ABSTRACT
CH A P T E R 13 Spectral Finite Element Formulation
We have seen in the previous chapters that wave propagation analysis involves solving for wavenumbers and wave coefficients, the number of which increases with increased degrees of freedom and the characteristic equation for solving wavenumbers becomes extremely complicated to solve and keep track of various wave modes. In Chapters 6-8, we used an ad hoc procedure based on wave kinematics to obtain the wave responses, but this approach is not feasible if the number of wave modes participating in the structure response is large. A better approach is to adopt matrix methodology, which allows formulation and solution of the wave propagation problem in a compact manner that can enable automation of the entire solution process. The Spectral Finite Element Method (SFEM) is one such method, which is ideally suited for solution of wave propagation problems as this method results in problems sizes that are orders of magnitude smaller than the corresponding conventional FE models. There are three different variants of SFEM, namely, the Fourier transform, wavelet transform, and Laplace transform, respectively, based on the type of integral transform used to transform the problem from the time to frequency domain. In the paragraphs to follow, we will explain their respective formulation in detail.
