A Chebyshev collocation multidomain method to solve the Reissner-Mindlin equations for the transient response of an anisotropic plate subjected to impact

Abstract

The transient response of an anisotropic rectangular plate subjected to impact is described through a Chebyshev collocation multidomain discretization of the Reissner-Mindlin plate equations. The trapezoidal rule is used for time-integration. 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 variables at the subdomain corners has to be solved. Back-substitution gives the variables at all other collocation points. The time history as represented by computed contour plots has been compared with analytical results and with photos produced by holographic interferometry. The agreements are satisfactory.

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

“…Each subdomain of the meridional cross-section is discretized using the Gauss-Lobatto-Chebyshev collocation points. The discretization is taken from a previous paper by the present author [3]. Thus, let there in each subdomain be M +1 points in the x-direction and N +1 points in the y-direction.…”

“…Also the patching is inspired by that of Kjellmert [3], and side points and corner points are treated separately, as in [3]. Some side points are on the outer boundary of the cross-section (the point p A ij belongs to subsection A, and p A ij ∈ S ext ), and others are shared by two subdomains (p A ij is physically the same point as p B kl for subdomains A and B (=A + 1), and some index pairs (i; j) and (k; l), and p A ij ∈ S int ).…”

“…In this way a number of planes are considered, one for each Fourier mode. In earlier research Kjellmert studied the impact response of a plate by integrating the Reissner-Mindlin plate equations [3]. They were discretized by a Chebyshev collocation multidomain method.…”

“…The tube variables are represented by Fourier coe cients on the meridional cross-section as in the paper by Ben-David and Bar-Yoseph [2]. The coe cients are discretized in the same way as the plate variables in the previous paper by Kjellmert [3], and the time integration is also the same as in that paper. In conclusion, the present tube procedure is an extension of a previous plate procedure, and parts of what will be said below have already been discussed in Kjellmert's previous paper [3].…”