TET_MESH
Routines for a Tet Mesh

TET_MESH is a MATLAB library of routines, using double precision arithmetic, containing routines that may be of use in constructing, describing, or modifying a mesh of tetrahedrons.

Linear and Quadratic Meshes

The simplest tet mesh, which we term an order 4 or linear mesh, uses four points to define each tetrahedron. A second type of mesh, known as an order 10 or quadratic mesh, uses ten points.

While an order 4 mesh can naturally be constructed directly from most sets of data points, a mesh of order 10 is not usually constructed directly from the data; at least in the simplest case, one wants the 6 extra nodes to be the midpoints of the sides determined by the 4 vertices.

Thus, an order 10 tet mesh is typically generated in two steps:

• generate an order 4 mesh, in which every tetrahedral vertex comes from a user data point;
• generate an order 10 mesh from the order 4 mesh, in which the new midside points are not user data points, but rather averages of pairs of tetrahedral vertices.

Related Programs:

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

GEOMETRY is a MATLAB library which includes a number of routines for making computations involving tetrahedrons.

GEOMPACK is a MATLAB library which contains a routine DTRIS3 that can compute the tet mesh for a set of 3D points, as well as the adjacency information.

KEAST is a MATLAB library which defines a number of quadrature rules for a tetrahedron.

MATLAB has a command delaunay3 that can compute the tet mesh for a set of 3D points.

NCC_TETRAHEDRON is a MATLAB library defining Newton-Cotes closed quadrature rules on a tetrahedron.

NCO_TETRAHEDRON is a MATLAB library defining Newton-Cotes open quadrature rules on a tetrahedron.

QUADRATURE_RULES_TET is a dataset directory of triples of files defining various quadrature rules on tetrahedrons.

TABLE_TET_MESH is a FORTRAN90 program which can compute the tet mesh for a given set of points.

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

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

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 an executable C++ program which can read in the node and tetra files defining a tet mesh and display a wireframe image using OPEN_GL.

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

TET_MESH_ORDER4 is a directory which contains a description and examples of a tet mesh using order 4 elements.

TET_MESH_ORDER10 is a directory which contains a description and examples of a tet mesh using order 10 elements.

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

TET_MESH_QUALITY is an executable MATLAB program which computes the quality of a tet mesh.

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.

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

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. Anwei Liu, Barry Joe,
   Quality Local Refinement of Tetrahedral Meshes Based on 8-Subtetrahedron Subdivision,
   Mathematics of Computation,
   Volume 65, Number 215, July 1996, pages 1183-1200.
4. Per-Olof Persson, Gilbert Strang,
   A Simple Mesh Generator in MATLAB,
   SIAM Review,
   Volume 46, Number 2, June 2004, pages 329-345.

Tar File:

A GZIP'ed TAR file of the contents of this directory is available. This is only done as a convenience for users who want ALL the files, and don't want to download them individually. This is not a convenience for me, so don't be surprised if the tar file is somewhat out of date.

• tet_mesh_m_src.tar.csh the commands used to create the GZIP'ed TAR file.
• tet_mesh_m_src.tar.gz the GZIP'ed TAR file.

Source code:

• i4_swap.m swaps two I4's.
• i4col_compare.m compares columns I and J of an I4COL.
• i4col_sort_a.m ascending sorts an I4COL.
• i4col_sort2_a.m ascending sorts the elements of each column of an I4COL.
• i4col_sorted_unique_count.m counts unique elements in an I4COL.
• i4col_swap.m swaps columns J1 and J2 of an I4COL.
• i4i4_sort_a.m ascending sorts a pair of I4's.
• i4i4i4_sort_a.m ascending sorts a triple of I4's.
• i4mat_transpose_print_some.m prints some of the transpose of an I4MAT.
• r8_swap.m switches two R8's.
• r8_uniform_01.m returns a unit pseudorandom R8.
• r8mat_det_4d.m computed the determiannt of a 4x4 R8MAT .
• r8mat_print.m prints an R8MAT.
• r8mat_print_some.m prints some of an R8MAT.
• r8mat_solve.m uses Gauss-Jordan elimination to solve an N by N linear system.
• r8vec_cross_3d.m computes the cross product of two R8VEC's in 3D.
• r8vec_length.m returns the Euclidean length of an R8VEC.
• r8vec_mean.m returns the mean of an R8VEC.
• r8vec_variance.m returns the variance of an R8VEC.
• s_len_trim.m returns the length of a string to the last nonblank.
• sort_heap_external.m externally sorts a list of items into ascending order.
• tet_mesh_neighbor_tets.m determines tetrahedron neighbors.
• tet_mesh_node_order.m determine the order of nodes in a tetra mesh.
• tet_mesh_order4_adj_count.m counts the number of nodal adjacencies.
• tet_mesh_order4_adj_set.m sets the nodal adjacency matrix.
• tet_mesh_order4_boundary_face_count.m counts the number of boundary faces.
• tet_mesh_order4_edge_count.m counts the number of edges.
• tet_mesh_order4_refine_compute.m computes a refined order 4 tet mesh.
• tet_mesh_order4_refine_size.m sizes a refined order 4 tet mesh
• tet_mesh_order4_to_order10_compute.m computes a quadratic tet mesh from a linear one.
• tet_mesh_order4_to_order10_size.m sizes a quadratic tet mesh from a linear one.
• tet_mesh_order10_example_cube_compute.m returns an example order10 tet mesh for a cube.
• tet_mesh_order10_example_cube_size.m sizes an example order10 tet mesh for a cube.
• tet_mesh_order10_to_order4_compute.m linearizes a quadratic tet mesh.
• tet_mesh_order10_to_order4_size.m sizes a linear mesh from a quadratic tet mesh.
• tet_mesh_quad.m approximates an integral over a tet mesh.
• tet_mesh_quality1.m returns the quality of each tet in a mesh.
• tet_mesh_quality2.m returns the quality of each tet in a mesh.
• tet_mesh_quality3.m returns the quality of each tet in a mesh.
• tet_mesh_quality4.m returns the quality of each tet in a mesh.
• tet_mesh_quality5.m returns the quality of each tet in a mesh.
• 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_order4_physical_to_reference.m converts physical points to reference points in a tetrahedron.
• tetrahedron_order4_reference_to_physical.m converts reference points to physical points in a tetrahedron.
• tetrahedron_quality1_3d.m "quality" of a tetrahedron in 3D.
• tetrahedron_quality2_3d.m "quality" of a tetrahedron in 3D.
• tetrahedron_quality3_3d.m computes the mean ratio of a tetrahedron.
• tetrahedron_quality4_3d.m computes the minimum solid angle of a tetrahedron.
• tetrahedron_reference_sample.m returns random points in the reference tetrahedron.
• tetrahedron_sample.m returns random points in 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.

Examples and Tests:

• tet_mesh_test.m, a sample calling program.
• tet_mesh_test001.m, tests R8MAT_SOLVE.
• tet_mesh_test002.m, tests TETRAHEDRON_ORDER4_PHYSICAL_TO_REFERENCE and TETRAHEDRON_ORDER4_REFERENCE_TO_PHYSICAL.
• tet_mesh_test003.m, tests TETRAHEDRON_ORDER10_TO_ORDER4.
• tet_mesh_test.out, the output from a run of the sample program.

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