Modelling Boundary Conditions in High-Order, Nodal, Time-Domain Finite Element Methods

Accurate modeling of boundary conditions is an important aspect in room acoustic simulations. It has been shown that the acoustics of rooms is not only dependent on the frequency characteristic of the complex boundary impedance, but also on the angle dependent properties of the impedance (``extended reaction''). This paper presents a computationally efficient method for modeling local-reaction (LR) and extended-reaction (ER) boundary conditions in high-order, nodal, time-domain finite element methods, such as the spectral element method (SEM) or the discontinuous Galerkin finite element method (DGFEM). The frequency and angle dependent boundary impedance is mapped to a multipole model and formulated in differential form. The solution of the boundary differential equations comes with minimal computational cost. In the ER model, wave splitting is applied at the boundary to separate the incident and reflected parts of the sound field. The directional properties of the incident sound field are determined from the incident particle velocity and the boundary conditions are adjusted continuously according to the wave angle of incidence. The accuracy of the boundary condition model is assessed by comparing simulations against measurements, where a significantly improved match between simulations and measurements is found when the ER model is used.