triangulation

triangulation, an Octave code which computes a triangulation of a set of points in 2D, and carries out various other related operations on triangulations of order 3 or 6.

The mesh is the collection of triangles. Each triangle is termed an "element". The points used to define the shape of the triangle (the corners, and sometimes a few more points) are called the "nodes".

Routines are available to:

Since triangulations are often used to define a finite element mesh, which in turn defines a sparse matrix, there are routines available which can define the sparse compressed column arrays needed for a sparse matrix associated with a mesh of order 3 or 6. The special case of the Taylor-Hood mixed element is also handled, which is essentially an order 6 grid counted twice and an order 3 grid that only uses the vertices of the order 6 grid.

Licensing:

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

Languages:

triangulation is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

Related Data and Programs:

triangulation_test

distmesh, an Octave code which takes the definition of a 2d region, and fills it up with a set of nodes, and triangulates those nodes to make a triangulation of the region. the region may be nonconvex and may include holes; the user may request a specific density for the nodes, and may require certain points to be in the set of nodes.

mesh_bandwidth, an Octave code which returns the geometric bandwidth associated with a mesh of elements of any order and in a space of arbitrary dimension.

mesh2d, an Octave code which can automatically create a triangular mesh for a given polygonal region, by Darren Engwirda.

test_triangulation, an Octave code which can set up a number of triangulation test problems.

triangulation_boundary_edges, an Octave code which reads data defining a triangulation, determines which edges lie on the boundary, organizes them into connected components, and writes this information to a file.

triangulation_boundary_nodes, an Octave code which reads data defining a triangulation, determines which nodes lie on the boundary, and writes their coordinates to a file.

triangulation_corner, an Octave code which patches triangulations so that no triangle has two sides on the boundary.

triangulation_delaunay_discrepancy, an Octave code which measures the amount by which a triangulation fails the local delaunay test;

triangulation_display, an Octave code which displays the nodes and elements of a triangulation on the MATLAB graphics screen;

triangulation_histogram, an Octave code which computes histograms of data over a triangulation.

triangulation_l2q, an Octave code which reads data defining a 3-node triangulation and generates midside nodes and writes out the corresponding 6-node triangulation.

triangulation_mask, an Octave code which takes an existing triangulation and deletes triangles and their corresponding nodes as requested by the user.

triangulation_order3, a directory which contains a description and examples of order 3 triangulations.

triangulation_order6, a directory which contains a description and examples of order 6 triangulations.

triangulation_orient, an Octave code which reads data defining a triangulation, makes sure that every triangle has positive orientation, and if not, writes a corrected triangle file.

triangulation_plot, an Octave code that reads data defining a triangulation and creates a postscript image of the nodes and triangles.

triangulation_q2l, an Octave code which reads data defining a 6-node triangulation, and subdivides each triangle into 4 3-node triangles, writing the resulting triangulation to a file.

triangulation_quad, an Octave code which estimates the integral of a function over a triangulated region.

triangulation_quality, an Octave code which reads data defining a triangulation and computes a number of quality measures.

triangulation_rcm, an Octave code which reads data defining a triangulation, determines an ordering of the nodes that will reduce the bandwidth of the adjacency matrix, and writes the new triangulation information to a file.

triangulation_refine, an Octave code which reads data defining a triangulation, replaces each triangle by four congruent smaller ones, and writes the new triangulation information to a file.

triangulation_to_gmsh, an Octave code which reads a file of node coordinates and a file of elements defined by node indices, and creates a corresponding gmsh mesh file.

triangulation_triangle_neighbors, an Octave code which reads data defining a triangulation, determines the neighboring triangles of each triangle, and writes that information to a file.

Reference:

  1. Franz Aurenhammer,
    Voronoi diagrams - a study of a fundamental geometric data structure,
    ACM Computing Surveys,
    Volume 23, Number 3, September 1991, pages 345-405.
  2. Paul Bratley, Bennett Fox, Linus Schrage,
    A Guide to Simulation,
    Second Edition,
    Springer, 1987,
    ISBN: 0387964673.
  3. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
    Computational Geometry,
    Springer, 2000,
    ISBN: 3-540-65620-0.
  4. 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.
  5. Albert Nijenhuis, Herbert Wilf,
    Combinatorial Algorithms for Computers and Calculators,
    Second Edition,
    Academic Press, 1978,
    ISBN: 0-12-519260-6,
    LC: QA164.N54.
  6. Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu,
    Spatial Tessellations: Concepts and Applications of Voronoi Diagrams,
    Second Edition,
    Wiley, 2000,
    ISBN: 0-471-98635-6,
    LC: QA278.2.O36.
  7. Joseph ORourke,
    Computational Geometry,
    Second Edition,
    Cambridge, 1998,
    ISBN: 0521649765,
    LC: QA448.D38.
  8. 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 04 July 2023.