Product: Abaqus/Standard
In this example we calculate the acoustic near field scattered from a sphere when impinged by a plane wave. The example illustrates the use of a simple absorbing boundary condition in conjunction with acoustic continuum elements. The results are compared with a classical solution.
A rigid spherical obstacle of radius 
= 0.1 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
is the scattered acoustic pressure, 
is the coefficient of the incident plane wave, 
are Legendre polynomials, 
are spherical Bessel functions of the first kind, 
are spherical Hankel functions of the first kind, 
is the acoustic wave number, 
is the speed of sound, and 
is the frequency. The orientation is shown in Figure 1.11.11–1; 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.Figure 1.11.11–2 shows the finite element mesh using seven layers of AC3D15 elements (252 in total), with an outer radius of 
= 0.4 m and a circumferential angle of 10°. Since the problem is axisymmetric, this is sufficient to resolve the field. Planar incident wave loads of unit reference magnitude are applied to the inner surface using *INCIDENT WAVE INTERACTION, LOAD CASE=1, 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. A spherical radiation condition is imposed using the *SIMPEDANCE option, applied to the outer surface. The acoustic properties of this problem are chosen as follows: 
= 2.0736 GPa, 
= 1000 kg/m3, so that the acoustic wave speed is 
= 1440 m/s. The analysis is run using the *STEADY STATE DYNAMICS, DIRECT procedure in the range from 30 to 9000 Hertz.
The finite element results for the scattered pressure in the near field, at 
, are shown in Figure 1.11.11–3, where they are compared with the analytical values. The real and imaginary parts of the solutions show excellent agreement.
Model that uses AC3D15 elements with the Bayliss et al. boundary condition.
Bayliss, A., M. Gunzberger, and E. Turkel, “Boundary Conditions for the Numerical Solution of Elliptic Equations in Exterior Regions,” SIAM Journal of Applied Mathematics, vol. 42, no.2, pp. 430–451, 1982.
Junger, M., and D. Feit, Sound, Structures, and Their Interaction, The MIT Press, 1972.
