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approximation

interpolation/approximation method for accurate evaluation of higher order derivatives of shape functions

Hello,

I  want to solve some problem from solid mechanics by means of meshless methods. Do you know some interpolation/approximation method which is able to accurate evaluate 3rd and 4th derivations of the shape functions and is not difficult to implement? MLS or PIM have problem with accuracy in these derivatives, as is known.
Thank you very much for your advices.

A Problem in Nature Of XFEM Approximation


There is a basic problem in NATURE of approximation of discontinuity in element using XFEM.I tried to illustrate that below:
for example, If you consider an element with strong discontinuity( like fracture or contact) which usually we use heaviside function as enrichment function. we expect two parts of element deform independently because of nature of problem.

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