A spherical cavity of unit radius in an unbounded acoustic medium is subjected to a uniform radial velocity on its inner surface. The analytical solution for the acoustic pressure is of the form

where is the acoustic pressure, is the fluid density, is the speed of sound, is the bulk modulus, is the acoustic wave number, is the frequency, is the radial coordinate, is the normal unit vector pointing into the fluid, and is the prescribed inward particle velocity on the spherical boundary. The ratio of the pressure to the velocity on the boundary is called the acoustic impedance (see Junger and Feit, 1972); for the zeroth (breathing) mode the impedance is given byThe imaginary part of the impedance is given by where is the magnitude of the impedance.

The absorbing boundary condition used here is the first-order condition of Bayliss, Gunzberger, and Turkel (1982):

where is the radius of the spherical truncation boundary of the finite element mesh. This boundary condition is theoretically exact for all vibration frequencies in the breathing mode. Since conservation of linear momentum requires and the *IMPEDANCE PROPERTY option enforces the conditionthe first-order Bayliss et al. boundary condition can be enforced in the *STEADY STATE DYNAMICS procedure using the *IMPEDANCE PROPERTY, TYPE=TABULAR option, with table values given by andIn ABAQUS this spherical radiation impedance can also be defined using the *IMPEDANCE PROPERTY, TYPE=SPHERE option. In this case only the radius of the terminating spherical mesh boundary needs to be given.

Figure 1.10.3–1 shows the finite element mesh using a single layer of 15 ACAX8 elements, with ASI3A coupling elements on the inner surface and the *IMPEDANCE PROPERTY option applied to the outer faces of the ACAX8 elements. Uniform radial velocities are applied to the inner surface using the *TRANSFORM option (to spherical coordinates) before specifying the magnitudes. The dimensions and acoustic properties of this problem are chosen to facilitate comparison with the analytical value of the impedance, With and , , and the acoustic wave number is equal to the analysis frequency in Hertz, Moreover, if the imposed normal velocity , the value of the pressure variable on the inner boundary will equal the value for the impedance, This normal velocity is imposed using the *BOUNDARY option in conjunction with the *TRANSFORM option, as shown in acousticimpsphere_acax8_bayliss.inp. Since the degrees of freedom representing the tangential velocity components have no stiffness associated with them, they must be constrained to prevent solver problems.

The outer boundary is placed at to provide a good aspect ratio for the elements; its value has no other significance for this problem. For comparison, the same mesh was reanalyzed using the plane wave absorbing condition, the default absorbing boundary condition in the *IMPEDANCE PROPERTY procedure. This plane wave absorber is equal to the limiting value of the Bayliss et al. first-order condition as and is theoretically exact for absorption of planar wavefronts normally incident on a planar truncation boundary.

A steady-state dynamic analysis is performed in ABAQUS/Standard over a range of frequencies from 0 to 20 Hz. The transient simulations are also performed in ABAQUS/Explicit with an excitation frequency of 1 Hz. Different excitation frequencies can be tested by changing the parameters defined in the input files.

The steady-state analysis was also performed in ABAQUS/Standard using three-dimensional and axisymmetric acoustic infinite elements. These acoustic infinite elements use infinite-direction basis functions that are based on radiating modes of a sphere (see “Acoustic infinite elements,” Section 3.3.2 of the ABAQUS Theory Manual). Consequently, the impedance for the spherical breathing mode is modeled accurately by these elements, and the results obtained using acoustic infinite elements should closely match the results obtained using the *IMPEDANCE PROPERTY, TYPE=SPHERE option. However, for more complex wave shapes the acoustic infinite elements are expected to give more accurate results, owing to the richness of the basis functions used to model the variation of the acoustic field in the infinite direction. The infinite element mesh required to model this problem consists only of acoustic infinite elements on the spherical surface, with acoustic-structural interface elements added (with the same mesh topology) to facilitate imposition of the velocity boundary condition at the surface of the sphere. The reference point for the infinite elements is located at the center of the sphere. For this problem a single input file, acousticimpsphere_acin.inp, includes sections of the sphere modeled with each of the three-dimensional acoustic infinite elements: ACIN3D3, ACIN3D4, ACIN3D6, and ACIN3D8. Two additional input files, acousticimpsphere_acinax2.inp and acousticimpsphere_acinax3.inp, model the problem using semicircular meshes of ACINAX2 and ACINAX3 elements, respectively.

For the transient analyses in ABAQUS/Explicit the Bayliss et al. boundary condition is enforced using the *IMPEDANCE PROPERTY, TYPE=SPHERE option. On the inner radius of the mesh the shell elements in the three-dimensional case and beam elements in the axisymmetric case are connected to the acoustic domain with the *TIE option. The transient analysis is also performed in ABAQUS/Explicit using three-dimensional and axisymmetric acoustic infinite elements instead of the *IMPEDANCE option. Two additional input files, acousticimpsphere_acin3d4_xpl.inp and acousticimpsphere_acinax2_xpl.inp, model the three-dimensional and axisymmetric problems, respectively.