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2. Is a mesh required in meshfree methods?

N. Sukumar's picture

In meshfree (this is more in vogue than the term meshless) methods, two key steps need to be mentioned: (A) construction of the trial and test approximations; and (B) numerical evaluation of the weak form (Galerkin or Rayleigh-Ritz procedure) integrals, which lead to a linear system of equations (Kd = f). In meshfree Galerkin methods, the main departure from FEM is in (A): meshfree approximation schemes (linear combination of basis functions) are constructed independent of an underlying mesh (union of elements).

However, since a Galerkin method is typically used in solid mechanics applications, (B) arises and the weak form integrals need to be evaluated. Three main directions have been pursued to evaluate these integrals:

1. Background cells (elements in a finite element mesh are suitable) are used. In this case, the finite element mesh is used as the background cells for the sole purpose of numerical integration. This choice leads to inaccuracies in the patch test (a separate topic unto itself).

2. As an alternative, nodal integration methods that do not require finite elements as background cells have also been proposed. However, even in these at least a local mesh or Voronoi cell construction is required. Is this a mesh or not ?

3. Lastly, explicitly carrying out the numerical integration over the intersection of nodal basis function supports has also been carried out. Again, if one wants to draw an analogy, these regions can be viewed as background cells.

Depending on one's perspective, the use of a mesh in meshfree methods is seen to be fuzzy. The distinctions are more a matter of semantics. At the end of the day, one must consider if the method can provide significant advantages over finite elements for particular class of problems and/or solve problems that heretofore have not been tractable by finite elements or other numerical methods.

To construct robust approximations for unstructured nodal data, it is intuitive to think that there must be a connection between geometry-and-approximation (notably for non-uniform discretizations). If a Galerkin method is used and numerical integration must also come into play, then it appears one can not discard geometry or at least the need for a local mesh altogether. Only when the strong form is directly used as in a finite-difference method or a radial basis function (collocation) method, the absence of a mesh is evident: discretize your domain using scattered nodes and solve a PDE. Additions and/or amendments are welcome.

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Dear Sukumar,

Thank you very much for your excellent posts both here and on meshfreemethods blogspot. Below are my opinions.

- I think local Voronoi mesh used in nodal integration is also a mesh. However, to mesh and adjust in a local domain are not time-consuming tasks such as in FEM. Moreover, with strain smoothing technique, this method can give very good results.

- If one uses Galerkin methods and does not want to deal with meshes in numerical integration, he can use moving integration subdomains. In this method, a set of patches Ωk covers the entire problem domain is used with its corresponding distribution function Ψk .

The patches can overlap each other or not (but there is no gap). A simple geometry for Ωk often used is quadrature (in 2D). If the centers of these quadratures coincide with given nodes, then Ωk(moving quadratures at node Xk) must be covered by the support domain of node Xk. They also can be the same.

The Shepard function is one of the distribution functions, which partition the unity.

Using these techniques leads to integrate on subdomains (moving quadratures) in stead of the entire domain when computing each nodal stiffness/force matrix. With linear problems, just one Gauss point is needed for each subdomain, and it is very interesting that it is the given node.

 That is one of truly meshfree methods for Galerkin weak form. No mesh is needed. 


abshaw's picture

I would like to add few comments. Firstly as we know that all difficulties present in FEM are consequence of use of elements (mesh) which are essentially the building blocks of FEM. Meshfree method is introduced as an alternative scheme to overcome these difficulties. However, even in Meshfree methods based on weak projection of the governing PDE, one requires a background mesh. Consequently,


  1. In the refinement process one needs to modify the background mesh. Is this not numerically intensive?
  2. Since Meshfree shape functions are rational we need higher order quadrature rule for numerical integration. This again increases the computation time.
  3. Support size of window function is one of the very tricky issue in Meshfree method. User requires some numerical experiment to fix the optimum support size.

 Now the question is “is it possible to bypass the above mentioned difficulties?” 

Issue of integration:

One may go for collocation technique. But this may lead to stability problem. I think one solution could be the use of Cosserat continuum theory where balance laws are formulated based on some deformable directors. Using the local form of balance laws one may easily avoid the burden of numerical integration. Moreover it is already numerically demonstrated that Cosserat theory does not suffer any stability problem.


Issue of support size

One may use a window function that having automatically adjustable support size depending on the order of consistency. One immediate choice could be the use of non-uniform rational B-splines (NURBS).  You may see this paper “A NURBS based error reproducing kernel method”, Appeared online in computational mechanics.


