Comparison of FEM and ESM in Modeling Outdoor Sound Propagating over a Smooth-Random Hard Surface

The propagation of outdoor noise over a semi-infinite domain is computationally challenging, especially for Finite Element Method (FEM) because the number of Degrees of Freedom (DOF) grows cubically with the domain size and resolution, and the resolution grows linearly with frequency or wavenumber. The memory needed to store the FEM matrices and the CPU time thus increase at least cubically. The Equivalent Source Method (ESM) is an alternative to the FEM and can be used to model sound transmission above an arbitrary reasonably smooth surface. The number of additional Equivalent Sources grows quadratically with the domain size and the resolution. We developed an algorithm to place the boundary points and the equivalent sources automatically. The complex magnitudes and phase of the small extra sources are solved using least-squares (LS). The relative error between the two methods for an arbitrary surface is less than 2%. The CPU time also favors the ESM over a significant frequency range, for which we were able to compute the FEM solutions.