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second derivative of MLS function

Can somebody help me with the second derivative of MLS function for three dimentional problems?


yawlou's picture

Hi Mike,

 You will find the 2nd derivatives for MLS shape functions in the following pdf document on page 21 (pdf page 12).

 I have not derived the 2nd derivatives before.  However, I have a derivation of the first derivatives in appendix A of my PhD dissertation( and the 2nd derivatives arise in a similar fashion.  I believe I arrived at the 1st derivative derivation by looking at the document above and also papers by Belytschko et al. (You might want to look in this paper [T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl. Meshless methods: An overview and recent developments. Computer Methods in Applied Mechanics and Engineering, 139:3-47, 1996])  It seems like there was another paper by Belytschko et al that showed the derivative derivation but I'm not remembering what it is right now.  If I find it I will reference it here.

 I hope this helps.



yawlou's picture

See the following pdf document.

This document includes the derivation of MLS shape functions and the derivation of 1st and 2nd derivatives of MLS shape functions.  Also, lots of introductory MLS information and terminology is provided.

If anyone finds typos, please feel free to let me know.  I hope this is a helpful introduction for beginners.





Thank you very much for the information!

Dear Mike

I have used 2nd derivatives of MLS shape function for solving planar elasticity problems. If it is useful for you, I can send MLS fortran program. Also I can help in developing this program for 3D programs.

Best regards



Dear Jafar

your program would be really usefull for me because i could compare the both approches. Unfortunaly i don't use Fortran. I hope that i can understand your program and rewrite it in Matlab.

Best Regards



s.faraji's picture




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