tet_mesh_l2q, a C++ code which converts a 4-node tetrahedral mesh ("tet mesh") into a 10-node tet mesh.

In particular, the code reads information describing a set of 3D points, and a 4 node ("linear") tet mesh of those points. It creates a "quadratic" tet mesh, which has the same number of tetrahedrons, but in which each tetrahedron is defined by 10 nodes. In the process, it adds just enough nodes, at the midsides of the edges of the original mesh, to define the new mesh. The data defining the new mesh is written out to files for further use.

The refinement of a single tetrahedron is easy. Every pair of nodes generates a new node whose location is the average of the locations of the original two nodes. The problem is that many tetrahedrons may share the same edge. The new node needs to be generated exactly once, and assigned a unique index, which all the tetrahedrons will share. The tricky part of this calculation is thus figuring out whether it is time to generate a new node, or whether the node has already been generated, in which case the appropriate index must be retrieved.


tet_mesh_l2q prefix
where prefix is the common file prefix:

The element definition file will list node indices. In C++, it may be more natural to use 0-based indices. This program will accept an element definition file that is 0-based or 1-based, and will convert a 1-based input file so that it becomes 0-based internal to the program. The detection of 1-based data is determined by the absence of the use of a 0 index, and the use of an index equal to the number of nodes. This is an implicit and fallible, but reasonable, way to handle this problem.

The input file prefix_nodes.txt contains the coordinate information for the 4-node tet mesh. Each data line contains the X and Y coordinates of a single node.

The input file prefix_elements.txt contains the tetrahedron information for the 4-node tet mesh. Each data line contains the indices of four nodes that form a tetrahedron.

The output file prefix_l2q_nodes.txt contains the coordinate information for the 10-node tet mesh. The data lines begin with the node information from nodes.txt, followed by the coordinates of the new nodes.

The output file prefix_l2q_elements.txt contains the tetrahedron information for the 10-node tet mesh. There are exactly as many tetrahedrons as before, but now each tetrahedron uses 10 nodes. Each line of the file contains the indices of 10 nodes that form the tetrahedron, listed in a particular order. The first four node indices are the same as for the linear tet mesh. The next six node indices may be thought of as being obtained as averages of pairs of the first four nodes. Symbolically, this may be thought of as:

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


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


tet_mesh_l2q is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Programs:

TET_MESH, a C++ code which includes a variety of routines for working with tetrahedral meshes.

TET_MESH_BOUNDARY, a C++ code 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_OPENGL, a C++ code which reads a tet mesh and displays the nodes and edges using OpenGL.


TET_MESH_Q2L, a C++ code which takes a 10-node tet mesh and makes a 4-node tet mesh.

TET_MESH_QUALITY, a C++ code which takes a 4-node tet mesh and computes the "quality" of the mesh.

TET_MESH_RCM, a C++ code which takes a tet mesh and relabels the nodes to reduce the bandwidth of the corresponding adjacency matrix.

TET_MESH_REFINE, a C++ code which can refine a tet mesh.

TET_MESH_TET_NEIGHBORS, a C++ code which computes the tetrahedral adjacency information.

TET_MESH_VOLUMES, a C++ code which computes the volume of each tetrahedron in a tet mesh;


  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:

Last revised on 26 April 2020.