Computational Improvements in the Boundary Element Method for Acoustics including Viscothermal Dissipation

Research output: Book/ReportPh.D. thesis

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A long range of problems in acoustical engineering necessitate the incorporation of both viscous and thermal dissipation in order accurately capture the real-world physics. These dissipative effects become particularly important when dealing with smaller geometrical dimensions in the acoustic domain, as seen in applications like acoustic transducers and hearing aids. The computational technique known as the boundary element method offers the ability to account for dissipation while avoiding the need for boundary layer meshing. This is opposed to the widely used finite element method for which boundary layer meshing is a must. In addition, the boundary element method is suitable for modeling unbounded domains, which is often of interest in acoustical modeling. However, the current formulation of the boundary element method including viscous and thermal losses, has two notable drawbacks. The first major limitation is the reliance on frequency-dependent boundary integrals, which makes the formulation unsuitable for scenarios involving multiple frequencies. The second major limitation is that the formulation depends on both sparse and dense matrices, each of which comes with its own problems. Although sparsity is most often associated with pleasantries in numerical analysis, the current formulation cannot utilize its properties to the fullest. In fact, the solution stage includes a series of sparse matrix-matrix products which cannot be guaranteed to be sparse, which in turn ruins the computational advantage that is usually contributed to sparsity. The dense matrices is a different beast and would instantly render the formulation unusable for large-scale problems. However, in the context of boundary element matrices, there exist ways to approximate the matrix-vector products of these types of matrices in a way that scales. The key is to then reformulate the problem in a way for which only the matrix-vector product is required. Solutions to the two issues at hand have been developed within the context of pure acoustical problems. This thesis extends the solutions in the context of viscous and thermal losses. In particular the multifrequency problem is resolved by extending a known reduced order series expansion boundary element method to include the boundary layer impedance boundary condition. This development required the derivation of the Taylor expansion of the tangential derivative of the Green’s function. The resulting model is more complex than the original due to the frequency-dependent coefficients in front of the integrals stemming from the boundary layer impedance condition. The solution in the context of large-scale simulations is a two-step process. The first step is an extensive reformulation of the underlying model that allows one to utilize the sparsity patterns of the so-called thermal and viscous modes in the solution phase. This was a necessary first step in the direction of large-scale simulation, as without this it would have not made sense to continue with this formulation. The final step was then to reformulate the problem in a way for which only the dense matrix-vector product with the so-called acoustical mode was required. Given this reformulation, it was then possible to approximate the matrix-vector product using standard techniques such as the fast multipole method or H-matrices. Using these, it was possible to solve problems of sizes far beyond what was previously possible.
Original languageEnglish
PublisherTechnical University of Denmark
Number of pages162
Publication statusPublished - 2023


  • Boundary element method
  • Fast multipole method
  • H-matrices
  • Rank-structure
  • Viscous thermal effects
  • Kirchhoff Decomposition
  • Boundary layer impedance
  • Reduced order model


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