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Constrained Moving Least Squares (CMLS) Method


I wish to ask where to find more references about application of constrained moving least squares method for imposing displacement bounday conditions in meshless methods. The most important advantage of CMLS over MLS is satisfying Kronecker delta function property, Hence such as finite element methods we can impose the essential boundary conditions in meshless method using CMLS approach.

Thank you,

Jafar Amani

PDF icon CMLS.pdf539.8 KB


Alejandro Ortiz-Bernardin's picture

Hello Jafar,

I don't know details on the method you ask but if it is similar to MLS and you only need linear approximation I would suggest that you search for Maximum-Entropy approach. It would be more efficient from the computational point of view and you can apply essential boundary conditions exactly as in FEM. You can get some information in these posts:



Alejandro A. Ortiz

Dear Alejandro

Thank you very much for creating the comment to my notes on the imposing displacement BC's in meshfree method.

My M.Sc thesis is mixed formulation and adaptive refinement in the new meshless method named Discrete Least Square Meshless (DLSM) that introduced by my thesis advisor Prof. M.H. Afshar. In this approach we use MLS approximation to construct shape functions. In mixed formulation displacements and stresses obtained simultaneously, therefore for imposing Dirichlet and Newmann BC’s only one penalty coefficient is used. Now my questions are,

1)  How determine amount of penalty coefficient?

2) Can I use maximum-entropy approximation instead of MLS for imposing BC’s without penalty coefficient in mixed formulation for any meshfree method?

Thank you

J. Amani

Alejandro Ortiz-Bernardin's picture

Dear Jafar,

These are my replies to your questions:

1)  Do you mean number of lagrange multipliers? if so, it should be equal to the number of essential boundary conditions you have.

2)  Yes, you don't need  to enforce essential boundary conditions with lagrange multipliers. Maximum-entropy basis functions vanish at the boundary naturally. That is because they are derived from a convex optimization problem (you may see the proof in the paper by Arroyo & Ortiz, 2006). Just apply them in the same way as in FE, and of course you can replace your MLS basis functions with max-ent basis functions with the advantange that you will not need to enforce essential boundary conditions. Also using MLS you need to invert the moment matrix for every evaluation point. On using max-ent you just need to solve a system of "n" nonlinear equations where "n" is the spatial dimension for every evaluation point. Thus, simpler and more robust.

Now, some questions for you:

1) What order is your approximation for stresses?

2) When you talk about one penalty coefficient, do you mean one lagrange multipliers?

3) Are you using Hellinger-Reissner variational principle? I don't see why you also need to use lagrange multipliers to enforce Neumman boundary conditions. Can you explain more on that. I just try to follow the u-sigma formulation (or Hellinger-Reissner variational principle) in Zienkiewicz's book and I don't think that you need to do that for Neumann BCs.



Dear Alejandro1)In  solving the differential equations governing the planar elasticity problems,  order of approximation of stresses is 2, therefore second derivative of MLS shape functions should be used. In paper with entitle: “Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations“ by Rahmani & Afshar was shown that second derivative of MLS shape functions have more discontinuity than first derivatives. Since in mixed formulation the resulting governing equations are of the first order, both the displacements and stress boundary conditions are of the Dirichlet type.  Thus in mixed formulation order of approximation decrease to 1, therefore higher accuracy is obtained. only in the paper by Prof. Atluri “Meshless Local Petrov-Galerkin (MLPG) mixed collocation method for elasticity problems” mixed formulations in meshless methods was used. 2) No, penalty coefficient method is differing from Lagrange multipliers. In Liu’s book “mesh free methods - moving beyond the finite element method” have been explained that in using Lagrange multipliers stiffness matrix may have nonsymmetrical, while in using penalty coefficient method in my approach stiffness matrix is symmetrical. Also the computational time maybe increase when using Lagrange multipliers specially in using sampling points with nodal points.    3)No, mixed method variously used in finite element method for analysis of plane elasticity problems. For stability the finite element spaces are required to satisfy the Ladyzhenskaya-Babuska-Brezzi (LBB) condition . But in mixed formulation only formulation was changed compared with previous formulation (Irreducible formulation). Results of mixed formulation are better compared with irreducible formulation when amount of penalty coefficients have been determined with some efforts. I want to Solve the penalty problem with:1)Change MLS approximation to max-ent approximation to avoid penalty coefficient.2) Change MLS approximation to CMLS approximation to avoid penalty coefficient. The formulation of CMLS from Liu’s book is attached.Which of the above can solve imposing BC’s in mixed formulation?Kindest regardJafar

Alejandro Ortiz-Bernardin's picture

Can you attach the file again? ... It seems there is no attachement.

Dear Alejandro

CMLS file is attached?


Dear Alejandro 

Excuse me. I was working in my thesis (Adaptive Refinement with mixed formulation) and CMLS approximation since past 5 weeks. My M.Sc. thesis finished. Now I want to use max-ent approach in discrete least square meshless method. I have some questions about max-ent approach?

1) Can I use max-ent approach instead of MLS for shape functions approximation in any meshfree methods?

2) Results of meshfree methods in irreducible formulation with MLS approximation for second derivatives of shape functions (stresses) have some error compared with analytical solution. Is this problem exist with max-ent approach?

3) Can you guide me in this work?

Best Regard

Jafar Amani

Alejandro Ortiz-Bernardin's picture

Dear Jafar,

1) In principle, whenever you can use MLS you can use maxent basis functions, as long as the approximation is linear. As you may notice in the papers by Prof. Arroyo and Prof. Sukumar, only linear approximations are currently developed since, for example, second-order approximation represents an non-feasible solution to the constrained maximization problem. However, I knew that Prof. Arroyo has developed a second-order maxent approximation which will be described in a subsequent paper to appear soon, I guess. 

2) I cannot answer this since I am not familiar with the problem you are trying to solve. If you need to compute second-order derivatives, then you can use the code that Professor Sukumar posted in his blog here in iMechanica. At the moment, I just have used first order derivatives, so I cannot say much about second-order derivatives.

3) Sure. I am still learning things about maxent, so I see in this an opportunity to learn more about the same.




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