TET_MESH
Routines for a Tet Mesh

TET_MESH is a MATLAB library which constructs, describes, or modifies 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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

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

Related Programs:

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

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

GEOMPACK, 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, a MATLAB library which defines a number of quadrature rules for a tetrahedron.

NCC_TETRAHEDRON, a MATLAB library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

NCO_TETRAHEDRON, a MATLAB library which defines Newton-Cotes open quadrature rules on a tetrahedron.

QUADRATURE_RULES_TET, a dataset directory which contains triples of files defining various quadrature rules on tetrahedrons.

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

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

TET_MESH_BOUNDARY, a MATLAB program which returns the nodes and faces of the boundary of a tetrahedral mesh, which themselves form a 3D triangular mesh or "TRI_SURFACE".

TET_MESH_DISPLAY, a MATLAB program which reads in the node and tetra files defining a tet mesh and displays a wireframe image.

TET_MESH_DISPLAY_OPENGL, a C++ program which can read in the node and tetra files defining a tet mesh and display a wireframe image using OpenGL.

TET_MESH_L2Q, a MATLAB program which converts a linear to quadratic tet mesh.

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

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

TET_MESH_Q2L, a MATLAB program which converts a quadratic to linear tet mesh.

TET_MESH_QUALITY, a MATLAB program which computes the quality of a tet mesh.

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

TET_MESH_REFINE, a MATLAB program which can refine a tet mesh.

TET_MESH_TET_NEIGHBORS, a MATLAB program which computes the tetrahedral adjacency information.

TET_MESH_VOLUMES, a MATLAB program which computes the volume of each tetrahedron in a tet mesh;

TETRAHEDRON_PROPERTIES, a MATLAB program which computes properties of a given tetrahedron.

TETRAHEDRONS, a dataset directory which contains examples of tetrahedrons;

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.

Source code:

• i4_swap.m swaps two I4's.
• i4_uniform.m returns a pseudorandom I4.
• 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.m prints the transpose of an I4MAT.
• i4mat_transpose_print_some.m prints some of the transpose of an I4MAT.
• i4vec_print.m prints an I4VEC.
• mesh_base_one.m detects and corrects 0-based indexing.
• 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.
• r8mat_transpose_print.m prints an R8MAT, transposed.
• r8mat_transpose_print_some.m prints some of an R8MAT, transposed.
• r8mat_uniform_01.m returns a unit pseudorandom R8MAT.
• 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_uniform_01.m returns a unit pseudorandom 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_example_set.m returns an example order 4 tet mesh for a cube.
• tet_mesh_order4_example_size.m sizes an example order 4 tet mesh for a cube.
• 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_set.m returns an example order 10 tet mesh for a cube.
• tet_mesh_order10_example_size.m sizes an example order 10 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.
• tet_mesh_search_naive.m naively searches a tet mesh.
• tet_mesh_search_delaunay.m searches a Delaunay tet mesh.
• tetrahedron_barycentric.m computes the barycentric coordinates of a point with respect to a tetrahedron.
• 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.m computes the volume of a tetrahedron.
• 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_test004.m, tests TET_MESH_NODE_ORDER.
• tet_mesh_test005.m, tests TETRAHEDRON_BARYCENTRIC.
• tet_mesh_test006.m, tests TET_MESH_NEIGHBOR_TETS.
• tet_mesh_test007.m, tests TET_MESH_SEARCH_NAIVE and TET_MESH_SEARCH_DELAUNAY.
• tet_mesh_test_output.txt, the output from a run of the sample program.

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