Modeling impedance boundary conditions and acoustic barriers using the immersed boundary method: The one-dimensional case

Abstract

Immersed boundary methods are heavily used in computational fluid dynamics, as an alternative to volumetric meshing, when a problem contains irregular geometric features. In wave-based architectural and room acoustics, the dynamics are simplified, but boundary conditions and acoustic barriers are usually described in terms of frequency-dependent impedance and transmittance functions. In this article, a formulation of the immersed boundary method is developed in the informative special case of one-dimensional linear acoustics. It relies on dual driving terms applied to the conservation of mass and momentum equations separately and is directly tunable against boundary impedances and barrier transmittances. It is shown how the driving terms may be combined to model either an impermeable frequency-dependent boundary condition or a barrier with a given transmittance. An explicit time-domain numerical method of finite-difference time-domain type is presented, and it is shown how the immersed boundary condition may be included, at minimal additional computational cost. Special attention is paid to the discrete approximation of the Dirac delta function, necessary in immersed boundary methods, as well as the discretisation strategy for frequency-dependent boundary and barrier conditions. Numerical results are presented. A complete derivation of numerical stability conditions for this immersed boundary method appears in an appendix.