A modal analysis approach using an Hybrid-Mixed formulation to solve 2D elastodynamic problems

The purpose of this work is to present a Hybrid-Mixed finite element formulation to solve elastodynamic plate problems in frequency domain. This Hybrid-Mixed model is derived [

2

] establishing, non-locally, the dynamic equilibrium, compatibility and constitutive relations based on the Galerking weighted residual method. Two different approximations fields are used in the element domain, namely the stresses and the displacements fields. A third approximation independent from the previous ones is also required for the displacement field on the static boundary of the elements. Once obtained the governing system, the stresses and the boundary displacements degrees of freedom are eliminated, thus resulting a new condensed system, were only the domain displacements degrees of freedom (

qV

) intervine:

1

$$ \left( {K - \omega ^2 M} \right)qv = Qv $$

The condensed system assume a form analogous to the one obtained with the conventional FEM[

1

], however with a different and richer physical meaning. In a first stage natural frequencies and shape modes are identified. Then, modal analysis technique is performed to uncouple the governing system equations, and to assess the relevance of each mode to a specific action. Less relevant modes are eliminated, thus reducing substantially the computational costs without significant lost of accuracy. The Hybrid-Mixed model is implemented using 4-node serendipian standard master elements to define the finite element shape, and non-nodal, Legendre polynomials as approximation functions. This functions are suited for this purpose because they allow to introduce in the finite element code closed form solutions to compute the coefficients of the structural matrices, avoiding time consuming numerical integration. Good results can be reach using macro finite elements since

h

-refinements are easy to performe simply by increasing the maximum degree of the polynomials in the approximation functions.

The efficiency and performance of the presented method is validated with the aid of numerical examples. Free vibrations natural frequencies are determined and transient response in undamped and viscous damped structures are studied.