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

A weak solution of the coupled, acoustic-elastic, wave propagation problem for a ßexible porous material is proposed for a 3-D continuum. Symmetry in the matrix equations; with respect to both volume, i.e. Ôporous frameÕÐÔpore ßuidÕ, and surface, i.e. Ôporous frame/pore ßuidÕÐÔnon-porous mediaÕ, ßuidÐstructure interaction; is ensured with only Þve unknowns per node; ßuid pore pressure, ßuid-displacement potential and three Cartesian components of the porous frame displacement Þeld. Taking BiotÕs 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 di¤erent surface conditions. The primary application of the proposed symmetric 3-D Þnite 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 Þnite elements, the noise transmission through a double panel with porous Þlling and di¤erent boundary conditions at the two panel boundaries are analysed.