TRIANGULATION_CORNER is a C++ program which tries to correct situations in which a triangulation includes corner triangles, that is, triangles which have two sides on boundary.
Especially in finite element settings, such a triangle is considered undesirable. Especially in the case when a linear (order 3) finite element is involved, the function approximation in the interior of the triangle will be entirely determined by boundary data. If, for instance, the solution is constrained to be zero on the boundary, then the finite element function will be zero throughout the interior of the corner triangle.
Presumably, the triangle has a neighbor triangle on the one non-boundary side. It is generally possible and reasonable to replace these two triangles but another two which fill the same quadrilateral, but which are separated by the other diagonal of the quadrilateral.
The initial situation at the corner is suggest by the following diagram:
|/ |/ A---C-- |\ |\ | \ | | \| B---D--
By rearranging the corner triangle and its neighbor, we have the more satisfactory triangulation:
|/ |/ A---C-- | /|\ | / | |/ | B---D--
triangulation_corner prefixwhere prefix is the common filename prefix:
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
TRIANGULATION_CORNER is available in a C++ version and a FORTRAN90 version and a MATLAB version.
MESH_TO_XML, a C++ program which reads information defining a 1D, 2D or 3D mesh, namely a file of node coordinates and a file of elements defined by node indices, and creates a corresponding XML file for input to DOLFIN or FENICS.
TABLE_DELAUNAY, a C++ program which reads a file of point coordinates in the TABLE format and writes out the Delaunay triangulation.
TRIANGLE, a C program which computes a triangulation of a geometric region.
TRIANGULATION, a C++ library which performs various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.
TRIANGULATION_BOUNDARY_NODES, a C++ program which reads data defining a triangulation, determines which nodes lie on the boundary, and writes their coordinates to a file.
TRIANGULATION_DELAUNAY_DISCREPANCY, a C++ program which measures the amount by which a triangulation fails the local Delaunay test;
TRIANGULATION_DISPLAY_OPENGL, a C++ program which reads files defining a triangulation and displays an image using Open GL.
TRIANGULATION_HISTOGRAM, a C++ program which computes histograms of data over a triangulation.
TRIANGULATION_L2Q, a C++ program which reads information about a 3-node (linear) triangulation and creates data defining a corresponding 6-node (quadratic) triangulation;
TRIANGULATION_MASK, a C++ program which reads a triangulation and calls a user-supplied routine to consider each triangle for deletion;
TRIANGULATION_NODE_TO_ELEMENT, a C++ program which reads files describing a set of nodes, their triangulation, and the value of one or more quantities at each node, and outputs a file that averages the quantities for each element. This operation in effect creates an "order1" finite element model of the data.
TRIANGULATION_ORDER3, a data directory which contains examples of TRIANGULATION_ORDER3 files, description of a linear triangulation of a set of 2D points, using a pair of files to list the node coordinates and the 3 nodes that make up each triangle;
TRIANGULATION_ORDER6, a data directory which contains examples of TRIANGULATION_ORDER6 files, a description of a quadratic triangulation of a set of 2D points, using a pair of files to list the node coordinates and the 6 nodes that make up each triangle.
TRIANGULATION_ORIENT, a C++ program which ensures that the triangles in an order 3 or order 6 triangulation have positive orientation;
TRIANGULATION_PLOT, a C++ program which makes a PostScript image of a triangulation of points.
TRIANGULATION_QUAD, a C++ program which estimates the integral of a function over a triangulated region.
TRIANGULATION_REFINE, a C++ program which can refine a triangulation.
TRIANGULATION_TRIANGLE_NEIGHBORS, a C++ program which reads data defining a triangulation, determines the neighboring triangles of each triangle, and writes that information to a file.
P03 is the square with a hole. The initial mesh has 1 corner triangle.
P05 is the horn. The initial mesh has 4 corner triangles.
P08 is the holey pie slice. The initial mesh has 2 corner triangles.
P09 is Jeff Borggaard's square with two hexagonal holes. The initial mesh has 3 corner triangles.
P10 is the unit square. The initial mesh has 4 corner triangles.
P11 is the L-shaped region. The initial mesh has 4 corner triangles.
Q11 is the L-shaped region, but with quadratric (6 node) triangles. The initial mesh has 4 corner triangles. Because we are using quadratic triangles, when we merge two triangles and swap the diagonal, the midside node on the common diagonal will no longer lie on a line, midway between the two vertices. TRIANGULATION_CORNER assumes this is undesirable, and recomputes the coordinates of such midside nodes, and outputs a modified node coordinate file as well as the new triangle file.
P12 is the John Shadid's H-shaped region. The initial mesh has 7 corner triangles.
P13 is the Sandia fork. The initial mesh has 6 corner triangles.
P14 is Marcus Garvie's Lake Alpha, with Beta Island. The initial mesh has 5 corner triangles. (Must be small!)
P15 is Sangbum Kim's forward step region. The initial mesh has 6 corner triangles.
P16 is Kevin Pond's elbow, a quarter of a circular annulus. The initial mesh has 1 corner triangle.
P17 is the rectangular channel with a Reuleaux triangle obstacle. The initial mesh has 4 corner triangles.
You can go up one level to the C++ source codes.