Matched interface and boundary (MIB) method for the vibration analysis of plates.

*(English)*Zbl 1307.74043Summary: This paper proposes a novel approach, the matched interface and boundary (MIB) method, for the vibration analysis of rectangular plates with simply supported, clamped and free edges, and their arbitrary combinations. In previous work, the MIB method was developed for three-dimensional elliptic equations with arbitrarily complex material interfaces and geometric shapes. The present work generalizes the MIB method for eigenvalue problems in structural analysis with complex boundary conditions. The MIB method utilizes both uniform and non-uniform Cartesian grids. Fictitious values are utilized to facilitate the central finite difference schemes throughout the entire computational domain. Boundary conditions are enforced with fictitious values - a common practice used in the previous discrete singular convolution algorithm. An essential idea of the MIB method is to repeatedly use the boundary conditions to achieve arbitrarily high-order accuracy. A new feature in the proposed approach is the implementation of the cross derivatives in the free boundary conditions. The proposed method has a banded matrix. Nine different plates, particularly those with free edges and free corners, are employed to validate the proposed method. The performance of the proposed method is compared with that of other established methods. Convergence and comparison studies indicate that the proposed MIB method works very well for the vibration analysis of plates. In particular, modal bending moments and shear forces predicted by the proposed method vanish at boundaries for free edges.

##### MSC:

74H45 | Vibrations in dynamical problems in solid mechanics |

74K20 | Plates |

65N06 | Finite difference methods for boundary value problems involving PDEs |

74S20 | Finite difference methods applied to problems in solid mechanics |

##### Keywords:

vibration of plates; matched interface and boundary method; free edge supports; high-order schemes
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\textit{S. N. Yu} et al., Commun. Numer. Methods Eng. 25, No. 9, 923--950 (2009; Zbl 1307.74043)

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