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The Future of Meshless Methods

Ettore Barbieri's picture

I joined imechanica almost a year ago and I've been frequently following its interesting discussions, even the most animated ones. I think that a place like this is ideal to foster the exchange of ideas in the scientific community;

Moreover it is fantastic as a simple student like me can interact and easily ask questions to the most important researcher in the field of mechanics.

Hence, I thought it would have been the right place to pose a question which I believe is quite controversial. The debate I would like to open is about the future of meshless methods, are they still valid? It is worth to keep investigation in this area?

Of course, my opinion is not unbiased, as my PhD topic is about meshless methods. But, as apprentice researcher, I think I should always bring myself into question, even, and more importantly, when I listen to opposite opinions.

In fact, recently I presented a work in a conference about an application of Element Free Galerkin and at the end I received a comment from a professor, who said he was surprised to see someone still working on meshless methods, since (according  to him) the recent trend is more orientated to XFEM.

That was certainly a good question.

In the coffee break we had a chat and he also said that all the big names in meshless methods (Onate, Belytschko and others) are giving up their studies in meshless methods in favour of other methods like XFEM, PUFEM or Generalized FEM.

 I replied that this is indeed a good point nevertheless it is an old debate in the scientific community. To my opinion meshless have such attractive features that it is worth to keep trying, so we shouldn't (for the moment) give up our efforts. Moreover there is still a lot of interest going on, as testified by the next world congress on computational mechanics WCCM8 to be held in Venice at the end of June, where there will be a session dedicated to meshless methods. Finally I told him that it's true that meshless methods have their issues (integration errors, computational costs etc...) but the other methods have their problems, too, so it's not said the last word and we have to wait for new works to see which technology will prevail in the future.

Then I realized that this was certainly a topic to discuss about on imechanica, so I firstly asked the opinion of Stephane Bordas (lecturer at University of Glasgow, also on imechanica) which I met last summer in Glasgow where I attended a Summer School on computational mechanics.

The answer was very interesting and I believe he went straight to the crucial point. We both agreed we should open a thread of discussion on imechanica, also he kindly allowed me to post his reply in this thread:



"I think meshfree still has advantages

1) higher order continuity

2) mesh distortion insensitivity


My key concern is the amount (or lack of it) of mathematical understanding (...) I think this is clearly the thing that is slowing down the process.


For instance, it would be great if we could use point collocation methods, because these methods can lead to higher accuracy for the same computational cost compared to Galerkin methods. However for now, we do not understand what is going on.


It is not entirely true that people are giving up, as T. Belytschko still publishes on meshfree and the finite point method was recently introduced. However, many people believe that XFEM will supersede meshfree methods, simply because it is closer to FEM. I think that XFEM, PUFEM, etc, can clearly capture discontinuities just as well as meshfree, but they still suffer, for now, from lack of continuity and accuracy near the singularities (crack tips for instance) and this is a general problem.


These methods, being FEM based are also sensitive to mesh distortion, except if you consider some of the new methods such as the Flexible XFEM that we are publishing papers about at the moment and the smoothed FEM."



Of course this opinion is comforting to me; nevertheless I'm also interested in adverse opinions, if there are any.

Thank you for your attention and I really look forward to listening to your opinions.


Mike Ciavarella's picture

There is so much confusion in the Literature. Now you can publish in any field you like.  There are enough journals that you can publish anything.

Another question is impact.  So you are clever to raise the question before even embarking on anythink. You are doing a referendum. 

Some subjects never dye. Leonardo da Vinci never published anything in Science, it is not clear why. Some people say he was secretive about Science, and not about Art. Maybe.  But at that time things were different.  He did consult other people apparently, but he had his own motivations.

Get inspired by Nature. Ted Belishtsko is of course n.1 in that area.  If you are not a student of his, it is UNLIKELY you will make too much impact.

I did a scholar search for you, read the first 3 or 4 papers, and let's then get back to discuss.

