On the modeling of narrow gaps using the standard boundary element method

Abstract

Numerical methods based on the Helmholtz integral equation are well suited for solving acoustic scattering and diffraction problems at relatively low frequencies. However, it is well known that the standard method becomes degenerate if the objects that disturb the sound field are very thin. This paper makes use of a standard axisymmetric Helmholtz integral equation formulation and its boundary element method (BEM) implementation to study the behavior of the method on two test cases: a thin rigid disk of variable thickness and two rigid cylinders separated by a gap of variable width. Both problems give rise to the same kind of degeneracy in the method, and modified formulations have been proposed to overcome this difficulty. However, such techniques are better suited for the so-called thin-body problem than for the reciprocal narrow-gap problem, and only the first is usually dealt with in the literature. A simple integration technique that can extend the range of thicknesses/widths tractable by the otherwise unmodified standard formulation is presented and tested. This technique is valid for both cases. The modeling of acoustic transducers like sound intensity probes and condenser microphones has motivated this work, although the proposed technique has a wider range of applications.

Sound source reconstruction using inverse boundary element calculations

“…A serious shortcoming is the failure when it is applied to bodies of thin shape or regular bodies with thin appendages. This thin-shape breakdown of the direct formulation was investigated extensively by Martinez 15 and recently by Cutanda et al, 16 and remedies ensuring a meaningful formulation for thin shapes were proposed.…”

Modeling sound scattering using a combination of the edge source integral equation and the boundary element method

“…Another challenge of the BEM is the singularity in the integral kernels, which becomes prominent for thin bodies and narrow gaps 11 as well as when the field point is located near the boundary of the scattering object. There have been different approaches to overcome that difficulty making use, for instance, of singular numerical integration as suggested by Cutanda et al 12 , splitting the integral using analytical removal of the singularity 11,13 , or using a polar coordinates transformation as presented by T. Terai 14 . The details of these techniques will not be discussed in this paper and the interested reader is referred to the cited work for more details.…”

Free surface Stokes flows obstructed by multiple obstacles

“…They include element subdivision [11], adaptive Gaussian integration [12], variable transformation techniques and semi-analytical integration based on series expansions and removal of singularities [13], besides others. In this work, the effects of ill-conditioning are not considered and improvements in the numerical integration scheme follow a method similar to that outlined in Cutanda et al [14], and consists of a combination of the adaptive Gaussian integration algorithm and the element subdivision approach. Cutanda's et al [14] approach of dealing with near-singular integrals was chosen due to its simplicity of formulation, although it is known that some of the other approaches, such as variable transformation and semi-analytical integration, are numerically more efficient.…”