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Where can I read about the basic ideas of the meshfree methods?

Zhigang Suo's picture

For someone with a background in solid mechanics and finite element methods, where should he go to read up on the elementary ideas of the meshfree methods?

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N. Sukumar's picture

Zhigang,

I think the best source for an introduction (not from the very basics though) to meshfree methods is via some of the review articles. The paper by Belytschko et al. (1996) and the report by Fries and Mathies (2004) are well-written and serve as good introductions; in my earlier post I have provided links to these articles. Many of the monographs on meshfree methods that are available might not be sufficiently detailed and gentle to be readily accessible to a beginner who is familiar with finite elements. I'll try to provide in brief the essentials as I see it:

1. In moving from FEM to meshfree approximations, it is beneficial to think in terms of approximating a function as a linear combination of basis functions. In FEM, shape functions are more commonly used---they are the local restriction of basis functions to an element. In meshfree methods, the basis functions are constructed by using the nodal coordinates and by assigning some means to define neighbor relationships (known as basis function supports). For instance, particular choices are made in Moving Least Squares (MLS) or natural neighbor interpolants (Voronoi diagram).

2. Once the approximation is defined, then a standard Galerkin method is adopted (plain vanilla version). As John has alluded to in his post, depending on the particular meshfree basis function that is under consideration, some additional modifications may be required to correctly impose essential boundary conditions. This applies to second-order PDEs (elasticity) as well as higher-order PDEs (thin-plate problems or gradient elasticity).

3. Assuming that 1. and 2. have been taken care of, then the next step is numerical integration of the weak form integrals. Unlike FEM, there is no mapping involved to compute meshfree basis functions. Typically, a higher-order Gaussian quadrature rule is used to compute the weak form. So, in essence a mesh structure is required in this case for the purpose of numerical integration. The derivatives of the basis functions are required (strain-displacement matrix, B), which are directly evaluated. Note that each Gauss point x may now have a variable number of neighbors; this is unlike the FEM where for each Gauss point inside an element (say a triangle), the number of neighbors is always three and hence the benefits of constructing an element stiffness matrix in the FEM.

4. Once modifications in the system matrices have been performed (to impose essential boundary conditions, if need be), then the linear system is solved: Ku = f

5. The approximation at any point is determined via: uh(x)= φa(x)ua (sum on a).

Of course, I have skipped many intermediate steps. So, please feel free to edit this response and fix (subscript/superscript is mangled) as well as improve my remarks.

phunguyen's picture

Hello Zhigang,

You could find in, http://www.civil.gla.ac.uk/~bordas/phu.html, my simple Matlab codes for EFG, enriched EFG for one and two dimentions problems.

By contacting Stephane Bordas at stephane.bordas@gmail.com, you are also be able to get a document in meshless methods with details on computer implementation.

Phu

 

Dear Zhigang,

You could also take a look at our recent meshfree review paper, which is written from a different angle to that of Thomas-Peter Fries, which you can find here:

 doi:10.1016/j.matcom.2008.01.003

Our goal is to present global-weak form based methods in a very simple way by explaining their implementation, through a very simple MATLAB code that we will send you.

 Let me know if you need more help, I or someone in my group will be happy to provide it,

Stephane 

 

Dr Stephane Bordas

http://people.civil.gla.ac.uk/~bordas

 

hello to every body

My thesis is about the lrpim method  apllying to the 2d beam analysis. but i could not construct the codes in matlab.if anyone has this codes written in matlab and send to the this mail I would be too happy. thank you

mahmutpekedis@hotmail.com

 

hello to every body

My thesis is about the lrpim method  apllying to the 2d beam analysis. but i could not construct the codes in matlab.if anyone has this codes written in matlab and send to the this mail I would be too happy. thank you

mahmutpekedis@hotmail.com

 If you are searching for a book, this book seems to be one of the best choices. G.R. Liu is a famous face in meshfree methods and the book is very attractive and integrative.

abbydutt's picture

where can I find that book ?... if you kindly post the link then it will be of much help.

abby_dutt

yawlou's picture

Hello Everyone,

Beginners may like the introductory document I put together regarding moving least squares basis functions (MLS).  These basis functions are common in meshfree methods.  In the document I include definitions of common terminology, the derivation of MLS basis functions, derivation of first derivatives, and derivation of 2nd derivatives.

Here is the link to the document  http://people.wallawalla.edu/~louie.yaw/otherfiles/introTo_MLS.pdf

regards,

Louie

Absolutely, the books of the Prof. G.R. Lui are a very important guide to understand MFREE METHODS.

 

 

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