: An overview and recent developments
- ejournals@cambridge
- les
9 versions »

T Belytschko, Y Krongauz, D
Organ, M Fleming, P … - Computer Methods in Applied Mechanics and Engineering,
1996 - Elsevier

... Methods Appl. Mech. Engrg. 139 (1996) 347
Meshless methods: An overview and recent
... Section 4 deals with the implementation of meshless
methods. ...
Cité 1236 fois -
- Recherche
sur le Web

concepts in meshless methods
- ejournals@cambridge
- les
2 versions »

SN Atluri, T Zhu - Int. J.
Numer. Methods Eng, 2000 -

... Meth. Engng. 47,
537–556 (2000) New concepts in meshless methods ... Page 3.
IN MESHLESS METHODS 539 Figure 1. The distinction
betweeen u i andu i ...
Cité 122 fois -
- Recherche
sur le Web

of boundary conditions for meshless methods
- ejournals@cambridge
- les
10 versions »

FC Günther, WK Liu - Computer
Methods in Applied Mechanics and Engineering, 1998 -

... Implementation of boundary conditions for
meshless methods ... Various ways of
boundary conditions for meshless methods have been
suggested, such as ...
Cité 96 fois -
- Recherche
sur le Web

On the
optimal shape parameters of radial basis functions used for 2-D meshless
- ejournals@cambridge
- les
3 versions »

JG Wang, GR Liu - Computer
Methods in Applied Mechanics and Engineering, 2002 -

... On the optimal shape parameters of radial basis
functions used for 2-D meshless
methods. ...
Meshless methods have achieved remarkable progress in recent
years. ...
Cité 84 fois -
- Recherche
sur le Web

based on collocation with radial basis functions
- les
5 versions »

X Zhang, KZ Song, MW Lu, X Liu -
Computational Mechanics, 2000 - Springer

methods based on collocation with radial basis functions ... In
collocation-based meshless are truly meshless
methods, and are very ef®cient. ...
Cité 83 fois -
- Recherche
sur le Web
- Cambridge
print holdings

Survey of
meshless and generalized finite element methods: A unified approach
12 versions »

I Babuška, U Banerjee, JE
Osborn - Acta Numerica, 2003 - Cambridge Univ Press

... In the
past few years meshless methods for numerically solving partial
tial equations have come into the focus of interest, especially in
the ...
Cité 89 fois -
- Recherche
sur le Web
- Cambridge
print holdings

and coupling of the ÿnite element and meshless methods
- les
2 versions »

A Huerta, S FernÃandez-MÃendez -
Int. J. Numer. Meth. Engng, 2000 -

... J. Numer.
Meth. Engng 2000; 48:1615–1636 Enrichment and coupling of the ÿnite
and meshless methods ... Meshless
methods are ideal for such a procedure. ...
Cité 81 fois -
- Recherche
sur le Web
- Cambridge
print holdings

[PDF] A review of
some meshless methods to solve partial differential equations

- les
6 versions »

CA Duarte - TICAM Report, 1995 -

Page 1. A Review of Some Meshless
Methods ... Meshless methods provide an
alternative for the analysis of this class of problems. ...

54 fois
- Autres
- Version
- Recherche
sur le Web

for shear-deformable beams and plates
- ejournals@cambridge
- les
8 versions »

BM Donning, WK Liu - Computer
Methods in Applied Mechanics and Engineering, 1998 -

... Engrg. 152 (1998) 47-71 Meshless
methods for shear-deformable beams and plates ...
methods inherently satisfy many of the problems
associated with FEM. ...
Cité 53 fois -
- Recherche
sur le Web

[PS] Hp clouds—a
meshless method to solve boundary-value problems

CA Duarte, JT Oden - TICAM Report, 1995 -

... These are among the reasons that interest
in so-called meshless methods
has grown rapidly in recent
years. Most meshless methods ...
Cité 160 fois -
- Version
- Recherche
sur le Web

Auteurs clés:  T
- Y
- D
- M
- P




michele ciavarella

Alejandro Ortiz-Bernardin's picture


You might want to see this other post as well:


Yes, get inspired by Nature.

