Unstructured 2D Triangular Mesh Generation in Matlab
Intro
For most of the Numerical simulations Unstructured Mesh are very common. Especially for Finite Element methods triangular meshes they offer the best trade-off between flexibility and precision of calculus. Today I am going to write about unstructured Mesh generation in MATLAB.
MESH2D – Automatic Mesh Generation
MESH2D is a toolbox of 2D meshing routines that allows for the automatic generation of unstructured triangular meshes for general 2D geometry.
An iterative algorithm is used that attempts to optimise the mesh topology/vertex position to achieve high quality triangulations.
In addition to the fully automatic settings, MESH2D allows the user to specify sizing information, allowing for varying levels of mesh resolution within the domain.
MESH2D includes a large set of demos/benchmarks. Please see the following:
- meshdemo – runs a standard set of demos.
- mesh_collection – a collection of user contributed geometries.
- facedemo – examples of geometry with connected faces.
It has one disavantage, due to the use of the qhull.mex routine (delaunay triangulator), despite of the slower standard matlab delaunay, it is only windows compatible.
Matlab PDE toolbox
Another solution is to use (and buy) the matlab MATLAB PDE Toolbox.
The PDE toolbox is designed to solve two-dimensional linear partial differential equations by finite element methods, but it also offers some simple CAD and mesh generation feature. By typing on the MATLAB command window:
pdetool
You will access the drawing tools which are pretty simple. It is possible to draw simple geometric primitives and combine them with boolean operations. After finished the 2D geometry by pressing the mesh command, you will realize unstructered mesh generation that can be used to solve pdes, but also to export on the workspace.
Triangle.m
Triangle.m is a MATLAB graphical user interface for creating 2D models and generating finite element triangulation grids. Grid generation is accomplished by an external call to the C-code Triangle.c, written by Jonathan Shewchuk at UC Berkeley. The features included in the MATLAB interface were created with the intent to handle 2D geophysical modeling applications. However, the generality of the interface should allow for multidisciplinary usage.
DistMesh
DistMesh is a simple MATLAB code for generation of unstructured triangular and tetrahedral meshes. It was developed by Per-Olof Persson and Gilbert Strang in the Department of Mathematics at MIT.
More infos can be found at the DistMesh Home page.
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Delaunay2_5D: a Straightforward Algorithm for Fast Surface Reconstruction
Filed under: Algorithms, Computational Geometry, Graphics
Description
Delaunay2_5D is an algorithm for fast surface reconstruction from scattered points cloud. The algorithm is designed to be quick and light. With this tool you’ll be able to reconstruct on your “little” laptop huge points cloud (several millions of points) which are generally considered a Workstation Job.
The project was made possible thanks to the collaboration of Luca Di Angelo and Di Stefano Paolo.
The algorithm uses a mesh growing approach: starting form a seed triangle a front propagation is performed. This technique is generally less accurate than the ones that use Delaunay Triangulation, on the other side they are terribly faster and lighter, so they are preferred for huge models
Perfomances
Models
A few models to test the algorithm and verify the perfomances:
| Model | Number of points | Call |
| Nicolò da Uzzano (the one in the picture) | 1M | Delaunay2_5D.exe |
| Thai Statue | 5M | Delaunay2_5D.exe i 3 p1 a 20 |
| Foot | 10k | Delaunay2_5D.exe i 2 p1 |
| Armadillo | 170k | Delaunay2_5D.exe |
| Happy Buddha | 500k | Delaunay2_5D.exe i3 p1 f5 a 25 |
| Dragon | 430k | Delaunay2_5D.exe i2 p1 f5 a 30 |
| Rolling Stage | 600k | Delaunay2_5D.exe |
| Pala(Turbine Blade) | 350k | Delaunay2_5D.exe i 3 f 20 p 4 |
| Neptune | 2M | Delaunay2_5D.exe i 3 p1 a 20 |
| Blade | 850k | Delaunay2_5D i 5 p 2 f 20 a 25 |
| Pulley | 580k | Delaunay2_5D a 25 |
Delaunay2_5D take as input a binary file. To build your own input file see the usage below.
Usage
How to build an input file and how input parameter works can be found here:
Release
We currently have only one Win32 version of Delaunay2_5D.
Download for Win32
Release History:
- Delaunay2_5D v1.0 ( 01/02/2010)
- Delaunay2_5D v1.1 ( 15/02/2010)(It is now possible to insert the input file name as parameter)
Demo version
The demo version differs from the original in the output mode. It has no output parameter so it forces tha user to suppress STL file creation and to visualize the reconstructed surface in output window.
