A spectral method to solve the equations of linear elasticity for the transient response of a tube subjected to impact

The transient response of a tube subjected to impact is described through Fourier{Galerkin and Chebyshev collocation multidomain discretizations of the equations of linear elasticity. The trapezoidal rule is used for time integration. For each Fourier mode the spatial collocation derivative operators are represented by matrices, and the subdomains are patched by natural and essential conditions. At each time level the resulting governing matrix equation is reduced by two consecutive block Gaussian eliminations, so that an equation for the complex Fourier coe cients at the subdomain corners has to be solved. Back-substitution gives the coe cients at all other collocation points. An inverse discrete Fourier transform generates, at optional time levels, the three components of the displacement eld. Through this method the long-term evolution of the eld may be calculated, provided the impact time is long enough. The time history as represented by computed contour plots has been compared with photos produced by holographic interferometry. The agreements are satisfactory