Tough job.

Actually, I do agree that the work of Ted Belytschko is remarkable in many aspects. From my point of view (ok, I am biased, since I spent 6 years at Northwestern...), I think the biggest strength of his ideas is their simplicity.

 My general trend of thought is to think simple and avoid complexities when they are not required (and generally, they are not). 

If I can solve the problem I have in mind within 200 lines of codes, I will do it this way and avoid going for too large complexities.

Another good advice that was given to me once by a very experienced researcher is to keep a line of focus in your mind. As was said by Michele, you can publish anything. It is not difficult to get your ideas published, especially if they are not revolutionary.

Keep in focus what your goal really is, and use all your energy to reach it. Avoid going in orthogonal directions. Although that can sometimes bring you closer to your goal, it can also tend to divert your attention.

This is just my two cents. :)





Dr Stephane Bordas

Mike Ciavarella's picture

michele ciavarella

The main irritations with Galerkin methods that use unstructured meshes are

1) mesh generation

2) convergence issues due to poor meshes

3) inability to deal with large deformations without remeshing 

I don't these these problems are going away any time soon.   So meshfree or mesh insensitive methods are needed.

X-FEM may be useful for certain classes of problems where there are few boundaries.  But try to represent a complicated CAD model with XFEM and you will soon run into the huge complications.

My approach is to wait and see.  Every method has its issues - hopefully some of you will be able to come up with the holy grail method of solving complex PDEs. 

-- Biswajit 

susanta's picture

-Can Meshfree methods be used for 3D frames, e.g. three beams connected at a single point and the other end of each beam are supported.



PhD Student
Structures Lab
Department of Civil Engineering
IISc, Bangalore 560012

Meshfree methods can easily be applied to 3D frames provided you  position the material points appropriately.  In fact a lot of the verification of meshfree methods is done on 1D structures with two degrees of freedom (which is the 2D version of the frame structure that you have proposed).  However, why would you want to use meshfree methods for such a structure when good old Bubvon-Galerkin finite element methods are available?  The method you choose to solve a problem should be chosen on the basis of its appropriateness for the problem at hand.

-- Biswajit 

Ettore Barbieri's picture

Dear All,

Firstly, thank you so much for showing interest in the topic.

Thanks Alejandro for the link, very useful, I already knew the website which has now merged into imechanica, and it contains very good posts by experts in meshless like Dolbow, Rabczuk and Sukumar. For a very recent review on meshless methods I would like also to add the following:

" N.V. Phu, T. Rabczuk, S. Bordas, M. Duflot: Meshless Methods: A review and computer implementation aspects, Mathematics and Computers in Simulation, in press (2008)"

Thanks Mike for the references, you even did a scholar search for me, very much appreciated! I've already read the "classical" paper on EFG and MLPG by the groups of Belytschko and Atluri, it is right what you said about embarking in something that it might not be of impact, but at the time I started my PhD  I had no doubt meshless were the recent trend. Now i still think that, but trends change quickly and you might end up doing something that in the meanwhile has became superseded (I hope not)!

Thanks also Biswajit for his comment, you made a good point that I think is shared by many researchers, especially the ones who have dealt with crack propagation and large deformation using standard FE!:)

Please, do not stop commenting, I'm very satisfied so far by the number of reads and the comments, so I'd like to keep discuss on it.

you can Use "Meshfree" and also "Mesh Free" as keyword.

this book is also good for introduction

Mesh Free Methods: Moving Beyond the Finite Element

GR Liu - 2003 -

I think if you do a GOOD REVIEW 
and choose something that has important usage for others in the scientific
society independent of its main subject popularity would lead to good
impact research. there are many papers not interesting in very popular
topics and there are many papers in less popular topics which have more
attended. pay attention to EFFECT more than POPULARITY. i believe it makes
more impact. all of mentioned methods have their advanatges and disadvantages
but  you should choose applications more compatible with the nature of meshfree method. even FDM could have advantages in comparision to others in specific



Dear Ettore,


Many thanks. This is an interesting topic. I agree that we pass this

discussion to imechanica, it is exactly what I would have proposed, in fact!



