Time-Domain Spectral BFS Plate Element With Lobatto Basis for Wave Propagation Analysis

Abstract

A computationally efficient spectral Kirchhoff plate element is presented for time-domain analysis of wave propagation at high frequencies in thin isotropic plates. It employs a $C^1$-continuous spectral interpolation based on the modified bi-Hermite polynomials using the Gauss–Lobatto–Legendre (GLL) points as a basis with selective collocation of rotational and twisting degrees of freedom (DOFs) at element edge and corner nodes. The lowest order version of the proposed element reduces to the classical Bogner–Fox–Schmit (BFS) element for Kirchhoff plates. The GLL basis allows diagonalisation of the mass matrix using the nodal quadrature technique, which enhances the computational efficiency. The numerical properties of the proposed element are comprehensively evaluated, including the conditioning of the system matrices. Moreover, the effect of employing different numerical integration schemes and nodal sets is examined in both static and free vibration analyses. The effectiveness of the proposed element in wave propagation problems is evaluated by comparing its performance to the converged solutions achieved using the BFS element with a very fine mesh. Results demonstrate that the current element, without and even with mass matrix diagonalisation delivers exceptional accuracy while also exhibiting faster convergence and enhanced computational efficiency than the existing Kirchhoff plate elements.