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numerical integration in meshfree method

Dear everybody

I am beginner in Meshfree method. I have some questions
in numerical integration in the weak form formulation. I would appreciate if
you help me.

In "meshless
methods in solid mechanics" written by chen et al, numerical
integration was summarized as: Nodal Integration, Background Mesh
and Wigner–Seitz Cell.

On the other hand, in the
paper “Meshless methods: A review and …” by Prof.Bordas most frequently used
techniques include: Direct nodal integration, Stabilized nodal integration, Stress
point integration, Support-based integration and Background mesh or cell

 It is mentioned nodal integration method would
result in a spatial instability, and a stabilization procedure is then needed. Also
it is mentioned by Prof.Bordas this method is based on voronoi diagram. On the
other hand it is pointed in the book, the Winger-Seitz method is another
approach independent from nodal integration. We know Winger-Seitz is special
kind of Voronoi diagram and it is pointed in the book that winger-seitz
approach eliminates the need of a background mesh and
provides an exact solution to the area or volume.

1) What is the
difference between nodal integration and winger-seitz methods?

2) Does the instability
problem eliminate by using winger-seitz method?

3) What is the best
method among mentioned methods? (Is it background mesh?)

4) In the background
mesh, how can we define Jacobian value?

Best regards


 I find Jacobian is similar to the isoparametric element in FEM and it is equal to the area of each background cell. however I have problem in the definition of Gauss points. we know these points in FE defines in mapped elements but how does these points define in meshfree methods? in Mfree starin matrix obtains in global coordinate, so is mapping necessary or the gauss points in global coordinate will be substitute in gauss quadrature? Please help me,



zahur_ullah's picture

If you are using background cells, so the definition of gauss points is the same as FE elemnts but here the only difference is that nodes can be independent of these cells. 

If you are new to the meshless methods, I will recommend you first go throught some available code, Here are some  


The EFG code from prof Stéphane P.A. Bordas


A 1D and 2D code for the Element-free Galerkin method by J.
Dolbow and T. Belytschko can be download from

And its complete description is given in their paper


1D and 2D EFG code from Canh Le


Zahur Ullah 

Durham University

Thanks alot for your help

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