I think meshfree still has advantages

1) higher order continuity

2) mesh distortion insensitivity


My key concern is the amount (or lack of it) of mathematical understanding,

which is why I am writing a proposal now on this point with two of my

colleague mathematicians, I think this is clearly the thing that is slowing

down the process.


For instance, it would be great if we could use point collocation methods,

because these methods can lead to higher accuracy for the same computational

cost compared to Galerkin methods. However for now, we do not understand

what is going on.


It is not entirely true that people are giving up, as T Belytschko still

publishes on meshfree and the finite point method was recently introduced.

However, many people believe that XFEM will supersede meshfree methods,

simply because it is closer to FEM. I think that XFEM, PUFEM, etc, can

clearly capture discontinuities just as well as meshfree, but they still

suffer, for now, from lack of continuity and accuracy near the singularities

(crack tips for instance) and this is a general problem.


These methods, being FEM based are also sensitive to mesh distortion, except

if you consider some of the new methods such as the Flexible XFEM that we

are publishing papers about at the moment, and the smoothed FEM.


I hope this canget the discussion going...





Dr Stephane Bordas

Mike Ciavarella's picture


Meshfree Methods: Recent Advances

Yagawa, G. | Fujisawa, T.
Nippon Kikai Gakkai Ronbunshu A Hen (Transactions of the Japan
Society of Mechanical Engineers Part A)(Japan). Vol. 16, no. 3,
pp. 330-337. Mar. 2004

In finite element methods (FEM), which are widely used in the
structure analysis, the formation of meshes becomes barriers
frequently. The meshfree methods are classified to the meshless
method (in narrow sense), the particle method, and the FEM-based
meshless method. Various methods had been proposed. SPH (Smooth
Particle Hydrodynamics), DEM (Diffuse Element Method),
EFGM(Element-free Galerkin Method), hp-meshless cloud method,
MLBIE(Meahless Local boundary Equation Method), NBNM (Node-by-node
Meshless Method), Gridless Method (Gridless Euler/Navier-Stokes
solution), etc. as meshless methods, PAF(Particle-and-force)
method, MAC (Marker-and-Cell) method, RKPM (Reproducing Kernel
Particle Method), etc. as particle methods, and FCM (Finite Cover
Method), GFEM (Generalized FEM), PUFEM (Partition of Unity FEM),
X-FEM, FMM (Free Mesh Method), NBN-FEM (Node-by-node FEM), etc. as
FEM-based methods, are categorized. The automatic formation
techniques for meshes are outlined. Cooperative works between the
academic theoretical research and the development of practical
software is requisite.



michele ciavarella

You're right!!


Concerning classification, the review by TP Fries is an excellent source of information.

Basically, you can categorize the methods (all within the framework of weighted residuals, please stop me if you do not agree with this statement).

 1) Type of trial and test functions (approximation)

2) Type of numerical integration (local, global)

3) Eulerian or Lagrangian kernel 

 Ex. Trial = test = Dirac delta => point collocation

Tial =test = Moving Least Squares + global weak form => EFG 

In this framework, the idea of Atluri (MLPG) is, I believe, nice, because it puts a general framework around meshfree methods. From the citations, I can tell that it is not the opinion of all researchers.

 Any thoughts? 




Dr Stephane Bordas

Mike Ciavarella's picture

dear Stephane


can you send me the references you cite?   I am not that expert as you may think (and the average imechanician, remember, is a student so about the same as me!!)  Both Atluri and TP Fries I don't know the reference.  Please attach rather than send links if possible.




michele ciavarella

Classification and Overview of Meshfree Methods Classification and ... 

is the link to TP Fries' report, which I consider the best recent review on the topic. I think I'm not the only one thinking that...


