DISTMESH
A Simple Mesh Generator in MATLAB

DISTMESH is a MATLAB program which generates and manipulates unstructured meshes in 2D, 3D and general ND, by Per-Olof Persson.

The code is relatively simple, and the user is able to define a variety of geometric shapes, and desired mesh densities.

DISTMESH can be a very quick and flexible means of computing a set of points in a region. However, keep in mind the following considerations:

• Especially if you are have specified some fixed points which must appear in the mesh, it is possible for DISTMESH to return multiple instances of the same point. For finite element applications, in particular, this can result in catastrophe. The program TABLE_MERGE can fix this.
• The nodes produced by DISTMESH are not ordered or sorted in any way whatsoever.
• Because the nodes are not ordered in any way, the triangular elements produced by DISTMESH will typically contain nodes with widely ranging indices. For finite element applications, this can result in a system matrix with an unnecessarily high bandwidth. The program TRIANGULATION_RCM can fix this, after the fact.
• The triangles produced by DISTMESH are not necessarily oriented; they are just as likely to have positive or negative orientation. Some finite element programs insist that all triangles have positive orientation. The program TRIANGULATION_ORIENT can fix this, after the fact.
• When DISTMESH is trying to approximate a boundary, particularly a long straight boundary, it is possible for several points that really belong on the boundary to be slightly out of line. This means that they will be used to form a triangle, of very small area, and terrible conditioning. This can result in perplexing problems near the boundary.
• Because DISTMESH uses tolerances, it is possible for some nodes to lie outside the boundary of the region; it is possible for some triangles on the boundary to lie partially outside the region; it is possible for triangles that lie partly on the boundary, but entirely within the region, to be deleted, leaving a triangular hole.
• Once you have the mesh, you may want to know which nodes lie on the boundary. You'll want this information, for instance, if you need to impose boundary conditions on such nodes. You can get a list of the boundary nodes using the program TRIANGULATION_BOUNDARY_NODES.

Usage:

[ p, t ] = distmesh_2d ( fd, fh, h, box, iteration_max, fixed );

• fd, the name of a distance function defining the region;
• fh, the name of a mesh density function;
• h, the nominal mesh spacing;
• box, defining a box that contains the region;
• iteration_max, the maximum number of iterations;
• fixed, a list of points which must be included in the mesh, or '[]', if no fixed points are given.
• p, (output), a list of node coordinates;
• t, (output), a list of node indices forming triangles;

Licensing:

DISTMESH is Copyright (C) 2004 Per-Olof Persson.

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

If you use DISTMESH in any program or publication, please acknowledge its authors by citing the reference.

Related Data and Programs:

DIST_PLOT, a MATLAB program which creates a color contour plot of the distance functions that are used by DISTMESH.

DISTMESH_3D, a MATLAB program which is a subset of the DISTMESH routines, exclusively for 3D problems.

MESH2D, a MATLAB library which can automatically create a triangular mesh for a given polygonal region, by Darren Engwirda.

POINT_MERGE, a MATLAB library which considers N points in M dimensional space, and counts or indexes the unique or "tolerably unique" items.

TABLE_IO, a MATLAB library which reads and writes files using the TABLE format; these routines are used by DISTMESH when creating some output files.

TABLE_MERGE, a MATLAB program which removes duplicate points from a TABLE file; it can also remove points that are "close" to each other;

TEST_TRIANGULATION, a MATLAB library which defines some test regions for triangulation.

TRIANGLE, a C program which computes a triangulation of a geometric region, by Jonathan Shewchuk.

TRIANGULATION_BOUNDARY_NODES, a MATLAB program which reads data defining a triangulation and determines which nodes lie on the boundary.

TRIANGULATION_DISPLAY_OPENGL, a C++ program which reads files defining a triangulation and displays an image using Open GL.

TRIANGULATION_L2Q, a MATLAB program which reads data defining a linear triangulation and adds midpoint nodes to create a quadratic triangulation.

TRIANGULATION_MASK, a MATLAB program which is compiled with a user routine that defines a region; it then reads data defining a triangulation and removes all triangles that are outside the region. This is one way to clean up an unconstrained Delaunay triangulation of a nonconvex region.

