Open Access

ARTICLE

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

Xiaozhong Jin¹, Gang Li², N. R. Aluru³

1 Graduate Student, Department of Mechanical and Industrial Engineering
2 Doctoral Student, Department of Mechanical and Industrial Engineering
3 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.

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