Impedance-to-scattering matrix method for large silencer analysis using direct collocation

Large silencers used in industry usually have a very large cross section at the inlet and outlet. Higher-order modes will populate the inlet and outlet even at very low frequencies. A three-dimensional analysis tool, such as the finite element method or the boundary element method, must incorporate certain forms of modal expansion in order to consider the higher-order modes in the transmission loss computation. In this paper, the impedance matrix obtained from the sub-structured boundary element method is converted into the scattering matrix that relates the higher-order modes. Since there are always more boundary elements at the inlet and outlet than the total number of propagating modes, a least-squares procedure is used to convert the element-based impedance matrix into the modal expansion-based scattering matrix. The transmission loss of the silencer can then be computed from the scattering matrix if a certain form of the incident wave is assumed. Furthermore, a slightly rearranged form of the scattering matrix may also be used to combine subsystems in series connection.