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A Meshless Collocation Method Based on the Differential Reproducing Kernel Approximation

Shih-Wei Yang1, Yung-Ming Wang1, Chih-Ping Wu1,2, Hsuan-Teh Hu1
Department of Civil Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC
Corresponding author. Fax: +886-6-2370804, E-mail address:

Computer Modeling in Engineering & Sciences 2010, 60(1), 1-40.


A differential reproducing kernel (DRK) approximation-based collocation method is developed for solving ordinary and partial differential equations governing the one- and two-dimensional problems of elastic bodies, respectively. In the conventional reproducing kernel (RK) approximation, the shape functions for the derivatives of RK approximants are determined by directly differentiating the RK approximants, and this is very time-consuming, especially for the calculations of their higher-order derivatives. Contrary to the previous differentiation manipulation, we construct a set of differential reproducing conditions to determine the shape functions for the derivatives of RK approximants. A meshless collocation method based on the present DRK approximation is developed and applied to the analysis of one-dimensional problems of elastic bars, two-dimensional potential problems, and plane elasticity problems of elastic solids to validate its accuracy and find the rate of convergence. It is shown that the present method is indeed a fully meshless approach with excellent accuracy and fast convergence rate.


Meshless methods, Reproducing kernels, Collocation methods, Deformation, Stress, Elastic solids.

Cite This Article

Yang, S., Wang, Y., Wu, C., Hu, H. (2010). A Meshless Collocation Method Based on the Differential Reproducing Kernel Approximation. CMES-Computer Modeling in Engineering & Sciences, 60(1), 1–40.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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