Product: Abaqus/Standard
This example analyzes the symmetric acoustic resonances of an elastic spherical shell immersed in an acoustic fluid of infinite extent using acoustic infinite elements.
A set of real-valued and complex-valued solutions are sought. An analytical solution for this problem is provided for comparison with the numerical results obtained.
The model consists of a layer of linear axisymmetric acoustic elements with an inner radius of 1 and an outer radius of 1.3. On the inner surface of the layer, linear axisymmetric shell elements of thickness 0.01 are coupled to the water layer. On the outer surface of the water layer, linear axisymmetric acoustic infinite elements are used. The units used are consistent with water (
and 
× 109) and with steel (
, 
× 1011, and 
). The frequency range of interest is 36 to 3000 cycles per second, which contains resonances in both the upper and lower branch of the system (see Chapter 10.3 of Junger and Feit, 1972).
Two Abaqus eigenanalysis procedures are used: regular eigenvalue extraction and complex eigenvalue extraction. In the former case the analysis considers the acoustic contributions due to the acoustic finite and infinite element mass and stiffness matrices but not the radiation damping. In addition, the infinite element stiffness matrices are rendered symmetric for compatibility with the real-valued eigensolver. Therefore, these modes are real valued and correspond to the analytical solutions computed using the acoustic accession to inertia only. In the complex eigenvalue extraction procedure the real-valued modes are used as a basis, and the entire finite and infinite element matrix contribution is projected onto this basis. Thus, the radiation damping term is included in the analysis. The shell resonant mode shapes are identical whether or not the shell is coupled to the fluid, which allows the numerically computed modes to be identified with the analytical solution by inspection.
The analytical results for the eigenfrequencies are calculated using the material properties described above.
Analytic and computed results agree well for both branches, as shown in Table 1.11.10–1. Mode shapes also correspond to the analytical solutions. The complex eigenvalue extraction results are more accurate, due to the inclusion of the nonsymmetric acoustic infinite element stiffness matrix and the important radiation damping term.