A finite element method for transmission in non-uniform ducts without flow: Comparison with the method of weighted residuals

Abstract

A finite element method is developed for the study of transmission of sound in non-uniform ducts without flow. The formulation is based on a weighted residual approach and eight noded isoparametric elements are used. Two computational schemes are described, one based on the Helmholtz equation obtained by combining the basic conservation equations and one based on the conservation equations themselves. The latter case is considered because in future extensions to problems involving mean flow a single governing equation is not readily obtainable except for irrotational flows. Both two-dimensional and circular duct geometries are considered. Comparisons are made with a Method of Weighted Residuals in the form of a Modified Galerkin Method in the two-dimensional case to assess both accuracy and computational efficiency. It is found that the finite element method produces results for transmission and reflection coefficients nearly identical to those from the Galerkin approach. Used to its best advantage the finite element method is of comparable efficiency.