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1. If I have meshfree shape functions that satisfy Kronecker-Delta, can I satisfy essential boundary conditions?

In fact, this is a common misconception with meshfree methods. Shape functions that satisfy Kronecker-Delta take a value of one at the node, and vanish at every other node in the domain. Finite element shape functions, for example, are usually designed with this property. This makes the satisfaction of essential boundary conditions relatively simple: we just set or fix the degree of freedom at the node to what it should be on the boundary. Unfortunately, this is usually not sufficient to impose essential boundary conditions with meshfree methods.

The issue is that meshfree shape functions associated with nodes located on the interior of the domain do not typically vanish on the boundary. So, what happens between nodes is just as important as what happens at the nodes. An excellent paper discussing the various options for imposing essential boundary conditions with meshfree methods is provided by Fernandez-Mendez and Huerta, Computer Methods in Applied Mechanics and Engineering, 193, pp. 1257-1275, 2004. At present, Nitsche's method is accepted as being the most robust for essential boundary conditions with meshfree methods. It should also be noted that with Natural-Neighbor interpolants, this is not an issue and the boundary conditions can be imposed just like they are with finite elements.

YongAn Huang's picture

you do a good job! Thank you very much.
Science Expands Ideals

Dear Prof. Dolbow

About this problem, do you know the Moving Kriging interpolation? which is used in Meshless methods and satisfied the Kronecker delta, furthur informarion you can be found the paper of Lei GU, 2003 within International Journal for Numerical Methods in Engineering about using the Moving Kriging in EFG method.

What you mean above about shape functions, I feel it is against what GU mentioned.

What do you think?




I agree with you Tinh.   I don't expect that Moving Kriging interpolation to yield good convergence in anything beyond a one-dimensional problem.  I don't think it's that surprising that the author did not perform even one convergence study.  

Dear Prof. Dolbow,

It means that  is this a non-conforming approach? Since the domain of influence and the integration region are not coincident? Is this  a reason?





    Certainly here the method is likely to be non-conforming.  Strictly speaking, the weight functions should vanish on the essential boundary to lie in the proper subspace.

    However, the fact that the domains of influence and integration are not the same is completely unrelated to this issue. 

Dear Prof. Dolbow,

Thanks  for your excellent  remarks.

May I ask questions here? For the conventional SPH method using the collocation method, shall I impose the ECB? From the literature reviews, I learned that most often the velocity constraints are directly imposed on the boundary particles without considering the ECB.  Is it correct or not? And, How about the stress constraints in this situation? Just imposing on the particles to get the corresponding particle accelerations?

Another queation is, how about the situations if using the Moving-least-squares method in the SPH (MLSPH) ?



Thank you for your question, as it provides an opportunity to clarify my post.  In it, I assumed that one is using a Galerkin method.  

So, it doesn't really apply to SPH with collocation.  While I'm not an expert on collocation methods, I'm pretty sure the EBCs are enforced directly at nodes on the boundary as you suggest. 

I'm not sure what you mean by stress constraints.  Do  you mean Neumann boundary conditions?  

For MLSPH, nothing changes if you're using collocation.  If you're using a Galerkin method with MLSPH, what I wrote above applies. 

Anand V Kulkarni's picture

Prof Dolbow  

While I was going  through your EFG program using MATLAB for 2D problem, I could not understand why classical anlytical equations for displacement were used in taking care of displacement boundary conditions(step no.11, forming q vector and G matrix).I feel that there is something wrong here. We resort to computational methods when analytical solutions are not available.What shall we do in such situations?


Anand V Kulkarni

Department of Mechanical Engineering

SDM College of Engineering and Technology

Dharwad-580 002


This is done to allow for a convergence study. It is a well-known (within the computational community) process of code verification.

In order to properly perform a convergence study in which a bulk error norm will be investigated, one has to know the solution in order to calculate the error.  Further, one has to set the boundary conditions to be consistent with the same solution.  Otherwise, convergence is by no means guaranteed.  

You're absolutely correct that in most engineering analysis of practical interest, we do not know the solution a-priori.  The code can easily be changed to handle such a case.  

Dr. Dolbow,

 I am working on element free galerkin method and want to study dynamic response of plate. I face difficulty in imposing essential boundary conditions. Can I use Chen's full transformation method ?





At this stage I believe the simplest, most robust method to be Nitsche's method.  It is a variationally consistent form of the penalty method, and works very well.  It does not require one to modify the shape functions at all, simply add boundary terms to the stiffness matrix and force vector.   

Thanks Dr. Dolbow



Engineer - Design,
GE - Aviation

Charles Augarde's picture

Dear John (and others)

Can you/anyone point me in the direction of explanations of Nitsche's method precisely for the purpose of imposing essential BCs for a meshless method such as EFG or MLPG? The best I can find is in the Encyclopedia of Computational Mechanics, Chapter 10.

Many thanks



I know I have no photo, yet. 


  My apologies for the delay in getting back to you on this.  There is a good paper by Antonio Huerta on this issue.  

  Fernandez-Mendez and Huerta (2004), "Imposing essential boundary conditions in meshfree methods," Computer Methods in Applied Mechanics and Engineering, v. 193, pp. 1257-1275.


Dear Prof. Dolbow,

I am using Meshless Petrov-Galerkin Method(MLPG) for non-rectangular domains. There is a flexibility of using any shape for weight function and also for local quadrature  domain. Here in my case case weight function domain and local quadrature domains are same.

 I want to know that  , for non-rectangular domains,how the different shapes of weight functions (specially circular or rectangular) will affect accuracy of the result  ?




  I have some experiences about the matter which you mentioned. I think you should consider the highest required rank of derivatives in your problem. Usually the third order spline satisfies the requirements. I have used different weight functions for my problem but the differences in result were negligible.



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