Simulation of structural dynamics using a non-polynomial method

Abstract

In structural dynamics, the modeling of steady-state thin plate bending is an important but, especially at high frequencies, computationally challenging problem. When solving the displacement of an elastic thin plate, a fourth order partial differential equation (Kirchhoff's plate equation) needs to be solved. In addition, two boundary conditions are needed in order to uniquely solve the problem. Polynomial methods, such as the finite element method (FEM) and discontinuous Galerkin method (DGM), are generally used to solve the plate dynamics. At higher frequencies the computational burden of a low order FEM becomes rapidly unbearable. Consequently, non-polynomial modeling methods are investigated because of their capability to solve the problem more efficiently than the standard FEM. The non-polynomial method used in this study is called the ultra weak variational formulation (UWVF). The UWVF uses finite element meshes and it is essentially an upwind DGM with a special choice of basis functions. To date, the UWVF has been successfully used in electromagnetism, acoustics and linear elasticity. We shall show, using a mixture of theory and numerical examples, that the UWVF is feasible for thin plate problems. For these problems the UWVF basis consists of plane wave and evanescent (corner) wave functions.