 The Meshless Local Petrov-Galerkin(MLPG) Method: A Simple Less-costly Alternative to the Finite …

is the link to Atluri's method, although the paper:

 A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

is probably the most cited.

 Sorry, did not have time to attach the pdf, but this should work.




Dr Stephane Bordas

Ettore Barbieri's picture

I agree with Stephane,

 the Fries' report gave me the first introduction to meshless methods when I started. I'm not an expert in MLPG, because I often use Element Free Galerkin or Reproducing Kernel Particle Method. I read the papers from Atluri, though, and it's definitely fascinating the idea of providing a general framework for meshless methods. Atluri says the all the most known meshless methods are particular cases of MLPG. This is due to the the local characteristics of MLPG, while the other are prevalently global approaches.

As Stephane said, it's true, not everyone thinks that. I couldn't say which one between EFG or MLPG is most used (or popular). If someone is expert in MLPG, please comment, maybe explaining the differences with the other methods.

But primary scope of my reply is the  book of Liu " Mesh Free Methods : Moving Beyond the Finite Element Methods".  It's the first book on MeshFree (Liu prefers the term MeshFree rather than Meshless because the word "free" gives a more positive idea than the word "less"). RoozbehSanaei also suggested this book in a previous reply.

If you're interested in the topic, don't miss this book. I want to post the preface, (I hope Liu wouldn't mind :), because in his words you can find the initial skepticism typical of someone who wants to migrate from an old well assested technolgy to a new one, fear that maybe it's shared also from researchers who come across these methods for the first time.

 " Topics related to modeling and simulation play an increasingly important role in building an advanced engineering system in rapid and cost-effective ways. For centuries, people have been using the finite difference method (FDM) to perform the task of modeling and simulation of engineering systems, in particular to solve partial differential equation systems. It works very well for problems of simple geometry. For decades, we have used techniques of finite element methods (FEM) to perform more-challenging tasks arising from increasing demands on flexibility, effectiveness, and accuracy in challenging prob- lems with complex geometry. I still remember, during my university years, doing a home- work assignment using FDM to calculate the temperature distribution in a rectangular plate. This simple problem demonstrated the power of numerical methods. About a year later, I created an FEM program to solve a nonlinear mechanics problem for a frame structural system, as my final year project. Since then, FEM has been one of my major tools in dealing with many engineering and academic problems. In the past two decades I have participated in and directed many engineering problems of very large scale with millions of degrees of freedom (DOFs). I thought, and many of my colleagues agreed, that with the advances of FEM and the computer, there were very few problems left to solve. Soon, I realized that I was wrong and for very simple reasons. When a class of problems is solved, people simply move on to solve a class of problems that are more complex and to demand results that are more accurate. In reality, problems can be as complex as we want them to be; hence, we can never claim that problems are solved. We solve problems that are idealized and simplified by us. Once the simplification is relaxed, new challenges arise. The older methods often cannot meet the demands of new problems of increasing complexity, and newer and more advanced methods are constantly born.

I heard about meshless methods in about 1993, while I was working at Northwestern University, but somehow I was reluctant to move into this new research area probably because I was quite happy with what I was doing using techniques of FEM. It was also partially because I was concentrating on the development of my strip element method (see the monograph by G. R. Liu and Xi, 2001). During 1995-1996, I handled a number of practical engineering problems for the defense industry using FEM packages, and encountered difficulties in solving mesh distortion-related problems. I struggled to use re-meshing techniques, but the solution was far from satisfactory. I then began to look for methods that can solve the mesh distortion problems encountered in my industrial research work.

I immediately started to learn more about meshless methods. I worked alone for about a year feeling as if I was walking in a maze of this new research area. I wished that I had a book on mesh free methods to guide me. I was excited for a time about the small progress I made, which motivated me to work day and night to write a proposal for a research grant from NSTB (a research funding agency of the Singapore government). I was lucky enough to secure the grant, which quickly enabled me to form a research team at the Centre for Advanced Computations in Engineering Science (ACES) working on element free methods. The research team at ACES is still working very hard in the area of mesh free methods. (...)

