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A variational multiscale stabilized and locking-free meshfree formulation for Reissner–Mindlin plate problems

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Abstract

In this study, a variational multiscale stabilized locking-free meshfree formulation is introduced for modeling Reissner–Mindlin plate problems under arbitrary plate thickness. Under this framework, the plate quantities are decoupled into coarse-scale and fine-scale components in the variational equations, where the fine-scale solution represents a correction to the residual of the coarse-scale equations that can be solved by an effective collocation-type approach with an approximation method meeting locking free conditions. The substitution of fine-scale solutions in the coarse-scale system leads to a residual-based Galerkin formulation. In the proposed framework, the reproducing kernel approximation, as well as the smoothed gradient and divergence, are adopted to ensure the bending exactness in the Galerkin formulation. The multiscale approach is also beneficial for problems exhibiting localized phenomena. The effectiveness of the proposed method is tested by solving a series of numerical examples and compared with classical methods.

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Acknowledgements

The support of this work by the Ministry of Science and Technology (MOST), Taiwan, under project contract number 110-2628-E-007 -008 to the author is greatly acknowledged.

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  1. Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan

    Tsung-Hui Huang

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Appendix 1

Appendix 1

In the free vibration test, the frequency of the plate can be determined by solving the eigenvalue problem shown in Eq. (102):

$$\left({\varvec{K}}-{\omega }^{2}{\varvec{M}}\right){\varvec{d}}=\bf{0}$$
(102)

where \({\varvec{K}}\) is the stiffness matrix, \(\omega \) is the frequency to be solved, and \({\varvec{M}}\) is the mass matrix computed by a nodal integration:

$${{\varvec{M}}}_{IJ}=\underset{\Omega }{\overset{}{\int }}{\varvec{\varPsi }}_{I}^{T}{\varvec{I}}{\varvec{\varPsi }}_{J}d\Omega \approx \sum_{L\in S}{\varvec{\varPsi }}_{I}^{T}\left({{\varvec{x}}}_{L}\right){\varvec{I}}{\varvec{\varPsi }}_{J}\left({{\varvec{x}}}_{L}\right){A}_{L}$$
(103)

In Eq. (103), \({\varvec{\Psi }}_{I}\) is the RK shape function shown in Eq. (31), and \({\varvec{I}}\) is a 3 by 3 diagonal inertia matrix, where \({I}_{11}=\rho t\) and \({I}_{22}={I}_{33}=\rho {t}^{3}/12\), with \(\rho \) the plate density and \(t\) the plate thickness.

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Huang, TH. A variational multiscale stabilized and locking-free meshfree formulation for Reissner–Mindlin plate problems. Comput Mech 69, 59–93 (2022). https://doi.org/10.1007/s00466-021-02083-5

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