Extraction of wave characteristics from wavelet-based spectral finite element formulation

In this paper, a spectrally formulated wavelet finite element is developed and is used not only to study wave propagation in 1-D waveguides but also to extract the wave characteristics, namely the spectrum and dispersion relation for these waveguides. The use of compactly supported Daubechies wavelet basis circumvents several drawbacks of conventional FFT-based Spectral Finite Element Method (FSFEM) due to the required assumption of periodicity, particularly for time domain analysis. In this work, a study is done to use the formulated Wavelet-based Spectral Finite Element (WSFE) directly for such frequency domain analysis. This study shows that in WSFE formulation, a constraint on the time sampling rate is paced to avoid spurious dispersion being introduced in the analysis. Numerical experiments are performed to study frequency-dependent wave characteristics (dispersion and spectrum relations) in elementary rod, Euler–Bernoulli and Timoshenko beams. The effect of sampling rate on the accuracy of WSFE solution for both impulse and modulated sinusoidal loading with different frequency content is shown through different examples. In all above cases, comparison with FSFEM are provided to highlight the advantages and limitations of WSFE.