mesh2d

mesh2d, an Octave code which generates unstructured meshes in 2D, by Darren Engwirda.

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

MESH2D is most useful because it allows a user to specify a shape or region, which the program will then fill with a triangular mesh. The density of the triangular mesh can be uniform, or the user can request that smaller triangles be used near certain features of the region. The program relies heavily on the features of the Delaunay triangulation, which chooses, among all possible triangulations of a set of points, that triangulation which best avoids small angles.

Interested users should refer to the copy of MESH2D that is made available through the MATLAB Central File Exchange. This copy is essentially my personal working copy, to which I may have added comments, small coding changes, and extra tests and examples.

The MESH2D routines include a call to a function called wbar(). The call to this function fails on my system, and since it seems to have no importance whatsoever, I commented it out. (I believe it is intended to generate a "wait bar", similar to the hourglass or beachball or wristwatch icons.

The MESH2D function "mytsearch()" was originally written to call MATLAB's "tsearch()" function. The tsearch() function has since been removed from MATLAB. Therefore, the call to tsearch() also causes MESH2D to fail!

One alternative is a file called tsearch_mex.c, which searches a triangulation to determine which triangle contains each point. It does not require that the triangulation be Delaunay. To use this function with MATLAB, you need to apply MATLAB's MEX compiler...if you have never used the MEX compiler before, you may have some difficulty, since you need to determine that you have the MEX compiler, that you have a C or C++ compiler on your system, that MEX knows where these compilers are, and that you know how to invoke MEX to compile the function. That should be something like

        mex tsearch_mex.c

(The second time you do something like this is, of course, a hundred times easier and only half as mysterious!)

A second alternative is to replace the call to tsearch() by a call to MATLAB's replacement function DelaunayTri; however, a simple substitution of one call for the other does not quite work. There is, apparently, some feature of tsearch() that is not available in DelaunayTri(). In particular, it may be that tsearch() did not require the triangulation to be Delaunay...

A third alternative is to replace the call to tsearch(x,y,t,px,py) by a call to tsearchn([x y], t, [px py] ), which seems to work.

Usage:

[ p, t ] = mesh2d ( vertices, edge, hdata, options );

where:

vertices, required input, a V by 2 list of (X,Y) coordinates of vertices of the boundary. If vertices is the only input argument, then it must be the case that the vertices are listed consecutively. Otherwise, assuming edge is supplied, the vertices can be given in any order.
edge, optional input, a V by 2 list of pairs of indices in the vertices array that constitute the edges of the polygonal boundary. If vertices is actually already in order, then edge, if specified, would contain the values [1,2; 2,3; 3,4; ... ; V,1].
hdata, optional input, a structure containing element size information (see below).
options, optional input that allows the user to modify the default behavior of the solver (see below).
p, (output), an N by 2 list of node coordinates. The number of nodes generated, N, is determined in part by the size of the edges along the boundary, and by other user input such as the maximum element size, and the user size function, if supplied.
t, (output), an M by 3 list of node indices, forming counterclockwise triangles. The number of triangles created depends on the number of nodes created.

hdata, the element size information. This structure, if supplied, can include the following information:

hdata.hmax, the maximum allowable global element size.
hdata.edgeh, an array of element sizes on specified geometry edges, where e1 is an index into the edge array. The edgeh component would have the form [e1,h1; e2,h2; ...], where the user has specified a certain number of sizes.
hdata.fun, the name of a function preceded by an AT sign, which is the user-defined size function. fun must have the form
h = fun ( x, y, args{} )
where x and y are vectors of point coordinates, and args is an optional addition set of input set in hdata.args. The function returns the user-desired elementsize at the given points.
hdata.args = {arg1, arg2, ...} contains additional arguments, if any, for hdata.fun.

options allows the user to modify the default behavior of the solver. This structure, if supplied, can include the following information:

options.mlim is the convergence tolerance. The maximum relative change in edge length per iteration must be less than this value, which defaults to 0.02.
options.maxit, the maximum allowable number of iterations, which defaults to 20.
options.dhmax, the maximum allowable (relative) gradient in the size function, which defaults to 0.3.
options.output, a "logical" variable which displays the mesh and the mesh statistics upon completion, and defaults to "TRUE", that is, 1.

Licensing:

Copyright (c) 2009, Darren Engwirda
All rights reserved.

Redistribution and use in source and binary forms, with or without 
modification, are permitted provided that the following conditions are 
met:

    * Redistributions of source code must retain the above copyright 
      notice, this list of conditions and the following disclaimer.
    * Redistributions in binary form must reproduce the above copyright 
      notice, this list of conditions and the following disclaimer in 
      the documentation and/or other materials provided with the distribution
      
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 
ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE 
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF 
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS 
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN 
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE 
POSSIBILITY OF SUCH DAMAGE.

Languages:

mesh2d is available in a MATLAB version and an Octave version.

Related Data and Programs:

mesh2d_test

distmesh, an Octave code which carries out triangular or tetrahedral mesh generation, by Per-Olof Persson and Gilbert Strang.

hand_mesh2d, an Octave code which reads in points which outline a human hand, and calls mesh2d(), which creates a fine triangular mesh of the region outlined by the points.

mesh2d_write, an Octave code which demonstrates how node and element data from MESH2D can be written to files.

test_triangulation, an Octave code which defines some test regions for triangulation.

triangle, a C code which computes a triangulation of a geometric region, by Jonathan Shewchuk.

triangulation, an Octave code which performs various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.

Author:

Darren Engwirda

Reference:

Darren Engwirda,
Locally-optimal Delaunay-refinement and optimisation-based mesh generation,
Ph.D. Thesis,
School of Mathematics and Statistics,
The University of Sydney, September 2014.

Source Code:

centroid_mesh.m returns the centroids of the triangles that make up a mesh.
checkgeometry.m checks a geometry input for MESH2D.
circumcircle.m computes the center and radius of the circumcircle of a triangle.
connectivity.m assembles connectivity data for a triangular mesh, including the unique mesh edges, the triangle neighbors, and the boundary edges.
dist2poly.m finds the distance between a point and a polygon.
findedge.m locates the edges that contain a sequence of points.
fixmesh.m checks a triangular mesh for consistency.
inpoly.m determines whether a point is inside a polygon.
mesh2d.m generates a triangular mesh for a polygon, which can include cavities.
mesh2d_to_xml.m writes the MESH2D node and element mesh data to an XML file for use by FENICS.
meshfaces.m generates a triangular mesh for a polygonal region which can include cavities.
meshpoly.m used by MESH2D or MESHFACES to mesh a region.
mydelaunayn.m computes the Delaunay triangulation of a set of points.
mytsearch.m finds a triangle which encloses a set of points in the plane.
quadtree.m decomposes a polygonal region using a quadtree.
quality.m determines the "quality" of a triangle.
refine.m refines a triangular mesh.
smoothmesh.m uses Laplacian smoothing on a triangular mesh.
tinterp.m carries out linear interpolation at points within a triangle, given function values at the vertices.
triarea.m computes the area of one or more triangles, assuming their vertices are given in counterclockwise order.
tsearch_mex.c, which carries out a search of the triangulation; Matlab's "tsearch()" function already does this, but is a) officially obsolete, resulting in lots of warnings; b) requires that the triangulation be Delaunay. In "mytsearch()", replace the call to "tsearch()" by a call to "tsearch_mex()". To use this file, it must be compiled within MATLAB (once) using a command like
```
            mex tsearch_mex.c
          
```

Last revised on 05 June 2023.