Volumetric drag

The model consists of a tube of fluid 4 m long with a constant cross-sectional area. The tube lies horizontally (along the x-axis) and has a sound source at x = 0 m, which is given in the form of an inward volume acceleration. From x = 0 m to x = 3 m, the acoustic material in the tube is air with a bulk modulus of 1.424 × 10⁵ N/m² and a density of 1.21 kg/m³. The region from x = 3 m to x = 4 m is filled with a dissipative material with the same bulk modulus and density of air but with a volumetric drag of 10,000 Ns/m⁴. The condition at x = 4 m is a closed end. The tube is modeled using 400 first-order or 200 second-order acoustic elements.

The speed of sound for these air constants is c = 343 m/s. At the highest frequency of 1100 Hz the wavelength is 0.312 m. The internodal interval (distance between nodes) for the meshes is always .01 m; therefore, at this frequency there are 30 first-order elements per wavelength or 15 second-order elements.

Both direct-solution and subspace-based steady-state dynamic analyses are performed in Abaqus/Standard over 3 frequencies ranging from 100 to 1100 Hz. The transient simulations are performed in Abaqus/Explicit using an excitation frequency of 100 Hz. Different excitation frequencies can be tested by changing the parameters defined in the input files. The transient analysis is also performed in Abaqus/Standard using the AC2D4 element for the purpose of providing a reference solution for Abaqus/Explicit.