Connection between geometry and approximation:

Sukumar, as you have told that there must be a relation between geometry and approximation. In this regard the iso-geomeric analysis concept proposed by Hughes et all is interesting (CMAME, vol 194, issue 39-41, pp.4135-4195).


I welcome any further comments on this.

I'll try to comment on most of your questions here.

1. In the refinement process one needs to modify the background mesh. Is this not numerically intensive?

It's not necessarily the case that the background mesh needs to be modified with refinement. Many methods do use a *fixed* background mesh. Further, some methods employ a variation on nodal integration, and do not rely on a background mesh at all. Finally, the modification of the mesh, when it is used, need not be numerically intensive.

2. Since Meshfree shape functions are rational we need higher order quadrature rule for numerical integration. This again increases the computation time.

Not all meshfree shape functions are rational functions. Sometimes people do use high-order Gauss quadrature, and that certainly can be expensive. However, I would contend that this is not a particularly smart way to handle things.

3. Support size of window function is one of the very tricky issue in Meshfree method. User requires some numerical experiment to fix the optimum support size.

This is a valid point. However, some of the "better" meshfree methods do not have this problem. For example, with NEM, there is no support size question.

N. Sukumar's picture

All your points are valid; a few more supplements to what John has mentioned.  Yes, issues such as numerical integration, `choice of support sizes' exist. Hence the attempt by many to move to stable nodal integration methods (at least for large deformation). A recent paper in ijnme (early view) provides a nice background on the subject, points our the issues, and proposes a solution.  Yes, the isogeometric concept is appealing (given its ties to b-splines/nurbs) from the geometry-approximation perspective.  If one wants freedom from a `mesh' one has to pay the price elsewhere . . . or at least be willing to do so. The link to the paper I mentioned is here

B.Banerjee's picture

I would like to comment on your use of “local form of Cosserat Balance laws “for avoiding Numerical Integration. I think this is only valid when the kinematic quantities are the functions of time only. For Cosserat surface and Cosserat rod approximations, kinematic quantities are functions of space variables also, so any weak projections needs Numerical integration, and one cannot avoid it. The only way to avoid it is to go for Collocation scheme.


Recently theory of Cosserat Points have been explored for nonlinear elasticity, where kinematic quantities are functions of time only, so that the resulting balance equations are nonlinear ODE’s instead of PDE’s. But the central difficulty lies in the unknown form of inhomogeneous strain energy functions, which cannot be obtained trivially for any class of materials. For homogeneous Cosserat Points, where deformation gradient is constant, the theory is well established, avoids the cost of Numerical Integration, which is even true for any linear finite elements. So, I think the issue of nodal integration can only be avoided if one uses the strong form of solutions.  Please correct me if I am wrong in any of the above statements.

Amirtham Rajagopal's picture

Where can i find useful information on patch test for Mesh free method.



Henry Tan's picture

Dear Sukumar,

Can you elaborate on the first direction for the evaluation of the integral, which uses background cells?

Why this choice leads to inaccuracies in the patch test?

Can you suggest some references for further reading? Thanks a lot.

N. Sukumar's picture

Henry,  The inaccuracies stem from : (1) The shape functions are rational functions (derivatives will be more oscillatory) and hence Gauss quadrature can only do so much; and more importantly, (2) The intersection of basis function supports (region over which they are non-zero) do not coincide with the `background cell' that is used for integration.  This is pertinent since to form the stiffness matrix one has to compute terms like \int \phi_i,x \phi_j,x dx in one dimension. The second item is particularly damaging . . .  think of integrating (using Gauss quadrature) a continuous function that is zero from (-1,-0.1 ) and is a polynomial from (-0.1,1). If you use the `cells' : (-1,0) and (0,1) to integrate this function, Gauss quadrature will perform poorly. Of course, if you choose (-1,-0.1) and (-0.1,1) as your cells you can recover the exact answer. Here's a link to John's paper (gzipped ps) that would provide more background, examples, and a clearer picture.

hiader k. mahbes's picture

Thanks Sir this is goog informatiom.

Dear Prof. Sukumar

I've applied Voronoi diagram for refinement process in which it's functional contains residuals of domain and boundaries. Which error estimator can be useful in adaptive refinement of meshfree methods?

best regards, 


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