A reciprocal identity method for large silencer analysis

Conventional techniques used in the boundary element method for evaluating muffler transmission loss have been limited by the cutoff frequency of the inlet and outlet ducts. Even though the boundary element method itself is a truly three-dimensional analysis tool, it has not been effectively used on large silencers due to the large inlet and outlet cross sections. In this paper, a numerical technique based on the reciprocal identity and the boundary element impedance matrix is proposed as a post-processing filter to extract the transmission loss of large silencers at all frequencies. Each reciprocal identity couples two different sound fields on the same silencer geometry. The first sound field has the analytical modal expansion in the inlet and outlet ducts, while the second sound field is the boundary element solution associated with a random boundary condition set. Depending on how many modes exist in the inlet and outlet ducts at a certain frequency, a minimum number of random boundary condition sets must be applied to the boundary element model. The boundary element impedance matrix provides more than enough such solution sets for the reciprocal identity coupling. The overdetermined system is then solved by a least-squares procedure. The proposed method is verified by comparing to the analytical solutions of a simple expansion chamber and a round bar silencer.