# Potential of Room Acoustic Solver with Plane-Wave Enriched Finite Element Method
## 1 Introduction  
### Background  
Acoustic simulation methods are necessary tools for predicting impulse responses or frequency responses of room spaces in architectural acoustics design. These quantities are necessary to evaluate room acoustics with acoustical parameters such as reverberation times, and the clarity of speech or music. These can also be used for the visualization and auralization of sound fields. Wave-based numerical methods, which solve a wave equation or a Helmholtz equation numerically, are physically reliable simulation methods with the capability of capturing wave phenomena such as interference and diffraction, and also of modeling boundary effects accurately by sound diffusers and sound absorbers. The finite element method (FEM) [1; 2; 3; 4; 5], boundary element method (BEM) [6], and finite difference time domain (FDTD) [7; 8; 9] method exemplify the often-used numerical methods for room acoustic simulations. Although they entail a huge computational effort for acoustic simulations especially at kilohertz frequencies in a real-sized room, their application to room acoustics prediction is increasing gradually by virtue of the progress of computer technology and the continuous development of efficientmethods [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. In addition, some recent studies [16; 18; 22; 25] use extended-reaction boundary conditions to address both the frequency dependent and incident-angle dependent absorption characteristics of sound absorbers accurately, whereas many studies use the simplest local-reaction boundary conditions, which simplify the incident-angle dependence of surface impedance. Nevertheless, wave-based predictions are still time-consuming. Therefore, the development of more efficient methods or optimizing the performance has a marked impact in room acoustics field. This report presents a preliminary study to this end particularly using FEM in the frequency domain solving the Helmholtz equation with a few degrees of freedom (DOF). The huge computational effort necessary for performing reliable acoustic simulations in real-sized rooms using FEM stems from the large dimension of spaces, the broad frequency range of interest, the complicated boundary conditions, and an inherent error property of FEM. The volumes of architectural spaces such as offices, lecture rooms, and concert halls range from the order of 10 m\({}^{3}\) to 10,000 m\({}^{3}\). The human audible frequency is 20 Hz to 20 kHz. In addition, the sound-absorption characteristics of boundary materials, which depend on both frequency and incident-angle of sound, should be modeled accurately. However, FEM is well known to have inherent spatial discretization error called dispersion error, which is an error evaluated in sound speed or in wavelength. To maintain the error within an acceptable level, the discretization of spaces, i.e., mesh generation, must be performed with consideration of a rule of thumb, e.g., for linear elements spatial discretization of 10 elements per wavelength at least. This discretization rule imposes the use of a large FE models with many DOF for acoustic simulation of a real-sized room, making the solution of the problem prohibitively expensive. For instance, in earlier works [4; 5] conducted with high-order finite elements [26], the acoustics in a multi-purpose hall of 37,000 m\({}^{3}\) were simulated using an FEM model of 2,630,435 DOF at 125 Hz. The acoustics in a small hall with complicated diffusers were analyzed using an FEM mesh with 8,926,001 DOF. A recent study [27] conducting simulations at kilohertz frequencies used FEM models with the order of ten million DOF for both acoustic simulations of a simply shaped concert hall and the reverberation absorption coefficient measurement though the use of a dispersion-reduced scheme [28]. Furthermore, when using extended-reaction boundary conditions, the required elements increase because of the discretization of materials.