A thin spherical shell of radius = 0.1 m and thickness h = 0.001 m in an unbounded acoustic medium is subjected to an incident plane wave. The analytical solution for the acoustic scattered pressure is of the form

whereThe elastic pressure term uses the in-vacuo modal impedance of the shell, and the specific acoustic modal impedance,Definitions of the terms in the expressions above are found in Table 1.11.12–1. The orientation of the incident wave with respect to the sphere is shown in Figure 1.11.12–1; the incident field is defined as having zero phase at the origin, which lies at the center of the sphere. The analytical solution is derived in Junger and Feit, but its complex conjugate is used for comparison to conform to the Abaqus sign convention for time-harmonic problems.

The finite element mesh uses AC3D20 elements to model the fluid, with an outer radius of = 0.25 m and a circumferential angle of 10°. Since the problem is axisymmetric, this is sufficient to resolve the field. The shell is meshed with S8R elements, and this mesh is coupled to the acoustic mesh using a tie constraint. Planar incident wave loads of unit reference magnitude are applied to the inner acoustic and outer shell surfaces using *INCIDENT WAVE INTERACTION, REAL, with the standoff point defined at the center of the sphere and the source point defined at a point along the positive x-axis. Specifying the load in this way means that Abaqus will apply loads on the surface corresponding to an incident pressure field having a value of 1 + 0 × i at the standoff point. Two Abaqus models are created: in one, a spherical nonreflecting condition is imposed on the outer surface using the *SIMPEDANCE option; in the other, acoustic infinite elements are created and coupled to the acoustic finite elements using a tie constraint. The material properties used in this problem are shown in Table 1.11.12–2. The analysis is run using the *STEADY STATE DYNAMICS, DIRECT procedure in the range from 1500 to 5000 Hertz.