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The vibro-acoustic behaviour of poroelastic materials is often formulated as boundary value problems based on the continuum mechanics Biot's theory expressed as two coupled partial differential equations. This paper presents an extension of the Wave Based Method (WBM), a numerical technique to solve these vibro-acoustic boundary value problems in a computationally efficient manner.At present, the Finite Element Method (FEM) is the most commonly used prediction technique to deal with these Biot equations, but suffers from the disadvantage that the system matrices have to be recalculated for each frequency of interest due to the frequency-dependent equation parameters.This harms the inherent effectiveness of the FEM. Additionally, due to the discretisation into a large number of small finite elements and the high number of unknowns per node, the computational efforts involved practically restrict the use of FEM to low-frequency applications. The method discussed in this paper is based on an indirect Trefftz approach. Exact solutions of the three coupled waves, supported by Biot's equations, are used as basis functions in a solution expansion to approximate the field variables in a poroelastic boundary value problem. This approach leads to smaller systems of equations, enabling an efficient solution at higher frequencies.

SUMMARYIn this paper a fully anisotropic symmetric weak formulation of Biot's equations for vibro-acoustic modelling of porous elastic materials in the frequency domain is proposed. Starting from Biot's equations in their anisotropic form, a mixed displacement-pressure formulation is discussed in terms of Cartesian tensors. The anisotropic equation parameters appearing in the differential equations are derived from material parameters which are possible to determine through experimental testing or micro-structural simulations of the fluid and the porous skeleton. Solutions are obtained by applying the finite element method to the proposed weak form and the results are verified against a weak displacement-based formulation for a foam and plate combination.

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