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Volume integration in mehsless methods

I use the meshless Galerkin method for 3D simulation of plate movement. This method requires the volume integration over a test domain. Could somebody help me with the volume integration? I wonder if there are standard or open source algorithms that can be used for this problem.

I would appriciate any help.

yawlou's picture

Hi Mike,

 If you are referring to nodal integration then you will likely want to look at the following references:

1.   J. S. Chen, C. T. Wu, S. Yoon, and Y. You. A stabilized conforming nodal integration
for Galerkin meshfree methods. International Journal for Numerical Methods in
Engineering, 50:435–466, 2001.

2.  J. S. Chen, S. Yoon, and C. T. Wu. Non-linear version of stabilized conforming nodal
integration for Galerkin mesh-free methods. International Journal for Numerical Meth-
ods in Engineering, 53:2587–2615, 2002.

3.  M. A. Puso, J. S. Chen, E. Zywicz, and W. Elmer. Meshfree and finite element nodal
integration methods. International Journal for Numerical Methods in Engineering,
74:416–446, 2008.

4.  Puso, M. A. and Solberg, J., 2006, A formulation and analysis of a
stabilized nodally integrated tetrahedral: Int. J. Numer. Methods Eng.,
67, 841–867.

The third reference is perhaps the most important because it discusses many techniques including the one described in the first two references.  Futhermore, nodal integration is susceptible to instabilities such as hourglass modes, spurious energy modes in an eigen analysis, and  locking in nearly or completely incompressible materials.  Because of this the third reference talks about how to "stabilize" nodal integration to remove the problem of instabilities.  The references sometimes explain the integration in a 2D context, but the theory is extendible to 3D.

I have implemented meshfee Galerkin nodal integration with stabilization for 2D domains using Voronoi diagrams as the domain of integration at each node.  You may see more information by looking at my dissertation I mentioned in response to your previous blog ( .

It is indeed difficult to implement this technique due to the requirements of volume integration.  It usually requires a lot of computational geometry, which is notoriously difficult.  One common way is to use a Voronoi diagram generator to get the volumes associated with each node.  There are such generators freely available on the web.  The difficult arises at the boundaries however.  The available Voronoi diagram generators do not clip the boundaries of the domain you are analyzing.  This aspect of the computational geometry (clipping the boundary, and the book keeping of clipping the voronoi cells, keeping track of edges, keep track of Voronoi cell facets) is as far as I've seen not available on the internet.  Obviously there are researchers that have such codes, but it is often difficult to get them to release these portions of their code that they have created in house.  Hence you may have to resort to creating your own routines.  When I did it in graduate school, I was only working in 2D, so I got a free Voronoi generator off the internet, then slugged through the process of writing my own routines to do the clipping of edges and process of calculating all of the other geometric information.

Based on this post I was searching the internet and found something that looks promising (Voro++), c++ routines for 3D voronoi generation with boundary specification also.  This may be a very good solution, see the following

For another Voronoi diagram generator you might find the following helpful.

You may also like to explore Cgal at

If you ever find a free Voronoi diagram generator that also allows you to clip the 3D boundary and then gives the output for each Voronoi cell in an organized fashion for use in a meshfree Galerkin formulation, please do let me know.  I would very much like to have something like that for future research.  I need to look more at Voro++ that I mentioned above, perhaps it does what I've been looking for.

I hope this helps.




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