Collocation methods for the solution of von-Kármán dynamic non-linear plate systems

Abstract

The von-Kármán nonlinear, dynamic, partial differential system over rectangular domains is considered, and numerically solved using both the Chebyshev-collocation and Legendre-collocation methods for the spatial discretization and the implicit Newmark- β scheme combined with a non-linear fixed point algorithm for the temporal discretization. As the system is non-linear, involving operators of different orders, different timescales, and may contain initial/boundary incompatible conditions at the domain's corners, it is our aim to highlight some of the difficulties inherent in its numerical treatment using collocation methods. We begin by examining the Chebyshev-collocation scheme considered in [Comput. Meth. Appl. Mech. Engrg. 193 (6–8) (2004) 575]. In that work, filtering was used to stabilize the full von-Kármán system, however the source of the instability was not discussed. We first show that IC/BC incompatibilities are not the culprits and demonstrate empirically that the source of the instability was the numerical treatment of the linear cross-derivative terms found in the in-plane equations. We prove mathematically that although the continuous linear system is well-posed, the Chebyshev-collocation solution of problems involving cross-derivatives and second-order time derivatives is unstable. Instead of adding sufficient filtering to counter the energy growth due to the linear terms, we employ the Legendre-collocation method. We show empirically and prove that the Legendre-collocation scheme for the linear system has the necessary characteristics to remain stable. Nevertheless, the collocation treatment of the non-linear terms still causes the solution to be unstable at long time. We thus return to advocating a filtering solution, but in this case only filtering the non-linear terms to compensate for the inherent aliasing error due to our collocation treatment of these terms. We conclude by comparing the results of filtered Chebyshev-collocation solution, non-linear terms filtered Legendre-collocation solution, and a simplified von-Kármán (solution omitting the inertial terms in the in-plane equations) as presented in (loc. cit).