Contributor version
The contributor version returns an STL binary file of the reconstructed surface. It can be obtained with a small donation as encouragement to improve the code itself. The project was realized without any kind of economical help from any academic or commercial organizations.
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Delaunay2_5D: Fast Surface Reconstruction– Contributor version is reserved to donators. The amount of the donation is 30 euro.
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| Once you have done this, please email us: giaccariluigi@msn.com As soon as possible (in a few days) you will receive our new release (and all future updates) of Delaunay2_5D. Alternatively, you can bestow using our banking coordinates:
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| Name : | Luigi Giaccari | ||
| IBAN (International Bank Account Number) : | IT 07 E05550 77710 000000483947 | ||
Further informations can be obtained on my skype contact: luigi.giaccari
Release notes
The Software provided has to be considered “as is” and it is without any kind of warranty. The authors deny any kind of warranty concerning the code as well as any kind of responsibility for problems and damages which may be caused by the use of the code itself including all parts of the source code.
Contacts
Delaunay2_5D is always under development…
For any problem, suggestion, question, bug report, whatever….
- giaccariluigi@msn.com
- skype contact: luigi.giaccari
Popularity: 1% [?]
Delaunay2_5D: Usage
Filed under: Algorithms, Computational Geometry, Graphics
Contents
When can it be used
Delaunay2_5D is designed for densely sampled surfaces, both open and closed one. The surface must be not too much rough and possibly with higher sampling in high curvature feature.
…and when can not
Delaunay2_5D wont work on:
- Sliced data: when the distance among slices is much higher than the distance among point of the same slice.
- Rough surfaces
- Not orientable surfaces
- Sharp edges coming from cad models (scanned ones should be meshed).
How to use it
Input format
After struggling on hundreds of points cloud formats I decided to feed the algorithm with a simple binary file. The file is composed by an integer indicating the number of points and a list of doubles indicating the coordinates.
The number of point is the first data and it is an “int” type. Points are stored this way: X1,Y1,Z1,X2,Y2,Z2…….
These are the Matlab commands to generate such file:
fid=fopen(name,'wb');%open input file for Delaunay2_5D fwrite(fid, np, 'int');%first data=the numbers of points; fwrite(fid,p,'double');%points; fclose(fid);%close the file
You can download tools to use D2_5D on matlab here: Download Now
And this are the C++ ones:
pFile =fopen("D2_5DInput.dat", "wb");//open input file for Delaunay2_5D nwritten=fwrite(&N, sizeof(int), 1, pFile);//first line (the number of points) nwritten=fwrite(&p, sizeof(double), N*3, pFile);//Write the points coordinate fclose(pFile);//close the file
Launching the .exe
In Matlab you can use:
!Delaunay2_5D.exe i 2 p 1 f 0 a 90
where the letters after the executable path are the input parameters described in the next section.
In C++, given an array p of Nx3 elemnts representing the points coordinate:
system("Delaunay2_5D.exe i 2 p 1 f 0 a 90 ");
Input parameters
The basic call is:
A professional use of Delaunay2_5D requires some knowledge about its input parameters.
- ‘i’=Number of meshing iteration. Default=1. Too high meshing iteration can slow down the algorithm and sometimes generates bad triangles. On the other side, it helps to fill big holes, so rise up the number of operation only if you find undesired holes in the output.
- ‘f’= enables a nearest neighbour filter. A nearest neighbour graph is computed, the meandist among points calculated. Points with distance less than meandist/f will be deleted. f must be >1, a 0 value will turn off the filtering. Use this parameter only if you have duplicated points in the input or points too close too each other.
- ‘p’=The “strength” of the post-processor. Set to zero turn off post-processing operations. Set to >0 attempts to fill small holes in the output. Default value is 1.
- ‘file’= the input file name. Ex Delaunay2_5D.exe file Bunny.dat . If omitted the input filename must be later inserted as console input.
- ‘w’: output window,(only contributor version) set to 0 suppress output window
- ‘o’: output file:(only contributor version) set to 0 returns an stl file. Set to 1 returns a binary file (this mode is for the developper). Set to 2 returns both of them.
There are several others option but for now it is clearer not considering all of them. A complete documentation will be published shortly.
I the meantime, to run some tests, you can use settings suggested in the model section.
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