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On the Equivalence Between Least-Squares and Kernel Approximations in Meshless Methods

Xiaozhong Jin1, Gang Li2, N. R. Aluru3
Graduate Student, Department of Mechanical and Industrial Engineering
Doctoral Student, Department of Mechanical and Industrial Engineering
Assistant Professor, Department of General Engineering and Beckman Institute, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801

Computer Modeling in Engineering & Sciences 2001, 2(4), 447-462. https://doi.org/10.3970/cmes.2001.002.447

Abstract

Meshless methods using least-squares approximations and kernel approximations are based on non-shifted and shifted polynomial basis, respectively. We show that, mathematically, the shifted and non-shifted polynomial basis give rise to identical interpolation functions when the nodal volumes are set to unity in kernel approximations. This result indicates that mathematically the least-squares and kernel approximations are equivalent. However, for large point distributions or for higher-order polynomial basis the numerical errors with a non-shifted approach grow quickly compared to a shifted approach, resulting in violation of consistency conditions. Hence, a shifted polynomial basis is better suited from a numerical implementation point of view. Finally, we introduce an improved finite cloud method which uses a shifted polynomial basis and a fixed-kernel approximation for construction of interpolation functions and a collocation technique for discretization of the governing equations. Numerical results indicate that the improved finite cloud method exhibits superior convergence characteristics compared to our original implementation [Aluru and Li (2001)] of the finite cloud method.

Cite This Article

Jin, X., Li, G., Aluru, N. R. (2001). On the Equivalence Between Least-Squares and Kernel Approximations in Meshless Methods. CMES-Computer Modeling in Engineering & Sciences, 2(4), 447–462.



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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