Finite element implementation of a variational formulation for acoustic radiation and scattering exemplified by the circular disk

Abstract

A previously derived variational principle [A. D. Pierce and X. -F. Wu, J. Acoust. Soc. Am. Suppl. 1 74, S107 (1983)] for the acoustic perturbation on the surface of a body, either vibrating or acting as a rigid-body scatterer, is a possible starting point for finite element formulations. The present paper studies the particular example of a circular disk that is presumed rigid and axisymmetrically excited; surface pressure is assumed to vary linearly between mesh points. An integrated Green's function that represents the signal at a field annulus due to a source annulus is pretabulated and interpolated in the computation scheme. The formulation yields a system of simultaneous equations for the pressures at the mesh points. Computation of coefficient matrix elements requires a twofold integration, with the axisymmetric annular Green's function in the integrand, and with the integration limits corresponding to the inner and outer edges of two annular regions. Numerical techniques for the integrations are discussed and the effects of mesh spacing and of numerical interpolation of the annular Green's function are explored. Results are compared with computations involving spheroidal wavefunctions and with computations using a Rayleigh-Ritz scheme based on the same variational formulation. [Work supported by ONR.]