A spectrally formulated finite element for analysis of wave propagation in layered composite media
Publisher Summary ::: Wave propagation in asymmetrically stacked composite laminate, because of high frequency impact loading, is studied using a new spectral finite element (SFE) formulation. This novel element is developed using matrix methodology similar to finite element (FE) formulation and has an exact dynamic stiffness matrix, obtained by exactly solving the governing Navier equation in the frequency-domain. As the mass distribution is modeled exactly, the element gives the exact frequency response of each layer. Thus, one element may be as large as one complete layer; and as a result, the system size is nominal compared to the conventional FE system sizes. The fast-Fourier transform (FFT) and Fourier series are used for inversion to the time/space domain. The element captures the essential wave propagation phenomena successfully and demonstrates Rayleigh wave in the surface and Stoneley wave in composite layer interfaces.
In this paper we study the multivariate ANOVA decomposition for $1$-periodic functions on the torus. In particular we use the integral projection operator that leads to the classical ANOVA decomposition. Relationships between the Fourier coefficients of the function and its ANOVA terms lead to special frequency index sets and give an understanding of the decomposition working in the frequency domain. Moreover, we consider the truncated ANOVA decomposition and provide error bounds for approximation in $\mathrm{L}_\infty$ and $\mathrm{L}_2$. We present an approximation method based on the truncated decomposition with regard to a superposition dimension $d_s$.