I experienced difficulties in the process of learning mesh free methods, because no single book was thus far available dedicatedto the topic. I therefore hope my effort in writing this monograph can help researchers, engineers, and students who are interested in exploring mesh free methods. My work in the area of mesh free methods has been profoundly influenced by the works of Professors Belytschko, Atluri, W. K. Liu, and many others working in this area. Without their significant contributions to this area, this book would not exist."



Ettore Barbieri's picture

As a preliminary concept, Liu calls a meshfree method whenever a predefined mesh is not necessary, at least in field variable interpolation. Moreover an ideal meshfree method is one which doesn't need a mesh throughout the whole process of solving the problem.

Liu suggests therefore in his book a classification based on:
1) if requires a background mesh for the integration of a global weak form (not truly meshless methods, like EFG)

2) if requires a background mesh for the integration of a local weak form (truly meshfree, like MLPG)

3) no mesh at all, like Collocation methods and finite difference methods, but they are less stable. No integration required.

4) Particle methods that require a predefinition of particles for their volumes or masses, like SPH.

THis discussion is quite interesting, I am a fresh PhD student who also started some work on meshless methods. Here is classification of meshless methods I can suggest, although of course it might be not complete one.

Firstly, I would suggest classification based on the formulation of the problem:

Based on the weak form:


  • Diffuse element method
  • Element free Galerkin
  • Reproducing Kernel Particle Method
  • Point Interpolation Method
  • Meshless Local Petrov-Galerkin Method
  • Local Radial Point Interpolation Method
  • HP-Cloud Method
  • Partition of Unity Method
  • Finite Point Method

As a remark - all of these methods are based on the global weak forms, except  Meshless Local Petrov-Galerkin Method and Local Radial Point Interpolation Method which are based on the local weak forms.

Based  on the strong form:


  • Collocation Method
  • Vortex Method
  • Generalized Finite Difference Method
  • Smoothed Particle Hydrodynamics


Secondly, its possible to make classification based on the function representation (approximation):

Moving Least Squares (MLS)


  • Diffuse Element Method
  • Element-Free Galerkin
  • Meshless Local Petrov-Galerkin Method

Point Interpolation


  • Point Interpolation Method
  • Radial Point Interpolation Method

Integral Representation


  • Reproducing Kernel Particle Method
  • Smoothed Particle Hydrodynamics



  • HP-Cloud Method
  • Partition of Unity Method

In general I would say its quite complicated to do a complete classification because as Dr. Bordas mentioned all of these methods are build in various fashion using a set of building blocks.


Regards, Slawa.

abshaw's picture

This discussion is indeed very interesting.  I am not an expert but have done some work on
mesh-free method in my PhD. When I started working on mesh-free method I have
been asked almost same question like “why do we need mesh-free method?”, “Is it
a substitute of FEM”, “If it is not so easy to implement then what is the use
of it?”. I remember once I had an opportunity to attend one lecture by Prof. J
N Rdddy on higher order shear deformation theory. At the end of the lecture I asked
him about his opinion on mesh-free method. Whatever he said if I write in one
sentence then it would be ‘it is useless’.  On the other hand you may find a large body of
literature where it is claimed that mesh-free method is a revolution.  

Now the question is who is correct? I think
none of them. These two statements are very extreme.  The truth lies somewhere in between. We need
mesh-free method (more precisely particle method) for a certain class of
problems (crack propagation, fragmentation, very large deformation etc.) where
FEM does not work either due to continuum nature or some other computational
difficulties. In that sense mesh-free method is the complement of FEM.  Combination of these two is also a good idea. You
may find many papers where this issue has been addressed.

