A generalized axisymmetric boundary element model for complex acoustic problems

Abstract

Aerospace structures frequently involve axisymmetric components. Sound radiation from vibrating axisymmetric structures and scattering from such structures is the main topic of this paper. An indirect boundary integral representation has been selected for that purpose. It enables to solve acoustic problems for both open or closed surfaces. Special attention is devoted to the treatment of generalized boundary conditions: both pressure, normal velocity, normal admittance and transfer admittance boundary conditions can be constrained along both faces of the mean surface but additionally the non-axisymmetric nature of these conditions can be accounted for using a Fourier decomposition along the circumferential direction. A Galerkin boundary element method has been selected for the implementation. Numerical examples are presented in order to demonstrate the efficiency of the proposed approach. Introduction The solution of acoustic problems in the 'low' frequency range usually relies on discrete techniques like finite element (FE) or boundary element (BE) methods. These methods offer a great flexibility for handling complex geometries as usually involved in industrial applications'. By essence, Finite element methods are primarly directed to interior (cavity) problems while boundary element methods are equally applicable to interior and/or exterior problems. This feature results from the use of a boundary integral representation enabling the exact treatment of the socalled 'Sommerfeld' radiation condition. Both direct and indirect integral representations are available for deriving associated boundary element models. In this paper, the attention is focused on the handling of axi-symmetric geometries combined with generalized non-axisymmetric boundary conditions. These boundary conditions can be either pressure, velocity or generalized admittance constraints. They can be independently constrained along both sides of the boundary surface. Additionally provisions are made in order to handle a generalized transfer relation between acoustic variables (pressure and normal velocity) on both sides of the boundary surface. The discrete models relies on an indirect boundary integral representation and a variational solution scheme. The indirect scheme is particularly well suited for handling thin structures where the thickness is assumed to be small versus the acoustic wavelength. The formal derivation of the axi-symmetric BEM model relies on the prealable setting of a generalized 3-D BEM model. Numerical results related to typical interior and exterior axisymmetric problems are compared to results provided by alternative FE or BE methods including fully 3-D solutions. Problem's statement The 3-D acoustic problem is formulated with reference to an unbounded domain V. The boundary surface S is related to a structure which is in contact with the acoustic fluid on both sides. The acoustic medium is assumed to be homogeneous (density p and sound speed c are constant within the domain). In the frequency domain, one supposes that the acoustic pressure P at a point x of V and time t can be expressed as P(x,t) = p(x).e°. The problem is then to find a complex valued function p that verifies the Helmholtz equation in V: