A rational kernel function selection for Galerkin meshfree methods through quantifying relative interpolation errors

Abstract

Although kernel functions play a pivotal role in meshfree approximation, the selection of kernel functions is often experience-based and lacks a theoretical basis. As an attempt to resolve this issue, a rational matching between kernel functions and nodal supports is proposed in this work for Galerkin meshfree methods, where the quadratic through quintic B-spline kernel functions are particularly investigated. The foundation of this rational matching is the design of an efficient quantification of relative interpolation errors. The proposed relative interpolation error measures are not problem-dependent and can be easily and efficiently evaluated. More importantly, these relative interpolation error measures effectively reflect the variation of the real interpolation errors for meshfree approximation, which essentially control the solution accuracy of the Galerkin meshfree formulation with consistent numerical integration. Consequently, an optimal selection of kernel functions that match the nodal supports of meshfree approximation can be readily realized via minimizing the relative interpolation errors of meshfree approximation. The efficacy of the proposed rational matching between kernel functions and nodal supports is well demonstrated by meshfree numerical solutions.