A nodal spacing study on the frequency convergence characteristics of structural free vibration analysis by lumped mass Lagrangian finite elements

Li, Xiwei; Wang, Dongdong; Xu, Xiaolan; Sun, Zhuangjing

doi:10.1007/s00366-022-01668-9

Cited by 3 publications

(1 citation statement)

References 40 publications

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“…( 8) into Eq. ( 41), and after a lengthy but straightforward derivation by performing the standard Taylor expansion [31][32][33], the following frequency error estimates are, respectively, obtained for the bending, shear and twist modes corresponding to IGA-FI:…”

Section: Accuracy Deterioration For Isogeometric Frequency Analysis W...mentioning

confidence: 99%

Coarse Mesh Superconvergence in Isogeometric Frequency Analysis of Mindlin–Reissner Plates with Reduced Integration and Quadratic Splines

Lin

Hou

et al. 2022

Acta Mech. Solida Sin.

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A frequency accuracy study is presented for the isogeometric free vibration analysis of Mindlin–Reissner plates using reduced integration and quadratic splines, which reveals an interesting coarse mesh superconvergence. Firstly, the frequency error estimates for isogeometric discretization of Mindlin–Reissner plates with quadratic splines are rationally derived, where the degeneration to Timoshenko beams is discussed as well. Subsequently, in accordance with these frequency error measures, the shear locking issue corresponding to the full integration isogeometric formulation is elaborated with respect to the frequency accuracy deterioration. On the other hand, the locking-free characteristic for the isogeometric formulation with uniform reduced integration is illustrated by its superior frequency accuracy. Meanwhile, it is found that a frequency superconvergence of sixth order accuracy arises for coarse meshes when the reduced integration is employed for the isogeometric free vibration analysis of shear deformable beams and plates, in comparison with the ultimate fourth order accuracy as the meshes are progressively refined. Furthermore, the mesh size threshold for the coarse mesh superconvergence is provided as well. The proposed theoretical results are consistently proved by numerical experiments.

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Section: Accuracy Deterioration For Isogeometric Frequency Analysis W...mentioning

confidence: 99%

Coarse Mesh Superconvergence in Isogeometric Frequency Analysis of Mindlin–Reissner Plates with Reduced Integration and Quadratic Splines

Lin

Hou

et al. 2022

Acta Mech. Solida Sin.

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A Unified Meshfree Path to Arbitrary Order Hermite Finite Elements for Euler-Bernoulli Beams

Wu¹,

Wang²,

Hou³

2023

Int. J. Str. Stab. Dyn.

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The fourth-order governing equation of Euler-Bernoulli beams necessitates the employment of C1 Hermite shape functions in finite element analysis. However, unlike the C0 finite element shape functions that can be easily formulated as Lagrangian polynomials in a unified manner, the Hermite shape functions of Euler-Bernoulli beam elements are usually constructed case by case. In this work, a simple and unified meshfree path is proposed to develop arbitrary order Hermite shape functions of Euler-Bernoulli beam elements. This approach is realized by proving that the Hermite reproducing kernel meshfree shape functions degenerate to the interpolatory Hermite finite element shape functions under a proper choice of support sizes. The proposed approach provides an explicit formalism for Hermite finite element shape functions, which enables an easy construction of arbitrary order Hermite beam elements. Accordingly, in addition to the conventional cubic Hermite beam elements, explicit shape functions, mass and stiffness matrices for quintic and septic Hermite beam elements are presented. Meanwhile, the frequency errors and general higher order mass matrix formulation for these Hermite beam elements are elaborated as well. These theoretical results are consistently demonstrated by numerical examples with respect to both static and free vibration analysis.

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Optimized Quadrature Rules for Isogeometric Frequency Analysis of Wave Equations Using Cubic Splines

Hou

et al. 2023

Int. J. Appl. Mechanics

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An optimization of quadrature rules is presented for the isogeometric frequency analysis of wave equations using cubic splines. In order to optimize the quadrature rules aiming at improving the frequency accuracy, a frequency error measure corresponding to arbitrary four-point quadrature rule is developed for the isogeometric formulation with cubic splines. Based upon this general frequency error measure, a superconvergent four-point quadrature rule is found for the cubic isogeometric formulation that achieves two additional orders of frequency accuracy in comparison with the sixth-order accuracy produced by the standard approach using four-point Gauss quadrature rule. One interesting observation is that the first and last integration points of the superconvergent four-point quadrature rule go beyond the domain of conventional integration element. However, these exterior integration points pose no difficulty on the numerical implementation. Subsequently, by recasting the general four-point quadrature rule into a three-point formation, the proposed frequency error measure also reveals that the three-point Gauss quadrature rule is unique among possible three-point rules to maintain the same sixth-order convergence rate as the four-point Gauss quadrature rule for the cubic isogeometric formulation. These theoretical results are clearly demonstrated by numerical examples.

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A nodal spacing study on the frequency convergence characteristics of structural free vibration analysis by lumped mass Lagrangian finite elements

Cited by 3 publications

References 40 publications

Coarse Mesh Superconvergence in Isogeometric Frequency Analysis of Mindlin–Reissner Plates with Reduced Integration and Quadratic Splines

Coarse Mesh Superconvergence in Isogeometric Frequency Analysis of Mindlin–Reissner Plates with Reduced Integration and Quadratic Splines

A Unified Meshfree Path to Arbitrary Order Hermite Finite Elements for Euler-Bernoulli Beams

Optimized Quadrature Rules for Isogeometric Frequency Analysis of Wave Equations Using Cubic Splines

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