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discontinuities in mesh free methods


I wish to ask where to find literature about introducing discontinuities in the shape functions to simulate cracks in mesh free methods.

I found the visibility criterion, the diffraction method and the transparency method referred in the (I think vey good) survey by Fries and Matthies "classification and overview of meshdree methods" but nothing else.

thank you,


N. Sukumar's picture


For modeling discrete cracks, partition of unity based methods might be preferable (rather than meshfree). You can see this post and the references therein for further details on this subject. A quick search on webofscience will also provide you with many other references on both meshfree and pu-based methods.


I'm working in extending discrete least squares meshless (DLSM) method on modelling crack propagation.

Regards this I have a question:

What is the general form of partial differential equation governing a elasticity problem when existing a crack in the domain?

Best Regards


Jinxiong Zhou's picture


Maybe the XFEM is superior to meshfree methods for treatment of discontinuities, but the strategy of enrichment is problem-dependent. Also, when the elements are cut through by the discontinuities, the subdivision of the element, in general, is needed and should be treated properly. I prefer some implicit treatment strategies of moving discontinuities, such as level set method you mentioned in another post, and also the phase field method. These methods avoid the explicit tracking of the discontinuities and the procedures are general. Nevertheless, an additonal cost of solving an level set equation or phase field equation is needed. So the choice of various method is a individual taste.


I just want to comment here that in fact the X-FEM has been used in conjuction with level sets to model many physical features (cracks, free surfaces, phase interfaces, etc).  

With the level set method, in particular, even in the finite difference community you will see special treatment of the numerical scheme near interfaces.   I believe it is generally accepted that methods which smear the interface across several grid cells are simply not as accurate.  

Hello Professor John E. Dolbow

I am a PhD student in electrical engineering I am preparing a thesis on the study of nonlinear phenomena in electrical devices by meshfree methods.
I wish to ask where to find literature  in the case of nonlinearities with  fixed point method and meshfree Galerkin method if possible.

I would be very  thankful.

Robert Gracie's picture



The discussion of modelling fracture by the phase field method is interesting to me.  Can you direct me to most influencial article on the subject.  Also do you know of any review article that compared the explicit discontinuity treatment with the implicit treatment for a large class of fracture problems (brittle, quasi-brittle, ductile, etc)




Jinxiong Zhou's picture


There are several papers about use of phase field to simulate dynamic crack propagation, particularly in brittle materials. Alain Karma have done very excellent research work on this topic. One of their paper is entitled "phase-field model of Mode 3 dynamic fracture". You can find the paper via arXiv: cond-mat/0105034, 2001.

Robert Gracie's picture

Rob, you may like to look at

T. Rabczuk, T. Belytschko, Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Meth. Engng 2004; 61:2316–2343

Thank you for all your insightful replies.

The Cracking particles don't really solve my problem because I would need something that is:

- as much independent as possible from sampling density (of the particles)

- unified tratment for non convex object and artifical discontinuities (it seems to me that only the visibility criterion fullfills this need)

- very efficient to compute, to be applied in a 3D interactive context.


I'm currently trying to find a way to extend the visibility criterion to something less prone to unwanted discontinuities.

Thank you again,



Is it possible using PIM shape function with EFG method?

abshaw's picture

Usual form of EFG is based on MLS approximation of the target function. However, to my knowledge any approximation scheme can be used.

As you know, the PIM shape functions produce a discontinue approximation and it is suitable for methods with local discretization such as MLPG. But I am not sure about it.

abshaw's picture

Generally PIM is not conforming and produces discontinuity in the approximation as domain moves. However there is a way to restor conformability. You may see this reference Liu et al., 2005, "A linearly conforming point interpolation method (LC-PIM) for 2D mechanics problems", International Journal of Computational Methods, 2, No. 4,


You should look at papers of Timon Rabczuk using the following link

I can summarize the basis ideas of the method. I assume that you already know the EFG. Using the local partition of unity concept, we incorporate discontinuous function (Heaviside function H(x)) into the EFG approximation to model the discontinuity due to the crack. So, nodes with domain of influence cut by the crack  are enriched by the H(x). It means:

u(x) = phi_I(x)u_I +  phi_J H(x) a_J, phi : MLS shape functions as usual

It is the second part of the approximation who handles the crack. 

For linear elastic fracture mechanics, since you know the singular field at the crack tip, you can also add them into the approximation. So, nodes whose support contain the crack tip are enriched by the asymptotic functions describing the singular field around the tip.

For numerical integration, the background integration cells are used. To get better accuracy, cells cut by crack are divided into triangles (2D) as done in XFEM.

They called the method, XEFG (eXtended EFG). The method can be applied to nonlinear materials, large deformation and cohesive cracks. The problem of crack iniation based on loss of hyperbolicity can be handles with ease.

Hope that it was clear.



Roozbeh Sanaei's picture

Sandia's Tahoe may be useful package in this regard.  For Atomistic-Continum Bridging. quasicontinuum is multiscale package in this field too.


Zhigang Suo's picture

Please also see a related thread:  The future of meshless methods.

Interest method, thank you.смешарики скачать

maxrobert's picture

 Hi, i'm new in the meshfree method. I'm doing my master dissertation in an Auto Adaptive Interface Treatment for the EFGM in eletromagnectic problems. I believe that it can be used for cracks too, just adjust the method. I'm brazilian, and presented in poster format in MOMAG 2010. Each interface has a precision, and for each interface contact it has test points that the mean relative errors between the test points compared to the interfaces precision decide wich interface has to be discreted again. Well, sorry for my english, i didn't practice for a long time haha. I believe that i'll finish my work between July and October. If it could help you, tell me. Thanks.

I'm working on modelling of induction heating in electromagnetics with meshfree methods (EFG method).
I want to have explanations on the techniques used in the modelling of
different regions in presence (air, ferromagnetic material,...) in the
global domain of study.
thank you

I am not sure about it, overoll

I'm working on modelling of induction heating in electromagnetics with meshfree methods (EFG method).
I want to have explanations on the techniques used in the modelling of different regions in presence (air, ferromagnetic material,...) in the global domain of study.
thank you

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