Reproducing kernel mesh-free collocation analysis of structural vibrations

Abstract

Purpose

Although high-order smooth reproducing kernel mesh-free approximation enables the analysis of structural vibrations in an efficient collocation formulation, there is still a lack of systematic theoretical accuracy assessment for such approach. The purpose of this paper is to present a detailed accuracy analysis for the reproducing kernel mesh-free collocation method regarding structural vibrations.

Design/methodology/approach

Both second-order problems such as one-dimensional (1D) rod and two-dimensional (2D) membrane and fourth-order problems such as Euler–Bernoulli beam and Kirchhoff plate are considered. Staring from a generic equation of motion deduced from the reproducing kernel mesh-free collocation method, a frequency error measure is rationally attained through properly introducing the consistency conditions of reproducing kernel mesh-free shape functions.

Findings

This paper reveals that for the second-order structural vibration problems, the frequency accuracy orders are p and (p − 1) for even and odd degree basis functions; for the fourth-order structural vibration problems, the frequency accuracy orders are (p − 2) and (p − 3) for even and odd degree basis functions, respectively, where p denotes the degree of the basis function used in mesh-free approximation.

Originality/value

A frequency accuracy estimation is achieved for the reproducing kernel mesh-free collocation analysis of structural vibrations, which can effectively underpin the practical applications of this method.

Citation

Qi, D., Wang, D., Deng, L., Xu, X. and Wu, C.-T. (2019), "Reproducing kernel mesh-free collocation analysis of structural vibrations", Engineering Computations, Vol. 36 No. 3, pp. 734-764. https://doi.org/10.1108/EC-10-2018-0439