TET_MESH_QUALITY
Interactive Program for
Tet Mesh Quality

TET_MESH_QUALITY is a MATLAB program, using double precision arithmetic, which computes and prints a variety of quality measures for a given tet mesh of a set of points in 3D.

The tet mesh is defined by a node file containing the coordinates of nodes, and a tetra file containing quadruples of node indices. For more information on this format, see order 4 Tet Mesh format.

The quality measures computed include:

• Measure 1, the minimum, over all tetrahedrons, of 3 times the radius of the insphere divided by the radius of the circumsphere.
• Measure 2, the minimum, over all tetrahedrons, of 2 times sqrt ( 6 ) times the radius of the insphere divided by the length of the longest side of the tetrahedron.
• Measure 3, the minimum, over all tetrahedrons, of 12 * ( 3 * volume )**(2/3) / (sum of squares of edge lengths);
• Measure 4, the minimum solid angle in any tetrahedron;
• Measure 5, a uniformity measure, the ratio of the maximum to minimum tetrahedron volume;

Each quality measure is defined as the minimum of its value over each tetrahedron. The maximum and best possible value is 1, and the minimum and worst value is 0.

Usage:

tet_mesh_quality ( node_file, tetra_file )

reads the tet mesh information from node_file and tetra_file, computes the quality measures, and prints them.

Related Programs:

CVT_TET_MESH is a FORTRAN90 library which uses CVT methods to compute a tet mesh in a region.

GEOMPACK3 is a FORTRAN90 library which includes a routine DTRIS3 to compute the tet mesh of a set of points in 3D.

MATLAB includes a command delaunay3 for computing the tet mesh of a set of points in 3D.

TABLE_QUALITY is an executable MATLAB program, which computes quality measures for an arbitrary (untriangulated) pointset in any dimension.

TABLE_TET_MESH is an executable FORTRAN90 program for computing the tet mesh of a set of points in 3D.

TEST_TET_MESH is a FORTRAN90 library defining a few test regions for the generation of a tet mesh.

TET_MESH is a MATLAB library of routines for computations with a tet mesh.

TET_MESH_DISPLAY is an executable 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.

TET_MESH_L2Q is an executable MATLAB program which converts a linear to quadratic tet mesh.

TET_MESH_NEIGHBORS is an executable MATLAB program which computes the tetrahedral adjacency information.

TET_MESH_Q2L is an executable MATLAB program which converts a quadratic to linear tet mesh.

TET_MESH_QUALITY is also available in a C++ version and a FORTRAN90 version.

TET_MESH_RCM is an executable MATLAB program which takes a tet mesh and relabels the nodes to reduce the bandwidth of the corresponding adjacency matrix.

TET_MESH_REFINE is an executable MATLAB program which can refine a tet mesh.

TRIANGULATION_QUALITY is an executable MATLAB program which computes triangle-based quality measures for a triangulation of a set of points in 2D.

Reference:

1. Herbert Edelsbrunner,
   Geometry and Topology for Mesh Generation,
   Cambridge, 2001,
   ISBN: 0-521-79309-2,
   LC: QA377.E36.
2. David Field,
   Qualitative Measures for Initial Meshes,
   International Journal of Numerical Methods in Engineering,
   Volume 47, 2000, pages 887-906.
3. Barry Joe,
   GEOMPACK - a software package for the generation of meshes using geometric algorithms,
   Advances in Engineering Software,
   Volume 13, 1991, pages 325-331.
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

Source files you may copy include:

• tet_mesh_quality.m determines quality measures for a tet mesh.
• dtable_data_read.m reads data from a double precision table file.
• dtable_header_read.m reads the header from a double precision table file.
• file_column_count.m counts the number of columns in the first line of a file.
• file_row_count.m counts the number of row records in a file.
• i4mat_transpose_print.m prints an I4MAT, transposed.
• i4mat_transpose_print_some.m prints some of the transpose of an I4MAT.
• i4vec_histogram.m makes a histogram of an I4VEC.
• i4vec_print.m prints an I4VEC.
• itable_data_read.m reads data from an integer table file.
• itable_header_read.m reads the header from an integer table file.
• r8mat_det_4d.m computes the determinant of a 4 by 4 matrix.
• r8mat_solve.m uses Gauss-Jordan elimination to solve an N by N linear system.
• r8mat_transpose_print.m prints an R8MAT, transposed.
• r8mat_transpose_print_some.m prints some of an R8MAT, transposed.
• r8vec_cross_3d.m computes the cross product of R8VEC's in 3D.
• r8vec_length.m returns the Euclidean length of an R8VEC.
• r8vec_mean.m computes the mean of an R8VEC.
• r8vec_print.m prints an R8VEC.
• r8vec_variance.m computes the variance of an R8VEC.
• s_len_trim.m returns the length of a string to the last nonblank.
• s_word_count.m counts the number of "words" in a string.
• tet_mesh_node_order.m returns the order of each node in a tetrahedral mesh.
• tet_mesh_quality1.m returns a tet mesh quality factor.
• tet_mesh_quality2.m returns a tet mesh quality factor.
• tet_mesh_quality3.m returns a tet mesh quality factor.
• tet_mesh_quality4.m returns a tet mesh quality factor.
• tet_mesh_quality5.m returns a tet mesh quality factor.
• tetrahedron_circumsphere_3d.m computes the circumsphere of a tetrahedron in 3D.
• tetrahedron_edge_length_3d.m returns edge lengths of a tetrahedron in 3D.
• tetrahedron_insphere_3d.m finds the insphere of a tetrahedron in 3D.
• tetrahedron_quality1_3d.m "quality" of a tetrahedron in 3D.
• tetrahedron_quality2_3d.m "quality" of a tetrahedron in 3D.
• tetrahedronal_quality3_3d.m computes the mean ratio of a tetrahedron.
• tetrahedron_quality4_3d.m computes the minimum solid angle of a tetrahedron.
• tetrahedron_volume_3d.m computes the volume of a tetrahedron in 3D.
• timestamp.m prints the current YMDHMS date as a time stamp.

Tests and Examples

CUBE is a simple example of 8 nodes in a cube. Test files you may copy include:

• cube_nodes.txt, the node file.
• cube_tetras.txt, the tetrahedron file.
• cube_quality.txt, the report from the program.

P01_00050 is a 3x1x1 channel, containing 50 nodes. Files you may copy include:

• p01_00050_nodes.txt, the node file.
• p01_00050_tetras.txt, the tetrahedron file.
• p01_00050_quality.txt, the report from the program.

P01_00400 is a 3x1x1 channel, containing 400 nodes. Files you may copy include:

• p01_00400_nodes.txt, the node file.
• p01_00400_tetras.txt, the tetrahedron file.
• p01_00400_quality.txt, the report from the program.

P01_03200 is a 3x1x1 channel, containing 3200 nodes. Files you may copy include:

• p01_03200_nodes.txt, the node file.
• p01_03200_tetras.txt, the tetrahedron file.
• p01_03200_quality.txt, the report from the program.

You can go up one level to the MATLAB source codes.