Low-Rank Iteration Schemes for the Multi-Frequency Solution of Acoustic Boundary Element Equations

The implicit frequency dependence of linear systems arising from the acoustic boundary element method necessitates an efficient treatment for problems in a frequency range. Instead of solving the linear systems independently at each frequency point, this paper is concerned with solving them simultaneously at multiple frequency points within a single iteration scheme. The proposed concept is based on truncation of the frequency range solution and is incorporated into two well-known iterative solvers - BiCGstab and GMRes. The proposed method is applied to two acoustic interior problems as well as to an exterior problem in order to assess the underlying approximations and to study the convergence behavior. While this paper provides the proof of concept, its application to large-scale acoustic problems necessitates efficient preconditioning for multi-frequency systems, which are yet to be developed.

Quantification of Numerical Damping in the Acoustic Boundary Element Method for Two-Dimensional Duct Problems

Spurious numerical damping in the collocation boundary element method is considered for plane sound waves in two-dimensional ducts subjected to rigid and absorbing boundary conditions. Its extent is quantified in both conditions based on a damping model with exponential decay, and meshes of linear and quadratic continuous elements are studied. An exponential increase of numerical damping with respect to frequency is found and the results suggest an upper bound for given element-to-wavelength ratios. The quantification of numerical damping is required for evaluation of meshes covering a large number of waves, and real damping phenomena can be prevented from overestimation.

A Three-Dimensional Acoustic Boundary Element Method Formulation with Viscous and Thermal Losses Based on Shape Function Derivatives

Sound waves in fluids are subject to viscous and thermal losses, which are particularly relevant in the so-called viscous and thermal boundary layers at the boundaries, with thicknesses in the micrometer range at audible frequencies. Small devices such as acoustic transducers or hearing aids must then be modeled with numerical methods that include losses. In recent years, versions of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) including viscous and thermal losses have been developed. This paper deals with an improved formulation in three dimensions of the BEM with losses which avoids the calculation of tangential derivatives on the surface by finite differences used in a previous BEM implementation. Instead, the tangential derivatives are obtained from the element shape functions. The improved implementation is demonstrated using an oscillating sphere, where an analytical solution exists, and a condenser microphone as test cases.