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In recent years, the boundary element method has shown to be an interesting alternative to the finite element method for modeling of viscous and thermal acoustic losses. Current implementations rely on finite-difference tangential pressure derivatives for the coupling of the fundamental equations, which can be a shortcoming of the method. This finite-difference coupling method is removed here and replaced by an extra set of tangential derivative boundary element equations. Increased stability and error reduction is demonstrated by numerical experiments.

The modern scope of boundary element methods (BEM) for acoustics is reviewed in this paper. Over the last decades the BEM has gained popularity despite suffering from shortcomings, such as fictitious eigenfrequencies and poor scalability due to its dense and frequency-dependent coefficient matrices. Recent research activities have been focused on alleviating these drawbacks to enhance BEM usability across industry and academia. This paper reviews what is commonly known as direct BEM for linear time-harmonic acoustics. After introducing the boundary integral formulation of the Helmholtz equation for interior and exterior acoustic problems, recommendations are given regarding the boundary meshing and treatment of the non-uniqueness problem. It is shown how frequency sweeps and modal analyses can be carried out with BEM. Further extensions for efficient modeling of large-scale problems, including fast BEM and solutions methods, are surveyed. Additionally, this review paper discusses new application areas for modern BEM, such as viscothermal wave propagation, surface contribution analyses, and simulation of periodically arranged structures as found in acoustic metamaterials.

The Helmholtz equation is a reliable model for acoustics in inviscid fluids. Real fluids, however, experience viscous and thermal dissipation that impact the sound propagation dynamics. The viscothermal losses primarily arise in the boundary region between the fluid and solid, the acoustic boundary layers. To preserve model accuracy for structures housing acoustic cavities of comparable size to the boundary layer thickness, meticulous consideration of these losses is essential. Recent research efforts aim to integrate viscothermal effects into acoustic boundary element methods (BEM). While the reduced discretization of BEM is advantageous over finite element methods, it results in fully populated system matrices whose conditioning deteriorates when extended with additional degrees of freedom to account for viscothermal dissipation. Solving such a linear system of equations becomes prohibitively expensive for large-scale applications, as only direct solvers can be used. This work proposes a revised formulation for the viscothermal BEM employing the Schur complement and a change of basis for the boundary coupling. We demonstrate that static condensation significantly improves the conditioning of the coupled problem. When paired with an iterative solution scheme, the approach lowers the algorithmic complexity and thus reduces the computational costs in terms of runtime and storage requirements. The results demonstrate the favorable performance of the new method, indicating its usability for applications of practical relevance in thermoviscous acoustics.