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approximations for meshfree methods

Xiujun Fang's picture

"Meshfree methods" is the general name of the numerical methods in which approximations(or trial functions) are constructed with arbitrary nodes without mecessity of forming mesh. Up to now, there are many kinds of meshfree methods, such as SPH, EFGM, MLPG, RKPM, NEM,DEM,PUM,h-p Clouds,MPM,FPM...... But the classification of them is not easy. Now I am writing an introduction to meshfree methods and find that most of them can be classified from the view of approxmations' construction which is also the salient differece between medhods based on mesh(such as FEM). I draw a sketch to express the different way to construct the approximations for meshfree methods and relations between them. Any comments and advices will be appreciated!


hmdthr's picture

I am a new user of ls-dyna and femb 27 and i like to model with sph method. But my problem is that i don't know how to model sph element in femb 27. For example generating *ELEMENT_SPH keyword was not found after looking for it in software. May i ask you to help me and guide me to model sph in FEMB

Xiujun Fang's picture

Dear Hamid

Thanks for your attention. The posts in ' Ask iMechanica' often have few replies. Maybe choosing a proper post area is as important as bringing up a well-organized question. I have heard of femb from you for the 1st time, but I believe that someone knows it! Good luck! 


Dan Cojocaru's picture

It seems you did a very good job. I think a natural continuation would be a list of advantages and weaknesses for each of the methods.


I am new to meshfree methods and so, appreciate your chart even more!

Continuing with Dan's suggestion, ideally, one would like to have an image-map clicking different areas of which will pop-up browser windows that show the features, advantages and limitations of that particular method.

Incidentally, this kind of chart is precisely what I meant by concept maps here.

(BTW, Windows users: MS Photo Editor on my machine had problems opening up this a 2 MB jpg file due to its large size, but IE 6 can open it OK together with scroll bars.)

Is there a version, or an article that has a review of all these methods, assuming that they are simular?

Xiujun Fang's picture

Dear all,

there have already been some excellent reviews of meshfree methods,such as:

1)Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless method: an overview and recent developments. Comput Methods Appl Mech Engrg, 1996, 139. 3-47

2)Shaofan Li, Wing Kam Liu. Meshfree and particle methods and applications. Applied Mechanics Review, 2002, 55(1):1-34    

(later they published the book: Shaofan Li, Wing Kam Liu. Meshfree particle methods. Springer-Verlag Berlin and Heidelberg, 2004)

Dan's suggestion is an important topic about the methods. But how to evaluate all of the meshfree methods maybe a more difficult problem. In general, a numerical method in computational mechanics includes two ingredients: 1) how to construct the approximation; 2) how to formulate the discrete equations and solve them. Generally mesh-based methods and meshfree methods are distiguished from the angle of the former, while to list the advantages and shortcomings of each of them need to consider both 1) and 2).  Maybe more efforts are needed in the area.

(btw, Ajit's idea about concept map is really interesting. ) 

hiader k. mahbes's picture

I am from Iraq, now I finished the second course of PhD course, I study to use

Meshless FEM in Mechanical field. Thanks on this useful chart. 


 Have you programs on msehless method?

hiader k. mahbes's picture

my email is    

Dear Xiujun

Why DLSM method presented by Afshar et al (2006) is not shown in the chart? It is based on strong form formulation used for analysis solid mechanic and fluid mechanic problems.


What are the main differences with MLPG and FEM. Which are advantages and disadvantages?


any one have you got any Abaqus tuotrials about shaft meshing..



Angelo Simone's picture


there is mistake in your diagram (left hand side). Quoting from a review paper by Belytschko et al. [1], "The XFEM and GFEM are basically identical methods: the name generalized finite element method was adopted by the Texas school [37, 71, 72] in 1995-1996 and the name extended finite element method was coined by the Northwestern school [10, 75] in 1999."



[1] Belytschko T., Gracie R. and Ventura G. A review of extended/generalized finite element methods for material modeling, 2009, Modelling Simul. Mater. Sci. Eng. 17 043001

[10] Belytschko T and Black T 1999 Elastic crack growth in finite elements with minimal remeshing Int. J. Numer. Methods Eng. 45 601-20

[37] Duarte C A and Oden J T 1996 An hp adaptive method using clouds. Comput. Methods Appl. Mech. Eng. 139 237-62

[71] Melenk J M and Babuska I 1996 The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 39 289-314

[72] Melenk J M 1995 On generalized finite element methods PhD Thesis University of Maryland

[75] Moes N, Dolbow J and Belytschko T 1999 A finite element method for crack growth without remeshing Int. J. Numer. Methods Eng. 46 131-50

Can anyone recommend books or papers that use RBF in meshless method? I will like to know more about how to use RBF in meshless methods to solve PDE. Matlab code on this subject will also be appreciated. 

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