The steady-state analyses are performed using the *STEADY STATE DYNAMICS, DIRECT procedure. The analyses are performed for two frequencies: the frequency corresponding to to test the effectiveness of the radiation boundary conditions, and the frequency corresponding to (in two dimensions) or (in three spatial dimensions) to test the limits of the acoustic mesh discretization (less challenging frequencies for the boundary conditions). Two steps are used. The impedance boundary condition is applied in the first step (for both frequencies) using the *IMPEDANCE option. In the second step these impedances are removed (using *IMPEDANCE, OP=NEW), and impedances of the same value are applied using the *SIMPEDANCE option. The results with both options are identical.

Each analysis shows good agreement between the reference solution and the solutions using the absorbing boundary conditions or the acoustic infinite elements on the smaller test meshes. The lower frequency is a more challenging test for the boundary conditions, particularly in two dimensions, where the boundary conditions are only asymptotic. Figure 1.10.4–2 through Figure 1.10.4–5 show the pressure amplitudes on the surface of the shell as a function of angular position. The angle is measured from the -axis, as shown in Figure 1.10.4–1. Each figure shows three curves; where differences between them are visible, the circular and elliptical condition solutions overlay the reference solution, and the spherical and prolate spheroidal conditions deviate by small amounts. The results from the acoustic infinite elements are similar.

The transient analyses are performed with an excitation frequency of 100 Hz using two steps and a fixed time increment of 2.5 × 10^–5 units, approximately one two-hundredth of the wavespeed. The analyses are run for 0.003 time units in the first step, where the impedance boundary condition is applied using the *IMPEDANCE option. This time period is long enough for the wave to reach the boundary of the test meshes. In the second step the simulation runs for another 0.003 units, this time with the boundary condition applied using the *SIMPEDANCE option. The total simulation time, 0.006 units, is not long enough for the wavefronts emanating from the structure to reach the boundary of the reference meshes, so the impedance conditions there do not play a role in the simulation. Figure 1.10.4–6 through Figure 1.10.4–9 show the pressure amplitudes in the acoustic domain at selected times. In each case the reference solution is on the left. Very good agreement is achieved in all cases; the three-dimensional boundary conditions perform slightly better, for reasons discussed in “Coupled acoustic-structural medium analysis,” Section 2.9.1 of the ABAQUS Theory Manual.

The ABAQUS/Explicit transient analyses performed using acoustic infinite elements give very similar results to the ABAQUS/Explicit transient analyses performed using the *IMPEDANCE option. The test using two-dimensional acoustic infinite elements gives very similar results to the test using the TYPE=CIRCULAR parameter on the *IMPEDANCE PROPERTY option, while the test using axisymmetric acoustic infinite elements gives very similar results to the test using the TYPE=SPHERE parameter on the *IMPEDANCE PROPERTY option. Figure 1.10.4–10 shows the pressure amplitudes on the surface of the shell at an angle of 90° to the vertical for the two-dimensional analysis (for a clear comparison the ABAQUS/Standard transient analysis results are also included). The pressures are seen to match closely. The tests with acoustic infinite elements do not include any acoustic elements to model the fluid surrounding the structure. However, the additional computational cost due to the acoustic infinite elements offsets any savings obtained from not including the acoustic elements. For this example it was found that the acoustic infinite element analyses were about 12% more expensive than the corresponding analyses without acoustic infinite elements; i.e., with acoustic elements and the *IMPEDANCE option.