs to visualize a tetahedral mesh in 3D:

• DISTMESH_3D includes a routine called simp_plot_3d which is specially designed to display all or part of a linear 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_ORDER10 is the 8 vertices of a cube, plus 19 midside points, making a mesh of 6 tetrahedrons of order 10.

• cube_order10_nodes.txt, the node file;
• cube_order10_tetras.txt, the tetra file;
• cube_order10.png, a PNG image of the tetrahedral mesh.

ONEONEEIGHT_ORDER10 is 118 nodes forming 70 order-10 tetrahedrons (derived by applying TET_MESH_L2Q to the order-4 dataset "TWENTY_ORDER4".)

• oneoneeight_order10_nodes.txt, the node file;
• oneoneeight_order10_tetras.txt, the tetra file;
• oneoneeight_order10.png, a PNG image of the tetrahedral mesh.

TETRA_RHOMBIC_ORDER10 is the 4 vertices of a tetrahedron, plus the 6 midside points, making a mesh of 1 tetrahedron of order 10. This tetrahedron has some nice properties under a subdivision algorithm.

• tetra_rhombic_order10_nodes.txt, the node file;
• tetra_rhombic_order10_tetras.txt, the tetra file;
• tetra_rhombic_order10.png, a PNG image of the tetrahedral mesh.

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