A least squares recursive gradient meshfree collocation method for superconvergent structural vibration analysis

• Published in: Computational mechanics 2021-11, Vol.68 (5), p.1063-1096
• Main Authors: Deng, Like, Wang, Dongdong, Qi, Dongliang
• Format: Article
• Language: English
• Subjects: Basis functions; Classical and Continuum Physics; Collocation methods; Computational Science and Engineering; Engineering; Error analysis; Least squares; Meshless methods; Original Paper; Shape functions; Structural vibration; Theoretical and Applied Mechanics; Vibration; Vibration analysis
• ISSN: 0178-7675; 1432-0924
• DOI: 10.1007/s00466-021-02059-5

A least squares recursive gradient meshfree collocation method is proposed for the superconvergent computation of structural vibration frequencies. The proposed approach employs the recursive gradients of meshfree shape functions together with smoothed shape functions in the context of least squares formulation, where both meshfree nodes and auxiliary points are taken as the collocation points. It turns out that this least squares formulation can effectively suppress the spurious modes arising from a direct meshfree collocation formulation using recursive gradients. Meanwhile, a detailed theoretical analysis with explicit frequency error measure is presented for the least squares recursive gradient meshfree collocation method in order to assess the frequency accuracy of structural vibrations. This analysis discloses the salient basis degree discrepancy issue regarding the frequency accuracy for the least squares meshfree collocation formulation, and it is shown that this issue can be essentially resolved by the proposed least squares recursive gradient meshfree collocation method. In fact, the proposed method leads to superconvergent vibration frequencies when odd degree basis functions are used, i.e., the frequency convergence rate is improved from ( p - 1 ) for the standard least squares meshfree collocation to ( p + 1 ) for the proposed approach in case of an odd p th degree basis function. This desirable frequency superconvergence of the proposed least squares recursive gradient meshfree collocation method is congruously demonstrated by numerical results.