TRIANGULATION_ORDER3, a data directory which discusses order 3 triangulations; The node and triangle files output by DISTMESH are an example of such triangulations.

TRIANGULATION_ORIENT, a MATLAB program which reads data defining a triangulation, makes sure that every triangle has positive orientation, and if not, writes a corrected triangle file.

TRIANGULATION_PLOT, a MATLAB program which plots a triangulation.

TRIANGULATION_RCM, a MATLAB program which reads data defining a triangulation, and uses the Reverse Cuthill McKee algorithm to reorder the nodes so as the reduce the bandwidth of the corresponding adjacency matrix. This can be very helpful for cases where the data is to be handled by a frontal technique, or treated as a banded linear system.

TRIANGULATION_REFINE, a MATLAB program which reads data defining a triangulation and creates a refinement of the triangulation by subdividing each triangle.

Author:

Per-Olof Persson.

Reference:

1. http://math.mit.edu/~persson/
   Per-Olof Persson's web site.
2. Per-Olof Persson and Gilbert Strang,
   A Simple Mesh Generator in MATLAB,
   SIAM Review,
   Volume 46, Number 2, June 2004, pages 329-345.

Source Code:

• boundedges.m finds the surface edges in a triangular mesh.
• circumcenter.m computes the circumcenters of the triangles that form a triangulation of a set of nodes.
• dcircle.m returns the signed distance of one or more points to a circle.
• ddiff.m returns the signed distance of one or more points to a region defined as the set difference of two regions.
• dexpr.m returns the signed distance of one or more points to a region defined by a general symbolic expression.
• dintersect.m returns the signed distance to a region that is the intersection of two regions.
• distmesh_2d.m computes a mesh of a given 2D region.
• distmesh_nd.m computes a mesh of a given ND region.
• dmatrix.m returns the signed distance to a region by interpolation of known distance values on a Cartesian grid.
• dpoly.m returns the signed distance of one or more points to a polygon.
• drectangle.m returns the signed distance of one or more points to a rectangle.
• drectangle0.m, returns the signed distance of one or more points to a rectangle.
• dsegment.cpp (C++ file), a version of the algorithm for the signed distance of one or more points to a set of line segments, for use with a non-Windows version of MATLAB. This file must be compiled, and the corresponding MEX file must be available to define the MATLAB/C++ interface.
• dsegment.dll (binary file), a Windows DLL file, returns the signed distance of one or more points to a set of line segments.
• dsegment.m a pure MATLAB version of DSEGMENT, which returns the signed distance of one or more points to a set of line segments. This routine will be significantly slower than the MEXGLX or DLL versions.
• dsegment.mexglx (binary file), used on non-Windows machines to allow MATLAB to calculate the DSEGMENT algorithm (distance to a line segment) by calling a compiled C++ routine (dsegment.cpp).
• dsphere.m returns the signed distance of one or more points to a sphere.
• dunion.m returns the signed distance to a region that is the union of two regions.
• file_column_count.m counts the number of columns in a table file.
• file_row_count.m counts the number of rows in a table file.
• hmatrix.m computes the mesh size function by interpolation from values specified on a Cartesian grid.
• huniform.m computes a uniform mesh size function.
• i4mat_data_read.m reads the data of an I4MAT file.
• i4mat_header_read.m reads the header of an I4MAT file.
• i4mat_read.m reads an I4MAT file.
• i4mat_write.m writes an I4MAT file.
• meshdemo_nd.m demonstrates the use of the program for higher dimensional problems.
• post_2d.m performs postprocessing for output from DISTMESH_2D.
• protate.m rotates a set of points by a given angle.
• pshift.m shifts a set of points by a given increment.
• r8_epsilon.m returns the R8 arithmetic precision.
• r8mat_data_read.m reads the data of an R8MAT file.
• r8mat_header_read.m reads the header of an R8MAT file.
• r8mat_write.m writes an R8MAT file.
• s_len_trim.m returns the length of a string to the last nonblank.
• s_word_count.m counts the number of "words" in a string.
• simp_plot_2d.m displays a plot of the triangles that form a mesh in 2D.
• simp_plot_2d_demo.m reads and displays the node and triangle data for each problem.
• simp_plot_3d.m displays a plot of the tetrahedrons that form a mesh in 3D.
• simpqual.m computes the simplex quality of the mesh.
• simpvol.m computes the volume of a simplex.
• surftri.m finds the surface triangles in a tetrahedral mesh.
• timestamp.m prints the current YMDHMS time as a timestamp.
• triangulation_order3_plot.m writes a PostScript file containing an image of the mesh.
• uniformity.m computes the uniformity of the mesh.

