# Potential of Room Acoustic Solver with Plane-Wave Enriched Finite Element Method
## 4 Conclusions  
This report described a study of a room acoustics solver based on plane-wave-enriched FEM using low-order quadrilateral elements and local-reaction impedance boundaries. We discussed the potential of plane-wave-enriched FEM as a room acoustic solver to solve the issues on wave-based room acoustic simulations via performance examination on 2D real-scale room acoustic problems with realistic boundary conditions in comparison with the standard second-order accurate FEM. In particular, we examined its accuracy for multi-frequency analysis including fine frequency resolution  
Figure 12: SPL distributions at 2 kHz calculated using PUFEM with \(C=13\).  
Figure 13: Comparisons of SPL distributions between Ref (**upper**) and PUFEM (**lower**) at 125 Hz (**left**), 250 Hz (**center**) and 500 Hz (**right**).  
quantitatively by comparison with reference solutions calculated using a fourth-order accurate FEM. The tested frequency range is 20 Hz to 2.5 kHz with 1 Hz interval. We used two numerical room models of a single room and a coupled room, each with 13-20 times larger interior sound fields than those used in earlier studies, and also including frequency-dependent impedance boundary conditions of sound absorbers commonly used in room acoustic simulations i.e., Helmholtz resonators and porous absorbers. The numerical results clearly revealed advantages of using plane wave enriched finite elements against the standard finite elements in terms of the resulting accuracy, required DOF to obtain the same accurate results, and also the ease of mesh generation. More specifically, the two numerical examples demonstrated that PUFEM can predict a broadband frequency response at low to high frequencies accurately using a single coarse mesh with much less DOF than the standard FEM. The reduction in DOF reaches at least 1/100 of the standard FEM. In the single-room problem, PUFEM produced acceptably accurate results at all frequencies with only 16,271 degrees of freedom, whereas the standard FEM showed unacceptable results because of the inherent larger dispersion error property despite the use of mesh with 1,599,840 DOF. Similarly, in the coupled room problem, the PUFEM results obtained using coarse mesh with 26,129 DOF showed much lower error values than those with 2,448,000 DOF of the standard FEM. Results demonstrate that PUFEM using plane wave enrichment has noteworthy potential for increasing the applicability of wave-based room acoustic simulations. However, numerical results also indicate that an appropriate number of plane waves exists for enrichment. When inappropriate numbers of plane waves are added to nodal points, the PUFEM produces unstable results because of the resulting ill-conditioned linear systems. Therefore, we expect to develop the mode of adding plane waves properly in future studies so that multi-frequency analysis can perform efficiently and robustly. Furthermore, we showed the mesh size effect on PUFEM results in limited conditions. Therefore, our future works will include showing the effect in detail. Application of PUFEM solver to 3D room acoustic simulations will be presented in future reports.  
Conceptualization, T.O.; Investigation, T.O.; Methodology, T.O. and M.S.M.; Project administration, K.S.; Validation, T.O.; Visualization, T.O.; Writing--original draft, T.O.; Writing--review and editing, T.O., M.S.M., and K.S. All authors have read and agreed to the published version of the manuscript. This work was supported by JSPS KAKENHI Grant No. 17K14771. The second author is partially supported by JSPS Invitational Fellowship No. L19554. The authors declare that they have no conflict of interest related to this report or the study it describes.