Pseudo Spectral Methods Applied to Problems in Elasticity

Abstract

Pseudo spectral methods offer an attractive alternative to finite element procedures for the solution of problems in elasticity. Especially for simple domains, questions involving both two and three dimensional elasticity (Navier’s Equations or their non-linear generalisations) would seem to be reasonable candidates for a pseudo spectral approach. This paper examines some simple vibrational eigenvalue type problems, demonstrating how Navier’s equations can be recast into pseudo- spectral format, including first derivative boundary conditions representing zero traction. Fourier–Chebyshev methods are shown to give solutions with typical spectral accuracy, with the addition of pole conditions being necessary for the case of a two dimensional disc. There is also consideration given to time-stepping solutions of elastodynamic problems, especially those involving non-linear friction effects, the authors particular interest being the study of disc brake noise. It is shown that, at least for relatively simple cases, it is possible to model systems in such a way that animated graphical output can be provided as the system of partial differential equations is numerically integrated. This provides a useful tool for engineers to rapidly examine the effect of parameter changes on a system model.