Examples and Tests:

MESHDEMO_2D runs all the 2D tests:

• meshdemo_2d.m demonstrates the use of the program for 2D problems.
• meshdemo_2d_output.txt the output from a run of MESHDEMO_2D.

P01 is the circle:

• p01_demo.m runs the problem.
• p01_fd.m the signed distance function.
• p01_fh.m a uniform mesh density function.
• p01_mesh.png a PNG image of the mesh.
• p01_nodes.txt the mesh nodes of the problem.
• p01_elements.txt the mesh triangles of the problem.

P02 is the circle with a hole:

• p02_demo.m runs the problem.
• p02_fd.m the signed distance function.
• p02_fh.m a uniform mesh density function.
• p02_mesh.png a PNG image of the mesh.
• p02_nodes.txt the mesh nodes of the problem.
• p02_elements.txt the mesh triangles of the problem.

P03 is the square with a circular hole:

• p03_demo.m runs the problem.
• p03_fd.m the signed distance function.
• p03_fh.m a uniform mesh density function.
• p03_mesh.png a PNG image of the mesh.
• p03_nodes.txt the mesh nodes of the problem.
• p03_elements.txt the mesh triangles of the problem.

P04 is the hexagon with a hexagonal hole:

• p04_demo.m runs the problem.
• p04_fd.m the signed distance function.
• p04_fh.m a uniform mesh density function.
• p04_mesh.png a PNG image of the mesh.
• p04_nodes.txt the mesh nodes of the problem.
• p04_elements.txt the mesh triangles of the problem.

P05 is the horn:

• p05_demo.m runs the problem.
• p05_fd.m the signed distance function.
• p05_fh.m a nonuniform mesh density function.
• p05_mesh.png a PNG image of the mesh.
• p05_nodes.txt the mesh nodes of the problem.
• p05_elements.txt the mesh triangles of the problem.

P06 is the superellipse:

• p06_demo.m runs the problem.
• p06_fd.m the signed distance function.
• p06_fh.m a nonuniform mesh density function.
• p06_mesh.png a PNG image of the mesh.
• p06_nodes.txt the mesh nodes of the problem.
• p06_elements.txt the mesh triangles of the problem.

P07 is the bicycle seat:

• p07_demo.m runs the problem.
• p07_fd.m the signed distance function.
• p07_fh.m a uniform mesh density function.
• p07_mesh.png a PNG image of the mesh.
• p07_nodes.txt the mesh nodes of the problem.
• p07_elements.txt the mesh triangles of the problem.

P08 is the holey pie slice:

• p08_demo.m runs the problem.
• p08_fd.m the signed distance function.
• p08_fh.m a nonuniform mesh density function.
• p08_mesh.png, a PNG image of the mesh.
• p08_nodes.txt the mesh nodes of the problem.
• p08_elements.txt the mesh triangles of the problem.

P09 is Jeff Borggaard's square with two hexagonal holes:

• p09_demo.m runs the problem.
• p09_fd.m the signed distance function.
• p09_fh.m a uniform mesh density function.
• p09_mesh.png a PNG image of the mesh.
• p09_nodes.txt the mesh nodes of the problem.
• p09_elements.txt the mesh triangles of the problem.

P10 is the unit square:

• p10_demo.m runs the problem.
• p10_fd.m the signed distance function.
• p10_fh.m a uniform mesh density function.
• p10_mesh.png a PNG image of the mesh.
• p10_nodes.txt the mesh nodes of the problem.
• p10_elements.txt the mesh triangles of the problem.

P11 is the L-shaped region:

• p11_demo.m runs the problem.
• p11_fd.m the signed distance function.
• p11_fh.m a uniform mesh density function.
• p11_mesh.png a PNG image of the mesh.
• p11_nodes.txt the mesh nodes of the problem.
• p11_elements.txt the mesh triangles of the problem.