There are many reasons why still mesh-free
method is not so popular. The most important criterion, any numerical scheme
must have to be successful is ‘easy implementation’.  Today in order to use FEM to solve some
industrial problem user does not need have a rigorous knowledge of FEM.  Unfortunately it is not the case with
mesh-free method. What type of kernel? What is the optimum support size of
kernel function? How to choose the background mesh? What order of integration would
be sufficient for a given problem? These are something which required to be
fixed based on some numerical experiment.   Another
important thing what I observed (I could be entirely wrong), unfortunately
people are making mesh-free method more complicated.  That is also a reason why the method is losing
its acceptability by general user.

recent trend is more orientated to XFEM”, yes XFEM is a very good method. But
you have few cracks then XFEM works. For a fragmentation problem where there is
a multitude of cracks, really you need a method which is particle in nature.

What is the future of mesh-free method? Time will
say that. Wait and watch.

In fact all methods are more complementary than competitive!

Citation based evaluation? it is an old story!


It seems to me that the problem with meshless methods is just their immaturity. FEM have a lot of decades on the field, and people uses it for many things. People is not going to change their already working software and expertise just because there's something new on the air. Quite contrary, they're going to push  FEM and its variants to the limit for getting any job done.

However, I don't think meshless have too much to loose compared with FEM. I'm doing my Ph.D. on meshless MLPG and, what I have found is that they just lack some rounding and polishing, at least to the point where FEM is. Let me explain this clearly: if I open the shelve of books at my left, I will find many, many books of finite elements, detailing every kind of thing, even some which assume you have no prior knowledge of advanced calculus. They contain also detailed listings of routines in FORTRAN for doing this and that. And the organization of the entire software, and also a CD with the code. But there's not such a thing for meshless methods.

If you come down to code for meshless, to my knowledge, there's not anything close to NASTRAN, not to mention integrated-with-all-applications like ABACUS...

Hope this can be reverted on the future.



An interesting and timely discussion I think. 

What do people think of the final statements in this paper

To mesh or not to mesh. That is the question…
Computer Methods in Applied Mechanics and EngineeringVolume 195, Issues 37-4015 July 2006, Pages 4681-4696
Sergio R. Idelsohn, Eugenio Oñate


They raise the issue of the need to determine nodal connectivity in meshless methods and imply (I think) that this is conceptually close to the work of mesh generation. I think it is an issue ducked in some papers.


Ettore Barbieri, Thanks for your interesting topic.

Your topic is about comparison of meshfree method and other kinds of method such as XFEM, leading to a wide-range knowledge.

(1) There is nothing absolutely bad or good. A method can work wonderfully in some problems, but are out of order in others.

    Which is better? Can one compare to the other?

    If no judging criterion has been chosen, comments become almost impossilble.

(2) If large displacement is key characteristic of the problem you investigated, then meshfree method is an excellent choice. The  discrete nature of meshfree method allow itself to properly simulate discrete problems, or processes from continuous to discrete.

(3) Leaving the topic aside, what should the task of numerical investigators be? Is their task to simulate the phenomenon as similar as possible to the real physics one? I think no! The physics experiment has been obtained, so it is not wise to spend so much efforts to develop such a numerical method which doesn't exist theoretically due to non-existence of limits on knowledge developments. Numerical methods should stand on the same start-line with physics experiments: to explore the mechanism of phenomena.

(4) A good numerical investigator has to be a predictor. Every numerical method is like a game, whose game rules are supposed to be supplied by its developer. Here, the game rule stands for the mechanisms, simplified and approximate.

Alejandro Ortiz-Bernardin's picture

Hello Everyone, I have read another post about a discussion on software design in computational mechanics motivated by the status of open source code:

According to this post it seems that the future of software design in computational mechanics is migrating to OOP. One idea that is coming to my mind with respect to meshfree methods, motivated in part, by the previous comments in this meshfree blog is: if there are several meshfree methods according to the classifications in the above posts, can we think of a meshfree class which can be used to compute shape functions and its derivatives? With this type of design one could use the class independently of the "name" of the meshfree method and the procedures involved apart from the shape functions computation. I am curious to know if there is someone working on that.

Alejandro A. Ortiz

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