# Modal analysis of elastic vibrations of incompressible materials based on a variational multiscale finite element method

2019-10-04
Codina, Ramon
Türk, Önder
In this study, we extend the standard modal analysis technique that is used to approximate vibration problems of elastic materials to incompressible elasticity. In modal analysis, the second order time derivative of the displacements in the inertia term is utilized, and the problem is transformed into an eigenvalue problem in which the eigenfunctions are precisely the amplitudes, and the eigenvalues are the squares of the frequencies. While this approach is applied directly to compressible materials in different structural models, incompressible media pose the difficulty associated to the need of introducing the pressure (or mean stress) as a variable and to interpolate it in an adequate manner. In particular, when the problem is approximated using finite elements, the standard Galerkin formulation requires the use of interpolations for the displacement and the pressure that satisfy the classical inf-sup condition, often called Ladyzhenskaya-Babuska-Brezzi condition in this context. The alternative to use finite element interpolations satisfying the inf-sup condition is to resort to stabilized finite element formulations. However, particular care is needed when dealing with the eigenvalue problem since in general, stabilization techniques yield a quadratic eigenvalue problem even if the original one is linear. We introduce a finite element formulation we have developed for the Stokes eigenvalue problem that is based on the variational multiscale (VMS) concept, preserving the linearity, and accommodating arbitrary interpolations for the displacement and the pressure, into the modal analysis of incompressible elastic materials, using displacements and pressures as variables. We show that each mode of the modal analysis (amplitude and frequency) can be obtained from an eigenvalue problem that can be split into the finite element scale and the subgrid scale. The latter needs to be approximated, and we show that this approximation should depend on the frequency of the mode being considered. Since this frequency is unknown, an iterative procedure must be devised. The result is a problem for the finite element component of the displacement amplitude and the pressure which allows for any spatial interpolation. Several eigenvalues and eigenfunctions of the Stokes eigenvalue problem need to be computed to perform the modal analysis. The time approximation to the continuous solution is obtained taking a few modes of the whole set, those with higher energy. We present an example of the vibration of a linear incompressible elastic material showing how our approach is able to approximate the problem. It is shown how the energy of the modes associated to higher frequencies rapidly decreases, allowing one to get good approximate solution with only a few modes.

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Citation Formats
R. Codina and Ö. Türk, “Modal analysis of elastic vibrations of incompressible materials based on a variational multiscale finite element method,” presented at the European Numerical Mathematics and Advanced Applications Conference 2019, Egmond aan Zee, Hollanda, 2019, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/85824.