A 3-D, symmetric, finite element formulation of the biot equations with application to acoustic wave propagation through an elastic porous medium

摘要

A weak solution of the coupled, acoustic-elastic, wave propagation problem for a flexible porous material is proposed for a 3-D continuum. Symmetry in the matrix equations; with respect to both volume, i.e. 'porous frame'-'pore fluid', and surface, i.e. 'porous frame/pore fluid'-'non-porous media', fluid-structure interaction; is ensured with only five unknowns per node; fluid pore pressure, fluid-displacement potential and three Cartesian components of the porous frame displacement field. Taking Biot's general theory as starting point, the discretized form of the equations is derived from a weighted residual statement, using a standard Galerkin approximation and iso-parametric interpolation of the dependent variables. The coupling integrals appearing along the boundary of the porous medium are derived for a number of different surface conditions. The primary application of the proposed symmetric 3-D finite element formulation is modelling of noise transmission in typical transportation vehicles, such as aircraft, cars, etc., where porous materials are used for both temperature and noise insulation purposes. As an example of an application of the implemented finite elements, the noise transmission through a double panel with porous filling and different boundary conditions at the two panel boundaries are analysed.