A Third-Order Boundary Element Method for Exterior Acoustics with Applications to Scattering by Rigid and Elastic Shells

Abstract

In this work a novel third-order direct boundary element method for the Helmholtz equation is presented. The methodology is best understood within the context of the so-called [puod]fictitious eigenvalues difficulty[pucd], introduced by the typical direct boundary integral approach; this problem consists of the existence of fictitious resonance frequencies (i.e., frequencies of the adjoint interior problem) and is overcome by the CONDOR technique by Burton and Miller (i.e., a suitable linear combination of the Kirchhoff[puen]Helmholtz integral equation with that obtained by taking its normal derivative). This in turn yields the presence of a hypersingular kernel (arising in the integral expression for the normal derivative of the Kirchhoff[puen]Helmholtz integral equation), which is regularized by introducing a novel integral relationship (closely related to the equivalence between doublet layers and vortex layers in incompressible potential flows). This requires the evaluation of the tangential Laplacian of the unknown, and hence the use of a high-order discretization. A piecewise bicubic discretization of the boundary integral equation is used in this paper. The resulting equation contains only the nodal values of the unknown. Numerical applications to particularly taxing problems (such as high-frequency radiation and scattering problems of rigid and elastic shells) are included, and validated through comparison with analytical solutions. Numerical results show that the convergence rate is of orderh3(hbeing the typical element size), even for high-frequency analysis, indicating that this is a true third-order method.