Computational Mechanics

, Volume 53, Issue 2, pp 343–357 | Cite as

An improved meshless Shepard and least squares method possessing the delta property and requiring no singular weight function

  • Xiaoying Zhuang
  • Hehua Zhu
  • Charles Augarde
Original Paper


The meshless Shepard and least squares (MSLS) method and the meshless Shepard method are partition of unity based meshless interpolations which eliminate the problems by other meshless methods such as the difficulty in direct imposition of the essential boundary conditions. However, singular weight functions have to be used in both methods to enforce the approximation interpolatory, which leads to the loss of smoothness in approximation and locally oscillatory results. In this paper, an improved MSLS interpolation is developed by using dually defined nodal supports such that no singular weight function is required. The proposed interpolation satisfies the delta property at boundary nodes and the compatibility condition throughout the domain, and is capable of exactly reproducing the basis function. The computational cost of the present interpolation is much lower than the moving least-squares approximation which is probably the most widely used meshless interpolation at present.


Meshless Shepard shape function  Partition of unity Delta property Compatibility 



The authors gratefully acknowledge the support of Natural Science Foundation of China (NSFC 41130751), National Basic Research Program of China (973 Program: 2011CB013800), Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT, IRT1029), Shanghai Pujiang Talent Program (12PJ1409100) and Shanghai Chenguang Talent Program (12CG20).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Geotechnical EngineeringTongji UniversityShanghaiChina
  2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  3. 3.School of Civil and Resource EngineeringThe University of Western AustraliaCrawleyAustralia

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