A C<sup>1</sup>‐continuous time domain spectral finite element for wave propagation analysis of Euler–Bernoulli beams
Santosh Kapuria, Mayank Jain
Abstract
Abstract A C 1 ‐continuous time‐domain spectral finite element (SFE) is developed for efficient and accurate analysis of flexural‐guided wave propagation in Euler–Bernoulli beam‐type structures. A new C 1 ‐continuous spectral interpolation using the Lobatto basis is proposed, which is shown to eliminate the Runge phenomenon observed in the conventional higher order Hermite interpolation. It is also able to diagonalize the mass matrix, an attractive feature of existing C 0 ‐continuous SFEs, which enhances computational efficiency. The developed element is validated by comparing the results for natural frequencies of first 20 modes with analytical solutions, and its performance for wave propagation problems is assessed in comparison with converged ABAQUS solutions obtained with a very fine mesh using the classical beam element. It is shown that the present element yields excellent accuracy with much faster convergence, higher computational efficiency, and many‐fold reduction in computational time than the conventional FE for narrowband high‐frequency flexural guided wave propagation problems in both undamaged and damaged beams. It also shows excellent performance for wave propagation under broadband impact excitations and initial displacements. The C 1 ‐continuous interpolation proposed here will pave the way for developing several new SFEs for elastic‐ and piezoelectric‐laminated beams using advanced higher order laminated theories, which require C 1 ‐continuity of displacements.