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Convex-approximation schemes: Max-Entropy approximation scheme

Hi,

      I would like to know how sensitive is the max-entropy approximation scheme to the arrangement of the points in the domain. To make my question clear, in a square shaped domain, the points sitting the corners of the domain seem to face problems in max-entropy approximation. Determination of Lagrange-multiplier(s), which are required for the shape functions p(x), has been found to be dependent on the point distrubution in the domain. Kindly help in this regard. I am referring to the paper by Arroyo and Ortiz - International Journal for Numerical Methods in Engineering, 2006, 65:2167-2202.

With Regards,

Siva Prasad AVS.

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zahur_ullah's picture

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Hi Siva Prasad

 If I understand your problem correctly, you want to find out
the maxent shape functions on a point sitting on a corner of a 2D square domain?
As you know that in the case of maxent shape functions the Lagrange multipliers
will blow up as you approach the problem domain. I have also discussed this
problem with N. Sukumar and he suggested the following way to deal with this
problem.

 As in the case of maxent shape functions all the shape
functions of the interior nodes are zero at the boundary because of the weak Kronecker
delta property of the maxent shape function. Using this property, the boundary
shape function can be found by only considering each boundary as a 1D problem.

 Now in case of 1D boundary as in your case, the shape
function of all the boundary nodes will be zero at the corner so the only shape
function will be of the node at the corner, which will be 1.

 Zahur Ullah

Durham University,
UK

 

Dear Mr.Zahur Ullah,

       Thank you very much for your reply. I also need to take the derivative of the interpolant (the function being interpolated). For this I need the derivatives of the shape functions. How do I determine the gradient of shape functions at the corner nodes, if I manually assign 1 as the shape function for the corner node?

Alejandro Ortiz-Bernardin's picture

As in any meshfree method that uses radial basis functions, there is a high chance to obtain basis functions with large supports if the nodal arrangement is too irregular. Larger supports are computationally expensive and introduce numerical integration errors.

What is the problem that you are trying to solve?

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