P12 is the John Shadid's H-shaped region:

• p12_demo.m runs the problem.
• p12_fd.m the signed distance function.
• p12_fh.m a uniform mesh density function.
• p12_mesh.png a PNG image of the mesh.
• p12_nodes.txt the mesh nodes of the problem.
• p12_elements.txt the mesh triangles of the problem.

P13 is the Sandia fork:

• p13_demo.m runs the problem.
• p13_fd.m the signed distance function.
• p13_fh.m a uniform mesh density function.
• p13_mesh.png a PNG image of the mesh.
• p13_nodes.txt the mesh nodes of the problem.
• p13_elements.txt the mesh triangles of the problem.

P14 is Marcus Garvie's Lake Alpha, with Beta Island:

• p14_demo.m runs the problem.
• p14_fd.m the signed distance function.
• p14_fh.m a nonuniform mesh density function.
• p14_mesh.png a PNG image of the mesh.
• p14_nodes.txt the mesh nodes of the problem.
• p14_elements.txt the mesh triangles of the problem.

P15 is Sangbum Kim's forward step region:

• p15_demo.m runs the problem.
• p15_fd.m the signed distance function.
• p15_fh.m a nonuniform mesh density function.
• p15_mesh.png a PNG image of the mesh.
• p15_nodes.txt the mesh nodes of the problem.
• p15_elements.txt the mesh triangles of the problem.

P16 is Kevin Pond's elbow, a quarter of a circular annulus:

• p16_demo.m runs the problem.
• p16_fd.m the signed distance function.
• p16_fh.m the mesh density function.
• p16_mesh.png a PNG image of the mesh.
• p16_nodes.txt the mesh nodes of the problem.
• p16_elements.txt the mesh triangles of the problem.

P17 is a rectangular region with a Reuleaux triangle obstacle.

• p17_demo.m runs the problem.
• p17_fd.m the signed distance function.
• p17_fh.m the mesh density function.
• p17_mesh.png a PNG image of the mesh.
• p17_nodes.txt the mesh nodes of the problem.
• p17_elements.txt the mesh triangles of the problem.

P18 is a dumbbell shape, two circles connected by a rectangular channel.

• p18_demo.m runs the problem.
• p18_fd.m the signed distance function.
• P18_UNIFORM uses a uniform density:
  • p18_uniform_fh.m the mesh density function.
  • p18_uniform_mesh.png a PNG image of the mesh.
  • p18_uniform_nodes.txt the mesh nodes of the problem.
  • p18_uniform_elements.txt the mesh triangles of the problem.
• P18_NONUNIFORM uses a nonuniform density:
  • p18_nonuniform_fh.m the mesh density function.
  • p18_nonuniform_mesh.png a PNG image of the mesh.
  • p18_nonuniform_nodes.txt the mesh nodes of the problem.
  • p18_nonuniform_elements.txt the mesh triangles of the problem.

P19 is a triangle:

• p19_demo.m runs the problem.
• p19_fd.m the signed distance function.
• p19_fh.m the mesh density function.
• p19_mesh.png a PNG image of the mesh.
• p19_nodes.txt the mesh nodes of the problem.
• p19_elements.txt the mesh triangles of the problem.

P20 is a rough draft of the first floor of ICAM's Wright House:

• p20_demo.m runs the problem.
• p20_fd.m the signed distance function.
• P20_UNIFORM uses a uniform density:
  • p20_uniform_fh.m the uniform mesh density function.
  • p20_uniform_mesh.png a PNG image of the mesh, for 2 iterations and a mesh size H = 0.25.
  • p20_uniform_nodes.txt the mesh nodes of the problem.
  • p20_uniform_elements.txt the mesh triangles of the problem.
• P20_NONUNIFORM uses a nonuniform density:
  • p20_nonuniform_fh.m the nonuniform mesh density function.
  • p20_nonuniform_mesh.png a PNG image of the mesh, for 2 iterations and a mesh size H = 0.25.
  • p20_nonuniform_nodes.txt the mesh nodes of the problem.
  • p20_nonuniform_elements.txt the mesh triangles of the problem.

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